UBC Theses and Dissertations

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UBC Theses and Dissertations

Studies in utility theory Larsson, Stig Owe 1978

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STUDIES IN UTILITY THEORY by STIG OWE LARSSON B.Sc.(Hons, w i t h d i s t i n c t i o n ) S i r George W i l l i a m s U n i v e r s i t y , 1970 M . S c , U n i v e r s i t y o f A l b e r t a , 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES THE FACULTY OF COMMERCE AND BUSINESS ADMINISTRATION We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r , 1977 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f QCSVUA^ j u t x . J ? T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e Q ASLUA. 1 YWK i ABSTRACT S i n c e vonNeumann and M o r g e n s t e r n made t h e i r c o n t r i b u t i o n s , t h e e x p e c t e d u t i l i t y c r i t e r i o n (EUC) has been t h e most a c c e p t e d c r i t e r i o n i n d e c i s i o n t h e o r y . F o l l o w i n g t h e i r a x i o m a t i c approach j u s t i f y i n g EUC, s e v e r a l o t h e r s t u d i e s have been made s u g g e s t i n g the same c r i t e r i o n b u t under s l i g h t l y d i f f e r e n t a x i o m a t i c systems. However, c r i t i c s have found s e v e r a l s i m p l e d e c i s i o n problems ( c a l l e d paradoxes) which seem t o c o n t r a d i c t t h e c o n c l u s i o n s o f EUC; t h a t i s , the paradoxes c o n t r a d i c t one o r more o f t h e axioms made t o s u p p o r t EUC. The c r i t i c i s m s a r e based on e m p i r i c a l s t u d i e s made i n r e g a r d t o t h e parad o x e s . I t i s n o t always o b v i o u s , however, w h i c h axiom(s) i s n o t a c c e p t e d , s i n c e each approach t o EUC g i v e s a s e t o f s u f f i c i e n t r a t h e r t h a n n e c e s s a r y assumptions f o r EUC t o h o l d . I n P a r t I o f the t h e s i s a s e t o f axioms which a r e nec e s s -a r y f o r EUC t o h o l d i s s p e c i f i e d . Each o f t h e s e axioms c o n t a i n s a b a s i c a ssumption o f a d e c i s i o n maker's b e h a v i o u r . T h e r e f o r e by c o n s i d e r i n g t h e paradoxes i n terms o f t h e s e axioms, a b e t t e r u n d e r s t a n d i n g i s o b t a i n e d w i t h r e g a r d t o w h i c h p r o p e r t i e s o f EUC seem t o be c o n t r a d i c t e d by the paradoxes. i The c o n c l u s i o n o f t h i s study shows t h a t most p e o p l e con-t r a d i c t EUC because i t does not d i f f e r e n t i a t e between a "known" r i s k and an "unknown" r i s k . I n K n i g h t ' s t e r m i n o l o g y , t h e r e i s a d i s t i n c t i o n between d e c i s i o n making under r i s k and u n c e r t a i n -t y . Most e m p i r i c a l s t u d i e s show t h a t t h e s e d i f f e r e n c e s a r e o f such s u b s t a n t i a l p r o p o r t i o n s t h a t t h e r e i s a q u e s t i o n a b l e j u s t i f i c a t i o n f o r u s i n g the e x p e c t e d u t i l i t y c r i t e r i o n f o r d e c i s i o n making under u n c e r t a i n t y . A l t h o u g h many a l t e r n a t i v e s t o EUC f o r d e c i s i o n making under u n c e r t a i n t y e x i s t , t h e r e a r e v e r y few c r i t e r i a f o r d e c i s i o n problems w h i c h f a l l between r i s k and u n c e r t a i n t y , t h a t i s , p a r t i a l r i s k problems. Those e x i s t i n g a r e o f an ad hoc n a t u r e . As a n o r m a t i v e t h e o r y the EUC i s f a r s u p e r i o r t o any o f t h e s e c r i t e r i a i n s p i t e o f i t s l a c k o f d i s t i n c t i o n between r i s k and u n c e r t a i n t y . I n the second p a r t o f the t h e s i s an a l t e r n a t i v e n o r m a t i v e c r i t e r i o n i s s u g g ested f o r d e c i s i o n making under p a r t i a l r i s k and u n c e r t a i n t y . As an e x t e n s i o n o f EUC, t h i s c r i t e r i o n d i s -t i n g u i s h e s between r i s k and u n c e r t a i n t y . T h i s t h e o r y expands on E l l s b e r g ' s s u g g e s t i o n t h a t " a m b i g u i t y " i n f l u e n c e s one's p r e f e r e n c e among a s e t o f a l t e r n a t i v e s . I n t h i s e x t e n s i o n a more p r e c i s e d e f i n i t i o n o f " a m b i g u i t y " i s needed and one i s s u g g e s t e d h e r e as a r e l a t i o n on t h e i n n e r and o u t e r measure o f an e v e n t . The e x t e n s i o n o f EUC i s t h e n o b t a i n e d by c o n s i d e r i n g a more g e n e r a l s e t f u n c t i o n , termed P-measure, w h i c h would depend on a s e t ' s a m b i g u i t y r a t h e r than a p r o b a b i l i t y measure on the s e t s o f rewards. I t i s c o n c l u d e d by an a x i o m a t i c development t h a t the P-measure must be a n o n - n e g a t i v e mono-t o n i c s e t f u n c t i o n w h i c h i s n o t n e c e s s a r i l y a d d i t i v e . I t i s a l s o shown t h a t t h e s t a n d a r d paradoxes r e l a t e d t o paradoxes based on "known" v e r s u s "unknown" p r o b a b i l i t i e s may be e x p l a i n e d by t h i s method and would t h e r e f o r e s uggest an a l t e r n a t i v e t o EUC f o r d e c i s i o n making under p a r t i a l r i s k and u n c e r t a i n t y . i i i T a b l e o f C o n t e n t s Page A b s t r a c t i Table o f C o n t e n t s i i i Acknowledgements v I n t r o d u c t i o n 1 1.0 I n t r o d u c t i o n t o P a r t I 7 2.0 Assumptions and N o t a t i o n s 11 2.1 B a s i c n o t a t i o n s and assumptions 11 2.2 Approaches t o t h e axioms o f e x p e c t e d u t i l i t y t h e o r y 3.0 O r d e r i n g axiom 18 3.1 Statement o f Axiom I 18 3.2 I m p l i c a t i o n s o f the o r d e r i n g assumptions 20 3.3 O r d e r i n g p r o p e r t i e s i n t h e d i f f e r e n t approaches 28 3.4 A l t e r n a t i v e s t o Axiom I 30 4.0 A d d i t i v i t y axiom 33 4.1 Statement o f Axiom I I 33 4.2 I m p l i c a t i o n s o f Axiom I I 37 4.3 E m p i r i c a l s t u d i e s o f Axiom I I 47 4.4 R e l a t i o n t o o t h e r axiom systems 54 4.5 A l t e r n a t i v e s t o Axiom I I 62 5.0 S e p a r a b i l i t y axiom 64 5.1 Statement o f Axiom I I I 64 5.2 I m p l i c a t i o n s o f Axiom I I I 67 5.3 E m p i r i c a l s t u d i e s on A l l a i s paradox I I I 78 5.4 R e l a t i o n t o o t h e r axiom systems 84 5.5 A l t e r n a t i v e s t o Axiom I I I 88 6.0 P r o b a b i l i t y axiom 92 6.1 Statement o f Axiom IV 92 6.2 I m p l i c a t i o n s o f Axiom IV 93 6.3 E m p i r i c a l s t u d i e s based on Axiom IV 101 6.4 R e l a t i o n t o o t h e r axiom systems 10 7 6.5 A l t e r n a t i v e s t o Axiom IV 108 7.0 Summary o f P a r t I 109 i v Page 1.0 I n t r o d u c t i o n t o P a r t I I 117 2.0 The b a s i c assumptions 125. 2.1 Statement o f the axioms 126: 2.2 Comments on the assumptions 130 3.0 The P-measure axiom 133 3.1 Statement o f t h e axiom 134 3.2 I m p l i c a t i o n s o f Axiom IV 137; 3.3 Comparison o f d i f f e r e n t approaches u s i n g E l l s b e r g ' s paradox . . . . . 142 3.4 E v a l u a t i o n o f P-measures 149 4.0 Sequences o f reward f u n c t i o n s 152 4.1 Statement o f t h e axiom 15'2-4.2 I m p l i c a t i o n s o f t h e axiom ( 155 5.0 S t o c h a s t i c dominance axiom 160 5.1 Statement o f t h e axiom 160 5.2 I m p l i c a t i o n s o f Axiom VI 162 6.0 Summary o f assumptions and axioms and t h e b a s i c r e s u l t s 168 6.1 R e l a t i o n o f t h e P-measure approach t o a l t e r n a t i v e approaches 172 7.0 D e r i v a t i o n o f P-measure 178 7.1 A d d i t i v i t y o f the P-measure . . 179 7.2 E x i s t e n c e o f P-measure 185 7.3 Some p o s s i b l e P-measures 187 8.0 The P-measure as a p r o b a b i l i t y measure 193 8.1 Savage e x t e n s i o n s 194 8.2 M a t h e m a t i c a l p r o b a b i l i t i e s 19 8 9.0 E m p i r i c a l s t u d i e s r e l a t e d t o P-measure 206 10.0 Summary o f P a r t I I 214 Summary 21.7 R e f e r e n c e s 220 Appendix I 226 Appendix I I 2 34 Appendix I I I 241 V Acknowledgements I g r a t e f u l l y acknowledge my g r e a t debt t o my committee f o r t h e i r work i n making t h i s t h e s i s p o s s i b l e . I am g r a t e f u l t o Dr. K. MacCrimmon, my t h e s i s s u p e r -v i s o r j o i n t l y w i t h Dr. S. B r u m e l l e , f o r h i s guidance i n the d e c i s i o n t h e o r y i n b o t h the s e l e c t i o n o f the problems o f i n t e r e s t and i n the recommendation of r e l e v a n t papers i n the a r e a . I am a l s o v e r y g r a t e f u l t o D r J S. B r u m e l l e f o r h i s i n t e r e s t and s u g g e s t i o n s d u r i n g the development o f the t h e s i s and f o r h i s c a r e f u l c h e c k i n g i n t o t h e c o n s i s t e n c y o f the t h e o r y . I am a l s o g r a t e f u l t o D r s . D. Wehrung and W. Ziemba f o r t h e i r c a r e f u l r e a d i n g o f the m a t e r i a l and f o r t h e i r suggested changes t o improve i t . I am g r a t e f u l f o r the d i s c u s s i o n s w i t h Dr. P. Greenwood i n t h e e a r l y s t a g e s o f t h e t h e s i s w h i c h l e d up t o i t s p r e s e n t form. A s i d e from my committee, I would l i k e t o e x p r e s s my g r a t i t u d e t o Dr. M. Chew who f a i t h f u l l y r e a d the t h e s i s , and s uggested n e c e s s a r y changes, b o t h i n s t y l e and c o n t e n t . 1 INTRODUCTION The problem o f how t o make t h e -best- c h o i c e among some a l t e r n a t i v e s has i n t e r e s t e d p h i l o s o p h e r s , s t a t i s t i c i a n s , p o l i t i c i a n s , and m a t h e m a t i c i a n s f o r a l o n g t ime m a i n l y because everyone i s f a c e d c o n s t a n t l y w i t h t h i s problem. Even by r e f u s i n g t o choose a c h o i c e i s made. The t h e o r y o r s c i e n c e o f s t u d y i n g "how t o choose" o r "what t o choose" i s c a l l e d t h e "Theory o f D e c i s i o n Making". There e x i s t , o f c o u r s e , s u b d i v i s i o n s o f the Theory o f D e c i s i o n Making, and the f i r s t a ssumptions we s h a l l make a r e the f o l l o w i n g : i ) t h e s e t o f p o s s i b l e c h o i c e s i s known. We s h a l l denote t h i s s e t by A, and c a l l i t an a c t i o n s e t . i i ) t he s e t o f consequences o f our c h o i c e s i s known. The e x a c t consequences w h i c h w i l l o c c u r may not be known f o r any g i v e n a l t e r n a t i v e . T h i s s e t w i l l be denoted by R, and c a l l e d a reward s e t . I n t h e f i r s t p a r t o f t h e t h e s i s , we s h a l l a l s o make t h e f o l l o w i n g a s s u mption: i i i ) t h e r e e x i s t a s e t o f p r o b a b i l i t i e s (II) such t h a t f o r a g i v e n a c t i o n a ( b e l o n g i n g t o A ) , we may r e c e i v e the rewards r ^ , . . . , r n w i t h p r o b a b i l i t i e s Pi_/«*-*Pn where p n + ...+p = 1. ^1 n I t i s assumed t h a t an o r d e r i n g e x i s t s on A; t h a t i s , we can s p e c i f y a p r e f e r e n c e among the a l t e r n a t i v e s w h i c h b e l o n g t o A. The t h e o r y o f d e c i s i o n making under u n c e r t a i n t y i s con-c e r n e d i n p a r t w i t h what p r o p e r t i e s the o r d e r i n g on A ought t o 2 have. F o r example i f we a r e asked t o s t a t e our p r e f e r e n c e o r d e r i n g f o r a l t e r n a t i v e s a, b and c, i s i t r e a s o n a b l e t o have t h e p r e f e r e n c e a t o b and b t o c b u t c t o a? I f our p r e f e r -ences were i n the o r d e r a, b, c, t h e n i f a l t e r n a t i v e c were n o t o f f e r e d would i t be r e a s o n a b l e t o assume t h a t b i s p r e f e r r e d t o a? I n n e a r l y a l l T h e o r i e s o f D e c i s i o n Making i t i s assumed t h a t t h e o r d e r i n g on A can a l s o be s p e c i f i e d by a r e a l - v a l u e d f u n c t i o n f on A. We s h a l l c a l l t h i s f u n c t i o n an e v a l u a t i o n f u n c t i o n . That i s , i f a l t e r n a t i v e a i s p r e f e r r e d t o a l t e r -n a t i v e b t h e n f ( a ) i s l a r g e r t h a n f ( b ) . T h e r e f o r e t h e ques-t i o n s "How t o choose" and "What t o choose" a r e e q u i v a l e n t t o s p e c i f y i n g t h e p r o p e r t i e s f i s assumed t o have. One such f u n c t i o n i s s p e c i f i e d by t h e e x p e c t e d u t i l i t y c r i t e r i o n . T h i s c r i t e r i o n s p e c i f i e s two c o n d i t i o n s on t h e o r d e r i n g on A. F i r s t l y , i f a and b are two a l t e r n a t i v e s be-l o n g i n g t o A such t h a t a r e s u l t s i n outcome r ^ f o r c e r t a i n and b r e s u l t s i n outcome r 2 f o r c e r t a i n , t h e n t h e r e e x i s t s a r e a l v a l u e d f u n c t i o n U on R whose n u m e r i c a l v a l u e s have t h e same o r d e r i n g as t h e p r e f e r e n c e o r d e r i n g on A. The f u n c t i o n U w i l l be c a l l e d a u t i l i t y f u n c t i o n . S e c o n d l y , i f i t i s assumed t h a t a l t e r n a t i v e c g i v e s t h e rewards r ^ , . . . , r w i t h p r o b a b i l i t i e s p ^ P n r e s p e c t i v e l y , t h e n t h e f u n c t i o n f on A i s d e f i n e d by the e x p e c t e d v a l u e o f the u t i l i t y f u n c t i o n , t h a t i s f ( c ) = P j U d r ^ ) + p 2 U ( r 2 ) +...+ p _ U ( r n ) . 3 There e x i s t a t l e a s t two c o n c e p t s o f t h e meaning o f a " u t i l i t y f u n c t i o n " i n the economic l i t e r a t u r e . The f i r s t one s t a r t s by assuming an o r d e r i n g on t h e reward s e t where the reward s e t i s u s u a l l y d e f i n e d as the r e a l l i n e , and t h e o r d e r -i n g i s d e f i n e d by the n a t u r a l o r d e r i n g on t h e r e a l l i n e . Any i n c r e a s i n g f u n c t i o n on the r e a l l i n e i s , t h e r e f o r e , o r d e r p r e s e r v i n g . I f a c t i o n a g i v e s r i s e t o the p r o b a b i l i t y measure P(»,a) on the r e a l l i n e , and s i m i l a r l y a c t i o n b the p r o b a b i l i t y measure P(«,b), an assu m p t i o n i s th e n made t h a t t h e r e e x i s t s an o r d e r - p r e s e r v i n g f u n c t i o n U such t h a t a i s p r e f e r r e d t o b i f J u ( r ) d P ( r , a ) * J\J (r) dP ( r ,b) . As e a r l y as 1738 B e r n o u l l i s u g g e s t e d t h a t U (r). = l o g r An o t h e r example t h a t has some a p p e a l i s the f o l l o w i n g : f 1 r ^ a U(r) ={ r < a where a i s any r e a l number. Then, a i s p r e f e r r e d t o b i f P ( r £ a,a) > P ( r > a,b) . That i s , a c t i o n a i s p r e f e r r e d t o a c t i o n b i f the p r o b a b i l i t y o f r e c e i v i n g a t l e a s t a i s g r e a t e r f o r a c t i o n a t h a n f o r a c t i o n b. Other examples assume t h a t U i s a d i f f e r e n t i a b l e f u n c t i o n such t h a t 4 > 0 , < 0. dr d r 2 The f i r s t c o n d i t i o n assumes t h a t the u t i l i t y i n c r e a s e s w i t h w e a l t h , and the second c o n d i t i o n assumes t h a t f o r a f i x e d i n c r e a s e i n w e a l t h our u t i l i t y d e c r e a s e s as our w e a l t h i s i n c r e a s e d . F or example, l o g r s a t i s f i e s t h e s e c o n d i t i o n s . F o r an a n a l y s i s o f some o f t h e more s p e c i a l i z e d u t i l i t y f u n c t i o n s , see P r a t t (1964) . T h i s method assumed, t h e r e f o r e , an o r d e r i n g on R and extended i t t o A by e x p e c t e d u t i l i t y , by d i r e c t l y assuming t h e e x i s t e n c e o f the u t i l i t y f u n c t i o n . The second approach t o e x p e c t e d u t i l i t y t h e o r y was sug-g e s t e d i n d e p e n d e n t l y by Ramsey (1926) and vonNeumann-Mor g e n s t e r n (1947) by showing t h a t the e x p e c t e d u t i l i t y c r i t e r -i o n can be j u s t i f i e d on the b a s i s o f a s e t o f r e l a t i v e l y s i m p l e assumptions o r axioms on the d e c i s i o n maker's o r d e r i n g on A. That i s , i f each one o f the axioms i s a c c e p t e d as r e a s o n a b l e , then they j o i n t l y i m p l y t h a t t h e o r d e r i n g must s a t i s f y the e x p e c t e d u t i l i t y c r i t e r i o n . The r e a s o n f o r c o n s i d e r i n g a s e t o f axioms i m p l y i n g the e x p e c t e d u t i l i t y c r i t e r i o n i s t h a t h o p e f u l l y the axioms a r e s i m p l e enough f o r the d e c i s i o n maker t o r e a l i z e a l l t h e i r i m p l i c a t i o n s . He can, t h e r e f o r e , d e t e r -mine whether h i s p r e f e r e n c e o r d e r i n g can be s p e c i f i e d by the e x p e c t e d u t i l i t y c r i t e r i o n . I n t h i s c a s e , however, the u t i l i t y f u n c t i o n i s d e r i v e d f o r each d e c i s i o n maker, and a d d i t i o n a l assumptions on the 5 u t i l i t y f u n c t i o n can u s u a l l y n o t be made. F o r example, i f a u t i l i t y f u n c t i o n i s d e r i v e d based on Savage's C1954) s e t o f axioms, i t i s i m p o s s i b l e for'. U t o a l s o s a t i s f y d 2U n d r z s i n c e U must be bounded. The d i f f i c u l t i e s w i t h t h i s approach a r e t h a t i n most c a s e s i t i s i m p o s s i b l e t o r e a l i z e a l l the i m p l i c a t i o n s o f any one axiom.. Many o t h e r a u t h o r s have s i n c e d e v e l o p e d t h e i r own s e t s o f axioms i m p l y i n g the e x p e c t e d u t i l i t y c r i t e r i o n i n the hope t h a t t h e i r s e t o f axioms might• be more c o n v i n c i n g . I n c o n t r a s t , the c r i t i c s o f t h e e x p e c t e d u t i l i t y c r i t e r -i o n have s u g g e s t e d r e l a t i v e l y s i m p l e d e c i s i o n problems f o r w h i c h many i n d i v i d u a l s ' p r e f e r e n c e s c o n t r a d i c t t h e c r i t e r i o n . Because o f t h i s a p p a r e n t c o n t r a d i c t i o n t h e s e problems, a r e u s u a l l y c a l l e d "paradoxes". From some o f t h e "paradoxes" i t i s n o t o b v i o u s whether e x p e c t e d u t i l i t y as a c r i t e r i o n i s b e i n g r e j e c t e d , o r s i m p l y one o f the suggested axioms. That i s , t h e axioms s u g g e s t e d a r e s u f f i c i e n t r a t h e r t h a n n e c e s s a r y f o r t h e e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d . C l e a r l y the s e t o f p o s s i b l e axioms one can s p e c i f y as s u f f i c i e n t , f o r t h e e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d i s v e r y l a r g e and we must t h e r e f o r e a s s e s s t h e m e r i t o f one s e t o v e r a n o t h e r . I n P a r t I o f t h e t h e s i s t h e main c o n c e r n w i l l be: what does the e x p e c t e d u t i l i t y c r i t e r i o n i m p l y o f our p r e f e r e n c e among the a l t e r n a t i v e s , o r , i n m a t h e m a t i c a l terms, we s h a l l 6 c o n s i d e r n e c e s s a r y r a t h e r t h a n . s u f f i c i e n t c o n d i t i o n s . In a d d i t i o n we s h a l l s p e c i f y a s e t o f axioms on our p r e f e r e n c e among the a l t e r n a t i v e s w h i c h a r e b o t h n e c e s s a r y and s u f f i c i e n t f o r t h e e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d . The r e l a t i o n s h i p t o o t h e r approaches w i l l then be c o n s i d e r e d . F o r t h e e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d one o f the assumptions i s t h a t the p r o b a b i l i t i e s o f the consequences f o r each c h o i c e i s known o r , a t l e a s t , the p r e f e r e n c e among our c h o i c e s i s c o n s i s t e n t w i t h t h e e x i s t e n c e o f p r o b a b i l i t i e s f o r w h i c h the e x p e c t e d u t i l i t y may be c a l c u l a t e d . C r i t i c s o f the e x p e c t e d u t i l i t y c r i t e r i o n have suggested t h a t a knowledge o f the p r o b a b i l i t i e s o f t h e consequences o c c u r r i n g i n a d d i t i o n t o t h e i r s p e c i f i c v a l u e s , ought t o i n f l u e n c e the p r e f e r e n c e o f t h e o r d e r i n g . As i s , the e x p e c t e d u t i l i t y c r i t e r i o n does not h o l d f o r a l t e r n a t i v e s where p r o b a b i l i t i e s a r e not known. T h i s c a s e , t h e r e f o r e , i n c l u d e s n e a r l y a l l p r a c t i c a l d e c i s i o n s i t u -a t i o n s . A second d i f f i c u l t y a r i s i n g from the a ssumption i s t h a t p r o b a b i l i t i e s may n o t e x i s t f o r r e c e i v i n g c e r t a i n rewards f o r some of t h e a c t i o n s . C l e a r l y the e x p e c t e d u t i l i t y c r i t e r i o n can not t h e n be used. Other a l t e r n a t i v e s must be used. I n P a r t I I o f the t h e s i s we s h a l l c o n s i d e r one such a l t e r n a t i v e . 7 1.0 I n t r o d u c t i o n t o P a r t I In P a r t I o f the t h e s i s we a r e concerned w i t h c o n d i t i o n s t h a t must be made on the o r d e r i n g so t h a t the e x p e c t e d u t i l i t y c r i t e r i o n h o l d s . V a r i o u s s e t s o f assumptions can be p o s t u l a t e d from w h i c h an e x p e c t e d u t i l i t y c r i t e r i o n can be c o n c l u d e d . We s h a l l con-s i d e r f i v e o f the most promin e n t approaches h e r e , i n c l u d i n g t h o s e o f vonNeumann-Morgenstern (1947) and Savage (1954). The d i f f e r e n c e s among the approaches a r e due t o p a r t i c u -l a r assumptions made about the p r o b a b i l i t y space and about the space on whi c h p r e f e r e n c e i s d e f i n e d . A l l t h e s e approaches a r e c o n cerned w i t h s u f f i c i e n t c o n d i t i o n s on the o r d e r i n g r a t h e r than n e c e s s a r y c o n d i t i o n s . F o r example, the Savage approach i m p l i e s t h a t the u t i l i t y f u n c t i o n under c o n s i d e r a t i o n must be bounded. T h i s i s c l e a r l y n o t a n e c e s s a r y c o n d i t i o n . Throughout t h e t h e s i s we s h a l l make t h e f o l l o w i n g d i s -t i n c t i o n : I f an assumption i s made w i t h r e g a r d t o the p r e f e r e n c e o r d e r i n g on A, i n d i r e c t l y o r d i r e c t l y , the assump-t i o n w i l l be c a l l e d an axiom. The s e t o f axioms we s h a l l c o n s i d e r can be s t a t e d i n a s i m p l i f i e d form as f o l l o w s : Axiom I . There e x i s t s a r e a l - v a l u e d f u n c t i o n f on the a c t i o n s e t A t h a t p r e s e r v e s t h e o r d e r i n g on A. That, i s , i f a i s p r e f e r r e d t o b, t h e n f ( a ) > f ( b ) , and i f we a r e i n d i f f e r e n t between a and b, then f ( a ) = f ( b ) . We s h a l l d i s c u s s t h i s axiom w i t h i t s i m p l i c a t i o n s i n d e t a i l i n s e c t i o n 3 . 8 Axiom I I . The f u n c t i o n f can be decomposed over the s e t o f e v e n t s i n t o a sum o f f u n c t i o n s h, each o f which depends o n l y on the p a y - o f f on t h a t p a r t i c u l a r e v e n t . F o r example, i f a r e s u l t s i n one o f the p a y - o f f s r , , . . . , r f o r t h e e v e n t s B.,...,B then I n 1' n f ( a ) = h ( r . , B . ) + ... + h ( r ,B ). l ± n n We s h a l l d i s c u s s t h i s axiom i n g r e a t e r d e t a i l i n s e c t i o n 4 . Axiom I I I . The f u n c t i o n h can a l s o be decomposed i n t o the p r o d u c t o f two f u n c t i o n s , W on B and U on R. That i s , Axiom I I I assumes h( r , B ) = where W(B) _i 0 f o r a l l B e One o f t h e i m p l i c a t i o n s o f b a r e two a c t i o n s whose p a y o f f s a l l s t a t e s o f n a t u r e t h e n U(r)-W(B); 3 . t h i s axiom i s t h a t i f a and a r e r and s r e s p e c t i v e l y f o r f (a) - f (b) = h ( r , f t ) - h ( s , f i ) - U(r)W(J_) - U(s)W(J_) = (U(r) - U(s) )W(ft) . T h e r e f o r e f ( a ) > f (b) i f and o n l y i f U(r) i s l a r g e r than U ( s ) . Hence an o r d e r i n g may be s p e c i f i e d on R such t h a t • f (a) > f (b). . i f and o n l y i f r i s p r e f e r r e d t o s ( i . e . , i f U(r) > U ( s ) ) . I n more g e n e r a l terms, h ( r , B ) > h(s,B) i f and o n l y i f r i s p r e f e r r e d t o s. T h e r e f o r e i f f ( a ) i s thought o f as our e v a l u a t i o n o f a l t e r n a t i v e a on Q, h(*,B) can be thought o f as our e v a l u a t i o n o f t h e a l t e r n a t i v e s on the s e t B. S i m i l a r l y , c o n s i d e r the d i f f e r e n c e h ( r , B ) - h ( r , C ) = U(r)(W(B) - W (€) ) and assume t h a t the reward r i s such t h a t U(r) > 0. Then h( r , B ) > h ( r , c ) i f and o n l y i f W(B) > W(c). C o n s i d e r , f o r example, t h a t i f r = $100 and B and C are some a r b i t r a r y e v e n t s , then h ( r , B ) = "our e v a l u a t i o n o f r e c e i v i n g $100 i f event B o c c u r s " and h(r,C) = "our e v a l u a t i o n o f r e c e i v i n g $100 i f event C o c c u r s " . Then our e v a l u a t i o n o f t h e f i r s t i s g r e a t e r (or p r e f e r r e d ) t o the second i f W(B) > W(C). C l e a r l y t h i s would be the case i f the l i k e l i h o o d o f B o c c u r r i n g i s g r e a t e r than the l i k e l i h o o d o f C o c c u r r i n g . T h e r e f o r e , i t seems n e c e s s a r y t o r e l a t e W(B) to the p r o b a b i l i t y o f B. Axiom IV s p e c i f i e s t h i s r e l a t i o n . Axiom IV. W(B) = y (B) . I n s e c t i o n 7 we s h a l l show t h a t t h e s e axioms a r e b o t h n e c e s s a r y and s u f f i c i e n t f o r t h e o r d e r i n g on A t o c o r r e s p o n d w i t h the o r d e r i n g i n d u c e d by t h e e x p e c t e d u t i l i t y . To summarize, the o b j e c t i v e i n the f i r s t p a r t o f the t h e s i s i s : 1) To s p e c i f y a n e c e s s a r y s e t o f axioms f o r the e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d ; 2 ) To i n d i c a t e the i m p l i c a t i o n s o f t h e s e axioms i n terms o f the s t a n d a r d d e c i s i o n problems where the c h o i c e by many would c o n t r a d i c t t h e e x p e c t e d u t i l i t y c r i t e r i o n , and t o p r o v i d e a s h o r t summary o f e m p i r i c a l s t u d i e s w i t h r e g a r d t o t h e ac c e p t a n c e l e v e l o f each axiom; 3 ) To r e l a t e , when p r a c t i c a l , t h i s s e t o f axioms t o those made by o t h e r s . 11 2.0 Assumptions and N o t a t i o n s I n s e c t i o n 2.1 we s h a l l summarize t h e n o t a t i o n t e r m i n o l o g y and assumptions used t h r o u g h o u t P a r t I o f the t h e s i s . The same n o t a t i o n s a r e a l s o used i n P a r t I I o f the t h e s i s a l t h o u g h the assumptions a r e m o d i f i e d . I n s e c t i o n 2.2 we s h a l l s p e c i f y some o f t h e most common approaches t o t h e e x p e c t e d u t i l i t y t h e o r y . 2.1 B a s i c n o t a t i o n s and assumptions We s h a l l f i r s t assume t h a t t h e r e e x i s t s a reward s e t R, whose elements w i l l be denoted by r , s , t , v , e t c . R i s not n e c e s s a r i l y a s e t o f monetary rewards nor a r e t h e rewards n e c e s s a r i l y d e s i r a b l e . We do, however, assume t h a t R i s a w e l l - d e f i n e d s e t , i . e . , t h a t t h e elements a r e d e f i n i t e and d i s t i n c t . We a l s o assume t h e e x i s t e n c e o f a s t a t e space 0, c o n s i s t i n g o f elements ..oo, c a l l e d s t a t e s . T h i s s e t r e p r e s e n t s what may p o s s i b l y happen i n t h e f u t u r e , and can n o t be c o n t r o l l e d o r i n f l u e n c e d . There i s , i n a d d i t i o n , a reward f u n c t i o n X(-,a) from Q t o R, by wh i c h we mean reward X (a j,a) i s o b t a i n e d i f s t a t e oo o c c u r s and a c t i o n a i s chosen. L e t r denote a l l f u n c t i o n s from fl t o R b e i n g c o n s i d e r e d , i . e . , the elements o f F a r e X ( * , a ) , X ( * , b ) , e t c . The s e t {a,b,...} i s denoted by A and i s c a l l e d an a c t i o n s e t ; t h a t i s , T = { X ( - , a ) : a e A ) . The e x i s t e n c e o f t h e s e s e t s makes up t h e b a s i c i n g r e d i e n t s i n d e c i s i o n t h e o r y . There a r e a l s o some a d d i t i o n a l b a s i c m a t h e m a t i c a l assumptions w h i c h must be made. We s h a l l summarize t h e s e j o i n t l y i n Assumption 1. (For a d e f i n i t i o n o f 12 the t e r m i n o l o g y , see Appendix I I . ) Assumption 1. There e x i s t : i ) a p r o b a b i l i t y space ( f t , 0 , u ) , where ft i s the s e t o f s t a t e s , i i ) a measurable space (R,V), where R i s the s e t o f rewards o r outcomes, i i i ) an i n d e x s e t A, c a l l e d an a c t i o n space, such t h a t f o r each a e A t h e r e e x i s t s a reward f u n c t i o n X(•,a) from ft t o R, i v ) a a - a l g e b r a 8 o f ft such, t h a t 0 C 8 and f o r each a e A, the f u n c t i o n X ( * , a ) , ~ i s 0-measurable, and A v) a r e l a t i o n < on A. Because o f the im p o r t a n c e o f r e l a t i o n s and o r d e r i n g s i n ex p e c t e d u t i l i t y t h e o r y , we have summarized the d e f i n i t i o n s and p r o p e r t i e s o f the more common ones i n Appendix I . We s h a l l A a l s o use the r e l a t i o n <, whi c h we d e f i n e a s : A A A a < b i f and o n l y i f a ( b and n o t b < a. A A The r e l a t i o n s < and ^  a r e i n t e r p r e t e d as p r e f e r e n c e s among the a c t i o n s . That i s , i f a c t i o n a i s p r e f e r r e d t o A A a c t i o n b, we s i m p l y w r i t e b < a o r b ^ a. P r e f e r e n c e s e x i s t w i t h o u t knowing e x a c t l y what s t a t e w i l l o c c u r . W i t h t h i s a s s u m p t i o n , e x p e c t e d u t i l i t y t h e o r y i s concerned w i t h the c o n d i t i o n s and a d d i t i o n a l assumptions under which t h e r e e x i s t s a measure W on 6, such t h a t W(B) = u(B) i f B e 0 ; and f u r t h e r , g i v e n a p r e f e r e n c e on A ( t h a t i s a r e l a t i o n A < on A) under what c o n d i t i o n s and assumptions t h e r e e x i s t s a A r e a l v a l u e d f u n c t i o n U on R such t h a t a 4 b i m p l i e s Ju (X (GO,a) )dy ^ j U (X (GO ,b) ) dy . V a r i a t i o n s o f t h i s problem o c c u r when G = 3 / o r when the o r d e r i n g i s d e f i n e d on T or n r a t h e r than on A, where T = { X ( - , a ) : a e A} and n = { a l l p r o b a b i l i t y measures i n d u c e d on R, by the f u n c t i o n s i n r}• The assumptions n e c e s s a r y f o r the e x p e c t e d u t i l i t y c r i -t e r i o n t o h o l d a r e o f two t y p e s . The f i r s t s p e c i f i e s t h e m a t h e m a t i c a l a s s u m p t i o n s . F o r example, how l a r g e can 3 be, o r what a c t i o n s can be i n c l u d e d i n A? Those o f the second t y p e a r e f a r more i m p o r t a n t i n t h a t t hey s p e c i f y the d e c i s i o n maker's b e h a v i o u r . That i s , t he e x t e n t t o which h i s p r e f e r e n c e i n one s i t u a t i o n a l s o s p e c i f i e s h i s p r e f e r e n c e i n a n o t h e r . We a r e most i n t e r e s t e d i n the l a t t e r t y p e o f assumption. Both s e t s o f assumptions a r e , of c o u r s e , i n t e r -r e l a t e d and cannot i n g e n e r a l be s e p a r a t e d . However, f o r each assumption we s h a l l c o n s i d e r our main c o n c e n t r a t i o n t o be on the i m p l i c a t i o n s o f the o r d e r i n g . 2.2 Approaches t o the axioms o f e x p e c t e d u t i l i t y t h e o r y V a r i o u s s e t s of assumptions o r axioms can be p o s t u l a t e d from w h i c h an e x p e c t e d u t i l i t y c r i t e r i o n can be c o n c l u d e d . We s h a l l c o n s i d e r f i v e o f the most promin e n t approaches h e r e : 1. vonNeumann-Morgenstern (1947) 2. Marschak (1950) 3. Savage (1954) 4. Arrow (1971) 5. Luce and K r a n t z (1971) The axioms a re g i v e n i n Appendix I I I and we s h a l l m e r e l y summarize the b a s i c d i f f e r e n c e s h e r e . These d i f f e r e n c e s a r e due t o p a r t i c u l a r assumptions b e i n g made about the p r o b a b i l i t y space and about the space on which a p r e f e r e n c e i s d e f i n e d . Each approach r e p r e s e n t s a d i f f e r e n t a s sumption about the space o f p r e f e r e n c e o r d e r i n g s . vonNeumann-Morgenstern Axioms. The vonNeumann-Morgenstern (1947) approach does n ot make any assumptions d i r e c t l y on the u n d e r l y i n g p r o b a b i l i t y space ( f t , 0 , y ) , o r on r . I n s t e a d i t assumes t h a t I I , the s e t o f p r o b a b i l i t y measures i n d u c e d on R by members o f T, i s e q u a l t o the s e t o f a l l d i s c r e t e p r o b -a b i l i t y measures on R and t h a t the p r o b a b i l i t i e s on a l l s t a t e s a r e known, t h a t i s , 0 = 8 . I t assumes an o r d e r i n g on R, wh i c h i s th e n extended t o I I . Marschak Axioms. Marschak (1950) was t h e f i r s t t o adopt an approach which e s t a b l i s h e s an o r d e r i n g on t h e p r o b a b i l i t y measures. Samuelson (1952) , H e r s t e i n and M i l n o r (1954) and o t h e r / a u t h o r s have a l s o adopted t h i s f o r m u l a t i o n . The axioms c o n s i d e r e d i n the appendix a r e e s s e n t i a l l y the same as Jensen's (1967) axioms, and he has shown them t o i m p l y Marschak's axioms. I n t h i s a p proach, we a l s o i g n o r e the u n d e r l y i n g p r o b a b i l -i t y space s i n c e a l l assumptions a r e based on the i n d u c e d p r o b a b i l i t y measure. T h e r e f o r e , t h i s approach a l s o assumes t h a t the p r o b a b i l i t i e s o f a l l s t a t e s o f n a t u r e a r e known. Savage Axioms. Savage (1954) s t a r t s w i t h a measurable space ( f t , 3 ) . I n h i s approach an o r d e r i n g i s assumed on r such t h a t a p r o b a b i l i t y measure can be d e r i v e d on 3 . F u r t h e r , i t i s assumed t h a t u s i n g t h i s p r o b a b i l i t y measure the o r d e r i n g s a t i s f i e s the e x p e c t e d u t i l i t y c r i t e r i o n . I n t h i s c a s e , 0 = { f i , c f > } . That i s , p r o b a b i l i t i e s a r e o n l y known f o r t h e u n i v e r s a l s e t and the empty s e t . Arrow Axioms. Arrow (1971) b a s i c a l l y uses the Savage axioms; however, Arrow s p e c i f i e s the o r d e r i n g on A r a t h e r t h a n on r. Luce and K r a n t z Axioms. One argument c r i t i c i z i n g t h e Savage Axioms has been t h a t a l l the random v a r i a b l e s have been d e f i n e d on t h e same s t a t e space. For example, the s e t 0, o f s t a t e s o f the w o r l d a p p r o p r i a t e f o r c o n s i d e r i n g b e t t i n g on heads i n a c o i n f l i p i s q u i t e d i f f e r e n t from t h e s e t f_ appro-p r i a t e f o r c o n s i d e r i n g i n v e s t i n g i n a p a r t i c u l a r s t o c k . Luce and K r a n t z (1971) d e v e l o p e d an a x i o m a t i c system t o handle t h i s case by c o n s i d e r i n g the o r d e r i n g on t h e s e t r f l = {r_IB e 3 } p B where _ B c o n t a i n s the f u n c t i o n s i n r w i t h t h e i r domains r e s t r i c t e d t o B. We denote t h e s e f u n c t i o n s by X _ ( ' , a ) , a _ A. _> Some g e n e r a l r e l a t i o n s e x i s t between the d i f f e r e n t approaches. For example, Marschak and vonNeumann-Morgenstern b o t h assume t h a t a n u m e r i c a l p r o b a b i l i t y i s g i v e n , w h i l e t h e o t h e r s do n o t make t h i s a s s u mption. However, t h i s i s not a fundamental d i f f e r e n c e s i n c e by a d d i n g some axioms we can always d e r i v e a p r o b a b i l i t y based on p r e f e r e n c e . T h i s w i l l be d i s c u s s e d i n g r e a t e r d e t a i l i n s e c t i o n 6.3. The o t h e r approaches have a l r e a d y i n c l u d e d the axioms needed t o d e r i v e the p r o b a b i l i t y . Arrow's and Savage's axioms a r e n e a r l y i d e n t i c a l e x c e p t f o r the space i n which they a r e d e f i n e d , though one r e a d i l y t r a n s l a t e s i n t o the o t h e r by the r e a l v a l u e d f u n c t i o n which i s assumed t o e x i s t between A and V. The main d i f f e r e n c e i s t h a t t h e Savage axioms do not n e c e s s a r i l y i m p l y t h a t t h e p r o b a b i l i t y measure i s a - a d d i t i v e . T h i s w i l l be d i s c u s s e d i n s e c t i o n 6.3 and i s a l s o c o n s i d e r e d i n P a r t I I . There i s v e r y l i t t l e d i f f e r e n c e between the Luce and K r a n t z axioms and Savage's. F i r s t , Savage assumed t h a t a l l reward f u n c t i o n s have t h e same domain. Second, he assumed t h a t a d e c i s i o n does n o t a f f e c t t h e p r o b a b i l i t i e s o f the s t a t e s o f n a t u r e . The i n s i g n i f i c a n c e from a t h e o r e t i c a l v i e w p o i n t o f t h e s e d i f f e r e n c e s i n our framework i s i l l u s t r a t e d as f o l l o w s Suppose, f o r example, t h a t a,b e A a r e t h e a c t i o n s such t h a t t h e i r r e s p e c t i v e reward f u n c t i o n s a re d e f i n e d by X ( o j,a) = 1 heads o c c u r s when a c o i n i s f l i p p e d 0 o t h e r w i s e and f 1 i f 3 o c c u r s when a d i e i s r o l l e d X(u>,b) = 0 o t h e r w i s e , I n the Luce and K r a n t z approach we would o n l y need t o c o n s i d e r the s t a t e space {heads o c c u r r i n g , heads n o t o c c u r r i n g } f o r a, and f o r b we c o n s i d e r the s t a t e space {3 o c c u r r i n g , 3 n o t o c c u r r i n g } . I n Savage's approach we can n o t do t h i s . We would have t o c o n s i d e r a l l p o s s i b l e s t a t e s (heads, 1 ) , (heads, 2 ) , . . . , (heads, 6) and ( t a i l s , 1 ) , ( t a i l s , 2),...., ( t a i l s , 6) f o r each a c t i o n . I t i s , perhaps, more d i f f i c u l t t o e v a l u a t e t h e p r o b a b i l i t i e s i n the Savage case s i n c e we would have more s e t s The d i f f e r e n c e i s o n l y a q u e s t i o n o f how t o d e r i v e the s u b j e c t i v e p r o b a b i l i t i e s . There would o n l y be a fundamental d i f f e r e n c e between Savage's and Luce and K r a n t z ' s axioms i f t h e r e e x i s t e d a s e t o f f u n c t i o n s d e f i n e d on a s u b s e t o f Q. f o r which we can not c o n s t r u c t a f u n c t i o n d e f i n e d on a l l o f ft. w i t h t h e same d i s t r i b u t i o n on R.. '' S i n c e t h i s can always be done, the d i f f e r e n c e can be i g n o r e d . We s h a l l c o n s i d e r o t h e r s i m i l a r i t i e s between the s p e c i f i c axioms from the d i f f e r e n t a pproaches, as we r e l a t e them t o the axioms p r e s e n t e d h e r e . 3.0 O r d e r i n g axiom I n t h i s s e c t i o n we s h a l l c o n s i d e r the f i r s t o f the f o u r axioms n e c e s s a r y f o r t h e e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d . I n s e c t i o n 3.1 we s h a l l s t a t e the axiom, i n s e c t i o n 3.2 some of the i m p l i c a t i o n s o f the axiom a r e s t u d i e d , i n 3.3 i t s r e l a t i o n s h i p t o the f i v e s e t s o f s u f f i c i e n t axioms i s examined, and f i n a l l y i n s e c t i o n 3.4 a l t e r n a t i v e r e l a t e d axioms are e x p l o r e d . 3.1 Statement o f Axiom I I n the axioms g i v e n i n t h e i n t r o d u c t i o n , we have chosen t o d e f i n e the o r d e r i n g on t h e a c t i o n s e t A w h i c h i s the most g e n e r a l s e t we can choose f o r t h i s p u r p o s e . Axiom I . E x i s t e n c e axiom There e x i s t s a r e a l - v a l u e d f u n c t i o n f on A such t h a t f o r any a,b e A, A i f a N< b then f (a) <: f (b) , and A i f a < b t h e n f ( a ) < f (b) . The f u n c t i o n f s h a l l be c a l l e d the e v a l u a t i o n f u n c t i o n . S e v e r a l theorems can be found w h i c h s p e c i f y the n e c e s s a r y c o n d i t i o n s f o r the e x i s t e n c e o f f ; see, f o r example, Debreu (1954) and P e l e g (1970). We g i v e a s h o r t summary o f them i n Appendix I . The c o n d i t i o n s f o r f t o e x i s t . c a n be d i v i d e d i n t o A two t y p e s : ( i ) c o n d i t i o n s on t h e r e l a t i o n < and ( i i ) t o p -o l o g i c a l c o n d i t i o n s . For example, i f our p r e f e r e n c e s a re as f o l l o w s : b < a A and a l s o c < b then by assuming the e x i s t e n c e o f f , t h i s i m p l i e s t h a t c can n o t be p r e f e r r e d t o a. That i s , the r e l a -t i o n on A " i n h e r i t s " c e r t a i n p r o p e r t i e s o f the n a t u r a l o r d e r i n g o f r e a l numbers th r o u g h f . For most p r a c t i c a l p u r p o s e s , an i n d i v i d u a l u t i l i t y f u n c -t i o n must be d e r i v e d by some method. We s h a l l n o t go i n t o d e t a i l s as t o how t h i s i s g e n e r a l l y done. However, t h e r e a r e some i m p l i c a t i o n s on the o r d e r i n g which we need t o c o n s i d e r . In d e r i v i n g a u t i l i t y f u n c t i o n a s e t o f r e l a t i v e l y e a s i l y made c h o i c e s o f some a l t e r n a t i v e s i s d e t e r m i n e d i n such a way t h a t a u t i l i t y f u n c t i o n may be s p e c i f i e d . T h i s f u n c t i o n may not be the same as t h e u t i l i t y f u n c t i o n ( i f i t e x i s t s ) when we con-s i d e r t h e a l t e r n a t i v e s i n A . T h i s c r e a t e s d i f f i c u l t i e s from a p r a c t i c a l v i e w p o i n t , i . e . , an i n d i v i d u a l u t i l i t y f u n c t i o n may depend on the a l t e r n a t i v e w hich i s b e i n g c o n s i d e r e d . To a v o i d t h i s , we s h a l l assume t h a t A always c o n t a i n s those a c t i o n s needed f o r d e r i v i n g a u t i l i t y f u n c t i o n , and s e c o n d l y t h a t t h e e v a l u a t i o n f u n c t i o n i s always the same f o r a g i v e n a c t i o n . T h i s i m p l i e s t h a t we e v a l u a t e an a c t i o n on i t s own m e r i t s w i t h o u t c o n s i d e r i n g i t s r e l a t i o n s t o o t h e r a c t i o n s . In m a t h e m a t i c a l terms t h i s i s e q u i v a l e n t t o c o n s i d e r i n g o n l y < A the o r d e r i n g on s u b s e t s o f A w h i c h have been i n d u c e d by <. That i s , i f A_ C A t h e n t h e o n l y o r d e r i n g we s h a l l c o n s i d e r on A i s d e f i n e d by o 1 * r = <. n a x a _ o o ,or, i f A^ 3 A t h e n we o n l y c o n s i d e r o r d e r i n g s on A-^  such t h a t A A i ^ 4 = < C\ A. x A. O r d e r i n g s i n d u c e d i n t h i s way we s h a l l c a l l h e r e d i t a r y  o r d e r i n g s . The t o p o l o g i c a l i m p l i c a t i o n s a r e c o n cerned w i t h what i s meant by a sequence o f a c t i o n s a p p r o a c h i n g a g i v e n a c t i o n . T h i s i s c o n s i d e r e d i n Appendix I . Another t o p o l o g i c a l p r o -p e r t y on A i s the c a r d i n a l i t y o f the s e t o f i n d i f f e r e n c e c l a s s e s . I f A^ i s the s u b s e t of A such t h a t we a r e never i n -d i f f e r e n t between any two o f the a l t e r n a t i v e s i n A^, then th e c a r d i n a l i t y o f A^ must be l e s s than o r e q u a l t o the c a r d i n a l -i t y o f t h e r e a l numbers, o t h e r w i s e f c l e a r l y does not e x i s t . T h i s c o n d i t i o n we s h a l l assume always h o l d s i n s e c t i o n 4.2. From our p o i n t o f view we a r e most i n t e r e s t e d i n whether Axiom I i s a r e a s o n a b l e a ssumption f o r an i n d i v i d u a l t o make. T h e r e f o r e we a r e m a i n l y i n t e r e s t e d i n the i m p l i c a t i o n s on the o r d e r i n g . A 3.2 I m p l i c a t i o n s o f some assumptions c o n c e r n i n g < In t h i s s e c t i o n we s h a l l c o n s i d e r the i m p l i c a t i o n s o f the o r d e r i n g from a d e c i s i o n maker's p o i n t o f view. Hence t h e q u e s t i o n we s h a l l c o n s i d e r i s whether i t i s " r e a s o n a b l e " t o e x p e c t the d e c i s i o n maker t o have a t r a n s i t i v e p r e f e r e n c e o r a complete o r a p a r t i a l o r d e r i n g o r , f o r t h a t m a t t e r , i f i t i s a c c e p t a b l e t o assume t h a t the o r d e r i n g i s h e r e d i t a r y . We s h a l l c o n s i d e r t h e s e i s s u e s i n t u r n . T r a n s i t i v i t y . I n Appendix I t r a n s i t i v i t y i s d e f i n e d as a b i n a r y r e l a t i o n such t h a t i f the p r o p e r t y h o l d s between a and b and between b and c, i t must a l s o h o l d between a and c. Rather than d i s c u s s whether t h i s i s a r e a s o n a b l e o r unreason-a b l e assumption t o make we s h a l l c o n s i d e r a few examples where t r a n s i t i v i t y h o l d s and a few where i t does n o t . We s h a l l s t a r t w i t h some t r i v i a l examples, and th e n d i s c u s s some r e l a t e d t o some d e c i s i o n problems. I n d o i n g so, we hope f o r a b e t t e r u n d e r s t a n d i n g o f t h e i m p l i c a t i o n s o f Axiom I . Note t h a t Axiom I does not q u i t e i m p l y t r a n s i t i v i t y ; t h a t i s , i f t h e r e e x i s t p r e f e r e n c e s between a and b, b and c and a l s o between a and c, the p r e f e r e n c e s must be t r a n s i t i v e . Some of the most common examples o f t r a n s i t i v e r e l a t i o n s a r e the p h y s i c a l p r o p e r t i e s such as ' h e a v i e r t h a n ' , ' s m a l l e r t h a n ' , ' l o n g e r t h a n ' and so on. An example o f a n o n - t r a n s i t i v e r e l a t i o n i s ' f a t h e r o f ' . L e s s t r i v i a l examples o f t h e non-t r a n s i t i v i t y i n d e c i s i o n t h e o r y can be c a t e g o r i z e d i n t o e i t h e r o f t h e f o l l o w i n g two c a s e s . The f i r s t case o f n o n - t r a n s i t i v e o r d e r i n g o c c u r s when t h e reward s e t R i s formed by a C a r t e s i a n p r o d u c t and the o r d e r i n g i s i n d u c e d on R by an o r d e r i n g on each component i n t h e C a r t e s i a n p r o d u c t s e p a r a t e l y . The second case o f n o n - t r a n s i t i v e o r d e r i n g o c c u r s when the o r d e r i n g i s i n d u c e d by a p r o b a b i l i t y measure. We s h a l l i l l u s t r a t e b o t h t h e s e approaches below. To c o n s i d e r the f i r s t c a s e , suppose t h a t t h e reward s e t i s e q u a l t o the p r o d u c t o f t h r e e s e t s , R x S x T. The reward f o r a c t i o n a can th e n be w r i t t e n as t h e t r i p l e ( r _ , s _ , t _ ) where r a e R, s a e S and t a e T. I f . i n d i v i d u a l o r d e r i n g s are f i r s t d e f i n e d s e p a r a t e l y on R, S and T, t h e f o l l o w i n g p r e f e r e n c e may o c c u r f o r a c t i o n s a, b and c: R R R r a < r b r b < r c r a < r c S S s s a < S b s c < S b s c < s a T < t a % T < t c t c T < t a Hence a c t i o n b i s p r e f e r r e d i n two o f t h r e e consequences o v e r a, s i m i l a r l y c o v er b, and a o v e r c. T h e r e f o r e i f an o v e r a l l o r d e r i n g i s d e f i n e d on the a c t i o n s by assuming t h a t an a r b i t r a r y a c t i o n c i s p r e f e r r e d t o a n o t h e r i f the f i r s t a c t i o n i s p r e -f e r r e d on a t l e a s t two o f t h r e e r ewards, t h e n the i n d u c e d o r d e r i n g on the o v e r a l l reward s e t i s n o t t r a n s i t i v e as our example i l l u s t r a t e d s i n c e b i s p r e f e r r e d t o a, c t o b but a i s p r e f e r r e d t o c. An i l l u s t r a t i o n o f t h i s type o f i n t r a n s i t i v i t y i s as f o l l o w s : L e t our a c t i o n s be a c h o i c e between a p p l e p i e , b l u e -b e r r y p i e and c h e r r y p i e , and l e t t h e s e l e c t i o n be based on t a s t e , f r e s h n e s s and s i z e . I t i s easy t o see how t h i s may g i v e the c o n t r a d i c t i o n by l e t t i n g r a s t a n d f o r the t a s t e o f t h e a p p l e p i e , s = f o r i t s f r e s h n e s s , and t f o r i t s s i z e ; C I cl s i m i l a r l y d e f i n e r ^ , s^ and t ^ f o r the b l u e b e r r y p i e and r , s c and t f o r the c h e r r y p i e and use t h e o r d e r i n g g i v e n above. A v a r i a t i o n o f t h i s would be t o c o n s i d e r t h r e e teams, A, B, and C, each team h a v i n g t h r e e p l a y e r s w h i c h must p l a y a g a i n s t each p l a y e r on a n o t h e r team. The p l a y e r s a r e ranked from 1 t o 9 (9 i s the b e s t and 1 the w o r s t ) . Hence i f the p l a y e r s on the teams A, B and C a r e ranked ( 8 , 1 , 6 ) , (3,5,7) and (4,9,2) r e s p e c t i v e l y and i f each p l a y e r meets e v e r y o t h e r p l a y e r on the o p p o s i n g team, we note t h a t A d e f e a t s B (w i n n i n g 5 games out o f 9 ) , B d e f e a t s C but C d e f e a t s A. A second c l a s s o f o r d e r i n g t h a t c o n t r a d i c t s t r a n s i t i v i t y o c c u r s when the o r d e r i n g i s based on p r o b a b i l i t i e s . As an example c o n s i d e r the o r d e r i n g s p e c i f i e d by the r e l a t i o n A a > b i f and o n l y i f y{X(-,a) > X(-,b)} > 1/2. The q u e s t i o n o f t r a n s i t i v i t y a r i s i n g here i s the f o l l o w i n g : I f y(X > Y) > 1/2 and y(Y > Z) > 1/2, i s i t a l s o t r u e t h a t y (X > Z) > 1/2? The answer i s no as the f o l l o w i n g i l l u s t r a t i o n shows: L e t X(u)iy.a) = 1 to e Q 5 0) E B 0 0) £ B -1 0) e D 2 £ B 4 to £ B-D where D C B. I f we assume t h a t y(B) = 1/2 - £ and y(D) = 2 E , where 0 < £ < 1/4, then y(X > Y) = y(B) = 1/2 + e, and y(Y > Z) = y(B) + y(D) = 1/2 + e, b u t y (X > Z) = 2E < 1/2. A A A Hence b < a- and c <•. b b u t a < c. - . i_ A n o t h e r i l l u s t r a t i o n i s the f o l l o w i n g game. Two o f the f o u r d i c e shown i n F i g . 3 a r e r o l l e d and the d i e w i t h the h i g h e r number wins (Gardner, 1974) . 0 3 2 5 4 0 4 . 3 3 3 2 2 2 1 1 1 4 4 3 3 6 6 5 5 F i g . 3 . 1 A n o n - t r a n s i t i v e s e t of d i c e Note t h a t d i e a w i l l win over d i e b w i t h the p r o b a b i l i t y 2 / 3 ; and s i m i l a r l y b wins over c, and c wins over d. However, d wins over a w i t h the p r o b a b i l i t y 2 / 3 . For an i n t e r e s t i n g study of t r a n s i t i v i t y , see May ( 1 9 5 4 ) . He c o n s i d e r e d s e v e r a l examples of n o n - t r a n s i t i v e r e l a t i o n s , e.g., the case of a p i l o t f a ced w i t h the c h o i c e of flames or red-hot metal, red-hot metal or f a l l i n g , or f a l l i n g or flames. In the examples above there e x i s t s no o r d e r p r e s e r v i n g , r e a l - v a l u e d f u n c t i o n on the s e t A of a l t e r n a t i v e s . There i s a s t r o n g argument f o r a c c e p t i n g t r a n s i t i v i t y which i s des-c r i b e d i n R a i f f a ( 1 9 6 1 ) . The e x i s t e n c e of ' -the e v a l u a t i o n - f u n c t i o n i s a means by which we can determine a v a l u e f o r each a l t e r n a t i v e . In most p r a c t i c a l cases we assume t h i s i s a f u n c t i o n of some monetary v a l u e , i . e . , we would be w i l l i n g to pay more f o r the choice of a p a r t i c u l a r given a l t e r n a t i v e r a t h e r than another one. R a i f f a used t h i s argu-ment to support the p r o p o s i t i o n t h a t t r a n s i t i v i t y i s a reason-a b l e assumption. For example, suppose p r e f e r e n c e s are non-t r a n s i t i v e , i . e . , a i s p r e f e r r e d t o b b i s p r e f e r r e d t o c and c i s p r e f e r r e d t o a, R a i f f a t h e n argues t h a t i f t h e i n d i v i d u a l i s g i v e n a l t e r n a t i v e a, he would be w i l l i n g t o pay a s m a l l amount t o s w i t c h t o c s i n c e c i s p r e f e r r e d t o a. Once c i s o b t a i n e d , R a i f f a a g a i n assumes f o r the same re a s o n t h a t one would be w i l l i n g t o pay a s m a l l amount t o s w i t c h t o b, from b t o c, and so on, w i t h a s m a l l amount b e i n g p a i d f o r each s w i t c h . . S i n c e i t i s o b v i o u s t h a t one would n o t be w i l l i n g t o c o n t i n u e p a y i n g a s m a l l amount f o r each s w i t c h , R a i f f a c o n c l u d e s t h a t t r a n s i t i v i t y must be s a t i s f i e d by the o r d e r i n g . However, assuming t h a t a l l a l t e r n a t i v e s can be compared on a monetary s c a l e i s tantamount t o assuming the e x i s t e n c e o f an e v a l u a t i o n f u n c t i o n . T h e r e f o r e R a i f f a assumes the o r d e r i n g i s t r a n s i t i v e i f a r e a l - v a l u e d f u n c t i o n e x i s t s . P a r t i a l o r d e r i n g v s . complete o r d e r i n g , one theorem 1 i n Appendix I whi c h used a p a r t i a l o r d e r i n g r a t h e r t h a n a complete o r d e r i n g i s , due t o P e l e g ( 1 9 7 0 ) . However, the i d e a b e h i n d a p a r t i a l o r d e r i n g i s due t o Aumann ( 1 9 6 2 ) who f i r s t p roved an ex p e c t e d u t i l i t y theorem w i t h o u t u s i n g a complete o r d e r i n g , b u t s i n c e P e l e g ' s theorem i s a l i t t l e more g e n e r a l we quoted h i s . The q u e s t i o n Aumann r a i s e d was: "Does r a t i o n a l i t y demand t h a t an i n d i v i d u a l make d e f i n i t e p r e f e r e n c e comparisons between a l l p o s s i b l e a c t i o n s ? " As an example he g i v e s h i s p r e f e r e n c e s : " I p r e f e r a cup o f cocoa t o a 7 5 - 2 5 l o t t e r y o f c o f f e e and t e a , but r e v e r s e my p r e f e r e n c e i f the r a t i o i s 2 5 - 7 5 . " However, can a break-even p o i n t be dete r m i n e d between "a l o t t e r y and the cocoa? For our purpose t h i s i s i r r e l e v a n t s i n c e w i t h the assump-t i o n o f t h e e x i s t e n c e o f f , a complete o r d e r i n g may be i n d u c e d on A which c o r r e s p o n d s t o any p a r t i a l o r d e r i n g f o r t h o s e elements w h i c h can be compared. T h e r e f o r e , t h e c o n d i t i o n s on f a r e i d e n t i c a l i n b o t h c a s e s . H e r e d i t a r y o r d e r i n g . W i t h a l l h e r e d i t a r y o r d e r i n g s we assumed t h a t an a l t e r n a t i v e may be removed w i t h o u t a f f e c t i n g the r e l a t i v e o r d e r i n g o f the r e m a i n i n g a l t e r n a t i v e s . I f , f o r example, A = {a,b,c} and a i s p r e f e r r e d t o b, then i f a l t e r n a t i v e c i s n o t p o s s i b l e , i . e . , A = {a,b} t h e n a ought t o be s t i l l p r e f e r r e d t o b. I n some c r i t e r i a i n d e c i s i o n making under u n c e r t a i n t y t h i s i s not the c a s e . C o n s i d e r , f o r example, Savage's r e g r e t c r i t e r i o n i n the f o l l o w i n g example. We a r e g i v e n a c h o i c e between a c t i o n s a, b and c w i t h the p a y o f f s as i n the f o l l o w i n g p a y o f f m a t r i x . Event A c t i o n Space B B a 3 6 b 0 10 c 6 0 The r e g r e t m a t r i x i s formed by s u b t r a c t i n g each e n t r y i n each column from the l a r g e s t e n t r y i n the column and hence would g i v e t h e m a t r i x : E v e n t A c t i o n Space B B a 3 4 b 6 0 c 0 10 The maximum r e g r e t f o r a c t i o n s a, b and c would be 4, 6 and 10 r e s p e c t i v e l y and we o r d e r t h e a c t i o n s by c h o o s i n g t h e minimum A A o f t h e maximum r e g r e t s , hence t h e o r d e r i n g becomes c- <, b < a.. However i f a c t i o n c i s n o t i n c l u d e d t h e r e g r e t m a t r i x would A change such t h a t our o r d e r i n g would be a < b. Anot h e r example has been s u g g e s t e d by Luce and R a i f f a (1957) w h i c h c o n t r a d i c t s the h e r e d i t a r y o r d e r i n g . "A gentleman wandering i n a s t r a n g e c i t y a t d i n n e r time chances upon a modest r e s t a u r a n t w h i c h he e n t e r s u n c e r t a i n l y . The w a i t e r i n f o r m s him t h a t t h e r e i s no menu, b u t t h a t t h i s e v e n i n g he may have e i t h e r b r o i l e d salmon a t $2.50 o r s t e a k a t $4.00. I n a f i r s t - r a t e r e s t a u r a n t h i s c h o i c e would have been s t e a k , b u t c o n s i d e r i n g h i s unknown s u r r o u n d i n g s and the d i f f e r e n t p r i c e s he s e l e c t s t h e salmon. Soon a f t e r t h e w a i t e r r e t u r n s from the k i t c h e n , a p o l o g i z e s p r o f u s e l y , b l a m i n g the uncommunicative c h e f f o r o m i t t i n g t o t e l l him t h a t f r i e d s n a i l s and f r o g ' s l e g s a r e a l s o on the b i l l o f f a r e a t $4.50 each. I t so happens t h a t o u r hero d e t e s t s them b o t h and would always s e l e c t salmon i n p r e f e r e n c e t o e i t h e r , y e t h i s response i s , " S p l e n d i d , I ' l l change my o r d e r t o s t e a k . " A j u s t i f i c a t i o n Jfcjor t h e gentleman changing h i s o r d e r can be g i v e n as f o l l o w s . He e n t e r s an unknown r e s t a u r a n t w i t h o u t knowing th e q u a l i t y o f f o o d he e x p e c t e d t o be s e r v e d . He p l a y e d i t s a f e and d e c i d e d bad salmon i s b e t t e r t h a n s t e a k f o r t h e p r i c e s p e c i f i e d . Once he found o u t t h a t t h e r e s t a u r a n t a l s o has f r o g s * l e g s ( i . e . , i s a b e t t e r c l a s s r e s t a u r a n t ) , he d e c i d e d t h i s r e s t a u r a n t would n o t s e r v e a bad s t e a k o r a bad salmon. Hence h i s c h o i c e becomes a good s t e a k o r a good salmon f o r t h e s p e c i f i e d p r i c e s , and t h e r e f o r e h i s c h o i c e changes. Hence, he o b t a i n s more i n f o r m a t i o n by t h e a d d i t i o n o f t h e new a c t i o n . I n g e n e r a l when we speak o f h e r e d i t a r y o r d e r i n g we assume no a d d i t i o n a l i n f o r m a t i o n i s g i v e n by a d d i n g a n o t h e r a c t i o n . A n o t h e r common example s t a t e d i s t h e v o t i n g paradox. L e t 1/3 o f a l l v o t e r s have t h e p r e f e r e n c e o f the t h r e e c a n d i d a t e s a, b and c as c < b •<• a", a n o t h e r t h i r d o f t h e v o t e r s the p r e f e r e n c e a < c < b,, and t h e f i n a l t h i r d b: < a ••<•. c . Hence 2/3 o f t h e v o t e r s p r e f e r b t o c, and 2/3 p r e f e r c t o a, and 2/3 p r e f e r a t o b. T h i s i n t e r e s t i n g r e s u l t i s perhaps i n r e l a t i o n t o t h e h e r e d i t a r y a s s u m p t i o n . I f a d i d n o t r u n , b would w i n ; i f b d i d not r u n , c would w i n ; and i f c d i d not r u n , a would w i n . 3.3 O r d e r i n g p r o p e r t i e s i n t h e d i f f e r e n t approaches We have s t a t e d how each o f t h e d i f f e r e n t approaches we c o n s i d e r here i m p l i e s t h e e x p e c t e d u t i l i t y t h e o r y b u t uses d i f f e r e n t axiom systems t o prove an e x p e c t e d u t i l i t y theorem. Here we s h a l l compare t h e d i f f e r e n t approaches t o Axiom I . We f i r s t note t h a t : 1) vonNeumann-Morgenstern d e f i n e p r e f e r e n c e o r d e r i n g s oh the s e t R, 2 ) Marschak d e f i n e s p r e f e r e n c e o r d e r i n g s on the s e t o f a l l d i s c r e t e p r o b a b i l i t y measures on R, 3 ) Savage d e f i n e s p r e f e r e n c e o r d e r i n g s on the s e t o f a l l p o s s i b l e f u n c t i o n s from 0, t o R, 4 ) Arrow d e f i n e s p r e f e r e n c e o r d e r i n g s on the s e t o f a c t i o n s A, 5 ) Luce and K r a n t z d e f i n e p r e f e r e n c e o r d e r i n g s on an a r b i t r a r y s e t o f f u n c t i o n s . Each o f t h e s e axiom systems assumes a complete o r d e r i n g . The most g e n e r a l o f t h e s e assumptions i s Arrow's. F o r example, a and b may be two a c t i o n s b e l o n g i n g t o A such t h a t X(o),a) = X(u),b) f o r a l l to e 0. where f (a) ^ f (b) b u t c l e a r l y X(-,a) = X ( * , b ) . S i m i l a r l y two f u n c t i o n s X(',a) and X(*,b) may have the p r o p e r t y X(u>,a) ^ X (o j,b) f o r a l l to e Q b u t P(C,a) = P(C,b) f o r a l l C e ¥, i . e . , the f u n c t i o n s X(-,a) and X(*,b) i n d u c e the same p r o b a b i l i t y measure on the rewards but X(-,a) ^ X ( - , b ) . T h e r e f o r e , some d e c i s i o n makers would f i n d i t e a s i e r t o a c c e p t an o r d e r i n g on t h e reward s e t R r a t h e r t h a t on the a c t i o n s e t A, f o r example. The r e a s o n f o r t h i s i s o b v i o u s : the c a r d i n a l i t y o f t h e s e t f o r the most g e n e r a l c a ses a re as f o l l o w s : C(R) « C(IT) 4 C(T) < C(Tb) < C (A) , where C(R) denotes the c a r d i n a l i t y o f R, and s i m i l a r l y f o r C ( I I ) , C ( r ) , and so on. That i s , i f we assume an o r d e r i n g on A, we can induce an o r d e r i n g on r w i t h a s u b s e t o f A, and s i m i l a r l y , i f we assume an o r d e r i n g on a p r o b a b i l i t y d i s t r i -b u t i o n on R, we can in d u c e an o r d e r i n g on R w i t h a s u b s e t o f I I . T h e r e f o r e , t h e g e n e r a l assumption o f an o r d e r i n g on A i s a s t r o n g e r assumption than an o r d e r i n g on R. There a l s o e x i s t a d d i t i o n a l assumptions on t h e c a r d i n a l i t y . F o r example, vonNeumann-Morgenstern assume t h a t between e v e r y R two rewards t h e r e e x i s t s a n o t h e r , i . e . , i f r ^ , ^ e R, r ^ < ^ , R R t h e r e e x i s t r e R such t h a t r , < r < r 0 . Marschak assumes o 1 o ^ t h a t a l l p r o b a b i l i t y measures on R b e l o n g t o I I . Savage assumes t h a t a l l f u n c t i o n s from ft t o R b e l o n g t o r. A l l t h e s e assump-t i o n s a r e n o t n e c e s s a r y . I t i s s u f f i c i e n t t o assume t h a t t h e f u n c t i o n f e x i s t s on A, and the o r d e r i n g may be r e p r e s e n t e d by the n u m e r i c a l v a l u e o f t h e f u n c t i o n . 3 . 4 A l t e r n a t i v e s t o Axiom I Most n o r m a t i v e t h e o r i e s o f d e c i s i o n - m a k i n g a c c e p t Axiom I ; t h a t i s , t h e e x i s t e n c e o f a r e a l - v a l u e d e v a l u a t i o n f u n c t i o n . Minimax, maximax and e x p e c t e d v a l u e s , f o r example, a l l s a t i s f y Axiom I . S i m i l a r l y many o f t h e c r i t e r i a i n f i n a n c e such as the pay-back method, the n e t - p r e s e n t v a l u e , and the i n t e r n a l r a t e o f r e t u r n a l l assume Axiom I . As a m a t t e r o f f a c t , v e r y few a l t e r n a t i v e s t o Axiom I can be p r e s e n t e d . Two d i f f e r e n t approaches a r e d i s c u s s e d below. S t o c h a s t i c u t i l i t y t h e o r y . Some e m p i r i c a l s t u d i e s have shown t h a t many pe o p l e a r e n o t c o n s i s t e n t i n r e p e a t e d c h o i c e s i t u a t i o n s . That i s , sometimes they p r e f e r a t o b and some-t i m e s the o p p o s i t e . T h i s has l e d t o what i s c a l l e d the " s t o c h a s t i c u t i l i t y t h e o r y " . In t h i s t h e o r y , t h e axioms are s t a t e d i n terms o f p r o b a b i l i t i e s o f c h o i c e . F o r i n s t a n c e , Debreu's axioms (1958) a r e as f o l l o w s , where a, b, c and d are a r b i t r a r y a c t i o n s : Axiom 1: S i s a s e t ; p i s a f u n c t i o n from S x S t o (0,1) such t h a t p(a,b) + p(b,a) = 1. Axiom 2: (p(a,b) < p ( c , d ) ) i m p l i e s (p(a,c) < p ( b , d ) ) . Axiom 3: I f p(b,a) < q < p(c,a) where q i s any r e a l number, t h e n t h e r e i s a,d e S such t h a t p(d,a) = q. The i n t e r p r e t a t i o n o f p(a,b) i s t h a t a i s p r e f e r r e d t o b, a p r o p o r t i o n p(a,b) o f the t i m e . Debreu proves t h a t t h e s e axioms i m p l y t h a t t h e r e e x i s t s a r e a l - v a l u e d f u n c t i o n V on S such t h a t p(a,b) < p(c,d) i f and o n l y i f V(a) - V(b) < V(c) - V ( d ) . C l e a r l y the f u n c t i o n V can n o t be c o n s i d e r e d as an e v a l u a t i o n f u n c t i o n o f a c t i o n s s i n c e t h i s i m p l i e s t h a t i f V(a) > V(b) then p(a,b) = 1, and hence p must be a f u n c t i o n t o {0,1}. From our d i s c u s s i o n on o r d e r i n g i n d u c e d by p r o b a b i l i t y , i t f o l l o w s t h a t i f p(a,b) > 1/2 and p(b,c) > 1/2 i t i s not n e c e s s a r y t h a t p ( a ,c) > 1/2, i . e . , i f a c t i o n a i s p r e f e r r e d more o f t e n t o a c t i o n b, and i f b i s p r e f e r r e d more o f t e n t o a c t i o n c, i t does not f o l l o w t h a t a i s p r e f e r r e d more o f t e n t o a c t i o n c. M u l t i v a r i a n t e v a l u a t i o n f u n c t i o n s . A nother a l t e r n a t i v e t o Axiom I i s t o assume f as a f u n c t i o n from A t o E n (the n-dimension E u c l i d i a n s p a c e ) . I n mapping i n t o E n more v a r i -a t i o n s can be d e f i n e d on the o r d e r i n g than on E as we do not need t o s a t i s f y t r a n s i t i v i t y . C o n s i d e r , f o r example, the 2 r e l a t i o n < on E d e f i n e d by ( a , 3 ) < ( y , 5 ) i f and o n l y i f B 2 Y 2 < a 2S 2. F o r t h i s r e l a t i o n a r e a l - v a l u e d f u n c t i o n s a t i s -f y i n g Axiom I does n o t e x i s t . As an example t o show t h a t n-dimension E u c l i d i a n space i s sometimes a p p r o p r i a t e , one might c o n s i d e r a group o f i n d i v i d u a l s where each one s a t i s f i e s Axiom I on the a c t i o n se An e v a l u a t i o n f u n c t i o n f can t h e n be d e f i n e d as f ( a ) = (f.. (a) , f 9 (a) , . . . , f (a)) where f • ( a ) i s the i i n d i v i d u a l ' s _ ^ n i e v a l u a t i o n f u n c t i o n . Sometimes t h i s f u n c t i o n i s t h e n reduced t o E 1 by A n n 1) a < b i f T f. (a) < £ • (b) . i = l 1 i = l 1 (Hence t h i s i s reduced t o Axiom I and i s used i n f i n a n c e f o r n e t - p r e s e n t v a l u e . ) A 2 ) a < b i f f (a) L f (b) where L i s the r e l a t i o n " l e x i o g r a p h i c a l l y l a r g e r " . T h i s approach i s used i n s o c i a l c h o i c e s . 4.0 A d d i t i v i t y Axiom I n t h i s s e c t i o n we s h a l l d i s c u s s Axiom I I , s t a r t i n g by s t a t i n g the axiom i n s e c t i o n 4.1. The axiom has some v e r y s t r o n g i m p l i c a t i o n s and i n s e c t i o n 4.2 we s h a l l c o n s i d e r some "paradoxes" t h a t have been proposed i n r e l a t i o n t o i t . I n s e c t i o n 4.3 we s h a l l summarize some e m p i r i c a l s t u d i e s a s s o -c i a t e d w i t h some o f t h e s e p a r a d o x e s , and i n s e c t i o n 4.4 we c o n s i d e r the r e l a t i o n s h i p t o o t h e r axiom systems. F i n a l l y , i n s e c t i o n 4.5 some a l t e r n a t i v e s t o Axiom I I a r e s t a t e d . 4.1 Statement o f Axiom I I L e t us assume t h a t a g i v e n a c t i o n a e A r e s u l t s i n one o f the f o l l o w i n g p o s s i b l e rewards: r r i f the event I n B^,...,B n o c c u r s r e s p e c t i v e l y . Axiom I I as s t a t e d i n t h e i n t r o d u c t i o n a s s e r t s t h e e x i s t e n c e o f a r e a l - v a l u e d f u n c t i o n h, s a t i s f y i n g t h e i d e n t i t y f ( a ) = h ( r , ,B.) +. . .+ h ( r , B J . 1 1 n n As a g e n e r a l i z a t i o n o f t h i s axiom, we s h a l l a l l o w f o r t h e reward f u n c t i o n X(-,a) t o r e s u l t i n any o f an u n c o u n t a b l e number o f rewards on any s e t B e g . Note t h a t t h e o r d e r i n g we d e f i n e d on A c o u l d have been d e f i n e d on r, i f t h e r e i s a one-to-one f u n c t i o n between A and V, i . e . , we c o u l d have w r i t t e n f ( X ( - , a ) ) f o r f ( a ) . The r e a s o n f o r not d o i n g so was b o t h t o emphasize t h e f a c t t h a t we chose an a c t i o n a, wh i c h gave us the reward f u n c t i o n X(*,a) and a l s o the s i m p l i c i t y o f n o t a t i o n . However, a f u r t h e r u n d e r s t a n d i n g can be a c h i e v e d by c o n s i d e r i n g the axioms i n terms o f random v a r i a b l e s . C o n s i d e r the s e t T , i . e . , t h e s e t o f a l l f u n c t i o n s X_,(*,a), a e A where X_,(",a) i s the r e s t r i c t i o n o f X(-,a) t o B. Axiom I assumes t h a t a r e l a t i o n e x i s t s on T, o r a t l e a s t t h a t an o r d e r i n g may be i n d u c e d . A n a t u r a l e x t e n s i o n o f t h i s a ssumption would be t o assume t h a t t h e r e a l s o e x i s t s an o r d e r i n g on __, f o r any B e 3 , as Luce and K r a n t z d i d , and a r e a l - v a l u e d o r d e r p r e -s e r v i n g f u n c t i o n on T . Axiom I I makes t h i s a ssumption i n 1 3 terms o f a c t i o n s r a t h e r than reward f u n c t i o n s . Axiom I I t h e r e f o r e assumes f i r s t t h e e x i s t e n c e o f a u n i q u e l y d e f i n e d r e a l - v a l u e d f u n c t i o n h(B,a) on 3 x A, where h(B,a) may be r e g a r d e d as t h e e v a l u a t i o n f u n c t i o n o f t h e reward f u n c t i o n X_,(*,a). I t a l s o s p e c i f i e s the r e l a t i o n between f ( a ) and h ( B , a ) . I f , f o r example, a,b e A, and h(B,a) = h ( B , b ) , t h a t i s , we are i n d i f f e r e n t between a c t i o n a and a c t i o n b i f event B o c c u r s , and i f h(B,a) > h ( B , b ) , t h a t i s we p r e f e r a c t i o n a t o a c t i o n b i f event B o c c u r s , Axiom I I c o n c l u d e s t h a t a c t i o n a i s p r e f e r r e d t o a c t i o n b. I t i s c l e a r l y n o t n e c e s s a r y t o assume t h a t an o r d e r i n g e x i s t s on r . However, s i n c e h(B,a) i s assumed t o e x i s t , i t _> i s perhaps e a s i e s t t o c o n s i d e r h(B,*) as the e v a l u a t i o n f u n c t i o n o f an o r d e r i n g on r n . / 3 5 Axiom I I . A d d i t i v i t y axiom There e x i s t s a r e a l v a l u e d f u n c t i o n h on 6 x A such t h a t a) f (a) = h(f i , a ) b) f o r {B.} i = l , . . . such t h a t B. e 8 f o r a l l i and B. n B. = 0 f o r i ^ j , t h e n h ( U B . , a ) = X h(B.,a) f o r 1 3 1 i = l 1 a l l a e A c) f o r any B e 8 / and f o r any a,b e A such t h a t X B ( - , a ) = X B ( - , b ) , t h e n h(B,a) = h ( B , b ) . The n e c e s s i t y of p a r t b) o f t h i s axiom f o l l o w s from the f a c t t h a t the i n t e g r a l i s a - a d d i t i v e , t h a t i s , i f we assume {B.} i s a p a r t i t i o n o f 0, such t h a t B. e 0 and i f we assume f o r the time b e i n g (as we s h a l l prove i n l a t e r s e c t i o n s ) t h a t h ( B ± , a ) = j UX(aj,a)dy, B. l t h e n c l e a r l y i f f ( a ) i s e q u a l t o the e x p e c t e d u t i l i t y o f X(«,a) we have the f o l l o w i n g i d e n t i t i e s : ,a) = J f ( a ) = EUX(-  / UX(aj,a)dy f / i = l J UX ( o j,a)dy B.= X h(B. ,a) . i = l P a r t c o f t h e axiom s p e c i f i e s i n terms of i n t e g r a l t h a t i f two f u n c t i o n s a r e e q u a l they must have the same i n t e g r a l v a l u e . I n the normal approach t o i n t e g r a t i o n h(B,a) i s d e f i n e d f o r f u n c t i o n s w h i c h a r e c o n s t a n t on B. An e x t e n s i o n i s t h e n made t o f u n c t i o n s w h i c h o n l y t a k e f i n i t e l y many v a l u e s . F i n a l l y , e x t e n s i o n s a r e made t o f u n c t i o n s w h i c h t a k e uncount-a b l y many v a l u e s . Here we have chosen t o r e v e r s e t h i s approach s i n c e we have assumed f ( a ) a l r e a d y e x i s t s b e f o r e we e v a l u a t e i t . Some comments on Axiom I I . S e v e r a l q u e s t i o n s a r i s e from Axiom I I o f a r a t h e r t e c h n i c a l n a t u r e i n r e g a r d t o 7 h ( B ^ , a ) . The r e a s o n f o r t h i s i s t h a t h i s n o t n e c e s s a r i l y a n o n - n e g a t i v e f u n c t i o n and i t may, t h e r e f o r e , be i m p o r t a n t i n what o r d e r the B^ 1s a r e s e l e c t e d . F o r example, assume t h a t B^,B 2,... i s a p a r t i t i o n o f B such t h a t t h e sequence h ( B l f a ) + h ( B 2 , a ) + h ( B 3 , a ) . . . i s e q u a l t o 1 - 1/2 + 1/3 - 1/4 + ... . Then by c h a n g i n g the o r d e r o f the B^.'s, we c o u l d have the sequence 1 + 1/3 - 1/2 + 1/5 + 1 / 7 - 1/4 + ... and a l t h o u g h b o t h sequences co n v e r g e , t h e y do n o t converge t o the same v a l u e . T h e r e f o r e by assuming t h a t h ( B i , a ) = J_ h ( B i , a ) we assume t h a t t h e summation i s a c o n s t a n t v a l u e n o t o n l y f o r any p a r t i t i o n o f B, b u t a l s o f o r any rearrangement o f a g i v e n p a r t i t i o n . T h i s assumption i s e q u i v a l e n t t o assuming n t h a t t h e sequence 2__h(B.,a) converges a b s o l u t e l y (see, f o r i = l 1 example, W. Rudin (1969), pp.68-69): Theorem. a) I f ^ _ a n converges f o r a l l r e a r r a n g e m e n t s , then they a l l converge t o the same sum. b) 2-. a n c o n v e r 9 e s f o r a H rearrangements i f and o n l y i f a n converges a b s o l u t e l y . 4.2 I m p l i c a t i o n s o f Axiom I I There a r e s e v e r a l i m p o r t a n t i m p l i c a t i o n s o f t h i s axiom. For example, i t i m p l i e s the e x i s t e n c e o f a u t i l i t y f u n c t i o n , i f some r e g u l a r i t y c o n d i t i o n s a r e assumed. Other i m p l i c a t i o n s a re o f e q u a l i m p o r t a n ce i n t h a t t hey s p e c i f y v e r y s t r o n g c o n d i t i o n s on t h e p r e f e r e n c e o r d e r i n g on A. We s h a l l f i r s t s p e c i f y t h e s e m a t h e m a t i c a l i m p l i c a t i o n s and s e c o n d l y i l l u s t r a t e the " p a r a -doxes" which c o n t r a d i c t t h e axiom. Lemma 4.2.1. h(0,a) = 0 P r o o f . L e t B^ = 0, B^, i=2,3,... be a sequence o f s e t s such t h a t B i 0 Bj = 0 , i ^ j , then oo oo h( L ) B . , a ) = h(0,a) + _C-h(B.,a) i = l 1 i=2 C O C O and s i n c e B = l ) B, = \ j B. i=2 . i = l oo h(O.B i,a) = l h ( B ,a) i=2 hence s i n c e |h( l j B. ,a) | < oo t h e n i = l 1 h ( c j ) ,a) = 0. Lemma 4.2.2. h(*,a) i s f i n i t e l y a d d i t i v e . P r o o f . T h i s f o l l o w s d i r e c t l y s i n c e h(<f>,a) = 0, Lemma 4.2.3. h i s c o n t i n u o u s from below. I f D i C D i + 1 and D ± _ 6 f o r i = l , 2 , . . . t h e n 0 0 h( O D i,a) = l i m h( D . , a ) . i = l i->«> 1 P r o o f . ( T h i s f o l l o w s from s t a n d a r d measure t h e o r e t i c a l r e s u l t s . ) i - 1 D e f i n e B. = (D. - ( J D.) where B 1 = D n. 1 1 _=1 n n Hence U B. = \ J D. f o r a l l n, and a l s o B. C\ B . = 0 ,t i ^ j i = l i = l 1 1 OO CO oo T h e r e f o r e h ( U D. , a) = h ( 0 B . , a ) = _ C.h(B.,a). i = l i = l 1 i = l 1 f n n n S i n c e h ( U D.,a) = h( U B.,a) = _ _ _ h ( B . , a ) , i = l 1 i = l 1 i = l 1 n n we have l i m h( U D. ,a) = l i m "___.h(B. ,a) = ,n->-oo i = l n-^ -oo i = l 1 l h ( B , , a ) = h ( O B . , a) i = l 1 i = l 1 h( U D. ,a) The p r o p e r t i e s we have d i s c u s s e d so f a r are based on the second p a r t o f the axiom. There i s a l s o one pr o p e r t y based on the t h i r d p a r t which i s important to the theory. To prove t h i s , however, an a d d i t i o n a l assumption i s needed, t h a t i s , t h a t a l l constant f u n c t i o n s belong t o T. Assumption 2 . I f Y(w) = r f o r a l l w e f t and f o r any r e R, then there e x i s t s an a e A such t h a t X(w,a) = Y(w). T h i s assumption i s not necessary f o r the expected u t i l i t y c r i t e r i o n to hold and we s h a l l not assume t h a t i t holds i n g e n e r a l . One i m p l i c a t i o n o f t h i s assumption i s t h a t there e x i s t s a r e a l - v a l u e d f u n c t i o n U on R, as we s h a l l show i n Lemma 4 . 2 i 4 . I f assumption 2 i s not made, however, we must assume t h a t the f u n c t i o n U e x i s t s on R, which we do i n axiom I I I . Lemma 4 . 2 . 4 I f assumption 2 h o l d s , the e v a l u a t i o n f u n c t i o n f induces a u t i l i t y f u n c t i o n U on R. Proof. Assume X(w,a) = r and X(w,b) = r f o r a l l we- ft and a,b e A. P a r t c i n Axiom I I i m p l i e s h(ft,a) = h(ft,b) or f ( a ) = f ( b ) . T h e r e f o r e , the f u n c t i o n U(r) = f( a ) i s a u n i q u e l y d e f i n e d f u n c t i o n on R. Although many other r e s u l t s can be proved these a re s u f f i c i e n t f o r our development here. 40 I m p l i c a t i o n s from a d e c i s i o n maker's, v i e w p o i n t . Lemma 4.2.1 i m p l i e s t h a t the. number o f s u b s e t s o f Q. t h a t a r e c o n s i d e r e d when an a c t i o n i s e v a l u a t e d i s i m m a t e r i a l f t h a t i s , we ought t o o b t a i n the same v a l u e o f f ( a ) i n d e p e n d e n t l y o f what ev e n t s a r e c o n s i d e r e d as l o n g as t h e y j o i n t l y i n c l u d e a l l p o s s i b i l i t i e s . An example o f a c r i t e r i o n w h i c h does not s a t i s -f y t h i s a s sumption i s t h e P r i n c i p l e o f I n s u f f i c i e n t Reason. The c r i t e r i o n was f i r s t f o r m u l a t e d by La P l a c e a p p r o x i m a t e l y two hundred y e a r s ago, and can be p a r a p h r a s e d a s : " I f no e v i d e n c e e x i s t s t h a t one o f t h e e v e n t s i n a p a r t i t i o n i s more l i k e l y t o o c c u r t h a n the o t h e r s , t h e n the e v e n t s s h o u l d be c o n s i d e r e d e q u a l l y l i k e l y t o o c c u r . " T h i s p r i n c i p l e i s not always a c c e p t e d because o f a p p a r e n t con-t r a d i c t i o n s such as t h e f o l l o w i n g : suppose we f l i p a c o i n t w i c e , t h e n f o u r s t a t e s can o c c u r (H,H), (H,T), (T,H), o r (T,T), hence the p r o b a b i l i t y o f (H,H) must be 1/4 by the P r i n c i p l e o f I n s u f f i c i e n t Reason. On the o t h e r hand, we can d i v i d e the sample space i n t o (H,H) (not (H,H)), and t h e n the p r o b a b i l i t y o f (H,H) must be 1/2 o r we must have e v i d e n c e t h a t t h e s e e v e n t s a r e n o t e q u a l l y l i k e l y . Of c o u r s e , most d e c i s i o n makers would a c c e p t t h e p r o b a -b i l i t y o f (H,H) b e i n g 1/4 as i n t h i s case a l l s t a t e s can be l i s t e d and t h e r e i s no r e a s o n t o assume one i s more l i k e l y t o o c c u r t h a n a n o t h e r . I n more g e n e r a l s i t u a t i o n s the q u e s t i o n a r i s e s as t o which p a r t i t i o n ought t o be made. Anoth e r i m p l i c a t i o n a r i s e s from t h e a d d i t i v i t y assumption o f h w h i c h i s r e l a t e d t o what i s c a l l e d t h e S u r e - T h i n g P r i n c i p l e by Savage (1954) o r what i s c a l l e d t h e S t r o n g Indep dence Axiom by Samuelson (1952). S e v e r a l paradoxes a r e based on the a d d i t i v i t y a ssumption and stem from the f o l l o w i n g ob-s e r v a t i o n : L e t a and b be two a c t i o n s such t h a t X_,(-,a) = X_,(',b) f o r some B e 6 . S i n c e (B,B) form a p a r t i t i o n o f ft. we have from Axiom I I t h a t f ( a ) = h(B,a) + h(B,a) f (b) = h(B,b) + h ( B , b ) , and a l s o h(B,a) = h ( B , b ) . Thus, f ( a ) > f ( b ) i f and o n l y i f h(B,a) > h ( B , b ) , t h a t i s when two a c t i o n s a r e compared we need o n l y e v a l u a t e them on t h e s e t s on which t h e y o b t a i n d i f f e r e n t r ewards. The d e c i s i o n problem which g i v e s r i s e t o a c o n t r a d i c t i o n i n the a l t e r n a t i v e s i s t h e n d e s i g n e d so t h a t most p e o p l e make a c h o i c e such t h a t f ( a ) > f ( b ) b u t h(B,b) + a c o n s t a n t > h(B,a) + a c o n s t a n t . The paradoxes t h a t b e s t i l l u s t r a t e t h i s p o i n t a r e E l l s b e r g ' s Paradox I I , A l l a i s 1 paradox, and MacCrimmon's paradox, each o f which i s d e s c r i b e d below. E l l s b e r g ' s Paradox I I . D. E l l s b e r g (1961) d e s c r i b e d the f o l l o w i n g d e c i s i o n p r oblem. C o n s i d e r an u r n c o n t a i n i n g 90 b a l l s , o f which 30 a r e known t o be r e d , and the r e m a i n i n g 60 a r e an unknown m i x t u r e o f b l a c k and y e l l o w b a l l s . One b a l l i s t o be drawn a t random from the u r n , and we a r e asked t o s t a t e o ur p r e f e r e n c e between a and b, and a l s o between c and d where a, b, c and d a r e d e f i n e d as f o l l o w s : a: R e c e i v e $1,000 i f a r e d b a l l i s drawn R e c e i v e $0 o t h e r w i s e 4 2 b: R e c e i v e $ 1 , 0 0 0 i f a b l a c k b a l l i s drawn R e c e i v e $ 0 o t h e r w i s e c: R e c e i v e $ 1 , 0 0 0 i f a r e d o r y e l l o w b a l l i s drawn R e c e i v e $ 0 o t h e r w i s e d: R e c e i v e $ 1 , 0 0 0 i f a b l a c k o r y e l l o w b a l l i s drawn R e c e i v e $ 0 o t h e r w i s e The paradox can most e a s i l y be c o n s i d e r e d by i l l u s t r a t i n g the problem i n the form o f a d e c i s i o n m a t r i x . 6 0 B a l l s 3 0 Red b a l l s B l a c k b a l l s Y e l l o w b a l l s a 1 , 0 0 0 0 0 b 0 1 , 0 0 0 0 0 1 , 0 0 0 1 , 0 0 0 1 , 0 0 0 Note t h a t t h e l a s t e v e n t " y e l l o w b a l l s " does n o t d i s c r i m i n a t e between the a l t e r n a t i v e s a and b nor between the a l t e r n a t i v e s c and d and hence can be i g n o r e d . I g n o r i n g t h i s e v e n t , we know t h a t a i s then i d e n t i c a l t o c and b i s i d e n t i c a l t o d; hence a c h o i c e o f a over b would r e q u i r e a c h o i c e o f c over d. T h i s i n t u i t i v e argument i s the b a s i s o f Axiom I I whi c h i m p l i e s i t s c o n c l u s i o n . I n terms o f Axiom I I , l e t us denote t h e ev e n t t h a t a r e d b a l l i s drawn by R, the ev e n t t h a t a b l a c k b a l l i s drawn by B, and t h e ev e n t t h a t a y e l l o w b a l l i s drawn by Y. T h e r e f o r e we can e x p r e s s f by Axiom I I , p a r t a, a s : f ( a ) = h(R,a) + h(B,a) + h(Y,a) f (b) = h(R,b) + h(B,b) + h(Y,b) f (c) = h(R,c) + h(B,c) + h(Y,c) c d 1 , 0 0 0 0 f (d). - h(.R fd) + h.(B fd) t h(Y,d) By Axiom I I , p a r t b, t h e f o l l o w i n g i d e n t i t i e s must h o l d : h(R,a) + h(B,a) = h(R,c) + h(B,c) h(R,b) + h(B,b) = h(R,d) + h(B,d) h(Y,a) = h(Y,b) h(Y,c) = h(Y,d) T h e r e f o r e i f f ( a ) > f ( b ) t h i s i m p l i e s h ( B L / a ) + h(B_,a) > h ( B l f b ) + h ( B 2 , b ) t h e r e f o r e h(Bltc) + h ( B 2 , c ) > h ( B l f d ) + h ( B - f d ) and hence h ( B l f c ) + h ( B 2 , c ) + h ( B 3 , c ) > h ( B l f d ) + h ( B 2 , d ) + h(B 3,d) o r e q u i v a l e n t l y f (c) > f (d) . Of c o u r s e an even s t r o n g e r r e s u l t i s i m p l i e d s i n c e f ( a ) - f(.b) = f ( c ) *- f ( d ) , r a t h e r t h a n j u s t the i n e q u a l i t i e s . T h i s i m p l i e s t h a t i f we change t h e amount we r e c e i v e i f a 4 4 b l a c k b a l l i s drawn u n t i l we a r e i n d i f f e r e n t between b and c, t h e n t h e amount we r e c e i v e i f a b l a c k b a l l i s drawn from a l t e r n a t i v e d must be changed e x a c t l y the same t o make us i n d i f f e r e n t between a l t e r n a t i v e s c and d. The paradox a r i s e s i f someone chooses a o v e r b and then d o v e r c. E l l s b e r g (1961) suggested t h a t t h i s i s a l i k e l y p r e f e r e n c e o r d e r i n g s i n c e a known p r o b a b i l i t y o f w i n n i n g i s p r e f e r r e d t o an unknown change i f t h e r e i s no r e a s o n t o b e l i e v e t h e unknown change has a h i g h e r p r o b a b i l i t y o f w i n n i n g . A l l a i s Paradox. T h i s p r o b l e m was f i r s t p r o p osed by A l l a i s (195 3 ) . A g a i n we have t o choose between a and b and a l s o between c and d, where a: $1 m i l l i o n w i t h a p r o b a b i l i t y o f 1.00 b: $5 m i l l i o n w i t h a p r o b a b i l i t y o f 0.10 $1 m i l l i o n w i t h a p r o b a b i l i t y o f 0.89 $0 w i t h a p r o b a b i l i t y o f 0.01 c: $1 m i l l i o n w i t h a p r o b a b i l i t y o f 0.11 $0 w i t h a p r o b a b i l i t y o f 0.89 d: $5 m i l l i o n w i t h a p r o b a b i l i t y o f 0.10 $0 w i t h a p r o b a b i l i t y o f 0.90. So f a r i n our assumptions we have not c o n s i d e r e d p r o b a b i l i t i e s b u t o n l y e v e n t s . T h e r e f o r e , we s h a l l r e w r i t e the A l l a i s p r o ^ • blem i n the form o f 100 l o t t e r y t i c k e t s numbers 1-100 w i t h p r i z e s as g i v e n i n the p a y o f f m a t r i x below: T i c k e t #1 T i c k e t s #2-11 T i c k e t s #12-100 a $1 m i l l i o n $1 m i l l i o n $1 m i l l i o n b 0 $5 m i l l i o n $1 m i l l i o n c $1 m i l l i o n $1 m i l l i o n 0 d 0 $5 m i l l i o n 0 The e v e n t o f d r awing a t i c k e t w i t h any o f t h e numbers 12-100 t h e r e f o r e does n o t d i s c r i m i n a t e between a and b nor between c and d. T h e r e f o r e by the same arguments used w i t h the E l l s b e r g paradox we have the r e l a t i o n To be c o n s i s t e n t w i t h Axiom I I , a p r e f e r e n c e o f a t o b, i t must t h e r e f o r e i m p l y a p r e f e r e n c e o f c t o d. Of c o u r s e , i t may be argued t h a t s i n c e we m o d i f i e d the problem i t i s n o t n e c e s s a r i l y i d e n t i c a l t o t h e o r i g i n a l one. T h i s i s t r u e s i n c e t h e assumptions we have made do n o t i m p l y t h a t t h e y a r e e q u a l . We s h a l l d i s c u s s t h i s p o i n t i n s e c t i o n 4 . 6 . A l l a i s (19 53) s u g g ested t h a t when one a l t e r n a t i v e g i v e s a c e r t a i n t y (or near c e r t a i n t y ) o f o b t a i n i n g a v e r y d e s i r a b l e consequence, one s h o u l d s e l e c t i t even i f i t e n t a i l s p a s s i n g up a l a r g e r amount h a v i n g a l ower p r o b a b i l i t y . When, however, t h e chances o f w i n n i n g a r e s m a l l and c l o s e t o g e t h e r , one s h o u l d t a k e the o p t i o n t h a t p r o v i d e s t h e l a r g e r p a y o f f . Hence f (a) - f (b) = f (c) - f (d) . i n the above a l t e r n a t i v e , he assumes t h a t we ought t o choose a t o b but d t o c, which, c o n t r a d i c t s Axiom I I . MacCrimmon' s Paradox-" I . A l t h o u g h b o t h the E l l s b e r g and A l l a i s paradoxes can be used t o i l l u s t r a t e some o f t h e d i f f i -c u l t i e s o f Axiom I I , the E l l s b e r g paradox a l s o i l l u s t r a t e s t h e d i f f i c u l t i e s o f a s s i g n i n g a s u b j e c t i v e p r o b a b i l i t y t o t h e e v ent B, d e f i n e d as {a b l a c k b a l l i s drawn}. I n o t h e r words, we can argue t h a t t h e e v e n t B e 3 - 0 . We s h a l l s t u d y t h i s a s p e c t o f the E l l s b e r g paradox i n more d e t a i l i n s e c t i o n 6.2 and i n the second p a r t o f t h e t h e s i s . MacCrimmon (MacCrimmon and L a r s s o n , 1975) has d e s i g n e d a paradox w h i c h a t t e m p t s t o c a p t u r e b o t h A l l a i s 1 and E l l s b e r g ' s arguments i n t o one p r o -blem. That i s , i t has t h e p r o p e r t y o f unknown v e r s u s known p r o b a b i l i t i e s and a l s o n e a r l y s u r e as w e l l as v e r y s m a l l chances o f w i n n i n g . C o n s i d e r an u r n c o n t a i n i n g 100 b a l l s , 20 o f w h i c h are r e d ; t h e o t h e r 80 a r e e i t h e r b l a c k o r y e l l o w . The number o f b l a c k b a l l s i s between 1 and 5 i n c l u s i v e . One b a l l w i l l be drawn from th e u r n and i t s c o l o u r w i l l d e t e r m i n e th e p a y o f f r e c e i v e d . A g a i n , c h o i c e s a r e t o be made between a l t e r n a t i v e s a and b, and between a l t e r n a t i v e s c and d. a: $100,000 i f a r e d b a l l i s drawn $1,000,000 i f a y e l l o w b a l l i s drawn $0 i f a b l a c k b a l l i s drawn b: $0 i f a r e d b a l l i s drawn $1,000,000 i f a y e l l o w b a l l i s drawn $1,000,000 i f a b l a c k b a l l i s drawn c; $ 1 0 0 f 0 0 0 i f a r e d b a l l i s drawn $ 0 i f a y e l l o w b a l l i s , drawn $ 0 i f a b l a c k b a l l i s drawn d: $ 0 i f a r e d b a l l i s drawn $ 0 i f a y e l l o w b a l l i s drawn $ 1 , 0 0 0 , 0 0 0 i f a b l a c k b a l l i s drawn The problems can be s e t up i n a p a y o f f m a t r i x as f o l l o w s : 8 0 B l a c k o r Y e l l o w B a l l s 2 0 Red b a l l s 1 - 5 B l a c k b a l l s 7 5 - 7 9 Y e l l o w b a l l s a $ 1 0 0 , 0 0 0 0 $ 1 , 0 0 0 , 0 0 0 b 0 $ 1 , 0 0 0 , 0 0 0 $ 1 , 0 0 0 , 0 0 0 c $ 1 0 0 , 0 0 0 0. 0 d 0 $ 1 , 0 0 0 , 0 0 0 0 A g a i n , we can show t h a t f (a) - f (b) = f (c) - f (d) . A A T h e r e f o r e t h e p r e f e r e n c e a > b must a l s o i m p l y c > d. 4 . 3 E m p i r i c a l s t u d i e s o f Axiom I I Most o f the e m p i r i c a l s t u d i e s r e l a t e d t o Axiom I I have bee made i n t h e form o f t h e paradoxes s t a t e d i n s e c t i o n 4 . 2 . I t i s not i n t e n d e d t o make an e x t e n s i v e s u r v e y o f t h e s e s t u d i e s b u t r a t h e r t o g i v e some i n d i c a t i o n as t o how r e a d i l y Axiom I I has been a c c e p t e d . E l l s b e r g ' s paradox a l s o c o n t r a d i c t s Axiom IV, and t h e r e f o r e we postpone the d i s c u s s i o n t o s e c t i o n 6 . 3 . 48 E m p i r i c a l r e s u l t s , o f models, based on t h e A l l a i s paradox The. p a r t i c u l a r p r o b a b i l i t i e s and p a y o f f i n the A l l a i s paradox have been c a r e f u l l y d e s i g n e d i n b o t h monetary v a l u e s and p r o b a b i l i t i e s t o e l i c i t v i o l a t i o n s o f t h e u t i l i t y axiom. The model based on the A l l a i s paradox can be r e p r e s e n t e d by the p a y o f f m a t r i x : Event A c t i o n B l B 2 B 3 s u t r v t s u w r v w We showed i n s e c t i o n 4.2 t h a t f ( a ) > f ( b ) i m p l i e s f ( c ) > f ( d ) . S t u d i e s o f t h e common consequence, t h a t i s , t he reward i s the same on a s u b s e t o f .J? f o r b o t h a c t i o n s , have been made by MacCrimmon (1965), Moskowitz (1974), S l o v i c and T v e r s k y (1975) and MacCrimmon and L a r s s o n (19 75). Most o f t h e s e a u t h o r s use the A l l a i s p r o b lem by h a v i n g two p a i r s o f c h o i c e s w i t h s = u = t , r = w, and w i t h t h e p r o b a b i l i t y v a l u e s y(B^) = 0.01, y ( B 2 ) = 0.10, and y(B^) = 0.89. MacCrimmon and L a r s s o n v a r i e d t h e p r o b a b i l i t i e s t o d e t e r m i n e i f t h i s would i n f l u e n c e t h e d e c i s i o n maker. MacCrimmon (1965) s t u d y . MacCrimmon uses t h r e e common consequence problems. The rewards o f a l l t h e s e problems a r e s p e c i f i e d i n terms o f t h e o c c u r r e n c e o f e v e n t s r a t h e r t h a n i n terms o f p r o b a b i l i t i e s . I n two o f t h e problems, t h e e v e n t s a r e based on s t a n d a r d urns, f and from t h e events, most p e o p l e i n f e r t h e same p r o b a b i l i t i e s . I n t h e o t h e r problem, t h e o c c u r r e n c e o f e v e n t s was d e s c r i b e d o n l y v e r b a l l y ( e . g . , " v e r y u n l i k e l y " o r " l i k e l y " ) ; t h e s u b j e c t s were s e n i o r b u s i n e s s e x e c u t i v e s ; t h e problems appeared i n t h e c o n t e x t o f i n v e s t m e n t s f o r a hypo-t h e t i c a l b u s i n e s s ; and the rewards were p e r c e n t r e t u r n on t h e i n v e s t m e n t . The parameters o f t h e problems may be summarized as: P r o b l e m 1. v = 500% r e t u r n s on t h e c a p i t a l , s = u = t = 5% r e t u r n s on t h e c a p i t a l , r = w = b a n k r u p t c y . Problem 2. Same as Problem 1 e x c e p t f o r t h e p r o b a b i l i t i e s w h i c h were o n l y q u a l i t a t i v e . Problem 3. v = 75% r e t u r n s on the c a p i t a l , s = u = t = 35% r e t u r n s on the c a p i t a l , r = w = 0% r e t u r n s on the c a p i t a l . I n a l l t h r e e problems a s i g n i f i c a n t number o f t h e 36 s u b j e c t s chose an answer c o n t r a d i c t i n g Axiom I I (14, 15, and 13 r e s p e c t i v e l y ) , and hence c o n f i r m e d t h e c l a i m made by A l l a i s . O n ly n i n e s u b j e c t s conformed t o t h e e x p e c t e d u t i l i t y axioms i n a l l t h r e e problems (15 had one d e v i a t i o n , n i n e had two, and t h r e e had t h r e e ) . I n a d d i t i o n t o t h e i r own c h o i c e s , t h e sub-j e c t s were p r e s e n t e d w i t h r e s p o n s e s s u p p o s e d l y from s u b j e c t s i n a p r e v i o u s s e s s i o n and were asked t o c r i t i c i z e them. These r e s u l t s a r e even more c o m p e l l i n g s i n c e 29 o f t h e s u b j e c t s (on problem 1) agreed w i t h t h e A l l a i s - t y p e answer; o n l y seven a g r e e d w i t h t h e Axiom I I - t y p e answer. • . However, t h e h y p o t h e s i s t h a t i n c r i t i c i z i n g t h e s e r e s p o n s e s , s u b j e c t s tended t o p i c k t h e answer, t h a t most c l o s e l y c o r r e s p o n d e d t o t h e i r own answer, a c c o u n t s f o r 32 o f t h e c a s e s . Moskowitz ( 1 9 7 4 ) s t u d y . Moskowitz p r e s e n t e d each o f 1 3 4 s t u d e n t s w i t h t h r e e problems where the rewards s e t was d e f i n e d as the s e t o f p o s s i b l e grades i n a c o u r s e . For the f i r s t two problems, Moskowitz does n o t s p e c i f y the n u m e r i c a l grades he used i n the problems; i n the t h i r d problem t h e grades were l e t t e r g r a d e s . The parameters o f h i s problem t h r e e were: v = A grade, s = u = t = B+ grade, r = w = F grade, y ( B 1 ) = 0 . 0 1 , y ( B 2 ) = 0 . 1 0 , y ( B 3 ) = 0 . 8 9 . Moskowitz p r e s e n t e d t h e s e problems i n t h r e e d i f f e r e n t f o r m a t s : word, t r e e , and m a t r i x , as sug g e s t e d i n MacCriitvmon ( 1 9 6 7 ) . He a l l o w e d some s u b j e c t s t o d i s c u s s t h e problems i n a group w h i l e o t h e r s had t o pro c e e d i n d i v i d u a l l y . I n a d d i t i o n t o p r e s e n t i n g the problem, he p r e s e n t e d pro and con arguments and a f t e r w a r d had the s u b j e c t s choose a g a i n . O v e r a l l , Moskowitz found a r a t e o f v i o l a t i o n o f about 3 0 % . The t r e e r e p r e s e n t a t i o n was the most d i f f i c u l t w i t h a v i o l a t i o n r a t e r a n g i n g from 2 0 % t o 5 0 % ( a c r o s s the o t h e r c o n d i t i o n s ) . I n the word f o r m a t , the v i o l a t i o n r a t e was 1 7 % t o 4 0 % and i n the m a t r i x f o r m a t t h e v i o l a t i o n r a t e was 2 1 % t o 4 2 % . There were o n l y s l i g h t d i f f e r e n c e s i n problem t y p e s w i t h problem 1 h a v i n g a v i o l a t i o n range o f 1 7 % t o 4 6 % ; problem 2 h a v i n g a v i o l a t i o n range o f 1 7 % t o 4 0 % ; and problem 3 h a v i n g a v i o l a t i o n range o f 1 7 % t o 4 5 % . There was a s i g n i f i c a n t l y g r e a t e r i n c r e a s e i n c o n s i s t e n c y f o r d i s c u s s i o n groups v e r s u s n o n - d i s c u s s i o n groups, b u t b o t h groups' answers were more c o n s i s t e n t on the second p r e s e n t a t i o n . 51 S l o v i c and T y e r s k y (119,75) s t u d y . S l o v i c and T v e r s k y use the s t a n d a r d A l l a i s p roblem. Of t h e i r 29 c o l l e g e s t u d e n t s u b j e c t s , 17 chose t h e A l l a i s r e s p o n s e , and 12 chose c o n s i s -t e n t l y w i t h Axiom I I , , On a r e c o n s i d e r a t i o n , a f t e r r e a d i n g arguments i n f a v o u r o f each p o s i t i o n , 19 s u b j e c t s chose the A l l a i s r e s p onse and 10 chose c o n s i s t e n t l y w i t h t h e axiom!' Over t h e two p r e s e n t a t i o n s , 16 made t h e A l l a i s - t y p e c h o i c e s , w h i l e n i n e made t h e axiom-based c h o i c e s . I n a second e x p e r i -ment w i t h 49 s t u d e n t s u b j e c t s , t h e s u b j e c t s f i r s t r e a d and r a t e d arguments f o r and a g a i n s t the axioms, t h e n they made t h e i r own c h o i c e s . W i t h t h i s f o r m a t , c o n s i s t e n c y i n c r e a s e d . I n t h e i r a c t u a l c h o i c e s , o n l y 17 s u b j e c t s made A l l a i s - t y p e c h o i c e s , b u t 30 s u b j e c t s made axiom-based c h o i c e s . I n r a t i n g t h e i r arguments f o r t h e axioms, 25 s u b j e c t s r a t e d t h e A l l a i s argument h i g h e r w h i l e 21 s u b j e c t s r a t e d t h e axiom argument h i g h e r . A c r o s s t h e s e s t u d i e s , we see t h a t Axiom I I i s v i o l a t e d a t a s i g n i f i c a n t r a t e . The r a t e o f v i o l a t i o n ranges around 27% t o 42% e x c e p t f o r t h e h i g h l e v e l i n S l o v i c and Tversky-'-s — f i r s t e x p e r i m e n t . However, t h e r e seems t o be a c o n s i d e r a b l e v a r i a t i o n a c r o s s the s t u d i e s , and even w i t h i n a s i n g l e s t u d y . MacCrimmon and L a r s s o n (1975) s t u d y . MacCrimmon and L a r s s o n made a s t u d y where the reward s e t was money, wh i c h was v a r i e d t o det e r m i n e i f a bound e x i s t s f o r which the A l l a i s paradox i s not v i o l a t e d . I n t h e i r s t u d y t h e y c o n s i d e r e d the f o l l o w i n g v a r i a t i o n o f t h e paradox: a b c d S 5S s -5S. S 0 S 0 S S 0 0 0 0 0 0 I n t h i s s t u d y s took on the v a l u e s $1,000,000, $100,000, $10,000, and $1,000 and y ( B ^ + y (B 2) + y (B 3) took on t h e v a l u e s 1.00, 0.99, 0.50, and 0.11. E l e v e n d i f f e r e n t combin-a t i o n s o f t h e s e parameters were r e p r e s e n t e d p l u s two check p o i n t s as a measure o f t h e random component o f t h e c h o i c e s . F o r s i m p l i c i t y we s h a l l denote y (B^) + y ( B 2 ) + y(B^) = P 1 f o r t h e c h o i c e between a and b, and p 2 = p^- v ( B 3 ) f o r t h e c h o i c e between c and d. From t h e s e one can form 12 d i f f e r e n t s e t s o f two p a i r s o f b i n a r y l o t t e r i e s . F i g u r e s 4.1 p r o v i d e a summary o f the r e s u l t s f o r the 19 s u b j e c t s . The t a b l e s l i s t the number o f s u b j e c t s making t h e a and c c h o i c e i n the h i g h e r p s e t and t h e b and d c h o i c e i n t h e lower p s e t . F o r example, i n F i g u r e 4 . 1 ( i ) t h e r e were s i x s u b j e c t s c h o o s i n g a and b i n the s e t s = $1,000,000, p 1 = 1.00, and c and d i n t h e s e t s = $1,000,000, p 2 = 0.11 — t h i s i s the s t a n d a r d A l l a i s problem. Hence t h e r a t e o f v i o l a t i o n i s 33% w h i c h i s about th e same as the p r e c e e d i n g s t u d i e s . Not s u r -p r i s i n g l y , t h e h i g h e s t r a t e o f v i o l a t i o n o c c u r s f o r extreme p r o b a b i l i t y - p a y o f f v a l u e s . There i s not a h i g h e r r a t e o f s = $1,000,000 P i P 2 LOO 0,99 0,50 0.11 1.00 1 4 4 6 0.99 5 3 0.50 6 0.11 0 F i g u r e 4 . 1 ( i ) s = $100,000 s = $10,000 p l P 2 0.99 0.11 P l ' P 2 0.99 0.11 1.00 . 2 4 1.00 ' 4 4 0.99 2 0.99 6 F i g u r e 4 . 1 ( i i ) F i g u r e 4 . 1 ( i i i ) F i g u r e 4.1. Number o f s u b j e c t s i n c o n s i s t e n t w i t h u t i l i t y axioms f o r v a r i o u s l e v e l s o f monetary rewards and p r o b a b i l i t i e s . v i o l a t i o n t h a n the s t a n d a r d A l l a i s . problem^ b u t two o t h e r c o m b i n a t i o n s , s = $1,000,000, p^ = 0.50, p 2 = 0.11, and s = $10,000, p^ = 0.99, p 2 = 0 . 1 1 a l s o have s i x v i o l a t i o n s . I t s h o u l d be n o t e d however t h a t t h e s e v i o l a t i o n s a r e f o r s i g n i f i c a n t l y changed parameter v a l u e s from t h e s t a n d a r d problem. They found a s i g n i f i c a n t v i o l a t i o n a t t h e $10,000 p a y o f f l e v e l and ( s e p a r a t e l y ) a t t h e 0.50 p r o b a b i l i t y l e v e l . 4 . 4 R e l a t i o n s t o o t h e r axiom systems T h i s axiom i s by f a r t h e most i m p o r t a n t axiom i n the sense t h a t e m p i r i c a l e v i d e n c e shows t h a t i t i s the one most o f t e n v i o l a t e d , and a l s o t h a t the m a j o r i t y o f the paradoxes a r e based on i t . T h e r e f o r e we s h a l l c o n s i d e r t h e r e l a t i o n s h i p t o o t h e r axiom systems i n some d e t a i l . C o n s i d e r e d from a l t e r n a t i v e approaches, most o f the axioms do n o t show t h a t f ( • ) may be e x p r e s s e d as a summation o f the e v a l u a t i o n f u n c t i o n on a s u b s e t o f ft. However, they c l e a r l y i n d i c a t e the independence e v a l u a t i o n on a d i f f e r e n t s u b s e t o f Q,. L e t us f i r s t c o n s i d e r Arrow's approach. Arrow's Axiom A 2 s t a t e s t h a t i f B i s an a r b i t r a r y e v e n t b e l o n g i n g t o B , t h e n i f two a c t i o n s have the same rewards on B t h e y must be i n d i f f e r e n t g i v e n 8 . . T h i s i s c l o s e t o Axiom l i b , the d i f f e r e n c e b e i n g t h a t he does not assume t h e e x i s t e n c e o f the f u n c t i o n h; however, i f h does e x i s t , i t must have the p r o p e r t i e s o f Axiom l i b . S i m i l a r l y , A 4 i s r e l a t e d t o the f i r s t p a r t o f Axiom I I , a l t h o u g h i t i s n o t as s t r o n g as Axiom I I . I n t h i s case Axiom A 4 s t a t e s t h a t i f we c o n s i d e r any sequence {B^} o f s e t s i n g and i f an a c t i o n a i s p r e f e r r e d t o an a c t i o n b on each o f t h e s e s e t s ( t h a t i s i f we assume h e x i s t s , t h e n h(B^,a) >, h(B^,b) f o r a l l i ) then f ( a ) > f ( b ) . T h i s does n o t i m p l y t h a t h i s a d d i t i v e , o f c o u r s e . However, i t does i m p l y t h a t the p r e f e r e n c e on one ev e n t does n o t i n f l u e n c e the p r e f e r e n c e on a n o t h e r e v e n t i n g. vonNeumann-Morgenstern's approach. I n Appendix I , Axiom N M 2 i s g i v e n as R R r ^ > r 2 i m p l i e s r ^ > F ( a , r ^ , r 2 ) and R F ( c t , r ^ , r 2 ) > r 2 f o r a l l a e ( 0 , 1 ) . I n t u i t i v e l y F ( a , r ^ , r 2 ) can be c o n s i d e r e d as r e c e i v i n g r ^ w i t h the p r o b a b i l i t y a and r 2 w i t h the p r o b a b i l i t y (1-a) . L e t us now c o n s i d e r how t h i s axiom i s r e l a t e d t o Axiom I I . C o n s i d e r two reward f u n c t i o n s d e f i n e d by X(*,a) = r ] _ a n < 3 X(-,b) = r 2 f o r a l l u e ft. Then f o r any B e 6 , Axiom I I s t a t e s t h a t i f a c t i o n a i s p r e f e r r e d t o a c t i o n b th e n U ( r 1 ) = f ( a ) = h(B,a) + h(B,a)'» > U ( r 2 ) = f ( b ) = h(B,b) + h ( B , b ) . T h i s c l e a r l y i m p l i e s t h a t a t l e a s t one o f t h e i n e q u a l i t i e s h(B,a) z> h(B,b) o r h(B,a) h(B,b) must h o l d . I f we make the assumption t h a t b o t h i n e q u a l i t i e s h o l d (Axiom I I I i m p l i e s t h a t b o t h must h o l d ) , t h e n t h i s i m p l i e s Axiom NM2. To see t h i s , we s h a l l d e f i n e a reward f u n c t i o n c by r1 to E B X U , c ) =.| 1 r 2 w e B, Then i f c e A f (c) = h(B,c) + h(B,c) . By Axiom l i b h(B,c) = h(B,a) and h(B,c) = h(B,b) w h i c h i m p l i e s h(B,a) + h(B,a) > h(B,c) + h(B,c) =s. h(B,b) + h(B,b) . Savage approach. I n t h i s approach t h e r e a r e s e v e r a l r e l a t e d axioms. We s h a l l c o n s i d e r two o f t h e s e . Axiom S2: I f X B ( - , a ) = X R ( - , b ) , X B ( - , c ) = X B ( - , d ) , X-(-,a) = X - ( ' , c ) , X-(-,b) = X - ( - , d ) , r and X(•,a) * X ( * f b ) f r t h e n X(•,c) > X ( • , d ) . A g a i n i f we assume t h a t a r e a l - v a l u e d f u n c t i o n h on 0 x A e x i s t s such t h a t h(B,a) i n d i c a t e s an e v a l u a t i o n o f a l t e r -n a t i v e a on t h e e v e n t B, t h e n Savage's Axiom S2 i m p l i e s t h a t i f X 0 ( - , c ) = X n ( - , d ) then h(B,c) = h ( B , d ) . T h i s f o l l o w s d i r e c t l y from Axiom S2. However, i t i s n o t n e c e s s a r y t h a t f ( a ) = h(B,a) + h(B,a) h o l d s f o r Axiom S2 t o h o l d . F o r example f ( a ) = h(B,a) x h(B,a) would a l s o s a t i s f y Axiom S2. Axiom S3 i s a l s o r e l a t e d t o Axiom I I , where S3 i s g i v e n by Axiom S3; L e t X(«,a) = r ^ and X(«,b) = r 2 . I f X B ( - , c ) = X B ( - , a ) , X 3 ( - , d ) = X B ( - , b ) and X-(.,c) = X - ( - , d ) , r t h e n X(-,a) 4 X(-,b) i f and o n l y i f r X(-,c) < X(-,d) f o r a l l B c 0 such t h a t B i s n o t n u l l . Axiom S3 i s r e l a t e d t o Axiom I I i n the same way as NM2. To see t h i s we s h a l l r e l a t e S3 t o NM2. L e t X—(*,c) = r , . a 1 R n Then S3 may be s t a t e d as r ^ < r 2 i f and o n l y i f r ^ < { r e c e i v i n g r 1 i f B o c c u r s o r r e c e i v i n g r 2 i f B o c c u r s } . S i m i l a r l y i f Xg-( i,d) = r 2 t h e n r ^ < r 2 i f and o n l y i f _ n { r e c e i v i n g r ^ i f B o c c u r s o r r e c e i v i n g r 2 i f B o c c u r s } < r^. T h e r e f o r e t h e r e l a t i o n we have s t a t e d between NM2 and Axiom I I a l s o h o l d s f o r S3 and Axiom I I . Marschak approach. I n Marschak's approach, Axiom I I i s the axiom most c l o s e l y r e l a t e d t o Axiom M2. To show t h i s we can n o t s t a r t w i t h an a r b i t r a r y p r o b a b i l i t y space. A s p e c i a l case w i l l be c o n s t r u c t e d b u t s i n c e t h i s does n ot r e l a t e t o the axiom on p r e f e r e n c e , we f e e l f r e e t o do so. L e t (ft , 0,y) be an a r b i t r a r y p r o b a b i l i t y space where 0 i s l a r g e enough t o in d u c e a l l p r o b a b i l i t y measures i n M a r s c h a k 1 s approach. L e t t h e p r o b a b i l i t y measures P ( - , a ) , P(-,b) and P(-,c) be i n d u c e d by X ( * , a ) , X(-,b) and X(*,c) r e s p e c t i v e l y . L e t ( [ 0 , l ] , g , X ) be t h e p r o b a b i l i t y space w i t h £ t h e B o r e l l s e t s and X t h e Lebesque measure, and l e t {[0,1] x ft, 6 x 0 , y x X} denote t h e p r o d u c t space. We now d e f i n e two random v a r i a b l e s as f o l l o w s : X ( ( w ^ , ^ ) , d ) = X U , ,a) X U ^ b ) u>2 e [0 , a) u>2 e [a , 1 ] X ( ( w ^ , a ^ ) ,e) X(co 1,c) X(a3 1,b) £ " [0 , a) T h i s i m p l i e s f ( d ) - f ( e ) = f ( a ) - f ( c ) by Axiom I I s i n c e ' X f t x [ a , l ] ( * ' d ) = X f t x [ a , l ] ( ' ' e ) a n d h e n c e d > e i f a n d  o n l Y  i f a > c. R e w r i t i n g t h i s i n t o p r o b a b i l i t y measures we have P(-,d) = aP(-,a) + ( l - a ) P ( - , b ) P(-,e) = aP(-,c) + ( l - a ) P ( - , b ) aP(-,a) + ( l - a ) P ( - , b ) > aP(-,c) + ( l - a ) P ( - , b ) n i f and o n l y i f P(-,a) > P ( - , e ) . Hence i n t h i s sense t h e axioms a r e e q u i v a l e n t . Hagen (1965) c r i t i c i z e d t h i s c o n s t r u c t i o n i n t h e case o f t h e A l l a i s paradox. I n t h a t case o ur c o n s t r u c t i o n would be as f o l l o w s : Assume a c t i o n s e , f and g a r e d e f i n e d as f XU,e) = { $0 o> e B 5 x 1 0 6 a) e B X ( - , f ) = $1 x 1 0 6 X(-,g) = $0 and y-(B) = 10/11. L e t a = 11/100, and d e f i n e a c t i o n a, b, c, 4 and d by X ( ( a > ^ , o > 2) / a) X ( ( w ^ , o > 2 ) ib) X( ( t o ^ u ^ ) rc) = < X ( a ) l f : f ) X ( o ) 1 , f ) X(a> ,e) X ( o ) 1 , f ) X(w , f ) X U ,g) u 2 e [0,11/100) u > 2 e (11/100,1] c o 2 E [0,11/100] 0 3 2 e (11/100,1] w 2 £ [0,11/100] c o 2 e (11/100,1] X( ( a ) 1 , u 2 - ) ,d) = X ( o > 1 ,e) X ( o ) l f g) O J 2 e [0,11/100] o i 2 e (11/100,1] Hence we note t h a t a c t i o n s a, b, c, and d i n d u c e t h e same pro-b a b i l i t y measure on t h e reward s e t as i n the A l l a i s paradox and c l e a r l y X f i x [ 0 , 1 1 / 1 0 0 ] ( * ' a ) E X f t x [ 0 , 1 1 / 1 0 0 ] ( " ' c ) and t h e r e f o r e h(nx[0,11/100],a) = h ( n x [ 0 , 1 1 / 1 0 0 ] , c ) . S i m i l a r l y h(nx[0,11/100],b) = h(nx[0,11/100],d) h ( n x ( l l / 1 0 0 , l ] , a ) = h ( n x ( l l / 1 0 0 , l ] , b ) , and h ( n x ( l l / l 0 0 , l ] , c ) = h ( n x ( l l / 1 0 0 , l ] , d ) . Hence by a d d i t i v i t y f (a) - f (b) = f (c) - f (d) . I f we r e w r i t e t h i s r e s u l t i n terms o f p r o b a b i l i t y d i s t r i -b u t i o n s we have P(-,a) = 11/100 P ( - , f ) + 89/100 P ( - , f ) P(-,b) = 11/100 P(«,e) + 89/100 P ( * r f ) P ( -,c) = 11/100 P ( - , f ) + 89/100 P(-,g) P(-,d) = 11/100 P(-,e) + 89/100 P ( - , g ) . Hence f ( a ) > f ( b ) i f and o n l y i f f ( c ) > f ( d ) i m p l i e s 11/100 P(«,f) + 89/100 P ( - , f ) > 11/100 P(-,e) + 89/100 P ( -i f a n d o n l y i f 11/100 P ( - , f ) + 89/100 P(-,g) > 11/100 P(-,e) + 89/100 P(-,g) w h i c h i s a s p e c i a l case o f t h e s t r o n g independence axiom. Hagen's c r i t i c i s m o f t h i s c o n s t r u c t i o n — as we c o u l d l i k e w i s e have d e f i n e d X(*,b) — was: and i n t h i s case t h e i n d u c e d p r o b a b i l i t y d i s t r i b u t i o n would have been the same; however, Axiom I I can not be used t o determine a p r e f e r e n c e between a and b. Hence we a r e n o t s t a t i n g an e x a c t e q u i v a l e n c e between Axiom I I and t h e s t r o n g independence axiom. However, we do s t a t e t h a t t h e r e always e x i s t s a p r o b a b i l i t y space and a s e t o f f u n c t i o n s d e f i n e d such t h a t i f t h e s t r o n g independence axiom i s n o t found t o be a c c e p t a b l e , t h e n n e i t h e r i s Axiom I I a c c e p t a b l e . On t h e o t h e r hand, i f Axiom I I i s n o t a c c e p t a b l e f o r some f u n c t i o n s , t h e n n e i t h e r can t h e s t r o n g independence axiom be a c c e p t a b l e f o r t h e i n d u c e d p r o b a b i l i t y measure o f t h o s e f u n c t i o n s . Luce and K r a n t z approach. I n t h e i r approach they assume a p r e f e r e n c e e x i s t s on X_(«,a) f o r a l l B e (3, a e A. C l e a r l y t h e n Axiom l i b must h o l d ( i f h e x i s t s ) . S i m i l a r t o t h e Savage approach, Axiom LK4 s p e c i f i e s t h a t i f X D ( - , a ) > X D ( * , b ) t h a t i s , a c t i o n a i s p r e f e r r e d t o a c t i o n b on the s e t D, th e n i f D H B 0 and X R ( • ,a) = X B ( • ,b) then X ( (<3 , ui •J''/b) =< X(o> l f f) X(o), ,e) I D 2 e 10,89/100) u>2 e [89/100,1) X_, , . _,(-,a) > X^ . (*,b). T h i s a l s o i n d i c a t e s an independence o f e v a l u a t i n g an a l t e r n a t i v e on d i f f e r e n t s u b s e t s o f 3 . I n terms o f t h e f u n c t i o n h, we have t h a t h(D,a) £ h(D,b) and h(B,a) = h(B,b) i m p l i e s t h a t h(D O B,a) h(D 0 B,b) i f B f\ D = 0 As b e f o r e , however, i t does not i m p l y t h a t h i s a d d i t i v e . 4.5 A l t e r n a t i v e s t o Axiom I I There a r e two b a s i c a l t e r n a t i v e s t o t h i s axiom, e i t h e r r e s t r i c t i n g the s e t o f e v e n t s f o r which the f u n c t i o n h i s a d d i t i v e , o r i n c r e a s i n g t h i s s e t . The a l t e r n a t i v e s a r e t h e n based on t h e c a r d i n a l i t y o f s e t 3 . I n view o f Axioms I I I and IV, i t i s i m p l i e d t h a t t h e r e must e x i s t a measure W w h i c h i s extended from y on 0 . Hence t h i s would r e s t r i c t 3 from a m a t h e m a t i c a l p o i n t o f view s i n c e the e x t e n s i o n may n o t e x i s t . Hence i f we assume f o r example t h a t 3 i s the s e t o f a l l s u b s e t s o f Q. t h e a s s u m p t i o n t h a t a measure e x i s t s on 3 must be r e l a x e d i n the same way. I n Savage's (19 54) approach he assumed t h a t 3 i s e q u i v -a l e n t t o a l l s u b s e t s and a l s o t h a t t h e extended measure i s o n l y f i n i t e l y a d d i t i v e r a t h e r than a - a d d i t i v e . The o t h e r a l t e r n a t i v e would be t o reduce t h e s e t f o r which h i s a d d i t i v e , t h a t i s t h e r e e x i s t s a s t r i c t s u b s e t o f 3 f o r w h i c h h i s a d d i t i v e . T h i s would i m p l y t h a t the e x t e n -s i o n may n o t be a measure on 3 . T h i s approach w i l l be d i s -c u s s e d i n P a r t I I o f the t h e s i s . I f we assume f o r t h e moment t h a t 0 = {ft,<|>}, t h e n t h e r e e x i s t s e v e r a l a l t e r n a t i v e s t o Axiom I I . We s h a l l c o n s i d e r a few o f t h e s e . A l l o f them assume a f u n c t i o n on R c a l l e d a u t i l i t y f u n c t i o n and t h e n f i s s p e c i f i e d by the f o l l o w i n g r u l e s : The maximax c r i t e r i o n ( H u r w i t z y 1951). The e v a l u a t i o n f u n c t i o n i s s p e c i f i e d by t a k i n g the maximum u t i l i t y f o r each a c t i o n . The maximin c r i t e r i o n (Wald, 1950). The e v a l u a t i o n f u n c t i o n i s s p e c i f i e d by t a k i n g t h e minimum u t i l i t y f o r each consequence. The H u r w i t z a - c r i t e r i o n ( H u r w i t z , 1951). The e v a l u a t i o n f u n c t i o n i s s p e c i f i e d by a l i n e a r c o m b i n a t i o n o f t h e maximum and minimum u t i l i t y f o r each a c t i o n . There a l s o e x i s t a d d i t i o n a l a l t e r n a t i v e s w h i c h w i l l be d i s c u s s e d i n the i n t r o d u c t i o n t o P a r t I I o f t h e t h e s i s . 5.0 S e p a r a b i l i t y axiom I n t h i s s e c t i o n we s h a l l d i s c u s s Axiom I I I . The axiom i s f o r m a l l y s t a t e d i n s e c t i o n 5.1 and i t s i m p l i c a t i o n s a r e g i v e n i n s e c t i o n 5.2. E m p i r i c a l s t u d i e s a r e d e s c r i b e d i n s e c t i o n 5.3, i t s r e l a t i o n t o o t h e r axiom systems i s shown i n s e c t i o n 5.4, and f i n a l l y , t h e a l t e r n a t i v e s t o i t a r e d i s c u s s e d i n s e c t i o n 5.5. 5.1 Statement o f Axiom I I I I n the i n t r o d u c t i o n we s t a t e d t h i s axiom as t h e e x i s t e n c e o f f u n c t i o n s W and U such t h a t h ( r , p ) = U ( r ) W ( p ) . The major a s s e r t i o n u n d e r l y i n g Axiom I I I , t h e r e f o r e , i s t h a t we can s e p a r a t e the u t i l i t y o f a reward from the p r o b a b i l i t y o f r e c e i v i n g t h e reward. T h i s s e p a r a b i l i t y c o n c e p t was c a l l e d e t h i c a l n e u t r a l i t y by Ramsey (1926), a l t h o u g h i t was s t a t e d s l i g h t l y d i f f e r e n t l y . We n o t e d i n s e c t i o n 4.2 t h a t i f a l l c o n s t a n t f u n c t i o n s from ft t o R b e l o n g t o A, a u t i l i t y f u n c t i o n may be d e f i n e d on R. However, i f A s s umption 2 does n o t h o l d , t h e e x i s t e n c e o f the f u n c t i o n U on R must be assumed. The f i r s t p a r t o f Axiom I I I w i l l s t a t e the e x i s t e n c e o f such a f u n c t i o n , i n a d d i t i o n t o a r e a l - v a l u e d f u n c t i o n W on 3 . Axiom I I I t h e n r e l a t e s t h e f u n c t i o n h(B,a) t o t h e f u n c t i o n s U and W. B e f o r e the axiom can be s t a t e d , however, we s h a l l need some a d d i t i o n a l n o t a t i o n s and d e f i n i t i o n s . I n f o r m a l l y , a s i m p l e f u n c t i o n from ft t o the r e a l l i n e , , E i .(:sejepAppendix'j: I.I„ J o r ^ d e f i r i i -t i o n ) i s a f u n c t i o n which o n l y a s s i g n s f i n i t e l y many /values . F o r example: i f B^,...,B a r e d i s j o i n t s e t s b e l o n g i n g t o B whose 6 5 u n i o n i s ft, then Y(-) d e f i n e d by Y ( O J ) = ''a^ to e B^ l a to e B v n n i s a s i m p l e f u n c t i o n . L e t Z be t h e i n d e x s e t o f a l l s i m p l e f u n c t i o n s from ft t o E. Then, c o n s i s t e n t w i t h our p r e v i o u s n o t a t i o n , we s h a l l l e t Y ( - , z ) denote t h e s i m p l e f u n c t i o n c o r r e s p o n d i n g t o z e Z. k F o r each f u n c t i o n Y ( - , z ) and f o r any B = L j B.. we s h a l l d e f i n e a number h(B,z) by i = l 1 k h(B,z) = 21 a.W(B.), i = l 1 1 where W i s a s e t f u n c t i o n d e f i n e d i n Axiom I I I . T h i s n o t a t i o n may, o f c o u r s e , c r e a t e some c o n f u s i o n s i n c e i n Axiom I I a n o t h e r f u n c t i o n h(B,a) was assumed t o e x i s t . However, as we s h a l l show, i f UX(-,a) i s a s i m p l e f u n c t i o n where a e A the two f u n c t i o n s a r e i d e n t i c a l . We a l s o note t h a t i f Assumption 2 h o l d s , a complete o r d e r i n g may be i n d u c e d on R i n t h e f o l l o w i n g way: R r ^ > r 2 i f and o n l y i f U(r^) 2> U ( r 2 ) and R r ^ > r 2 i f and o n l y i f U ( r ^ ) > U ( r 2 ) . A g a i n , i f Assumption 2 i s n o t assumed, an o r d e r i n g on R can s t i l l be s p e c i f i e d i f U i s assumed t o e x i s t . T h e r e f o r e , a f t e r we assume the e x i s t e n c e o f U (Axiom I l i a ) , whenever we r e f e r t o an o r d e r i n g on R, we mean th e i n d u c e d o r d e r i n g on R by U. Axiom I I I . S e p a r a b i l i t y Axiom a) There e x i s t s a n o n - n e g a t i v e r e a l - v a l u e d a - a d d i t i v e f u n c t i o n W on 3 and a r e a l - v a l u e d measurable f u n c t i o n U on R such t h a t f o r any a e A, B e 3 and r e R, i f X (••,&•) = r , t h e n h(B,a) = W(B)U(r) . b) L e t r Q be any f i x e d reward i n R, and l e t R X(-,b) be any reward f u n c t i o n such t h a t X_.(oj,b) > r Jo O f o r a l l a) e B, f o r B e 3 • I f Z i s t h e s e t o f a l l o s i m p l e f u n c t i o n s z such t h a t Y_(*,z) <: UX_(*,b), then B B h(B,b) s a t i s f i e s h(B,b) = sup h(B,z) zeZ c) S i m i l a r l y f o r any reward f u n c t i o n X(-,c) such t h a t R r £ X^Cw /C ) f o r a l l oo e B i f Z, i s t h e s e t o f a l l O 13 J_ s i m p l e f u n c t i o n s z such that- Y_ (• , z) UX_,(-,c), t h e n 1 3 1 3 h(B,c) s a t i s f i e s h(B,c) = i n f h(B,z) zeZ, The f i r s t p a r t o f t h e axiom s p e c i f i e s h(B,a) f o r reward f u n c t i o n s w h i c h a r e c o n s t a n t on a s e t B i n 3 . S i n c e Axiom I I i m p l i e s t h a t h i s an a d d i t i v e f u n c t i o n , Axiom I l i a a l s o s p e c i f i e s h(B,a) f o r reward f u n c t i o n s w h i c h a r e s i m p l e f u n c t i o n s . Axiom I l l b and I I I c e x t e n d t h e d e f i n i t i o n o f h(B,a) t o an a r b i t r a r y reward f u n c t i o n b e l o n g i n g t o A. 5.2 I m p l i c a t i o n s o f Axiom I I I F i r s t we s h a l l show t h a t i f t h e r e e x i s t s a c o n s t a n t reward f u n c t i o n such t h a t UX(«,a) ^ 0 then i t i s n o t n e c e s s a r t o assume t h a t W i s a - a d d i t i v e s i n c e t h i s i s i m p l i e d . To see t h i s , l e t X(-,a) = r , and B^, i = - l , 2 , . . . , be any p a r t i t i o n o f B, such t h a t B^ e 8 f o r a l l i . We have by Axiom I I oo and by Axiom I I I , 0 0 WCB)U(r) = T W(B . ) U C r ) . i = l 1 T h i s i m p l i e s t h a t i f U ( r ) ^ 0, the n 0 0 W(B) = 2 WCB^ )^ . i = l Hence i f a c o n s t a n t f u n c t i o n UX(-,a) = UCr) ^  0 e x i s t s such t h a t a e A the n W must be a - a d d i t i v e . F o r s i m p l e f u n c t i o n s , Axioms I I and I l i a s p e c i f y the e v a l u a t i o n f u n c t i o n f . L e t t h e v a l u e s o f X(-,a) be e q u a l t o r , , r 5 , . . . , r and d e f i n e B. = {oj:X(co,a) = r.,oj e ft}. Then J- ^ II _L x {B^} i s a p a r t i t i o n of ft, and B^ e 3 f o r a l l i . Hence f (a) = Z hCB.,a) (by Axiom I I ) i = 21w(B i)U(r i) (by Axiom I I I ) . Hence i n t h i s case we have an e x p e c t e d u t i l i t y theorem i f W(B^) = 'u(B^); t h a t i s , t h e e v a l u a t i o n f u n c t i o n f i s s p e c i f i e d by the e x p e c t e d u t i l i t y o f the reward f u n c t i o n X(- ,a) . We , s h a l l a l s o c o n s i d e r some q u e s t i o n s o f c o n s i s t e n c y between Axiom I I and Axiom I I I as we must show t h a t t h e two axioms do not c o n t r a d i c t each o t h e r . A q u e s t i o n a r i s e s from the a d d i t i v i t y and s e p a r a b i l i t y axioms as t o whether o r n o t f i s u n i q u e l y d e f i n e d . F o r example i f r n = r ^ i n our p r e v i o u s example, would f have the same v a l u e i n d e p e n d e n t l y o f whether we c o n s i d e r the p a r t i t i o n B ^ , . . . , B ' _ ^ (by c o n s i d e r i n g = B,\J B , and B. = B. f o r i = 2 , . . . , n - 1 ) , o r the p a r t i t i o n 1 1 n 1 1 i i i i f B.,,...,B . I t i s o b v i o u s t h a t f would t a k e t h e same v a l u e 1 n i f and o n l y i f W i s an a d d i t i v e s e t f u n c t i o n . By c o n s i d e r i n g a c o u n t a b l e p a r t i t i o n o f any o f the s e t s B^, t h e same a r g u -ment would i m p l y t h a t W must be a - a d d i t i v e . I n the c a s e where X ( o j,a) > r Q f ° r a l l co e ft b u t i s n o t n e c e s s a r i l y a s i m p l e reward f u n c t i o n , we must a l s o show t h a t i f {B^} i s any a r b i t r a r y p a r t i t i o n o f a s e t B t h e n Axiom I I I does not c o n t r a d i c t the a d d i t i v i t y assumption i n Axiom I I , t h a t i s h(B,b) = X h ( B ± , b ) . i We s h a l l show t h i s by f i r s t c o n s i d e r i n g the c a s e when the p a r t i t i o n o n l y c o n t a i n s two d i s j o i n t s e t s B-, and B0. I f ¥ . i R ('rac) ( f o r a d e f i n i t i o n , see Appendix I I I ) 1 w a 2 i s a s i m p l e f u n c t i o n l e s s t h a n UX^ , . _ (-,b) th e n Y„ (>,a) i s a s i m p l e f u n c t i o n l e s s t h a n UX_ ('• ,b) and Y„ (',c) i s a B l B 2 s i m p l e f u n c t i o n l e s s than UX (•,b). S i m i l a r l y , i f 2 Y D (*/a) and Y_, (-,c) a r e s i m p l e f u n c t i o n s l e s s t h a n B l B 2 UX (-,b) and UX D (-,b) r e s p e c t i v e l y , t h e n Y D , , _ (-,ac) i s B 1 B 2 B]_ O B 2 a s i m p l e f u n c t i o n l e s s t h a n UX_ , (*,b). Hence t h e f i r s t B l u B 2 a s s e r t i o n i m p l i e s sup h ( B 1 \ J B 2 ,ac) < sup h(.B 1,a) + sup h(B 2,< and t h e second i m p l i e s sup hCB^Vj B 2 ,ac) > sup h(B^,a') + sup h ( B 2 , c ) , T h e r e f o r e h ( B 1 L j B 2 , b ) = h(Blfb) + h ( B 2 , b ) . By i n d u c t i o n the f o l l o w i n g e q u a l i t y must be s a t i s f i e d : n n h( U B. ,b). = 2 h ( B . ,b) i = l i = l x To show the g e n e r a l c a s e , we must show t h a t t h i s may be extended t o a c o u n t a b l e number of s e t s , t h a t i s OO 0 0 h ( V j B . , b ) = '2Lh(B.,b) f o r any p a r t i t i o n {B. } o f B i = l 1 i = l 1 1 h o l d s f o r an a r b i t r a r y a c t i o n b. To show t h i s , we note t h a t k |h(B,b) - 2! h(B. ,b) | ^  |h (B,b) - h (B, z) I + i = l k h(B,z) - 2 T h ( B . , z ) | + | h ( V j B . , z ) - h ( O B . f b ) i = l 1 i = l 1 i = l 1 by t r i a n g l e i n e q u a l i t y . L e t h(B,z) be the v a l u e a s s o c i a t e d w i t h a s i m p l e f u n c t i o n such t h a t |h(B,b). - h(B,z) | < £ / 3 and k k |h(. L ) B . ,?:) - h C V j B . , b ) | < e / 3 . i = l 1 i = l 1 k S i n c e |h(B,z) - £ h ( B . , z ) | d e c r e a s e s as k i n c r e a s e s , t h e n i = l 1 f o r k l a r g e enough k |h(B,z) - Z l h(B. ,z) | < 1/3 i = l 1 hence k - |h(B,b) - 21 h(B ,b) | < e i = l x f o r k s u f f i c i e n t l y l a r g e , t h e r e f o r e oo h(B,b) = T~ h(B. ,b) . m 1 What we have shown so f a r i s t h a t Axiom I I and Axiom I I I a r e c o n s i s t e n t w i t h each o t h e r ; t h a t i s , t h a t t h e d e f i n i t i o n o f h(B,a) does n o t c o n t r a d i c t the a d d i t i v i t y a s sumption i n Axiom I I f o r X(-,a) .^ r Q . S i m i l a r l y , t h e same r e s u l t h o l d s i f X(•,a) < r . o We a l s o have t o show t h a t t h e number h(B,b) -v.v •.' <~ i s independent o f the c h o i c e o f r , i . e . , i f any o t h e r reward r i s chosen r a t h e r t h a n r Q , h ( B , d ) i s u n i q u e l y d e f i n e d . To do so, R l e t r-^ be any o t h e r reward i n R, say < r Q . Then h (B O (X (• ,d) > r 1 ) ,a) = = h ( B A (X(.-,d) > r 0>' a> + M B H 0 ^ ^ XI-,d) < r Q ) , a ) . S i m i l a r l y h ( B r\ (X(- ,d) < r Q ) ,a) = = h ( B A (X(-,d) < r±) ,a) + M B H 0 ^ ^ X(-,d). < r Q ) , a ) . Hence h ( B A (XI-,d) > r Q ) - a ) + M B A. (X(>,d) < r Q ) , a ) = = h ( B A (X(- ,d) > r 1 l ,a) - h ( B A , Cr]_ X(-,d) < r Q ) ,a) + h ( B A (X(-,d) < r±)ra) + h ( B H (r± £ X(-,d) < r Q ) , a ) = = h ( B A (X (• ,d) £ r 1 ) ,a) + h ( B A (X(- ,d) < r 1 ) ,a) . T h i s i m p l i e s t h e n t h a t any reward r e R may be chosen i n Axiom I I I , p a r t s b and c. So f a r we have o n l y shown t h a t Axiom I I I does n o t v i o l a t e the p r e v i o u s axioms. We s h a l l now c o n s i d e r t h e i m p l i c a t i o n o f Axiom I I I by c o n s i d e r i n g some s i m p l e d e c i s i o n problems, which have been suggested a s "paradoxes" i n e x p e c t e d u t i l i t y t h e o r y . I n d o i n g so, W i s c o n s i d e r e d a p r o b a b i l i t y , as i n s e c t i o n 6.1. MacCriiriirion paradox I I (MacCrimmon and L a r s son, 1975) . "Two f r i e n d s on t h e i r way t o a r e s t a u r a n t d e c i d e t o o r d e r t h e c h e f ' s s p e c i a l o f t h e day a l t h o u g h n e i t h e r knows what i t i s . On t h e way i n , Tom makes H a r r y t h e f o l l o w i n g o f f e r : H a r r y i s to guess i f i t w i l l be meat o r f i s h - i f he i s r i g h t , Tom w i l l t r e a t them t o a b o t t l e o f t h e b e s t w h i t e wine. H a r r y guesses f i s h . The wine stewa r d o v e r h e a r s them t a l k i n g about t h e wine and t e l l s them t h a t i t i s o u t o f s t o c k but the b e s t r e d wine i s i n s t o c k . Tom t h e n changes t h e p r i z e t o a b o t t l e o f r e d wine. H a r r y changes h i s guess t o meat." A t f i r s t g l a n c e , t h i s seems t o be a c o n t r a d i c t i o n t o Axiom I I I i n the f o l l o w i n g way: L e t 0 = {(meat), (f i s h ) ,U, <f>} where (meat) = (the e v e n t t h a t meat i s the c h e f ' s s p e c i a l ) and s i m i l a r l y f o r ( f i s h ) . H a r r y g u e s s i n g f i s h i m p l i e s U ( w h i t e w i n e ) W { ( f i s h ) } > U ( w h i t e wine)W{(meat)}. I f U > 0 then W { ( f i s h ) } > W{(meat)} and t h e r e f o r e U ( r e d wine) W{ ( f i s h ) } > U ( r e d wine)W{ (meat)}. Hence by r e v i s i n g h i s guess t h i s would i n d i c a t e t h a t he thought meat was more l i k e l y . T hat i s W{ (meat) } > W { ( f i s h ) } c o n t r a d i c t s our p r e v i o u s c o n c l u s i o n . H a r r y ' s b e h a v i o u r may be p e r f e c t l y r a t i o n a l , however, i n t h a t we may be d e f i n i n g the reward s e t i n c o r r e c t l y . H a r r y does n o t c o n t r a d i c t Axiom I I I i f our d e f i n i t i o n o f t h e reward s e t i s e q u a l t o ( w h i t e wine w i t h f i s h ) , 7 3 Cwhite wine w i t h , meat) f (.red wine w i t h f i s h ) , ( r e d wine w i t h meat). S i n c e Tom o f f e r e d w h i t e wine e i t h e r (1) t h e n a t u r a l a s s o c i a t i o n would be f i s h so H a r r y s a i d t h a t , o r (2) s i n c e Tom o f f e r e d w h i t e wine t h i s i m p l i e d t h a t he e x p e c t e d f i s h so H a r r y guessed ; a c c o r d i n g l y , o r he changed t o meat because he t h o u g h t i t u n l i k e l y the c h e f would put on a f i s h s p e c i a l when he was o u t o f w h i t e wine. A n o t h e r paradox a p p a r e n t l y c o n t r a d i c t i n g t h e s e p a r a b i l i t y p a r t o f t h e axiom i s c a l l e d t h e Newcombe paradox. Newcombe paradox - N o z i c k (1969). C o n s i d e r the f o l l o w i n g s i t u a t i o n : Two c l o s e d boxes A and B a r e on t h e t a b l e i n f r o n t o f you. Box A c o n t a i n s $1,000. Box B c o n t a i n s e i t h e r n o t h i n g o r $1,000,000. You do not know w h i c h . You have a c h o i c e between two a c t i o n s : ( i ) Take what i s - i n b o t h boxes, ( i i ) Take o n l y what i s i n box B. A t some time b e f o r e t h i s o p p o r t u n i t y , a s u p e r i o r b e i n g made a p r e d i c t i o n about what you w i l l d e c i d e . The b e i n g i s "almost c e r t a i n l y " c o r r e c t . I f t h e b e i n g e x p e c t s you t o t a k e a c t i o n ( i ) , he w i l l l e a v e box B empty. I f he e x p e c t s you t o t a k e a c t i o n ( i i ) , he w i l l l e a v e $1,000,000 i n box B. I f he e x p e c t s you t o randomize your c h o i c e , f o r example by f l i p p i n g a c o i n , he w i l l l e a v e box B empty. I n a l l c a s e s , box A c o n t a i n s $1,000. Which a c t i o n would you choose? C o n s i d e r the f o l l o w i n g arguments: ( i ) E i t h e r t h e money i s i n box B o r i t i s n o t . I f t h e money i s i n box B and I t a k e b o t h boxes, I w i l l have $1,000 more th a n i f I had o n l y t a k e n Box B. A l t e r n a t i v e l y , i f t h e 7 4 money i s n o t i n box B f and I t a k e both, b o x e s f a t l e a s t I w i l l g e t $1,000, Hence, t a k i n g a l t e r n a t i v e ( i ) ( i . e . ' , s e l e c t i n g both, boxes, i s the b e t t e r s t r a t e g y . ( i i ) I f the b e i n g can guess w i t h "almost c e r t a i n t y " then I would o n l y t a k e box B, s i n c e i f I were t o t a k e b o t h , he would a l m o s t s u r e l y guess c o r r e c t l y and hence l e a v e box B empty. T h i s problem, and i t s a s s o c i a t e d arguments, was f i r s t pub-l i s h e d by N o z i c k (1969) and i s c a l l e d t h e "Newcombe paradox". The Newcombe problem d i f f e r s from o t h e r s i n s e v e r a l ways. P e r -haps most i m p o r t a n t l y , i t presumes t o s e t one o f t h e axioms i n o p p o s i t i o n t o the e x p e c t e d u t i l i t y c r i t e r i o n , r a t h e r t h a n t o a t t a c k one o f t h e axioms w i t h a c o u n t e r - a x i o m . Argument ( i ) i s based on the dominance axioms, t h a t i s , i f a l t e r n a t i v e ( i ) i s always b e t t e r t h a n a l t e r n a t i v e ( i i ) i n d e p e n d e n t l y o f what s t a t e o f n a t u r e o c c u r s , choose ( i ) . I n symbols we have t h a t i f X(o),a) < X(w,b) f o r a l l w e f t t h e n a < b. The argument f o r a l t e r n a t i v e ( i i ) i s based on an e x p e c t e d u t i l i t y f o r m u l a t i o n . Presumably we cannot have b o t h . I t i s u s e f u l t o a n a l y s e more d i r e c t l y how t h e dominance and e x p e c t e d u t i l i t y f o r m u l a t i o n s a p p a r e n t l y c o n t r a d i c t each o t h e r . L e t us l o o k f i r s t a t dominance, as e x p r e s s e d most d i r e c t l y i n Arrow's Axiom A 4 . Dominance i s a l m o s t u n i v e r s a l l y a c c e p t e d as a r e a s o n a b l e axiom t o use when i t a p p l i e s , and so i t would be h a r d t o choose i n c o n t r a d i c t i o n t o i t . C o n s i d e r the f o l l o w i n g way o f f o r m u l a t i n g t h e problem i n a p a y o f f m a t r i x : 7 5 $ 1 , 0 0 0 , 0 0 0 i n box B N o t h i n g i n box B ( i ) Take b o t h boxes $ 1 , 0 0 1 , 0 0 0 $ 1 , 0 0 0 ( i i ) Take o n l y box B $ 1 , 0 0 0 , 0 0 0 $ 0 I f we a c c e p t t h i s f o r m u l a t i o n o f t h e problem, i t would be d i f f i c u l t n o t t o t a k e a l t e r n a t i v e ( i ) because i t dominates a l t e r n a t i v e ( i i ) . However, t h i s f o r m u l a t i o n may be q u e s t i o n a b l e because i t f a i l s t o t a k e i n t o a c c o u n t t h e p r e d i c t i v e a b i l i t y o f t h e s u p e r i o r b e i n g . C o n s i d e r , i n s t e a d , t h e f o l l o w i n g p a y o f f m a t r i x f o r m u l a t i o n : B e i n g p r e d i c t s c o r r e c t l y B e i n g does n o t p r e d i c t c o r r e c t l y ( i ) Take b o t h boxes $ 1 , 0 0 0 $ 1 , 0 0 1 , 0 0 0 ( i i ) Take o n l y box B $ 1 , 0 0 0 , 0 0 0 $ 0 O b v i o u s l y i n t h i s f o r m u l a t i o n dominance does n o t a p p l y and one would t a k e a l t e r n a t i v e ( i i ) i f P ( b e i n g c o r r e c t ) = p, and p U ( $ l , 0 0 0 , 0 0 0 + ( l - p ) U ( $ 0 ) > p U ( $ l , 0 0 0 ) + ( l - p ) U ( $ l , 0 0 1 , 0 0 0 ) . F o r any r e a s o n a b l e u t i l i t y f u n c t i o n , and assuming t h a t p i s c l o s e t o 1 as i m p l i e d i n the problem, a l t e r n a t i v e ( i i ) would have the h i g h e r e x p e c t e d u t i l i t y . Thus, i f i t were n ot f o r the d i f f e r e n t . f o r m u l a t i o n s , one would have t h e p a r a d o x i c a l s i t u a t i o n i n w h ich dominance i m p l i e s one a c t i o n w h i l e the m a x i m i z a t i o n o f o f the e x p e c t e d u t i l i t y i m p l i e s a n o t h e r a c t i o n . The major d i f f e r e n c e , t h e n , between t h i s c h a l l e n g e t o the axioms and t h o s e c o n s i d e r e d e a r l i e r , i s t h a t t h e "Newcombe paradox" i s based on the way the problem i s f o r m u l a t e d , r a t h e r than i n t h e c h o i c e s o f f e r e d i n a s p e c i f i c f o r m u l a t i o n . Expec-t e d u t i l i t y t h e o r y r e q u i r e s an independence between the e v e n t s and t h e a c t i o n s . I n the "dominance f o r m u l a t i o n " o f t h e p r o -blem, the p r o b a b i l i t y o f e i t h e r e v e n t o c c u r r i n g i s n o t in d e p e n -dent o f our c h o i c e o f a c t i o n s and i s t h e r e f o r e i n a p p r o p r i a t e . W h i l e t h i s d i f f i c u l t y does n ot h o l d f o r t h e second f o r m u l a t i o n above, t h e second f o r m u l a t i o n does n ot t a k e i n t o a c c o u n t the amount i n the boxes and hence may seem i n c o m p l e t e . I n o r d e r t o g e t b o t h u n c e r t a i n elements i n t o t h e problem, we need t o form t h e compound e v e n t s : B e i n g p r e d i c t s c o r r e c t l y and p u t : B e i n g p r e d i c t s i n c o r r e c t l y and p u t : $ 1 , 0 0 0 , 0 0 0 i n box B $ 0 i n box B $ 1 , 0 0 0 , 0 0 0 i n box B $ 0 i n box B ( i ) Take b o t h boxes - $ 1 , 0 0 0 $ 1 , 0 0 1 , 0 0 0 -( i i ) Take o n l y box B $ 1 , 0 0 0 , 0 0 0 - - $ 0 The c r o s s e d o u t c e l l s r e p r e s e n t i m p o s s i b l e c o m b i n a t i o n s and an e x a m i n a t i o n o f the whole t a b l e shows t h a t dominance cannot be a p p l i e d . Hence one can choose o n l y box B and a c t i n a c c o r -dance w i t h e x p e c t e d u t i l i t y w i t h o u t v i o l a t i n g dominance. I f t h e r e a r e v e r y l a r g e , non-monetary s a t i s f a c t i o n s o f e x h i b i t i n g the " f r e e - w i l l " o f t a k i n g b o t h boxes o r o f b e a t i n g the b e i n g out o f $1,001,000 and showing him up i n the p r o c e s s , as a s s e r t e d by Asimov (Gardner, 1974, p.123), t h e n you might choose a c t i o n ( i ) . You would, however, be c h o o s i n g i t on an ex p e c t e d u t i l i t y b a s i s r a t h e r than on t h e b a s i s o f dominance. A l l a i s paradox I I ( A l l a i s , 1953) . There e x i s t s a s i m p l e g e n e r a l i z a t i o n o f t h e A l l a i s paradox I . C o n s i d e r , f o r example, two f u n c t i o n s d e f i n e d as X (*,a) = r , and X (*,b) = s where 1 3 D h(B,a) = h(D,b). T h i s i m p l i e s U(s)W(D) = U(r)W(B) o r e q u i v a l e n t l y U(s) = U(r) , i f W(D) + 0. Then, f o r any o t h e r f u n c t i o n s X„(«,c) = r and X„(-,d) = s hi r such t h a t ^|p-| = ^ (D) i t i s i m p l i e d t h a t h(E,c) = h(F,d) . T h i s g i v e s r i s e t o t h e paradox o f common r a t i o . W i t h Axiom IV we s h a l l see t h a t W(B) = y(B) f o r a l l B e 0 , and we w i l l use t h i s a s s u m p t i o n h e r e t o i l l u s t r a t e t h e common r a t i o paradox. I f we assume t h a t the reward s e t i s the r e a l l i n e and t h a t U(0) = 0 , t h e n the paradox o f common r a t i o o f p r o b a b i l i t i e s i m p l i e s t h a t : A p r e f e r e n c e o f a t o b i m p l i e s a p r e f e r e n c e o f c t o d where a, b, c, and d a r e d e f i n e d as f o l l o w s : a: $1 m i l l i o n w i t h a p r o b a b i l i t y o f 1.0 $0 o t h e r w i s e b: $5 m i l l i o n w i t h a p r o b a b i l i t y o f 0.8 $0 o t h e r w i s e 7 8 c: $1 m i l l i o n w i t h a p r o b a b i l i t y o f 0.05 $0 o t h e r w i s e d: $5 m i l l i o n w i t h a p r o b a b i l i t y o f 0.04 $0 o t h e r w i s e S i n c e 0.8/1 = 0.04/0.05, the r a t i o o f p r o b a b i l i t i e s i s common. T h i s t h e r e f o r e i m p l i e s t h a t i f f ( a ) > f ( b ) t h e n f ( c ) > f ( d ) . E m p i r i c a l s t u d i e s show t h a t t h i s i s n o t always so, i n t h a t p e o p l e commonly s e l e c t a and d. 5.3 E m p i r i c a l s t u d i e s on A l l a i s ' paradox I I I n the two paradoxes c o n s i d e r e d i n t h e p r e c e d i n g s e c t i o n ( i . e . , MacCrimmon and Newcombe), the i s s u e r e v o l v e d around a d e f i n i t i o n o f t h e problem r a t h e r t h a n an e m p i r i c a l i m p l i c a t i o n o f t h e axiom. Thus e m p i r i c a l s t u d i e s o f t h e s e problems would no t g i v e any f u r t h e r s u p p o r t f o r o r a g a i n s t t h e axiom. Hence we s h a l l o n l y c o n s i d e r e m p i r i c a l s t u d i e s o f t h e common r a t i o p roblem. The paradox i s u s u a l l y w r i t t e n i n t h e f o l l o w i n g form. L e t t h e reward space be the r e a l l i n e , and l e t a and (3 be two r e a l numbers. A c h o i c e i s t o be made between a l t e r n a t i v e s a and b: a: r e c e i v i n g r w i t h a p r o b a b i l i t y o f p b: r e c e i v i n g ar w i t h a p r o b a b i l i t y o f Bp. The same p r e f e r e n c e must t h e n h o l d f o r a l l v a l u e s o f p. L e t us t h e r e f o r e choose two v a l u e s o f p, p-^  and p 2 and f o r s i m p l i c i t y we s h a l l c a l l t h e f i r s t c h o i c e i n t h e above p r o -blem a^ o r b^, and t h e second c h o i c e a 2 o r b 2 , and hence a A A p r e f e r e n c e a^ > b^ must i m p l y a 2 > b 2 . Hagen's (1971) s t u d y . Hagen o b t a i n e d some e v i d e n c e from Norwegian t e a c h e r s when he used the problem w i t h t h e f o l l o w i n g parameter v a l u e s : r = 1 : m i l l i o n Norwegian k r o n e r , a = 5, p^ = .99, p 2 = .11, and 0 = 10/11. Because s u b j e c t s were asked f o r c h o i c e s i n o n l y one o f t h e s e t s , Hagen c o u l d compare o n l y the aggregate number o f a v s . b c h o i c e s i n the two s e t s ; he c o u l d not compare each i n d i v i d u a l s u b j e c t ' s c h o i c e s a c r o s s b o t h s e t s . Hagen found t h a t i n the f i r s t s e t , 37 s u b j e c t s o u t o f 52 s e l e c t e d a^ w h i l e i n the second s e t 37 s u b j e c t s o f 52 s e l e c t e d b 2 . Hence t h e r e was a p r e f e r e n c e f o r t h e a^ a l t e r -n a t i v e i n t h e f i r s t s e t b u t t h e b 2 a l t e r n a t i v e i n the second s e t . Thus, we can i n f e r t h a t i f s u b j e c t s had been p r e s e n t e d w i t h b o t h s e t s , the m a j o r i t y would p r o b a b l y have v i o l a t e d t h e e x p e c t e d u t i l i t y h y p o t h e s i s . I n a second e x p e r i m e n t , Hagen used the parameter v a l u e s : r = 10,000 Norwegian k r o n e r , a = 2, p1 = 1.00, p 2 = 0.02, and 0 = 1/2. I n the f i r s t s e t o f c h o i c e s , 4 7 o f t h e 52 s u b j e c t s s e l e c t e d a^ w h i l e i n the second s e t 23 o f t h e 52 s u b j e c t s s e l e c t e d b 2 . Even though the a 2 a l t e r n a t i v e was t h e more f a v o u r e d one i n t h e second s e t , t h i s p a t t e r n o f c h o i c e s a g a i n s u g g ested some v i o l a t i o n o f Axiom I I I . MacCrimmon and L a r s s o n (19 7 5 ) . MacCrimmon and L a r s s o n a t t e m p t e d t o s t u d y the e f f e c t o f v a r y i n g some o f the parameter v a l u e s c o n c e n t r a t i n g on t h e e f f e c t o f u s i n g d i f f e r e n t p a y o f f and p r o b a b i l i t y l e v e l s ( i . e . , v a l u e s o f r and p ) . They used two e x p e r i m e n t s . I n the f i r s t t h e y used p o s i t i v e p a y o f f s ; i n the second t h e y used n e g a t i v e p a y o f f s ( l o s s e s ) t o d etermine whether n e g a t i v e p a y o f f s r e s u l t e d i n major d i f f e r e n c e s i n b e h a v i o u r . The p a y o f f s were a l l h y p o t h e t i c a l but t h e s u b j e c t was asked t o a c t as i f each would a c t u a l l y be r e a l i s e d and t o 80 t r e a t each s e t i n d e p e n d e n t l y o f the o t h e r s . The parameter v a l u e s used were t h e f o l l o w i n g : On the p o s i t i v e e x p e c t e d v a l u e s e t s : a = 5 , 3 = 4/5, r t o o k on the v a l u e s $1,000,000; $100,000; $10,000; $1,000; $100; $10; and $1, w h i l e p took on the v a l u e s 1.00, 0.75, 0.50, 0.25, 0.10, and 0.05. On the nega-t i v e e x p e c t e d v a l u e s e t s : a = 5, 8 = 3/4, r took on the v a l u e s -$1,000; -$100; -$10; and -$1, w h i l e p took on the v a l u e s 1.00, 0.80, 0.20, and 0.04. Twenty-one d i f f e r e n t p o s i t i v e e x p e c t e d v a l u e s e t s were p r e s e n t e d , w i t h f o u r s e t s r e p e a t e d t o check f o r c o n s i s t e n c y . E i g h t d i f f e r e n t n e g a t i v e e x p e c t e d v a l u e s e t s were p r e s e n t e d w i t h two s e t s r e p e a t e d t o check f o r c o n s i s t e n c y . Not a l l com-b i n a t i o n s were p r e s e n t e d s i n c e a p i l o t s t u d y had a s c e r t a i n e d t h a t some c o m b i n a t i o n s ( e . g . , a low p o s i t i v e p a y o f f and a low p r o b a b i l i t y l e v e l ) l e d t o a l m o s t a l l s u b j e c t s c h o o s i n g t h e same a l t e r n a t i v e . ( T h i s i s i n t e r e s t i n g i n i t s e l f , b ut i s n o t the b e s t use o f l i m i t e d t ime f o r an experiment.) The 25 p o s i t i v e p a y o f f s e t s and 10 n e g a t i v e p a y o f f s e t s were p r e s e n t e d i n random o r d e r . The p a r t i c u l a r c o m b i n a t i o n s g i v e n can be seen from the graph o f t h e r e s u l t s i n F i g u r e 5.1. The numbers show f o r each p a y o f f - p r o b a b i l i t y c o m b i n a t i o n how many o f the 19 s u b j e c t s chose t h e h i g h e r p r o b a b i l i t y (a) a l t e r n a t i v e . So, f o r example, w i t h t h e c o m b i n a t i o n o f p a y o f f s and p r o b a b i l i t i e s used i n s e c t i o n 5.2 above, 15 s u b j e c t s chose a i n s e t 1 w h i l e o n l y two chose a 2 i n s e t 2. Not s u r p r i s i n g l y , t h e h i g h e s t l e v e l o f a c h o i c e s , f o r p o s i t i v e amounts, o c c u r s when t h e r e i s a sure 81 P * One s u b j e c t chose a, on one p r e s e n t a t i o n o f t h i s s e t , and b^ on the o t h e r p r e s e n t a t i o n . ** Two s u b j e c t s chose a, on one p r e s e n t a t i o n o f t h i s s e t and b, on the o t h e r p r e s e n t a t i o n . F i g . 5.1. The number o f s u b j e c t s , o ut o f 19, s e l e c t i n g t h e c o n t r a d i c t o r y p r e f e r e n c e s t o e x p e c t e d u t i l i t y c r i -t e r i o n i n the problem o f Common R a t i o o f P r o b a b i l i t i e s 82 chance o f g e t t i n g a l a r g e amount o f money. As t h e graph shows, when t h e p r o b a b i l i t y l e v e l s d e c r e a s e , o r when the money p a y o f f l e v e l s d e c r e a s e , then t h e r e i s a reduced tendency t o p i c k t h e a a l t e r n a t i v e ( i . e . , t h e one g i v i n g the lo w e r p a y o f f w i t h t h e h i g h e r p r o b a b i l i t y ) . When t h e p r o b a b i l i t y l e v e l s d e c r e a s e t h e r a t i o n a l e i s one o f v i e w i n g the p r o b a b i l i t y d i f f e r e n c e as i n -s i g n i f i c a n t and thus " g o i n g f o r b r o k e " on the l a r g e r p a y o f f . When t h e p a y o f f l e v e l s d e c r e a s e , t h e r a t i o n a l e i s one o f " g o i n g f o r b r o k e " s i n c e t h e amount you g e t f o r s u r e does not mean t h a t much t o you i n terms o f l i f e t i m e s e c u r i t y , e t c . The m a j o r i t y o f t h e s u b j e c t s would o n l y s e l e c t t h e a a l t e r n a t i v e when t h e r e was a c e r t a i n t y o f g e t t i n g a v e r y l a r g e p a y o f f ( i . e . , e i t h e r $1,000,000 o r $100,000). Even though each o f ou r s u b j e c t s made 25 ( p o s i t i v e p a y o f f ) c h o i c e s , t h e y seemed t o be q u i t e a l e r t t o the changes i n p a y o f f and p r o b a b i l i t y and hence chose d i f f e r e n t i a l l y . I t i s c l e a r , t h e n , t h a t t h e p a r t i -c u l a r parameter v a l u e s p l a y a major r o l e i n whether one v i o l a t e s the u t i l i t y independence c o n d i t i o n s . S i n c e a l m o s t a l l s u b j e c t s can be e x p e c t e d t o p r e f e r t h e b a l t e r n a t i v e f o r p a y o f f -p r o b a b i l i t y c o m b i n a t i o n s t o t h e l e f t and below t h e dashed l i n e , t h e n t h e r e would be no v i o l a t i o n o f t h e e x p e c t e d u t i l i t y c r i -t e r i o n , i f any o f t h e s e c o m b i n a t i o n s were compared t o each o t h e r . S i n c e i n r e a l c h o i c e s t h e s u b j e c t s would r a r e l y have a l t e r n a t i v e s w i t h p a y o f f s such t h a t t h e y would be t o t h e r i g h t o f t h e l i n e , we may q u e s t i o n whether t h e p o s s i b l e v i o l a t i o n s i n v e r y u n l i k e l y c a s e s have much r e l e v a n c e f o r u t i l i t y t h e o r y . Some c o u n t e r v a i l i n g e v i d e n c e , though, i s found by exam i n i n g t h e n e g a t i v e p a y o f f s . I n t h i s case t h e r e i s more a = $1,000,000 p \ 1. 00 .75 .50 .25 .10 .05 1. 00 1* 6 9 11 11 12 .75 7 5 5 6 .50 2 2 3 .25 0 1 . 10 1 . 05 1* ( i ) a = $1,000 a = -$1,000 P \ 1.00 .75 .50 P \ .80 .20 .04 1. 00 2* 6 4 1.00 1 1 3 . 75 4 . 80 1* 2 4 150 1* . 20 4 ( i i ) ( i i i ) F i g . 5 v a l u e s o f .2 Number o f s u b j e c t s i s c o n s i s t e n t w i t h Axiom I I I f o r d i f f e r e n t a-p i n v a r i a t i o n s o f A l l a i s paradox I I . *These s u b j e c t s were i n c o n s i s t e n t on the r e p e a t o f t h e same s e t , hence t h e y were o m i t t e d from the t a b u l a t i o n o f the remainder o f the t a b l e . . So the i n c o n s i s t e n c i e s i n Ta b l e ( i ) are out o f 17 s u b j e c t s , i n Ta b l e ( i i ) o u t o f 16 s u b j e c t s , and i n T a b l e ( i i i ) o ut o f 18 s u b j e c t s . 84 a m b ivalence i n s w i t c h i n g from a t o b as the p r o b a b i l i t y l e v e l o r s i z e o f t h e l o s s d e c r e a s e s s i n c e t h e l o s s l e v e l s were chosen t o be ones t h a t would be r e a l i s t i c f o r t h e s u b j e c t s . The c h o i c e s t h a t s u b j e c t s made a c r o s s two s e t s a r e shown f o r v a r i o u s c o m b i n a t i o n s i n F i g u r e 5.2. The v a l u e 12 i n the u p p e r - r i g h t c o r n e r o f F i g u r e 5 . 2 ( i ) t e l l s u s , f o r example, t h a t when t h e c h o i c e s o f a s u b j e c t f o r t h e p r o b a b i l i t y l e v e l 1.00 a r e com-p a r e d t o h i s c h o i c e s a t t h e 0.05 l e v e l , 12 o f t h e s u b j e c t s chose a i n one o f t h e s e t s and b i n t h e o t h e r , hence v i o l a t i n g Axiom I I I . Note t h a t t h o s e d i r e c t comparisons c o n f i r m what we o b s e r v e d i n F i g u r e 5.1, t h a t l a r g e p a y o f f and p r o b a b i l i t y l e v e l s l e a d t o a h i g h e r p r o p e n s i t y f o r v i o l a t i n g Axiom I I I . Note though t h a t t h e n e g a t i v e p a y o f f r e s u l t s i n d i c a t e a r e l a t i v e l y low l e v e l o f v i o l a t i o n . 5.4 R e l a t i o n t o o t h e r axiom systems I n Axiom I I I we s t a t e two p r o p e r t i e s , f i r s t , t h e s e p a r a -b i l i t y between the r ewards, and second, th e method of e v a l u -a t i n g h ( B , a ) . I n comparing t h e s e c o n c e p t s t o o t h e r p r e v i o u s l y mentioned systems we e n c o u n t e r some d i f f i c u l t i e s s i n c e n e a r l y a l l t hese axioms a r e needed t o e v a l u a t e h ( B , a ) . However the f i r s t p r o p e r t y o f s e p a r a b i l i t y i s e a s i e r i n some cases t o com-pare and we s h a l l do so h e r e . I n Axiom l i b we assumed t h a t i f X (*,a) = X_(*,b) then h(B,a) = h ( B , b ) . That i s , i f t h e two f u n c t i o n s a r e i d e n t i c a l on an event,, t h e n t h e " e v a l u a t i o n f o r the a c t i o n s on t h a t e v e n t must -be t h e same. The s e p a r a b i l i t y a ssumptions e x t e n d t h i s i d e a t o t h e case where i f X(-,a) i s a c o n s t a n t f u n c t i o n on the e vent B e 0, and X(-,b) has the same c o n s t a n t v a l u e on C e 0 then h(B,a) = h(C,b) i f and o n l y i f W(B) = W(C). S i m i l a r l y i f X B ( - , a ) = r± and X ( • , b) = r 2 a r e c o n s t a n t reward f u n c t i o n s on B I n each o f t h e approaches o f vonNeumann & M o r g e n s t e r n , Marschak, and Arrow i t i s assumed t h a t t h e p r o b a b i l i t i e s a r e g i v e n and we i g n o r e the e v e n t s g i v e n t o t h e s e p r o b a b i l i t i e s , i . e . , t h i s i m p l i e s t h a t h(B,a) i s o f t h e form h ( y ( B ) , a ) . T h i s i s c l e a r l y a s e p a r a t i o n between the event and i t s reward, s i n c e we a r e o n l y e v a l u a t i n g y (B) , and would be i n d i f f e r e n t t o any o t h e r e v e n t D w i t h the same reward f o r which y(D) = y ( B ) . However t h i s does n o t n e c e s s a r i l y i m p l y t h a t the p r o b a b i l i t y and the reward may be s e p a r a t e d . I n t h e i r approaches they i m p l y t h a t h(B,a) = h(C,b) i f and o n l y i f W(B) = W(C). To see t h i s c o n s i d e r t h e case where X_(*,a) = r, B e 0 , and W(B) = a, t h e n Axiom I I I i m p l i e s t h a t h(B,a) = h(B,b) i f and o n l y i f U ( r x ) = u ( r 2 ) . h (B,a) = aU(r) . 8 6 I f y i s any number between ( 0 , 1 ) , t h e n yh(B,a) = ayU(r) o r e q u i v a l e n t l y yh(B,a) must e q u a l t h e e v a l u a t i o n o f any a c t i o n c f o r w h i c h X D(«,c) = r where W(D) = ay. T h i s i n t u r n i m p l i e s t h a t i f X_(*,a) = r , X_(',c) = r , B,E e B, B A "E = 0, W(B) = a, W(E) = y. Then . . (l-6,)h(B,a) + '5h(E,c) .must be e q u i v a l e n t t o an a c t i o n d d e f i n e d by X D ( - , d ) = r and W(D) = (1-6)a + 6y. I n more g e n e r a l terms t h i s i m p l i e s t h a t i f P(«,a), P(*,b) and P ( * , c ) a r e t h r e e p r o b a b i l i t y measures b e l o n g i n g t o n such t h a t t h e m a t h e m a t i c a l i d e n t i t y 6P(-,a) + ( l - 5 ) P ( \ b ) = P ( - , c ) h o l d s we must a l s o have t h e p r e f e r e n c e 6P(.,a) + ( l - 6 ) P ( - , b ) = P(-,c) . T h i s p r e f e r e n c e i s e x p r e s s e d i n b o t h Marschak's and vonNeumann & M o r g e n s t e r n 1 s approaches. There have been some c r i t i c i s m s a g a i n s t t h i s f o r t h e f o l l o w i n g r e a s o n . Suppose we a r e o f f e r e d a p r i z e i f a r e d b a l l i s drawn from e i t h e r u r n I o r u r n I I , o f our c h o i c e . Urn I c o n t a i n s 50 b l a c k and 50 r e d b a l l s . Urn I I has been drawn a t random from a c o l l e c t i o n o f 101 u r n s , one o f w h i c h had 0 r e d and 100 b l a c k b a l l s , a n o t h e r had 1 r e d and 99 b l a c k b a l l s and so on up t o one h a v i n g 100 r e d and 0 b l a c k b a l l s . I f we assume t h a t 8 7 the p r o b a b i l i t y o f a r e d b a l l b e i n g drawn from any u r n i s e q u a l t o t h e number o f r e d b a l l s d i v i d e d by t h e t o t a l number o f b a l l s i n t h e u r n , we must be i n d i f f e r e n t as t o the c h o i c e o f u r n s s i n c e 1/101-0/100+1/101-1/100+1/101-2/100+...+1/101-100/100 = 1/2. However once the second urn has been chosen, t h e r e i s c l e a r l y a f i x e d number o f r e d b a l l s i n t h a t u r n , and t h e p r e f e r e n c e may n o t be e x a c t l y the same s i n c e the chances i n u r n I I a r e n o t e x a c t l y known. F o r example, i f u r n I I were chosen a t random from two u r n s , one w i t h 100 r e d b a l l s and one w i t h 0 r e d b a l l s , ought t h i s t o be e q u i v a l e n t t o an u r n w i t h 50 r e d and 50 b l a c k b a l l s when we know t h a t u r n I I i n f r o n t o f us can o n l y have e i t h e r 0 o r 100 r e d b a l l s ? The argument t h a t u r n I I c o n t a i n s 50 r e d b a l l s on t h e average can c l e a r l y n o t be used s i n c e t h i s i m p l i e s t h a t we have t h e c h o i c e r e p e a t e d l y r a t h e r t h a n once. I n Arrow's approach he assumes t h a t i f two p r o b a b i l i t y d i s t r i b u t i o n s a r e e q u a l they must have the same p r e f e r e n c e . Hence t h i s i s e q u i v a l e n t t o s a y i n g t h a t i f ctP(-,a) + ( l - a ) P ( - , b ) = P ( - , c ) the p r e f e r e n c e must a l s o be t h e same w h i c h we have a l r e a d y d i s c u s s e d . I n Savage's approach the s e p a r a b i l i t y i m p l i c a t i o n s a r e made by c h a n g i n g t h e reward on t h e s e t . R e c a l l Savage's Axiom Axiom S4: I f B , C C n and r ± e R f o r 1=1,2,3,4 ,1^ > r 2 , r 3 > r 4 t h e n f o r a c t i o n s a,b,c,d e A, d e f i n e d by the reward f u n c t i o n s X-(-,a) = r 2 X c ( - , b ) = x1 X^(-,b) = X-(-,c) = r 4 x c ( * ^ ) = r 3 X-(-,d) = r X ( - , b ) , t h e n X(-,c) < X ( - , d ) . Assume t h a t r 2 = r 4 , th e n h(B,a) = h(B,c) and h(C,b) = h(C,d) . I f h(B,a) + h(B,a) < h(C,b) + h(C,b) f o r a g i v e n U(r^) > U ( r 2 ) then t h e same e q u a l i t y must be t r u e f o r r e R such t h a t U ( r 3 ) > U ( r 2 ) . T h i s i m p l i e s t h a t h(B,c) < h(C,d) + c o n s t i f U ( r 3 ) > U ( r 2 ) where X B ( - , c ) = X («,d) = r 3 . That i s , t h e p r e f e r e n c e can be d e t e r m i n e d by comparing i f the reward i s above a f i x e d reward. X B ( - , a ) = r± X B ( - , c ) = r 3 r and i f X( • ,a) 4.. 5.5 A l t e r n a t i v e t o Axiom I I I Axiom I I assumes W(B) i s a r e a l - v a l u e d f u n c t i o n on 3 w h i c h i m p l i e s t h a t i f two reward f u n c t i o n s a r e d e f i n e d by X B ( - , a ) = r± and X B ( - , b ) = r 2 then h(B,a) > h(B,b) i f and o n l y i f U ( r 1 ) > U ( r 2 ) . F o r example, say t h a t we a r e i n v i t e d t o d i n n e r where we know t h a t e i t h e r c h i c k e n , beef o r f i s h i s t o be s e r v e d , and we d e c i d e t o b r i n g a b o t t l e o f wine, e i t h e r r e d , w h i t e , o r r o s e . We a l s o assume t h e f o l l o w i n g u t i l i t y o f t h e reward: C h i c k e n Beef F i s h a 1 - 1 1 b 0 1 - 1 c 0.5 0 -1 w h e r e * a c t i o n a i s t o b r i n g a b o t t l e o f w h i t e wine, a c t i o n b i s t o b r i n g a b o t t l e o f r e d wine, and a c t i o n c i s t o b r i n g a b o t t l e o f r o s e wine. T h e r e f o r e i f B i s the event beef i s s e r v e d we have h(B,a) = -1W(B) h(B,b) = 1W(B) h(B,c) = 0W(B) and hence the p r e f e r e n c e may be d e t e r m i n e d by comparing - 1 , 1, and 0. The assumption here i s t h a t W(B) i s independent o f our a c t i o n . That i s i f we c o n s t r u c t a m a t r i x i n d i c a t i n g t h e v a l u e s -of W(B) f o r each a c t i o n we would have 90 a A c t i o n s b c C h i c k e n W(C) W(C) W(C) Beef W(B) W(B) W(B) F i s h W(F) W(F) W(F) The v a l u e f ( a ) can then be found by m u l t i p l y i n g the row a o f the reward m a t r i x by column a i n e v e n t m a t r i x , i . e . , f ( a ) = h(C,a) + h(B,a) + h(F,a) = 1W(C.) + (-l)W(B) + 1W(F), and s i m i l a r l y f o r f ( b ) and f ( c ) . T h i s example i s t a k e n from R. C. J e f f r e y ' s book "The l o g i c o f d e c i s i o n " (19 65) where he d e v e l o p s a t h e o r y by a r g u i n g t h a t t h e e v a l u a t i o n o f W(B) ought t o be a f u n c t i o n from '3 x A r a t h e r t h a n o n l y g . Hence th e e v e n t m a t r i x can t a k e t h e form a b c C h i c k e n k l k 2 k 3 Beef h X 2 F i s h m l m 2 m 3 where k^, 1^ and i = l , 2 , 3 a r e n o n - n e g a t i v e real-numbers such t h a t k i + 1 i + m i = 1 i = l f 2 , 3 . 91 I t i s easy to see that the Newcomb paradox f a l l s into t h i s type of decision problem. The reward matrix would be as before, Being predicts c o r r e c t l y Being does not predict c o r r e c t l y a • U($1,000) U($l f001,000) b U($l f000,000) U($0) and the event matrix would be a b Being predicts c o r r e c t l y P a Being does not predict (1-P ) corr e c t l y a d-p b) where P & = pr o b a b i l i t y that the being predicts c o r r e c t l y given action a i s chosen. Hence f(a) = U($1,000)P + U($l,001,000)(1-P,) a a f(b) = U($1,000,000)P, + U($0)(1-P,), b b and therefore the preference would depend on P_ and P, . a D There i s some behavioural support for the concept that W(B) also depends on the action. For example, when betting during a game of roulette, some people would argue that they are always unlucky and w i l l therefore lose, while others are always lucky and w i l l win. That i s , the pr o b a b i l i t y of winning does not only depend on the ivory b a l l and the rou-l e t t e wheel but also on who does the betting. 92 6.0 P r o b a b i l i t y axiom F i r s t , ' w e s h a l l s t a t e the axiom i n s e c t i o n 6.1. I n s e c t i o n 6.2 we s h a l l c o n s i d e r t h e i m p l i c a t i o n s o f the axiom and i n s e c t i o n 6.3 we s h a l l e x p l o r e some o f the e m p i r i c a l e v i d e n c e r e g a r d i n g t h e paradoxes r e l a t e d t o t h i s axiom. I n s e c t i o n 6.4 we s h a l l compare t h i s axiom t o o t h e r systems, and f i n a l l y i n s e c t i o n 6.5 we d e a l w i t h some a l t e r n a t i v e s t o t h i s axiom. 6.1 Statement o f Axiom IV I n the i n t r o d u c t i o n Axiom IV was s t a t e d as W(B) = y (B). I n s e c t i o n 5, W was s p e c i f i e d as a measure on a a - a l g e b r a 8 c o n t a i n i n g 0 . S i n c e y i s o n l y d e f i n e d on 0, W may be thought o f as an e x t e n s i o n o f the measure y on 0 t o a measure W on 3 . The p r e v i o u s axiom i s s u f f i c i e n t t o s p e c i f y the e v a l u a t i o n f u n c t i o n as Hence t h e e v a l u a t i o n f u n c t i o n i s s p e c i f i e d as t h e e x p e c t e d u t i l i t y . However, f o r f ( - ) t o have some meaning f o r the d e c i s i o n maker, W must i n some way be connected t o the proba-b i l i t y o f t h e d i f f e r e n t s t a t e s o c c u r r i n g . I t i s e a s i l y seen t h a t i f W(B) i s d e f i n e d by W(B) = ay(B) a > 0, W would s t i l l be a a - a d d i t i v e measure and the e x p e c t e d u t i l i t y c a l c u l a t e d f (a) 93 by u s i n g W would g i v e the same o r d e r i n g f o r a l l p o s i t i v e v a l u e s o f a. For s i m p l i c i t y r a t h e r t h a n n e c e s s i t y , we s h a l l assume t h a t a = 1. Axiom IV. F o r any B e 0 , W(B) = y (B) . Axiom I I I assumes the e x i s t e n c e o f the s e t f u n c t i o n W on 3 . However, Axiom I I I does n o t g i v e us a method o f d e t e r m i n i n g the v a l u e s o f W(B). Axiom IV s p e c i f i e s t h o s e v a l u e s f o r a l l B e 0 , and i t a l s o g i v e s t h e r e q u i r e d c o n d i t i o n :o:f s p e c i f y i n g W f o r a l l B i n 6 . How t h i s i s done, we s h a l l d i s c u s s i n s e c t i o n 6 . 2 . Very few e m p i r i c a l s t u d i e s have been made t o v e r i f y the f a c t t h a t p e o p l e a c t as though y(B) = W(B). Of c o u r s e , t h e axiom can n o t be t e s t e d d i r e c t l y , and i t i s r a t h e r the p e o p l e ' s a b i l i t y t o e s t i m a t e W(B) f o r d i f f e r e n t v a l u e s o f y(B) which i s t e s t e d , o r t o see i f W i s a measure. We s h a l l d i s c u s s how t h i s i s done i n more d e t a i l i n s e c t i o n 6 . 3 . I n g e n e r a l , t h o s e s t u d i e s w h i c h have been made i n d i c a t e t h a t W(B) i s o v e r e s t i m a t e d i f y(B) i s " s m a l l " and u n d e r e s t i m a t e d i f y(B) i s " l a r g e " . 6 . 2 I m p l i c a t i o n s o f Axiom IV There have been some s u g g e s t i o n s t h a t Axiom IV s h o u l d not always be s a t i s f i e d . One o f t h e s e c r i t i c s , Menger (1950), has suggested t h a t s e t s w i t h " s m a l l p r o b a b i l i t i e s " a r e t o be r e g a r d e d as i m p o s s i b l e . T h i s s u g g e s t i o n c r e a t e s as many d i f f i c u l t i e s as i t s o l v e s . The main d i f f i c u l t y w h i c h a r i s e s i s the meaning o f " s m a l l p r o b a b i l i t i e s " a l t h o u g h some d e f i n i t i o n s e x i s t . I n s t a t i s t i c s a d e c i s i o n c r i t e r i o n i s o f t e n used w h i c h i g n o r e s p r o b a b i l i t i e s up t o 0.05. S i m i l a r l y i n c h a n c e - c o n s t r a i n e d programming we a r e o f t e n w i l l i n g t o i g n o r e s m a l l p r o b a b i l i t i e s , u s u a l l y 0.05 o r l e s s . Menger's sugges-t i o n s have some v a l i d i t y , based on e m p i r i c a l s t u d i e s . C o n s i d e r f o r example, t h e d e c i s i o n problem i n s e c t i o n 5.2. where the a l t e r n a t i v e s were c o n s i d e r e d as f o l l o w s : a: $1 m i l l i o n w i t h a p r o b a b i l i t y o f 1.0 $0 o t h e r w i s e b: $5 m i l l i o n w i t h a p r o b a b i l i t y o f 0.80 $0 o t h e r w i s e . Thus, the d i f f e r e n c e o f w i n n i n g between t h e a l t e r n a t i v e s i s a p r o b a b i l i t y o f 0.20. However, when we compare a l t e r n a t i v e s c and d where c: $1 m i l l i o n w i t h a p r o b a b i l i t y o f 0.05 $0 o t h e r w i s e d: $5 m i l l i o n w i t h a p r o b a b i l i t y o f 0.04 $0 o t h e r w i s e t h a t t h e d i f f e r e n c e i s o n l y . 0 . 0 1 . One argument w h i c h has been su g g e s t e d f o r c h o o s i n g d r a t h e r t h a n c i s t h a t t h e d i f f e r e n c e i n t he p r o b a b i l i t i e s o f w i n n i n g i s " s m a l l " enough t o i g n o r e e s p e c i a l l y s i n c e t h e p r o b a b i l i t i e s o f w i n n i n g a r e v e r y s m a l l . I n t h e case o f a v s . b the d i f f e r e n c e i n the p r o b a b i l i t i e s o f w i n n i n g i s 0.20, t o o " l a r g e " t o i g n o r e , and hence we may choose a. T h i s would i m p l y t h a t W i s not a l i n e a r f u n c t i o n o f y, (or e q u i v a l e n t l y VI i s not a d d i t i v e ) and hence t h i s p r e f e r -ence does n o t s u p p o r t t h e e x p e c t e d u t i l i t y c r i t e r i o n . The most i m p o r t a n t a s p e c t o f Axiom IV i s i t s u s e f u l n e s s i n d e r i v i n g t h e v a l u e s f o r W(B) where B e B- 6 . The method t y p i c a l l y used i s as f o l l o w s : I f two a c t i o n s a,b e A are d e f i n e d a c c o r d i n g t o the r to e D X(co,a) = { _ X(oo,b) = s CO £ D where U(r) > U ( s ) , then f ( a ) = U(r)W(B) + U(s) (l-W(B)) f ( b ) = U(r)W(D) + U(s) (l-W(D)) o r e q u i v a l e n t l y f ( a ) - f ( b ) = U ( r ) [ w ( B ) - W(D)J - U ( s ) j w ( B ) - W(D)} = [ u ( r ) - U ( s j ] [ w ( B ) - W(D)] . S i n c e by assumption U(r) > U ( s ) , i t f o l l o w s t h a t f ( a ) > f ( b ) i f and o n l y i f W(B) > W(D). I f i t i s assumed t h a t f o r e v e r y r e a l number ye ( 0 , 1 ) t h e r e e x i s t s a D £ 0 such t h a t u(D) = yi then i t i s e a s i l y seen t h a t W(B) can then be e s t i m a t e d as c l o s e l y as we w i s h i f A c o n t a i n s s u f f i c i e n t l y many comparable a c t i o n s , i n t h e f o l l o w i n g manner: A c t i o n a i s compared t o a c t i o n b: A 1) i f a > b y > W(B) A 2 ) i f b > a W(B) £ Y -I f case 1 h o l d s a s m a l l e r y can be s e l e c t e d , and a new comparison i s made. S i m i l a r l y f o r case 2, the p r o c e s s i s r e p e a t e d u n t i l W(D) has been e s t i m a t e d t o t h e degree o f a c c u r a c y d e s i r e d . Some d i f f i c u l t i e s a r i s e , however, as we attempt t o e x t e n d y t o 3. We s h a l l d i s c u s s some o f them below. The f i r s t i s s t r i c t l y m a t h e m a t i c a l and i s concerned w i t h t h e e x i s t e n c e o f the e x t e n s i o n o f y . For example, i f 6 i s a l l s u b s e t s o f Q, t h e r e may n o t e x i s t a measure on 3 w h i c h i s e q u i v a l e n t t o y on 0 . Hence we must make the a d d i t i o n a l a ssumption t h a t t h i s e x t e n s i o n can always be made. T h i s i m p l i e s t h a t we must r e s t r i c t t h e s e t 8 o r e q u i v a l e n t l y r e s t r i c t t h e number o f a c t i o n s i n A. F o r p r a c t i c a l p u r p o s e s t h i s c a n u s u a l l y be done, s i n c e 8 can i n g e n e r a l be g e n e r a t e d by 0 and a f i n i t e number o f s e t s and i n t h i s case t h e e x t e n s i o n s always e x i s t (see s e c t i o n 8 i n P a r t I I ) . A second d i f f i c u l t y a r i s e s i n the attempt t o d e t e r m i n e the v a l u e o f W(B). One r e a s o n i s due t o t h e assumptions o f s e p a r a b i l i t y between t h e p r o b a b i l i t i e s o f r e c e i v i n g a reward and t h e reward. T h i s can be b e s t i l l u s t r a t e d by an example i n DeGroot's (19 70) book. C o n s i d e r a c t i o n s a and b where: a: R e c e i v i n g $100 i f you w i l l be e x t e r m i n a t e d by a n u c l e a r war w i t h i n the n e x t t e n y e a r s , o r $0 o t h e r w i s e b: R e c e i v i n g $100 i f you become t h e p r e s i d e n t o f t h e U n i t e d S t a t e s w i t h i n the n e x t t e n y e a r s , o r $0 o t h e r w i s e . I t i s not s u r p r i s i n g t h a t most p e o p l e would p r e f e r b t o a and a l s o b e l i e v e t h a t t h e e v e n t { e x t e r m i n a t i o n by a n u c l e a r war} i s more l i k e l y t o o c c u r t h a n t h e e v e n t {becoming the p r e s i d e n t o f the U n i t e d S t a t e s w i t h i n the n e x t t e n y e a r s } . The problem a r i s e s because t h e rewards a r e not p r e c i s e l y d e f i n e d o r e x a c t l y s p e c i f i e d . I f rewards a r e not r e c e i v i n g s $100 b u t r a t h e r " r e c e i v i n g $100 and b e i n g e x t e r m i n a t e d " and " r e c e i v i n g $100 and b e i n g p r e s i d e n t " t h e c o n t r a d i c t i o n w i l l n o t o c c u r . I n t h i s case an o b v i o u s r e l a t i o n e x i s t s between t h rewards and t h e e v e n t s y i e l d i n g them. I n some c a s e s , however, i t may be p o s s i b l e t o d e t e r m i n e i f such a r e l a t i o n e x i s t s . A n o t h e r d i f f i c u l t y a r i s e s when we attempt t o determine th e v a l u e o f W(B). Assume t h a t W(D) has been d e t e r m i n e d t o e q u a l y . T h i s v a l u e may n o t t h e n s a t i s f y the a d d i t i v i t y p r o -p e r t y o f a measure. L e t us i l l u s t r a t e t h i s u s i n g t h e E l l s b e r g paradox I . E l l s b e r g paradox I . C o n s i d e r t h e f o l l o w i n g two u r n s : Urn I c o n t a i n s 100 b a l l s , e i t h e r r e d o r b l a c k though the numbe: o f each c o l o u r i s n o t known. Urn I I c o n t a i n s 50 r e d b a l l s and 50 b l a c k b a l l s . We a r e asked t o s t a t e a p r e f e r e n c e between a and b and a p r e f e r e n c e between c and d. a: Win $1,000 i f a r e d b a l l i s drawn from Urn I b: Win $1,000 i f a r e d b a l l i s drawn from Urn I I c: Win $1,000 i f a b l a c k b a l l i s drawn from Urn I d: Win $1,000 i f a b l a c k b a l l i s drawn from Urn I I . L e t us denote t h e e v e n t (drawing a r e d b a l l from Urn I) by {Rj} and s i m i l a r l y f o r { R J - J K {B-j.} and { B }. Hence i f someone s t r i c t l y p r e f e r s b t o a we can c o n c l u d e t h a t W(Ri;].) > WCRj). S i m i l a r l y , i f our p r e f e r e n c e i s d > c, we A A must have W f B ^ ) > WtBj) . I f bo t h p r e f e r e n c e s b > a and d > c ar e made, a c o n t r a d i c t i o n o c c u r s , i . e . , 1 = P ( n ) = VKRZ1) + W(R I : [) > W(R I) + W(B I) = y(ft) = 1. Thus, the a d d i t i v i t y o f W must be r e j e c t e d and hence i f our A p r e f e r e n c e i s b > a t h e n we must a l s o have the p r e f e r e n c e A c > d. T h i s problem can a l s o be i l l u s t r a t e d by our attempt a t s p e c i f y i n g WfR^). Assume t h a t the o n l y e v e n t s t h a t b e l o n g t o 9 a r e ° . , G , ( R ), and ( B ^ ) , and a l s o t h a t i f t h e r e i s a p r o -p o r t i o n p o f r e d b a l l s i n Urn I I , the p r o b a b i l i t y o f drawing a r e d b a l l i s p. W(R) can now be de t e r m i n e d by comparing t h e a c t i o n s : a: R e c e i v i n g $1,000 i f a r e d b a l l i s drawn from Urn I , c: R e c e i v i n g $1,000 i f a r e d b a l l i s drawn from an u r n w i t h the p r o p o r t i o n p o f r e d b a l l s . When the d e c i s i o n maker becomes i n d i f f e r e n t between a and b we l e t W(R ) = p^. I f we r e p e a t our ex p e r i m e n t f o r b l a c k b a l l s we s h a l l f i n d W(B^) = p 2» U n f o r t u n a t e l y most e m p i r i c a l s t u d i e s i n d i c a t e t h a t p^ + p 2 7* 1. I t i s n o t easy t o d e t e r -mine i f p^ o r p 2 o r b o t h s h o u l d change. A f o u r t h d i f f i c u l t y a r i s e s i f 9 does n ot c o n t a i n a s u f f i c i e n t number o f e v e n t s such t h a t W(B) ( B e g - 0 ) can be e s t i m a t e d w i t h s u f f i c i e n t a c c u r a c y by t h e method s u g g e s t e d . An a l t e r n a t i v e approach e x i s t s i f B c o n t a i n s a s u f f i c i e n t number o f s e t s . C o n s i d e r a g a i n the two a c t i o n s a, b e A, 99 d e f i n e d by X(oj,a) = 0 ) £ B 0) E B X(w,b) = r s to e D ai e D where U(r) > U ( s ) . An o r d e r i n g can then be d e f i n e d on the s e t s i n 8 i n the f o l l o w i n g way: A 8 I f a < b t h e n B ^ D. 8~ I f the o r d e r i n g ^ s a t i s f i e s a g i v e n s e t o f a s s u m p t i o n s , i t can be proved t h a t t h e r e e x i s t s a r e a l - v a l u e d f u n c t i o n P on 8^ w h i c h s a t i s f i e s the axioms o f a p r o b a b i l i t y measure. (For p r o o f see, f o r example, De F i n e t t i , (1964) o r Savage (1 9 5 4 ) ) . 8 We s h a l l summarize the c o n d i t i o n s on the o r d e r i n g < f o r the e x i s t e n c e o f P below: The r e l a t i o n ( " i s n o t more p r o b a b l e than") on e v e n t s must s a t i s f y the f o l l o w i n g axioms: Axiom o f o r d e r i n g . 8 8 1. I f C e 8 f B e 8 / t h e n e i t h e r C < B o r B < C; 8 2. F o r any s e t B e 8 , B ^ B. 8 8 6 3. I f D < B and B ^ C, t h e n D ^ C. 8 8 8 4. <j) < ft and f o r any e v e n t B,<j> ^  B ^ ft. 8 We d e f i n e < ( " i s s t r i c t l y l e s s p r o b a b l e than") i n the u s u a l way by: C < B i m p l i e s C < B and n o t B < C. Axiom o f m o n o t o n i c i t y . 1. I f B±r\ B 2 = 0, c 1 « 8 8 * B and C 9 < B 0 , t h e n C l U C 2 < B l ^ B 2 _ 8 B 2 . I f B 1 f \ B 2 = J2f, C x < B 1 and C 2 <-B , t h e n k B . C 1 O C 2 < B 1 O B 2 • B The r e l a t i o n ^ s a t i s f y i n g t h e above axioms i s c a l l e d a q u a l i t a t i v e p r o b a b i l i t y on t h e a l g e b r a B• Savage's axioms 1-5 i m p l y t h e s e axioms and i n h i s assumptions the a l g e b r a B c o n t a i n s a l l s u b s e t s o f ft. However, t h e s e axioms a r e n o t s u f f i c i e n t t o guarantee a - a d d i t i v i t y o f t h e c o r r e s p o n d i n g p r o b a b i l i t y measure. F o r a - a d d i t i v i t y we need the f o l l o w i n g axioms: Axiom o f monotone sequence. F o r e v e r y monotone sequence o f e v e n t s C * C and an e v e n t B such t h a t n B B Cn 4 B, f o r a l l n, t h e n C < B. As K r a f t , P r a t t and S e i d e n b e r g (1959) showed, t h e s e axioms are s t i l l n o t s u f f i c i e n t t o guarantee a p r o b a b i l i t y measure; we need an a d d i t i o n a l axiom such as t h e f o l l o w i n g one i n r e g a r d t o the p a r t i t i o n i n g o f e v e n t s : Axiom o f p a r t i t i o n i n g o f an e v e n t . E v e r y e v e n t can be p a r t i t i o n e d i n t o two e q u a l l y ' p r o b a b l e e v e n t s . V i l l e g a s (1964) showed under c e r t a i n a s s umptions t h a t t h i s i s e q u i v a l e n t t o an axiom t h a t t h e r e a r e no atoms, f o r example, 'Savage's axiom 6 i s o f t h i s form. Another approach t o d e r i v i n g t h e p r o b a b i l i t i e s on B has been suggested by Anscombe and Aumann (1964). T h e i r approach i s c l o s e l y r e l a t e d t o t h e assumption t h a t 0 c o n t a i n s a s u f f i c -i e n t number o f s e t s t o e x t e n d the measure t o B. We s h a l l n o t c o n s i d e r t h i s method h e r e . 6.3 E m p i r i c a l s t u d i e s based on Axiom IV I n t h i s s e c t i o n we s h a l l g i v e t h e r e s u l t s o f some e m p i r i c a l s t u d i e s w h i c h a r e r e l a t e d t o Axiom IV. F i r s t we p r e s e n t s t u d i e s r e l a t i n g t o t h e E l l s b e r g Paradoxes I and I I w h i c h may c o n t r a d i c t e i t h e r Axiom I I o r Axiom IV though Paradox I a t l e a s t seems t o c o n t r a d i c t Axiom IV r a t h e r t h a n I I . E m p i r i c a l r e s u l t s o f t h e type o f E l l s b e r g Paradox I . Most e m p i r i c a l s t u d i e s have used r e a l numbers as rewards (amounts o f money whi c h can be won) and t h e a c t i o n s c o n s i d e r e d have reward f u n c t i o n s o f t h e f o l l o w i n g form: where D e 0 b u t B e .6 - 0, and a any r e a l number. I f a = 1 A A an " E l l s b e r g - t y p e v i o l a t i o n " i s the c h o i c e b > a and d > c. MacCrimmon (19 65) s t u d y . I n a s t u d y w i t h 38 b u s i n e s s e x e c u t i v e s , MacCrimmon used a s e r i e s o f t h i s type o f problem where a m i x t u r e o f "known" p r o b a b i l i t i e s v e r s u s "unknown" 102 p r o b a b i l i t i e s was used t o determine i f b i a s e s e x i s t . The f i r s t p roblem c o n s i d e r e d was B = {the s t o c k p r i c e o f P i e r c e I n d u s t r i e s i s i n c r e a s i n g } D = {a r e d c a r d i s drawn from a s t a n d a r d deck} r = $1,000 a = 1. I n t h i s t y pe o f problem t h e c o n t r a d i c t i o n w i l l o n l y o c c u r i f t he p r o b a b i l i t y o f t h e e v e n t B i s c l o s e t o 1/2. For example, i f t h e economy i s on t h e upswing and n e a r l y a l l s t o c k s have i n c r e a s e d i n r e c e n t days, i t i s n o t v e r y l i k e l y t h a t an " E l l s b e r g - t y p e v i o l a t i o n " w i l l be o b t a i n e d . The second t y p e o f problem was s i m i l a r . I n t h i s case the e v e n t s B and D were d e f i n e d by {U.S. GITP i n c r e a s e s n e x t year} and {a c o i n l a n d i n g heads} r e s p e c t i v e l y . I n t h e f i r s t problem, 27 s u b j e c t s were c o n s i s t e n t w i t h the axioms. That i s , i f t h e y p r e f e r r e d a c t i o n a o v e r a c t i o n b i n the f i r s t s e t , then they p r e f e r r e d t h e a c t i o n d o v e r t h e a c t i o n c i n t h e second s e t . Seven s u b j e c t s had v i o l a t i o n s o f the e v e n t complement c o n d i t i o n ; f i v e o f t h e s e s u b j e c t s had E l l s b e r g - t y p e v i o l a t i o n s ( i . e . , t h e y p r e f e r r e d t h e "known" c a r d b e t b o t h t i m e s ) , w h i l e the o t h e r two p r e f e r r e d t h e s t o c k b e t b o t h t i m e s . Hence t h e r a t e o f v i o l a t i o n was 21%. I n the second problem t h e r e were 24 c o n s i s t e n t s u b j e c t s and 7 sub-j e c t s who had E l l s b e r g - t y p e v i o l a t i o n s . (The r e m a i n i n g 7 s u b j e c t s had some degree o f i n d i f f e r e n c e . ) Hence the r a t e o f v i o l a t i o n was 2 3%. When the s t a k e s were changed t o y i e l d $10 more on t h e "unknown" e v e n t ( i . e . , a = 1.01), t h e p r o p o r t i o n o f E l l s b e r g -t y pe v i o l a t i o n s dropped t o 12% (4 out o f 34) on the f i r s t 103 problem and 17% (6 o u t o f 35) on t h e second problem. Note h e r e , though, t h a t the c h o i c e o f the "unknown" s t o c k b e t i n b o t h s e t s cannot be c a l l e d a v i o l a t i o n w i t h t h e s e p a y o f f s s i n c e they pay more and would be t h e l o g i c a l c h o i c e i f t h e "known" and "unknown" e v e n t s were deemed e q u a l l y l i k e l y . F o r t h e f i r s t problem, the s u b j e c t s were a l s o p r e s e n t e d w i t h r e a s o n s s u p p o r t i n g c o n s i s t e n t ( i . e . , axiom-based) r e s p o n s e s and w i t h r e a s o n s s u p p o r t i n g E l l s b e r g - t y p e v i o l a t i o n s . N i n e t e e n s u b j e c t s judged th e c o n s i s t e n t argument the more r e a s o n a b l e , w h i l e 12 s u b j e c t s p r e f e r r e d t h e v i o l a t i n g argument. Thus o v e r a l l , t h e r e was a r a t e o f 39% a c c e p t i n g t h e E l l s b e r g - t y p e v i o l a t i o n . Among tho s e s u b j e c t s who had c o n s i s t e n t answers t h e m s e l v e s , t h e r a t e o f a c c e p t a n c e o f t h e E l l s b e r g - t y p e answer was 22% (6 o u t o f 2 7 ) , w h i l e among tho s e who had an i n c o n s i s t e n t answer t h e m s e l v e s , th e a c c e p t a n c e r a t e was 57% (4 o u t o f 7) . MacCrimmon and L a r s s o n (1975) s t u d y . I n t h e i r s t u d y , 19 s u b j e c t s were p r e s e n t e d w i t h two s e t s o f 11 a l t e r n a t i v e wagers and were asked t o rank the wagers i n each s e t i n o r d e r o f t h e i r p r e f e r e n c e . The s e t s d i f f e r e d i n terms o f p a y o f f s ; i n the f i r s t s e t r = $1,000, a = 1; i n the second s e t , r = $1,000, a = 1.01. We s h a l l o n l y c o n s i d e r the p a r t o f t h e i r "study w h i c h concerns the E l l s b e r g Paradox I . = {a r e d b a l l drawn from Urn 1} Ri;]. = {a r e d b a l l drawn from Urn I I } . F i f t e e n o f t h e 19 s u b j e c t s had an E l l s b e r g - t y p e v i o l a t i o n . A n o t h e r s u b j e c t r a nked t h r e e o f t h e a c t i o n s e q u a l , w i t h the f o u r t h one l e s s . Two s u b j e c t s r a n k e d a l l f o u r a c t i o n s e q u a l , 104 w h i l e t h e r e m a i n i n g s u b j e c t r a nked one o f t h e a c t i o n s h i g h e s t and the o t h e r l o w e s t w i t h the known p r o b a b i l i t i e s and t h e o t h e r ranked between them. Hence, o n l y t h r e e o f t h e 19 s u b j e c t s behaved c o n s i s t e n t l y w i t h the u t i l i t y axioms; t h e r e were 16 s u b j e c t s w i t h E l l s b e r g - t y p e v i o l a t i o n s . T h i s i s a v e r y h i g h v i o l a t i o n r a t e o f 84%. L o o k i n g a t i t a n o t h e r way, 16 s u b j e c t s p r e f e r r e d b t o a, and t h r e e s u b j e c t s were i n d i f f e r e n t . Of t h e s e 16 s u b j e c t s , o n l y one p r e f e r r e d c t o d ( i . e . , t h e b e t s on t h e complements were i n the r i g h t o r d e r ) . They c o n c l u d e from t h e s e r e s u l t s t h a t b e t s on an u r n w i t h a s p e c i f i e d com-p o s i t i o n seem t o be p r e f e r a b l e t o b e t s on an u r n w i t h an un-known c o m p o s i t i o n . E m p i r i c a l s t u d i e s o f E l l s b e r g Paradox I I . I n E l l s b e r g ' s paradox I I we a r e c o n c e r n e d w i t h t h r e e e v e n t s , say B ^ , B 2 , and B 3 , such t h a t B ^ rt B_. = 0 i ^ j and B 1 O B 2 O B 2 = ft. We a l s o assume t h a t B ^ e 0 b u t B 2 , B 3 8 - 0 , and as i n E l l s b e r g paradox I the rewards a r e "amounts o f money which can be won". Fo r s i m p l i c i t y we s h a l l denote y ( B ^ ) by p. The a c t i o n s w h i c h w i l l be c o n s i d e r e d have reward f u n c t i o n s d e f i n e d by: X(u,a) = X ( i u , c ) { r CJ e B , f XU,b) = / 0 a) e B ^ y { r 03 e B „ J r x ( o ) , d ) = / 0 03 £ B { 0 r o) e B 2 0 03 e B 2 03 £ B ^ 0) £ B ^ We a r e t h e n asked our p r e f e r e n c e between a and b and a l s o between c and d. An E l l s b e r g - t y p e v i o l a t i o n would th e n be A a > b and d > c. The E l l s b e r g problem can be d e s c r i b e d a s : p = 0.33, B.^  = the e v e n t t h a t (a r e d b a l l i s drawn) , B 2 = t h e event t h a t (a b l a c k b a l l i s drawn), B^ = the e v e n t t h a t (a y e l l o w b a l l i s drawn), and r = $1,000. S i n c e the number o f b a l l s i n the u r n may i n f l u e n c e t h e d e c i s i o n we s h a l l a l s o denote t h i s by n. S l o v i c and T v e r s k y (1975) s t u d y . They used t h e o r i g i n a l E l l s b e r g problem (p = 1/3, B 1 = {red b a l l } , B 2 = { b l a c k b a l l } , B 3 = { y e l l o w b a l l } , n = 90, and r = $1000- Of t h e i r 29 c o l l e g e s t u d e n t s , 19 made E l l s b e r g - t y p e v i o l a t i o n s when the problem was f i r s t p r e s e n t e d t o them. On a second p r e s e n t a t i o n o f t h e same problem, t h e r e were 21 E l l s b e r g - t y p e v i o l a t i o n s . I n a second group o f 49 s u b j e c t s who were p r e s e n t e d w i t h arguments b e f o r e making t h e i r c h o i c e s , 38 s u b j e c t s agreed w i t h an E l l s b e r g -t y p e o f argument and 39 made an E l l s b e r g - t y p e v i o l a t i o n i n t h e i r c h o i c e s . MacCrimmon and L a r s s o n ' s s t u d y (19 75). I n t h e i r s t u d y , t h e y v a r i e d t h e parameters p i n t h e problem. T h i s r e f l e c t s the "known" chances o f w i n n i n g w i t h e v e n t B^ and c o r r e s p o n d -i n g l y , t h e r e i s a 1-p chance o f w i n n i n g w i t h e v e n t "B 2 o r B^". They c o n s i d e r e d t h e f o l l o w i n g v a l u e s o f p: 0.20, 0.25, 0.30, 0.33, 0.34, 0.40, and 0.50. One would e x p e c t t h e h i g h e s t tendency t o make E l l s b e r g - t y p e v i o l a t i o n s around p = 1/3. To o b t a i n some i n f o r m a t i o n on p a y o f f l e v e l s they used r = $1,000 and r = $1,000,000. They used n = 100 b a l l s i n a l l cases e x c e p t one p r e s e n t a t i o n o f the o r i g i n a l E l l s b e r g problem w i t h n = 90 b a l l s (and p = 1/3) f o r d i r e c t comparison p u r p o s e s . I n the o r i g i n a l E l l s b e r g problem, 11 o f t h e 19 s u b j e c t s made th e E l l s b e r g - t y p e v i o l a t i o n ( i . e . , a , d ) . Only f i v e 106 i n d i v i d u a l s made c h o i c e s c o n f o r m i n g t o the axioms ( i . e . , a,c o r b , d ) . Hence t h e r e does seem t o be a h i g h r a t e o f v i o l a t i o n o f the n o t i o n t h a t p r o b a b i l i t i e s can be a s s i g n e d t o " u n c e r t a i n " e v e n t s . I n the form o f the problem we used ( w i t h n = 100 b a l l s ) , the r a t e o f v i o l a t i o n tended t o be even h i g h e r . For p = 0.33, o r 0.34, 70% o f t h e s u b j e c t s made E l l s b e r g - t y p e v i o l a t i o n s . So t h e r e was c o n s i d e r a b l e v i o l a t i o n f o r thes e p a r t i c u l a r p a r a r a m e t e r s . I n an urn w i t h 100 b a l l s w i t h p r o p o r t i o n p o f r e d b a l l s , we would e x p e c t t h a t when p i s c l o s e t o 0, a c h o i c e o f b,d and when p i s c l o s e t o 1, would e x p e c t a c h o i c e o f a and c. However, when p i s around 1/3, i n d i v i d u a l s may be somewhat i n d i f f e r e n t about t h e c h o i c e s and may then choose on the b a s i s o f how w e l l t h e chances a r e known. S i n c e the chances f o r a l t e r n a t i v e s a and d a r e s p e c i f i e d , we would e x p e c t a much h i g h e r p r o p o r t i o n o f such v i o l a t i o n s around t h i s v a l u e o f p. As p d e v i a t e s from 1/3, then a l t h o u g h the chances a r e s t i l l s p e c i f i e d w i t h a l t e r n a t i v e s a and b, one i s a c c e p t i n g a r a t h e r low chance o f w i n n i n g by t a k i n g a when p i s s m a l l o r d when p i s l a r g e . R e g a r d l e s s o f whether t h i s i s the r a t i o n a l e f o r the c h o i c e s , they d e f i n i t e l y o b s e r v e d t h i s k i n d o f b e h a v i o u r . I t was ob-s e r v e d t h a t 70% o f t h e c h o i c e s f o r p = 0.33 o r 0.34 a r e i n c o n -s i s t e n t w i t h t h e axioms, but f o r p = 0.20, t h i s p e r c e n t a g e drops t o 14% and f o r p = 0.50 i t drops t o 0%. Another s t u d y d i r e c t l y r e l a t e d t o Axiom IV was made by E. M. S h u f o r d (1959) who d e s i g n e d and c o n s t r u c t e d a s e t o f 20 x 20 m a t r i c e s o f s m a l l l i n e s w h i c h would e i t h e r be i n a 107 v e r t i c a l o r h o r i z o n t a l p o s i t i o n . The s u b j e c t s s t u d i e d t h e s e m a t r i c e s f o r a s h o r t p e r i o d o f time and were th e n asked what pe r c e n t a g e o f l i n e s were v e r t i c a l . Here th e e x a c t number o f v e r t i c a l l i n e s was known t o the e x p e r i m e n t e r , and t h e r e f o r e a s t u d y o f how a c c u r a t e t h e s u b j e c t s were i n d e t e r m i n i n g the f r a c t i o n o f v e r t i c a l l i n e s c o u l d be d e t e r m i n e d . S h u f o r d found t h a t s m a l l p e r c e n t a g e s were n e a r l y always o v e r e s t i m a t e d and l a r g e r p e r c e n t a g e s u n d e r e s t i m a t e d which would suggest t h e f o l l o w i n g g r a p h i c a l r e p r e s e n t a t i o n between W and y. W(B) 1.0 1.0 y(B) F i g . 6.3. R e l a t i o n between y(B) and W(B) • ... by e x p e r i m e n t o f E*. M. S h u f o r d T h i s does not suggest t h a t W ought not t o be a n o n - l i n e a r f u n c t i o n o f <y. I t o n l y i l l u s t r a t e s t h e d i f f i c u l t i e s o f o b t a i n i n g c o r r e c t p r o b a b i l i t i e s . 6.4 R e l a t i o n t o o t h e r axiom systems Marschak's, vonNeumann & Morgenstern's and Arrow's approaches assume a p r o b a b i l i t y measure as we have g i v e n f o r a l l s e t s i n g . T h e i r approaches t o t h e e x p e c t e d u t i l i t y t h e o r y would t h e r e f o r e have l i t t l e i n t e r e s t t o s e c t i o n 6. Savage's 108 and Luce & K r a n t z 1 s approaches a r e s l i g h t l y d i f f e r e n t . I n t h e i r approaches an o r d e r i n g i s i n d u c e d on a l l p o s s i b l e e v e n t s i n such a way t h a t a p r o b a b i l i t y measure may be s p e c i f i e d on those e v e n t s . That i s , the o r d e r i n g i n d u c e d on the events must s a t i s f y t h e c o n d i t i o n s s p e c i f i e d i n s e c t i o n 6.2. Savage's approach does not s a t i s f y t he axiom o f monotone sequence, and can, t h e r e f o r e , o n l y s p e c i f y a f i n i t e a d d i t i v e measure. 6.5 A l t e r n a t i v e t o Axiom IV An a l t e r n a t i v e approach would be t o remove t h e c o n d i t i o n t h a t W behave s t r i c t l y as a measure. F or example, F e l l n e r (1961) found t h a t i n some c a s e s when he t r i e d t o d e r i v e sub-j e c t i v e p r o b a b i l i t i e s t h a t W(B) + W(B) = y where y ^ 1. I f W(B) = ky(B) (u > 0) i t would n o t be d i f f i c u l t t o d e r i v e a t h e o r y s i m i l a r t o the e x p e c t e d u t i l i t y t h e o r y . However, he a l s o found t h a t k i s not a c o n s t a n t b u t i s a f u n c t i o n o f B. Hence Axiom I I would a l s o be c o n t r a d i c t e d . We s h a l l d i s c u s s h i s approach i n a d d i t i o n t o a new approach i n P a r t I I . 7.0 Summary o f P a r t I I n t h i s s e c t i o n we s h a l l show t h a t the axioms we have c o n s i d e r e d here a r e b o t h n e c e s s a r y and s u f f i c i e n t f o r the e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d . B e f o r e d o i n g s o , however, we s h a l l summarize the b a s i c assumptions and axioms made. Summary o f the assumptions I n Assumption 1 we assumed X(-,a) was a measurable f u n c t i o n w i t h r e s p e c t t o Q.. Axiom I I I assumes the e x i s t e n c e o f a measurable f u n c t i o n U on R, and hence we may now c o n s i d e r the composite f u n c t i o n UX(-,a) f o r a l l a. The most i m p o r t a n t assumption c o n c e r n i n g m e a s u r a b i l i t y i s t h a t UX(-,a) i s a measurable f u n c t i o n w i t h r e g a r d t o t h e B o r e l sete on t h e r e a l l i n e and 6 . T h i s f o l l o w s from the f a c t t h a t i f U and X are b o t h measurable then UX i s measurable (Halmos, 1950, Theorem B pp.162). Assumption 1. There e x i s t : i ) a p r o b a b i l i t y space ( f t , 0 , u ) , where ft i s the s e t o f s t a t e s , i i ) a measurable space ( R , ^ ) , where R i s the s e t o f rewards o r outcomes, i i i ) an i n d e x s e t A, c a l l e d an a c t i o n space, such t h a t f o r each a e A t h e r e e x i s t s a reward f u n c t i o n X ( - , a ) from ft t o R, i v ) a a - a l g e b r a 6 o f ft such t h a t Q C 8 and each f u n c t i o n X(«,a) i s B-measurable, and A v) a r e l a t i o n < on A. The f o l l o w i n g axioms were a l s o made w i t h r e s p e c t t o the d e c i s i o n maker's b e h a v i o u r . Axiom I . E x i s t e n c e Axiom There e x i s t s a r e a l - v a l u e d f u n c t i o n f on A such t h a t f o r any a,b e A, A i f a < b th e n f ( a ) < f ( a ) , and A i f a < b t h e n f (a) < f (b) . From a p r a c t i c a l v i e w p o i n t t h i s axiom s u g g e s t s t h a t t h e r e e x i s t s a monetary amount w h i c h we would be w i l l i n g t o pay o r r e c e i v e i n o r d e r t o o b t a i n o r s e l l each a l t e r n a t i v e . I t does not su g g e s t t h e amount we are w i l l i n g t o pay b u t o n l y t h e e x i s t e n c e o f an amount. I f t h e r e a r e a l t e r n a t i v e s w h i c h a r e "above r u b i e s " , o r i m p o s s i b l e t o s p e c i f y i n monetary r e w a r d s , t h i s axiom would n o t have a meaning and would n o t be a c c e p t a b l e . Axiom I I . A d d i t i v i t y Axiom There e x i s t s a r e a l - v a l u e d f u n c t i o n h on 3 x A such t h a t a) f (a) = h (ft,a) b) f o r {B.} i = l , . . . such t h a t B. e 3 f o r a l l i and oo B. r \ B . = 0 f o r i ^ j t h e n h ( U B. , a) = £ h ( B . , a ) 1 3 . 1 i = l 1 f o r a l l a e A c) f o r any B e 3, and f o r any a,b e A such t h a t X B ( - , a ) = X B ( - , b ) , then h(B,a) = h(.B,b). Axiom I I i m p l i e s t h a t t h e e v e n t s a r e independent of each o t h e r . However, we do n o t mean a s t a t i s t i c a l independence, b u t t h a t t h e e v a l u a t i o n o f an a c t i o n by c o n s i d e r i n g t h e reward f o r a g i v e n e v e n t i s not a f f e c t e d by t h e p o s s i b l e I l l rewards on t h e complement o f t h a t e v e n t . T h e r e f o r e , we can n o t argue t h a t we a r e w i l l i n g t o t a k e a s m a l l chance on a g i v e n event because we have a r e l a t i v e l y c o n s e r v a t i v e reward e l s e w h e r e . R e c a l l t h a t i n Axiom I l l b and c we use the n o t a t i o n k h(B,z) d e f i n e d by 21 W(B.)a. where z i d e n t i f i e s the s i m p l e i = l 1 1 k f u n c t i o n Y (co, z) = a. , co e B. f o r i = l , . . . , n and B. = B (see 1 1 i = l 1 s e c t i o n 5.1). Axiom I I I . S e p a r a b i l i t y Axiom a) There e x i s t s a n o n - n e g a t i v e r e a l v a l u e d a - a d d i t i v e f u n c t i o n W on 8 and a r e a l v a l u e d measurable f u n c t i o n U on R such t h a t f o r any a e A, B e 8 and r e R, i f X D ( • , a ) = r , then h (B,a) = W(B)U (r) . b) L e t r Q be any f i x e d reward i n R, and l e t X(-,b) R be any reward f u n c t i o n such t h a t X B(co,b) > r Q f ° r a l l co e B, f o r B e 8- I f Z i s t h e s e t o f a l l s i m p l e f u n c t i o n s z such t h a t Y_,(-,z) < U X n ( - , b ) , t h e n h(B,b) s a t i s f i e s h(B,b) = sup h (B,z) zeZ o c) S i m i l a r l y f o r any reward f u n c t i o n X ( - , c ) such t h a t R r > XT,(co,c) f o r a l l co e B, i f Z, i s t h e s e t o f a l l O o ± s i m p l e f u n c t i o n s z such t h a t Y B ( - , z ) UX (',c), t h e n I h(B,c) s a t i s f i e s 112 h (B,c) - i n f h (B, z) Axiom IV. F o r any B e 0 , W(B) = y ( B ) . Axioms I I I and IV s p e c i f y an independence between t h e reward and the p r o b a b i l i t y of r e c e i v i n g t h i s reward. T h i s i m p l i e s t h a t we have e q u a l s a t i s f a c t i o n from w i n n i n g $1,000 on t h e s t o c k exchange o r from w i n n i n g a l o t t e r y t i c k e t . T h i s may not always be t r u e , o f c o u r s e , s i n c e i n the f i r s t c ase we may f e e l a g i v e n s a t i s f a c t i o n because r e c e i v i n g $1,000 from b u y i n g and s e l l i n g s t o c k i n v o l v e s s k i l l , b u t w i t h a l o t t e r y t i c k e t , o n l y l u c k would d e t e r m i n e the w i n n e r . Of c o u r s e , i n t h i s c a s e t h e reward s e t may be m o d i f i e d so as t o i n c l u d e more than a monetary reward. I n t h i s c a s e , however, i t becomes v e r y d i f f i c u l t t o d e t e r m i n e any u t i l i t y f u n c t i o n . These axioms and assumptions a r e n e c e s s a r y and s u f f i c i e n t f o r the e x i s t e n c e o f a f u n c t i o n V on R such t h a t A a ^ b i m p l i e s VX(•,a)dW > and a > b i m p l i e s To show t h i s i t i s s u f f i c i e n t by Axiom I t o show t h a t f (a) Theorem 7 . 1 I f Axioms I , I I , I I I and IV h o l d t h e n t h e r e e x i s t s a r e a l - v a l u e d measurable f u n c t i o n V on R such t h a t f ( a ) = JVX(to,a)dW P r o o f . By Axiom I I I t h e r e e x i s t s a r e a l - v a l u e d f u n c t i o n U on R, so d e f i n e V ( r ) = U ( r ) . I f a e A and X(oo,a) = r f o r a l l weft then by Axioms I I , I I I and IV r e s p e c t i v e l y f (a) = h(ft,a) = W(ft)U(r) = U(r) = V ( r ) and f ( a ) = J ' u x(a),a)dW = f VX(oj,a)dW. T h e r e f o r e , t h e theorem h o l d s f o r c o n s t a n t f u n c t i o n s . Suppose i n s t e a d t h a t a e A and X(«,a) i s a s i m p l e f u n c t i o n , t h a t i s , t h e r e e x i s t s a p a r t i t i o n B ^ , i , .,B n o f ft such t h a t X_, (*,a) = r . i = 1 . . . . . . ,n. Then Axioms I l a , I I I I and t h e d e f i n i t i o n o f t h e i n t e g r a l i m p l y n f (a) = J h(B. ,a) i=1 1 n = X W(B. ) U ( r . ) i = l 1 1 - J UX(oj,a) dW WX(w,a)dW. L a s t l y , c o n s i d e r a e A f o r an a r b i t r a r y reward f u n c t i o n X(«,a). L e t rQ be any f i x e d reward i n R. Axioms I l a , I l l b and I I I c r e s p e c t i v e l y g u a r a n t e e t h a t f ( a ) = h ( { x ( - , a ) > r Q } , a ) + h ( { x ( - , a ) < r Q } , a ) where Z q i s t h e s e t o f a l l s i m p l e f u n c t i o n s t h a t Y(o>,z) < UX(to,a) f o r a l l co e'{X(-,a) > r } and Z N i s o i s t h e s e t o f a l l s i m p l e f u n c t i o n s such t h a t ¥(co,z) ? UX(oo,a) f o r a l l co e {X(-,a) < r }. Hence h ( { X ( - , a ) > r Q } , a ) i s e q u a l t o t h e supremum o f t h e i n t e g r a l o f a l l s i m p l e f u n c t i o n s l e s s t h a n U X ( . , a ) , What i s a l s o t r u e i s , o f c o u r s e , t h a t i f f i s t o be e x p r e s s e d as the e x p e c t e d u t i l i t y , t h e axioms must a l s o h o l d . Theorem 7.2. G i v e n t h e o b j e c t s i n Assumption 1, suppose t h a t t h e r e e x i s t s a measure v on 8 such t h a t v(B) = u(B) f o r a l l B e 0, and a r e a l - v a l u e d measurable f u n c t i o n V on R such t h a t i f a > b t h e n s i m i l a r l y f o r h({-X(-,a) < r Q } , a ) . T h i s i s t h e d e f i n i t i o n o f t h e i n t e g r a l f UX(-,a)dW and com p l e t e s t h e p r o o f . and i f a ) b then VX ( o),a)dv >, J Vx(a3,b)dV. Then Axioms I , I I , I I I , and IV are s a t i s f i e d w i t h W = V , u (•) = V„( •) and y U X ( o o , f ( - ) = /UX(a),-)dW. P r o o f . D e f i n e f (a) = J vx ( c o,a)dW, f o r a l l a e A. Then Axiom I i s o b v i o u s l y s a t i s f i e d . L e t i = l , 2 , . . . form a p a r t i t i o n o f ft such t h a t B^ e 8 f o r a l l i , then f o r any a e A f ( a ) = y"vx ( a>,a)dV .= 71 ^ , & ) dv. B i D e f i n e h(B.,a) = / VX(a>,a) dv, then c l e a r l y f (a) h(B. ,c B. l Hence Axiom I l a must h o l d , I f X B ( - , a ) = X B ( - , b ) t h e n VX(-,a) = VX(-,b) f o r a l l OJ e B. Hence y v x ( w , a ) d v = y v x ( c o , b ) d v o r h(B,a) = h(B,b) B B Hence Axiom l i b must h o l d , I f f o r any B e 3 , X (-,a) = r f o r some a e A then 116 h ( B , a ) = j'vx(w,a)a\> B = V ( r ) V ( B ) D e f i n e U ( r ) = V ( r ) a n d V ( B ) = W ( B ) t h e n A x i o m I l i a i s s a t i s f i e d . A x i o m s I l l b >• a n d I I I < & t h e n f o l l o w f r o m t h e d e f i n i t i o n o f t h e i n t e g r a l . A x i o m I V f o l l o w s f r o m t h e a s s u m p t i o n t h a t ^ ( B ) = v ( B ) = w-'(B) ' f o r ~ B c 0 . W e h a v e , t h e r e f o r e , s h o w n t h a t t h e a x i o m s s p e c i f i e d a r e b o t h n e c e s s a r y a n d s u f f i c i e n t f o r t h e e x p e c t e d u t i l i t y t h e o r y t o h o l d . T h u s , t h e t h e o r y o f m a x i m i z a t i o n o f t h e e x p e c t e d u t i l i t y c a n o n l y b e a c c e p t e d i f w e a g r e e w i t h e a c h o f t h e a x i o m s s p e c i f i e d h e r e , a n d t h e i r i m p l i c a t i o n s . H o p e f u l l y , w e h a v e i l l u s t r a t e d t h e m a i n d i f f i c u l t i e s i n v o l v e d w i t h e a c h o f t h e a x i o m s , g i v i n g a g r e a t e r u n d e r s t a n d i n g f o r t h e i r a c c e p -t a n c e o r r e j e c t i o n . T h e t w o m o s t d i f f i c u l t a x i o m s t o a c c e p t a r e A x i o m I I a n d A x i o m I V . W e h a v e , t h e r e f o r e , s p e n t t h e m o s t t i m e o n t h e m t o i l l u s t r a t e t h e i r i m p l i c a t i o n s a n d r e l a t i o n s t o o t h e r a x i o m s — a t l e a s t f o r A x i o m I I . A x i o m I V w i l l b e c o n s i d e r e d i n m u c h m o r e d e t a i l i n P a r t I I o f t h e t h e s i s . W e s h a l l a l s o c o n s i d e r A x i o m I I t h e r e , a n d s u g g e s t a n a l t e r n a t i v e t o t h e a d d i t i v i t y a x i o m . 117 1.0 I n t r o d u c t i o n t o P a r t I I D e c i s i o n making under p a r t i a l r i s k C l a s s i c a l d e c i s i o n making i s u s u a l l y c a t e g o r i z e d i n a c cordance w i t h the d e c i s i o n maker's knowledge o f t h e conse-quences (rewards) o f h i s a l t e r n a t i v e s ( a c t i o n s ) . S p e c i f i c a l l y , d e c i s i o n problems a r e c l a s s i f i e d as b e i n g under c e r t a i n t y , r i s k and u n c e r t a i n t y ( K n i g h t , 1921) . C e r t a i n t y means t h a t , one reward _ is;, - i s p e c i f i e d f o r each o f t h e d e c i s i o n maker's a l t e r -n a t i v e s . I n t h e case o f r i s k , t h e p r o b a b i l i t y o f t h e conse-quences i s known f o r a l l consequences and f o r a l l a c t i o n s . U n c e r t a i n t y means t h a t o n l y the s e t o f p o s s i b l e consequences i s known. E v a l u a t i o n o f t h e d e c i s i o n maker's a l t e r n a t i v e s We s h a l l assume as b e f o r e t h a t t h e r e e x i s t s an e v a l u a t i o n f u n c t i o n on the s e t o f a c t i o n s ; t h a t i s , a r e a l - v a l u e d f u n c -t i o n f ( - ) e x i s t s on A such t h a t i f one a c t i o n i s p r e f e r r e d t o a n o t h e r , the n u m e r i c a l v a l u e a s s o c i a t e d w i t h the f i r s t a c t i o n i s g r e a t e r t h a n t h a t o f t h e second. I t i s a l s o assumed t h a t t h e r e e x i s t s a . ^ comp^lete - o r d e r i n g on the rewards . which can be r e p r e s e n t e d by a r e a l - v a l u e d o r d e r - p r e s e r v i n g f u n c t i o n . We s h a l l c a l l t h i s f u n c t i o n a u t i l i t y f u n c t i o n . The u t i l i t y f u n c t i o n i s e a s i e r t o d e t e r m i n e t h a n the e v a l u a t i o n f u n c t i o n s i n c e t h e r e l a t i o n s h i p between a c t i o n s and consequences i m p l i e s t h a t t h e t o t a l number o f p o s s i b l e u t i l i t y f u n c t i o n s i s a s u b s e t o f t h e t o t a l number o f p o s s i b l e e v a l u -a t i o n f u n c t i o n s . I n d e c i s i o n t h e o r y under c e r t a i n t y , each a c t i o n g i v e s o n l y one p o s s i b l e reward and we assume t h a t t h e e v a l u a t i o n f u n c t i o n on an a c t i o n t a k e s the same v a l u e as the u t i l i t y f u n c t i o n f o r the c o r r e s p o n d i n g reward. In t h i s case the s p e c i f i c a t i o n o f a u t i l i t y f u n c t i o n i s e q u i v a l e n t t o t h e s p e c i f i c a t i o n o f the e v a l u a t i o n f u n c t i o n . In d e c i s i o n making under e i t h e r r i s k o r u n c e r t a i n t y i t i s o f t e n assumed t h a t t h e u t i l i t y f u n c t i o n can e a s i l y be s p e c i f i e d (or d e r i v e d by some r u l e ( s ) ) and the e v a l u a t i o n f u n c t i o n i s t h e n s p e c i f i e d i n terms o f the u t i l i t y f u n c t i o n . For example, i n d e c i s i o n making under r i s k s e v e r a l e v a l u a t i o n f u n c t i o n s have been s u g g e s t e d , a l t h o u g h t h e most a c c e p t e d i s t h e e x p e c t e d v a l u e o f t h e u t i l i t y f u n c t i o n ( B e r n o u l l i , 1738). I n P a r t I we c o n s i d e r e d the e v a l u a t i o n f u n c t i o n d e f i n e d by e x p e c t e d u t i l i t y i n some d e t a i l . Under u n c e r t a i n t y t h e r e has n o t been an e v a l u a t i o n f u n c t i o n s p e c i -f i e d w hich has been u n i f o r m l y a c c e p t e d . F i v e f u n c t i o n s a r e f r e q u e n t l y s u g g e s t e d ( a l t h o u g h more e x i s t ) . We s h a l l b r i e f l y s t a t e them h e r e . They a l l assume t h e e x i s t e n c e o f a u t i l i t y f u n c t i o n , and t h e e v a l u a t i o n f u n c t i o n i s s p e c i f i e d i n terms o f t h i s u t i l i t y f u n c t i o n . The maximax c r i t e r i o n ( H u r w i t z , 1951). The e v a l u a t i o n f u n c t i o n f o r each a c t i o n i s s p e c i f i e d by t a k i n g the maximum u t i l i t y o v e r a l l p o s s i b l e rewards of t h a t a c t i o n . The maximin c r i t e r i o n (Wald, 1950). The e v a l u a t i o n f u n c t i o n f o r each a c t i o n i s s p e c i f i e d by t a k i n g t h e minimum u t i l i t y o f a l l p o s s i b l e rewards f o r each consequence o f t h a t a c t i o n . 1 1 9 " The H u r w i t z a - ^ c r i t e r i o n (Hurwitz:, 1 9 5 1 ) . . The e v a l u a t i o n f u n c t i o n i s s p e c i f i e d b y a l i n e a r c o m b i n a t i o n o f t h e maximum and minimum u t i l i t y f o r each a c t i o n . The p r i n c i p l e o f i n s u f f i c i e n t r e a s o n ( B e r n o u l l i , 1738). The e v a l u a t i o n f u n c t i o n i s s p e c i f i e d as the average u t i l i t y f o r each a c t i o n . The e x p e c t e d u t i l i t y c r i t e r i o n (Savage, 1 9 5 4 ) . A p r o b a -b i l i t y d i s t r i b u t i o n i s " d e r i v e d " on t h e consequences. The e v a l u a t i o n f u n c t i o n i s t h e n e q u a l t o t h e e x p e c t e d u t i l i t y f o r each a l t e r n a t i v e . F o r a c r i t i c a l r e v i e w o f each o f t h e s e e v a l u a t i o n f u n c -t i o n s , see M i l n o r ( 1 9 5 4 ) . P a r t i a l R i s k Problems A f o u r t h c a t e g o r y o f d e c i s i o n problems can be s p e c i f i e d as f a l l i n g between r i s k and u n c e r t a i n t y ; t h a t i s , some know-le d g e o f the p r o b a b i l i t i e s o f rewards a r e known, b u t a p r o -b a b i l i t y d i s t r i b u t i o n can n o t be c o m p l e t e l y s p e c i f i e d . T h i s c a t e g o r y has been r e c o g n i z e d by d e c i s i o n t h e o r i s t s f o r some time and i s commented on i n s t a n d a r d t e x t b o o k s i n the a r e a . F o r example, "A common c r i t i c i s m o f such c r i t e r i a as the maximin u t i l i t y , minimax r e g r e t , H u r w i t z - a , and t h a t based on the p r i n c i p l e o f i n s u f f i c i e n t r e a s o n i s t h a t they a r e r a t i o n a l i z e d on some n o t i o n o f complete i g n o r a n c e . I n p r a c t i c e , however, t h e d e c i s i o n maker u s u a l l y has some vague p a r t i a l i n f o r m a t i o n c o n c e r n i n g t h e t r u e s t a t e . No m a t t e r how vague i t i s , he may not w i s h t o endorse any c h a r a c t e r i z a t i o n o f complete i g n o r a n c e , and so t h e h e a r t i s c u t o u t o f c r i t e r i a based on t h i s n o t i o n . " (Luce and R a i f f a , 1 9 5 7 , p . 2 9 9 ) 120 K n i g h t a l s o a s s e r t s t h a t such, o v e r a l l judgments may i n f l u e n c e d e c i s i o n : "The a c t i o n w h i c h f o l l o w s upon an o p i n i o n depends as much upon the amount o f c o n f i d e n c e i n t h a t o p i n i o n as i t does upon the f a v o r a b l e n e s s o f t h e o p i n i o n i t s e l f . . . F i d e l i t y t o t h e a c t u a l p s y c h o l o g y o f t h e s i t u a t i o n r e q u i r e s we must i n s i s t r e c o g n i t i o n o f t h e s e two s e p a r a t e e x e r c i s e s o f judgment, t h e f o r m a t i o n o f an e s t i m a t e and t h e e s t i m a t i o n o f i t s v a l u e . " ( K n i g h t , 1921, p.227) D e c i s i o n problems w h i c h f a l l i n t o t h i s c a t e g o r y a r e c a l l e d p a r t i a l r i s k problems ( o t h e r common names a r e d e c i s i o n making under p a r t i a l i g n o r a n c e , o r p a r t i a l u n c e r t a i n t y ) . A p a r t i a l r i s k problem may t h e r e f o r e be v e r y c l o s e t o a r i s k p roblem i f t h e p r o b a b i l i t y d i s t r i b u t i o n can n e a r l y be s p e c i f i e d and s i m i l a r l y i t may be v e r y c l o s e t o an u n c e r t a i n t y problem i f v e r y l i t t l e can be s p e c i f i e d o f t h e p r o b a b i l i t y d i s t r i b u -t i o n . We can t h e r e f o r e c o n s i d e r r i s k and u n c e r t a i n t y as extreme c a s e s o f p a r t i a l r i s k . I t i s n o t s u r p r i s i n g t h e r e f o r e t o f i n d t h a t the e x i s t i n g methods f o r s p e c i f y i n g an e v a l u a t i o n f u n c t i o n f o r a p a r t i a l r i s k problem f a l l between t h o s e used i n e v a l u a t i o n the r i s k and u n c e r t a i n t y problems. E v a l u a t i o n o f p a r t i a l r i s k problems There a r e b a s i c a l l y t h r e e methods used i n d e s c r i b i n g t h e e v a l u a t i o n f u n c t i o n on a l t e r n a t i v e s . They can be c l a s s i f i e d as f o l l o w s : 1) T r a n s l a t i n g t h e problem i n t o a r i s k problem. That i s , assuming t h e e x i s t e n c e o f a p r o b a b i l i t y d i s t r i b u t i o n o v e r t h e reward t h e e v a l u a t i o n f u n c t i o n i s th e n s p e c i f i e d as i n the case o f d e c i s i o n making under u n c e r t a i n t y . 121 2) Combining t h e e v a l u a t i o n f u n c t i o n s used f o r r i s k and u n c e r t a i n t y . 3) D e r i v i n g a " p r e f e r e n c e f u n c t i o n " on the u n c e r t a i n r a t h e r t h a n a p r o b a b i l i t y measure. We b r i e f l y d i s c u s s each method here t o i l l u s t r a t e t h e d i f f e r e n c e s from the approach we s h a l l d i s c u s s i n t h i s t h e s i s . E v a l u a t i o n Method 1. ... The e x a c t p r o b a b i l i t y d i s -t r i b u t i o n i s assumed t o e x i s t but i s not n e c e s s a r i l y known. A p r o b a b i l i t y measure has been s p e c i f i e d i n two r e l a t e d meth-ods. The f i r s t o f t h e s e i s a d e r i v a t i o n o f s u b j e c t i v e p r o -b a b i l i t i e s . A p r o b a b i l i t y d i s t r i b u t i o n i s d e r i v e d from t h e p r e f e r e n c e among a l t e r n a t i v e s . T h i s method has been used by Ramsey (1926), De F i n e t t i (1937), and Savage (1954). The sub-j e c t i v e p r o b a b i l i t y measure u s u a l l y has the same p r o p e r t i e s as Kolmogorov's p r o b a b i l i t y axiom (1933), e x c e p t t h a t sometimes f i n i t e a d d i t i v i t y i s assumed r a t h e r t h a n a - a d d i t i v i t y . The second method used t o d e r i v e a p r o b a b i l i t y i s t o assume a s e t o f w e i g h t s on t h e s e t o f p o s s i b l e p r o b a b i l i t y d i s t r i b u t i o n s and t o use t h e combined p r o b a b i l i t y measure t o c a l c u l a t e t h e e x p e c t e d u t i l i t y . T h i s method i s sometimes used i n s t a t i s t i c s (see, f o r example, Good, 1965). Once the p r o b a b i l i t y measure i s s p e c i f i e d t h e problem has been reduced t o a d e c i s i o n p r o b lem under r i s k . Hence any e v a l u a t i o n f u n c t i o n used f o r d e c i s i o n making under r i s k can a l s o be used i n t h e s e c a s e s . The most common e v a l u a t i o n f u n c t i o n i s t h e e x p e c t e d v a l u e o f t h e u t i l i t y f u n c t i o n . T h i s a pproach, t h e r e f o r e , does n o t d i f f e r e n t i a t e between r i s k and u n c e r t a i n t y . Savage r e c o g n i z e d t h a t t h i s approach might c r e a t e 122 d i f f i c u l t i e s ; " . . . t h e r e seem t o be some p r o b a b i l i t y r e l a t i o n s about w h i c h we f e e l r e l a t i v e l y " s u r e " as compared w i t h o t h e r s . . . T h e n o t i o n o f " s u r e " and "unsure" i n t r o d u c e d here i s vague, and my c o m p l a i n t i s p r e c i s e l y t h a t n e i t h e r t h e t h e o r y o f p e r s o n a l p r o b a b i l i t y , as i t i s d e v e l o p e d i n t h i s book, n o r any o t h e r d e v i c e known t o me r e n d e r s th e n o t i o n l e s s vague..." (Savage, 1954, pp.57-58,59) E v a l u a t i o n method 2. The second e v a l u a t i o n method combines the e v a l u a t i o n f u n c t i o n s used f o r r i s k and u n c e r t a i n t y . There a r e a l s o two b a s i c approaches f o r t h i s method. B o t h o f t h e s e approaches assume t h a t a s e t o f p o s s i b l e p r o b a b i l i t y d i s t r i b u t i o n s can be s p e c i f i e d on t h e consequences. The f i r s t (see Good, 1965) a s s i g n s a s e t o f w e i g h t s on t h e p o s s i b l e p r o b a b i l i t y d i s t r i b u t i o n s and the e x p e c t e d v a l u e i s c a l c u l a t e d u s i n g t h e combined p r o b a b i l i t y d i s t r i b u t i o n . The e v a l u a t i o n f u n c t i o n i s s p e c i f i e d as a l i n e a r c o m b i n a t i o n o f any o f t h e c r i t e r i o n under u n c e r t a i n t y and t h e e x p e c t e d u t i l i t y . There has o n l y been one s u g g ested so f a r and t h a t i s a l i n e a r com-b i n a t i o n w i t h w e i g h t s g r e a t e r t h a n z e r o , and sum t o one between e x p e c t e d u t i l i t y and t h e maximin (see E l l s b e r g , 1961). I t i s o b v i o u s t h a t h i s approach c o u l d a l s o have been used f o r any o f t h e o t h e r e v a l u a t i o n f u n c t i o n s f o r d e c i s i o n making under u n c e r t a i n t y . The c o e f f i c i e n t o f t h e l i n e a r c o m b i n a t i o n would depend on how " c l o s e " t h e problem would be t o a d e c i s i o n p r o -blem under r i s k v e r s u s u n c e r t a i n t y . We s h a l l d i s c u s s t h i s method i n g r e a t e r d e t a i l i n s e c t i o n 6.1. The second approach has been t o combine maximin w i t h the e x p e c t e d u t i l i t y t h e o r y i n the f o l l o w i n g way. F o r each a c t i o n a s e t o f p o s s i b l e d i s t r i b u t i o n s on the rewards i s d e t e r m i n e d , t h e n the e v a l u a t i o n f u n c t i o n i s s p e c i f i e d as the minimum o f a l l p o s s i b l e e x p e c t e d v a l u e s f o r a g i v e n a c t i o n . T h i s approach has been the most common (see, f o r example, Menges, 1966; Blum and R o s e n b l a t t , 1968; Randies and H o l l a n d e r , 1971). E v a l u a t i o n method 3 . The l a s t o f the t h r e e methods o f e v a l u -a t i o n i s s i m i l a r t o Savage's i n the sense t h a t a f u n c t i o n i s d e r i v e d on t h e reward space i n d i c a t i n g i n some way the l i k e l i -hood o f r e c e i v i n g t h e rewards. I n t h i s c a s e , however, t h i s f u n c t i o n may not be a p r o b a b i l i t y measure. The j u s t i f i c a t i o n f o r t h i s would be t h a t i t i s e a s i e r t o d e t e r m i n e a p r o b a b i l i t y d i s t r i b u t i o n o f rewards f o r some a c t i o n s than f o r o t h e r s . T h i s i s t h e r e f o r e an e x t e n s i o n o f the Savage approach i n the sense t h a t t h i s method r e c o g n i z e s t h e d i f f e r e n c e s between " s u r e " and "unsure" e v e n t s . T h i s f u n c t i o n w i l l be c a l l e d a p r e f e r e n c e f u n c t i o n r a t h e r than a p r o b a b i l i t y measure. Once the p r e f e r e n c e f u n c t i o n has been s p e c i f i e d , a d i f f i -c u l t y a r i s e s from how t o use t h i s t o s p e c i f y the e v a l u a t i o n f u n c t i o n . F e l l n e r (1961) s u g g e s t e d one method o f d o i n g t h i s . He assumed t h a t t h e p r e f e r e n c e f u n c t i o n must have the p r o p e r -t i e s t h a t a p r o b a b i l i t y measure on the rewards can be d e r i v e d from i t . The e x p e c t e d u t i l i t y was then c a l c u l a t e d u s i n g the d e r i v e d p r o b a b i l i t y measure. However, i n h i s approach, the u t i l i t y f u n c t i o n i s d i f f e r e n t from t h e u t i l i t y f u n c t i o n under r i s k s i n c e the u t i l i t y under p a r t i a l r i s k c o n t a i n s an element o f gambling. I n P a r t I I o f t h e t h e s i s we s h a l l s u g gest an a l t e r n a t i v e 124 method o f e v a l u a t i o n p a r t i a l r i s k problems. Of those methods so f a r d e s c r i b e d t h i s new method i s c l o s e s t t o F e l l n e r ' s approach. There a r e some fundamental d i f f e r e n c e s , however. F o l l o w i n g F e l l n e r , a p r e f e r e n c e f u n c t i o n i s f i r s t d e r i v e d on the p o s s i b l e rewards f o r each a c t i o n . Here, however, we do n o t assume t h i s p r e f e r e n c e can be t r a n s f o r m e d i n t o a p r o b a b i l -i t y measure on the s e t s . We a l s o s p e c i f y a s e t o f axioms w h i c h we b e l i e v e t h e d e c i s i o n maker ought t o f o l l o w from w h i c h an e v a l u a t i o n f u n c t i o n may be s p e c i f i e d . I n s e c t i o n 2 we s h a l l s p e c i f y t h e u n d e r l y i n g a ssumption axioms and n o t a t i o n we s h a l l use. We s h a l l a l s o s p e c i f y the problem i n more m a t h e m a t i c a l terms. I n the t h e o r y d e v e l o p e d here we s h a l l assume t h r e e b a s i c axioms f o r problems under p a r t i a l r i s k . These a r e s t a t e d and d i s c u s s e d i n s e c t i o n 3, 4 and 5 r e s p e c t i v e l y . I n s e c t i o n 6 we s h a l l summarize a l l the axioms and assumptions f o r easy r e f e r e n c e , and a l s o show t h a t an e v a l u a t i o n f u n c t i o n i s com-p l e t e l y s p e c i f i e d g i v e n t h e s e axioms and a s s u m p t i o n s . I n s e c t i o n 7 we s h a l l s t a t e an a d d i t i o n a l axiom t h a t s i m p l i f i e s the p r a c t i c a l a s p e c t s o f p a r t i a l r i s k problems b u t w h i c h i s n o t n e c e s s a r y f o r t h e t h e o r y . We s h a l l a l s o s u g g e s t some o t h e r s i m p l i f i c a t i o n s . I n s e c t i o n 8 we s h a l l c o n s i d e r the p r e f e r e n c e f u n c t i o n s we have d e r i v e d as a p r o b a b i l i t y measure. In s e c t i o n 9 we s h a l l g i v e some s u p p o r t f o r the t h e o r y from some e m p i r i c a l s t u d i e s which have been made, and f i n a l l y i n s e c t i o n 10 we s h a l l summarize t h e r e s u l t s . 125 2.0 The b a s i c assumptions I n t h i s s e c t i o n we s h a l l s p e c i f y the n o t a t i o n and some o f the b a s i c assumptions needed f o r the model d e v e l o p e d h e r e . Most o f the n o t a t i o n f o l l o w s t h a t used i n P a r t I . We s h a l l assume t h a t t h e r e e x i s t s a p r o b a b i l i t y space ( iT2,0, y) , a reward space (R,*F) and an a c t i o n s e t , A , w i t h a complete o r d e r i n g A > on A . For the time b e i n g we s h a l l l e t A be an a r b i t r a r y s e t A and > an a r b i t r a r y complete o r d e r i n g and r e s t r i c t t he s e t A A and the o r d e r i n g > as needed i n subsequent s e c t i o n s . F o r a g i v e n a c t i o n a e A , the reward f u n c t i o n i s a f u n c t i o n from ft t o R , and i s denoted by X(>,a). The s e t o f a l l 0 - m e a s u r a b l e reward f u n c t i o n s i n A w i l l be i n d e x e d by t h e s u b s e t A o f A , J o t h a t i s i f a e A Q t h e n X(-,a) i s measurable (we s h a l l modify, t h i s d e f i n i t i o n o f A q p r e s e n t l y ) . The s m a l l e s t a - a l g e b r a o f s u b s e t s o f ft, such t h a t a l l f u n c t i o n s X ( - , a ) , a e A , are measurable w i l l be denoted by 3 - N e c e s s a r i l y , t h e n , 0 C 3 . (The case o f 6 = 3 w i l l n o t be c o n s i d e r e d . ) W i t h t h i s n o t a t i o n we can summarize the s t a n d a r d c a t e -g o r i e s o f d e c i s i o n making as f o l l o w s : A g i v e n a c t i o n a e A , i s a d e c i s i o n under 1) c e r t a i n t y , i f a e A and t h e r e e x i s t s a B e 0 such t h a t X B ( o o,a) = r and y (B) = 1, 2) r i s k , i f a e A Q , 3) u n c e r t a i n t y , i f a e A , and i f f o r any s e t B c ft such t h a t B ^ 0, B ^ ft and B = {cu:X ( io,a) e C f o r some C e ¥}, then B i 0 , 4) p a r t i a l r i s k , i f a i s any a l t e r n a t i v e i n A . I n t h e s e d e f i n i t i o n s t h e c a t e g o r i e s 1 ) , 2 ) , and 3) a r e a l l i n c l u d e d i n 4 ) . T h i s c l a s s i f i c a t i o n i s , t h e r e f o r e , redun-dant i f an a c c e p t a b l e c r i t e r i o n may be found f o r p a r t i a l r i s k p roblems. Our aim i s t o s p e c i f y a c r i t e r i o n f o r p a r t i a l r i s k problems. The same c r i t e r i o n may a l s o , t h e r e f o r e , be used f o r u n c e r t a i n t y , r i s k , o r c e r t a i n t y problems. I n s e c t i o n 2 . 1 we s h a l l s t a t e t h r e e fundamental axioms which a r e t h e b a s i s f o r our development, and i n s e c t i o n 2 . 2 we s h a l l i l l u s t r a t e the d i f f e r e n c e between 6 and 0 , from a p r a c t i c a l d e c i s i o n making approach. 2 . 1 Statement o f the axioms I n t h i s s e c t i o n we s h a l l s p e c i f y t h r e e axioms and two assumptions which j o i n t l y guarantee t h e e x i s t e n c e o f a u t i l i t y f u n c t i o n U on R, and a l s o s p e c i f y the e x p e c t e d u t i l i t y c r i -t e r i o n f o r a c t i o n s b e l o n g i n g t o A Q . The f i r s t axiom we s h a l l s t a t e here i s s i m i l a r t o Axiom I i n P a r t I . That i s , i t assumes th e e x i s t e n c e o f a f u n c t i o n on A, w h i c h p r e s e r v e s t h e o r d e r i n g on A. Axiom I . There e x i s t s a r e a l - v a l u e d f u n c t i o n f ( • ) on A such t h a t A a > b i f and o n l y i f f ( a ) ^ f ( b ) The o b j e c t i o n s and a l t e r n a t i v e s t o t h i s axiom were d i s -c u s s e d i n P a r t I , and w i l l , t h e r e f o r e not be d i s c u s s e d h e r e . Other axioms w h i c h we s h a l l assume impose c o n d i t i o n s on the A o r d e r i n g > on A and the membership of A, .or e q u i v a l e n t l y , they s p e c i f y p r o p e r t i e s o f the f u n c t i o n f . I f f o r a l l r e R t h e r e e x i s t s an a £ A such t h a t o X(u),a) = r f o r a l l w £ ft, t h e n , i n g e n e r a l , a f u n c t i o n U may be d e f i n e d on R by U(r) = f ( a ) . A d i f f i c u l t y a r i s e s i f t h e r e a l s o e x i s t s a b £ A q such t h a t X(u),b) = X(u),a) f o r a l l w £ ft but f ( b ) ^ f ( a ) . I n t h i s case U i s n o t u n i q u e l y d e f i n e d . T h e r e f o r e , f o r U t o be a u n i q u e l y d e f i n e d f u n c t i o n we need b o t h the e x i s t e n c e o f a l l c o n s t a n t f u n c t i o n s (Assumption 1) and a l s o the assumption t h a t i f two c o n s t a n t reward f u n c t i o n s a re e q u a l t hey have the same p r e f e r e n c e (Axiom H i ) . Assumption 1. For each r e R, t h e r e e x i s t s an a e A Q such t h a t X(u),a) = r f o r a l l to £ ft. J o i n t l y Axiom H i and Assumption 1 a l s o i m p l y t h a t an o r d e r i n g may be i n d u c e d on R, s i n c e t h i s i s a weaker c o n d i t i o n than the e x i s t e n c e o f U. The o r d e r i n g we s h a l l c o n s i d e r on R i s d e f i n e d as f o l l o w s : I f X(co,a) = s and X(o),b) = r f b r a l l u) £ ft then R r J> s i f f (a) > f (b) , o r e q u i v a l e n t l y R r > s i f U(r) > U(s) . R For any o r d e r i n g > on R, dominance can be d e f i n e d among the reward f u n c t i o n s i n the u s u a l way: the reward f u n c t i o n X(«,a) dominates t h e reward f u n c t i o n X(*,b) w i t h r e s p e c t t o R > i f 128 X(u,a) X(u>,b) f o r a l l u e Q ( o r r e w a r d f u n c t i o n X(-,b) i s d o m i n a t e d by r e w a r d f u n c t i o n X ( - , a ) ) . T h a t i s , we p r e f e r t h e r e w a r d o f X(u>,a) t o t h e r e w a r d o f X(w,b) f o r a l l p o s s i b l e outcomes. I t seems r e a s o n a b l e t o assume t h a t i f X(«,a) d o m i n a t e s X ( * , b ) , a,b e A and t h e o r d e r i n g on R has been i n d u c e d by t h e c o n s t a n t r e w a r d f u n c t i o n s , t h e n f (a) > f (b)'. I n t h e f u r t h e r d e v e l o p m e n t we s h a l l assume t h a t when we d i s c u s s an o r d e r i n g on R o r dominance among r e w a r d f u n c t i o n s we s h a l l assume t h a t t h e o r d e r i n g has been i n d u c e d by t h e c o n -s t a n t f u n c t i o n s . We a r e now r e a d y t o s t a t e Axiom I I com-p l e t e l y . A x i o m I I . i ) I f f o r any r e R t h e r e e x i s t s a,b e A s u c h t h a t X(u>,a) = r and X(co,b) = r f o r a l l w e f t , t h e n f (a) = f (b) . A i i ) I f X(»,a) d o m i n a t e s X(*,b) t h e n a > b. A n a t u r a l e x t e n s i o n o f A x i o m H i w o u l d be t o assume t h a t i f X(w,a) = X(w,b) f o r a l l w e n t h e n f ( a ) = f ( b ) . T h i s f o l -l o w s , however, d i r e c t l y f r o m p a r t i i o f t h e Ax i o m as f o l l o w s : R I f X(.,a) = X( = ,b) t h e n X(o>,a) > X(u),b) f o r a l l co e and hence X(*,a) d o m i n a t e s X(-,b) by Axiom I l i i f ( a ) > f ( b ) . S i m i l a r l y X(-,b) d o m i n a t e s X(*,a) and hence f ( b ) > f ( a ) and t h e r e f o r e f ( a ) = f ( b ) . I f a e A Q and X(-,a) i s a c o n s t a n t r e w a r d f u n c t i o n t h e n U i s d e f i n e d by t h e e q u a l i t y UX(*,a) = f ( a ) . The f u n c t i o n U w i l l 129 R X ( w,a) > X (w,b) f o r a l l w e f t (or reward f u n c t i o n X(«,b) i s dominated by reward f u n c t i o n X ( * , a ) ) . That i s , we p r e f e r t h e reward o f X ( w,a) t o t h e reward o f X (w,b) f o r a l l p o s s i b l e outcomes. I t seems r e a s o n a b l e t o assume t h a t i f X(-,a) dominates X ( - , b ) , a,b e A and t h e o r d e r i n g on R has been i n d u c e d by t h e c o n s t a n t reward f u n c t i o n s , t h e n f ( a ) > f ( b ) . I n t h e f u r t h e r development we s h a l l assume t h a t when we d i s c u s s an o r d e r i n g on R o r dominance among reward f u n c t i o n s we s h a l l assume t h a t t h e o r d e r i n g has been i n d u c e d by t h e con-s t a n t f u n c t i o n s . We a r e now r e a d y t o s t a t e Axiom I I com-p l e t e l y . Axiom I I . i ) I f f o r any r e R t h e r e e x i s t s a,b e A such t h a t X(.a-,a) = r and X (w,b) = r f o r a l l w e f t , t h e n f (a) = f (b) . A i i ) I f X(-,a) dominates X(-,b) t h e n a > b. A n a t u r a l e x t e n s i o n o f Axiom H i would be t o assume t h a t i f X ( w,a) = X (w,b) f o r a l l w e f t t h e n f ( a ) = f ( b ) . T h i s f o l -l o w s , however, d i r e c t l y from p a r t i i o f t h e Axiom as f o l l o w s : R I f X(.,a) = X(.,b) t h e n X ( w,a) > X (w,b) f o r a l l w e f t and hence X(*,a) dominates X(*,b) by Axiom I l i i f ( a ) > f ( b ) . S i m i l a r l y X(-,b) dominates X(-,a) and hence f ( b ) > f ( a ) and t h e r e f o r e f ( a ) = f ( b ) . I f a e A Q and X(-,a) i s a c o n s t a n t reward f u n c t i o n then U i s d e f i n e d by t h e e q u a l i t y UX(-,a) = f ( a ) . The f u n c t i o n U w i l l and i s made f o r c o n v e n i e n c e . S i n c e t h e i n t e g r a l a l w a y s e x i s t s f o r bounded m e a s u r a b l e f u n c t i o n s fvx(o,*)av must e x i s t f o r a l l a ~ * a e A . o We s h a l l use t h e n o t a t i o n EUX(-,a) f o r / UX(io,a)dy. n o A x i o m I I I . I f b e A Q t h e n f ( b ) = E U X ( - , b ) . To summarize, A s s u m p t i o n 1, Axiom I and A x i o m H i guaran-t e e t h e e x i s t e n c e o f a u t i l i t y f u n c t i o n U on R, and a l s o an o r d e r i n g ^ on R. A s s u m p t i o n 2 i m p l i e s t h a t JUX(-,a)dy i s w e l l d e f i n e d f o r a l l a e A q and Axiom I I I t h a t t h e e x p e c t e d u t i l i t y c r i t e r i o n i s t h e a c c e p t e d c r i t e r i o n f o r d e c i s i o n making u n d e r r i s k . The axioms w h i c h f o l l o w i n s e c t i o n s 3-6 w i l l e x t e n d t h i s e v a l u a t i o n f u n c t i o n f rom A Q t o A . 2.2 Comments o n t h e a s s u m p t i o n s We have assumed t h e e x i s t e n c e o f a p r o b a b i l i t y s p a c e ( f t , 9 , y ) ; t h a t i s , t h e r e e x i s t e v e n t s (members o f 0) f o r w h i c h t h e p r o b a b i l i t i e s a r e known o r a t l e a s t t h e d e c i s i o n makers a r e w i l l i n g t o a c c e p t c e r t a i n e v e n t s f o r w h i c h t h e y b e l i e v e t h e y know t h e p r o b a b i l i t i e s . C o n s i d e r , f o r example, t h e f o l l o w i n g e v e n t s : = {heads o c c u r s when a c o i n i s t o s s e d } B 2 = {a f i v e o c c u r s when a d i e i s r o l l e d } B^ = {Dow-Jones w i l l c l o s e h i g h e r tomorrow t h a n t o d a y } B^ = { t h e t e m p e r a t u r e tomorrow w i l l r e a c h a maximum 131 v a l u e o f 10°C}. Some o f t h e s e may be c l a s s i f i e d as b e l o n g i n g t o 9 and o t h e r s as b e l o n g i n g t o f $ . Many p e o p l e would be w i l l i n g t o a c c e p t y ( B 1 ) = 1/2 and u ( B 2 ) = 1/6. The p r o b a b i l i t i e s f o r t h e eve n t s and B^ a r e such t h a t we may be l e s s w i l l i n g t o spec-i f y them e x a c t l y . (The assumptions do not p r e v e n t us from d o i n g so o f c o u r s e . F o r example a m e t e o r o l o g i s t may f e e l c o n f i d e n t about s p e c i f y i n g an e x a c t p r o b a b i l i t y f o r event B ^ . ) The i d e a t h a t some p r o b a b i l i t i e s can be " a c c u r a t e l y " s p e c i f i e d and o t h e r s n o t , has been s u g g e s t e d by Anscombe and Aumann (19 63 ) . They c o i n e d t h e phrase "horse l o t t e r y " f o r t h o s e e v e n t s f o r whi c h p r o b a b i l i t i e s may not be c o m p l e t e l y s p e c i f i e d . Those e v e n t s where a p r o b a b i l i t y can be s p e c i f i e d t h e y c a l l e d r o u l e t t e l o t t e r i e s . F e l l n e r (1961) a l s o s e p a r a t e d e v e n t s f o r whic h t h e p r o b a b i l i t y may be s t a t e d w i t h some a c c u r a c y and th o s e f o r whi c h t h i s i s not p o s s i b l e . He suggested t h a t s e t s w h i c h b e l o n g t o 0 a r e tho s e f o r which the s u b j e c t i v e probab-i l i t y may be s u p p o r t e d by f r e q u e n c y p r o b a b i l i t i e s a s , f o r example, t h e drawing o f c a r d s from a deck o f g u a r a n t e e d com-p o s i t i o n . Savage (1954) i n c l u d e d a l l s u b s e t s o f ft i n 0 . Hence he would h o l d t h a t a l l p r o b a b i l i t i e s can be s p e c i f i e d e x a c t l y . Good (1965) d i s a g r e e s t h a t t h e p r o b a b i l i t i e s can be e x a c t l y d e t e r m i n e d . He w r i t e s My own v i e w , f o l l o w i n g Keynes and Koopman, i s t h a t judgments o f p r o b a b i l i t y i n e q u a l i t i e s a r e p o s s i b l e b u t n o t judgments o f e x a c t p r o b a b i l i -t i e s ; t h e r e f o r e a B a y e s i a n s h o u l d have upper and lower b e t t i n g p r o b a b i l i t i e s . (Good, 1965; pp 5) I n the approach d e v e l o p e d here we agree w i t h Anscombe and Aumann (19 6'3,) t h a t t h e r e e x i s t b o t h "horse l o t t e r i e s " and " r o u l e t t e l o t t e r i e s " . We a l s o a c c e p t i n p a r t Good's argument t h a t p r o b a b i l i t i e s may n o t n e c e s s a r i l y be e x a c t l y d e t e r m i n e d , a l t h o u g h we a l l o w i t t o be s p e c i f i e d as p r e c i s e l y as one chooses by making a judgment o f a s u f f i c i e n t number o f p r o -b a b i l i t y i n e q u a l i t i e s . 133 3.0 The P-measure axiom In section 2 the expected u t i l i t y c r i t e r i o n was suggested for actions belonging to A q . In extending t h i s c r i t e r i o n i n the usual mathematical fashion to a l l of A , two d i f f i c u l t i e s a r i s e which are i n t e r r e l a t e d . One d i f f i c u l t y i s that the required extension of the measure y to the a-algebra 6 may not e x i s t . A property such as a - a d d i t i v i t y of the measure must often be s a c r i f i c e d . However, when such a property i s not s a t i s f i e d by the measure i t i s d i f f i c u l t to define expected value. In this section we s h a l l assume the existence of a set function on 8 which i s an extension of the measure y . The existence of such a function w i l l depend on what properties i t i s assumed to have. The properties we s h a l l specify in the following section, however, w i l l not put any r e s t r i c t i o n on 3 . In section 3.1 we s h a l l specify the f i r s t property of the extended measure. In section 3.2 we s h a l l discuss some of the more important implications of th i s property and also give an example i l l u s t r a t i n g the assumptions so far made. F i n a l l y , i n section 3.3 additional assumptions w i l l be made to enable us to determine the values of the function for.each B e B . 3.1 Statement o f t h e axiom The c o n c e p t o f Axiom IV can be i l l u s t r a t e d by the f o l -l o w i n g example. C o n s i d e r two reward f u n c t i o n s d e f i n e d by ( o e B j r o> e D _ and X( o),b) = < _ m e B I s a) e D where B,D e 9 , and U(r) > U ( s ) . From Axiom I I I i t f o l l o w s t h a t : f ( a ) = U ( r ) y ( B ) + U ( s ) y ( B ) and f (b) = U ( r ) y ( D ) + U ( i ) y (D) and hence f ( a ) - f ( b ) = [U(r) - U ( s ) ] [ y ( B ) - y ( D ) ] . T h e r e f o r e , f ( a ) > f ( b ) i f and o n l y i f y(B) > y(D) , so t h a t t h e p r e f e r e n c e between a and b can be d e t e r m i n e d by comparing t h e p r o b a b i l i t i e s y(B) t o y ( D ) . Thus, f o r a l t e r -n a t i v e s w i t h t h e same two p o s s i b l e rewards t h e a l t e r n a t i v e w i t h t h e l a r g e s t p r o b a b i l i t y o f r e c e i v i n g t h e h i g h e r o f the two rewards i s chosen. I f P i s the extended s e t f u n c t i o n o f y t o 6, i t seems r e a s o n a b l e t o assume t h a t P has the same p r o p e r t y . T h e r e f o r e i f P i s assumed t o be a p r o b a b i l i t y measure the assumption i s c l e a r l y c o n s i s t e n t w i t h the e x p e c t e d u t i l i t y c r i t e r i o n . T h i s assumption i s f o r m a l i z e d i n Axiom IV. S i n c e t h e e x t e n -s i o n may not s a t i s f y the r e q u i r e m e n t o f what i s g e n e r a l l y c a l l e d a p r o b a b i l i t y measure (see d e f i n i t i o n i n Appendix I I , page 2 35), we s h a l l c a l l t h e e x t e n s i o n a p r e f e r e n c e measure, o r s i m p l y a P-measure. We s h a l l f i r s t assume t h a t a l l reward f u n c t i o n s w i t h o n l y two rewards b e l o n g t o A. Assumption 3. For any D e 0 and f o r any r , s e R t h e r e e x i s t s : b e A such t h a t X(o3,b) = co E D 03 e D Axiom IV. There e x i s t s a s e t f u n c t i o n P d e f i n e d on 0 such t h a t a) i f B e 0 t h e n P(B) = y ( B ) , b) f o r any r , s £ R such t h a t U(r) > U ( s ) , and f o r any B,D £ 0 , l e t a,b be elements o f A f o r w h i c h the reward f u n c t i o n s a re r 03 e B X ( o j,a) = { _ and X(o3,b) = \ S 03 £ B r 03 £ D 03 £ D A Then a > b i f and o n l y i f P(B) > P(D) We showed one i m p l i c a t i o n o f the e x p e c t e d u t i l i t y c r i t e r i o n t o be t h a t i f two reward f u n c t i o n s a r e g i v e n by X ( to , c) r to £ B J r c o e D and X ( i o,d) = < s to e B I s c o e D A A w i t h B,D e 9 , then d > c, when a > b. T h i s f o l l o w s s i n c e A a > b i m p l i e s the p r o b a b i l i t y o f B i s g r e a t e r t h a n the p r o -b a b i l i t y o f D, so t h a t the p r o b a b i l i t y o f D must be g r e a t e r t h a n the p r o b a b i l i t y o f B ( s i n c e y (B) + y ( B ) = 1 i f B £ 0 ) A and hence d > c. I f , however, the p r o b a b i l i t i e s a r e n o t known, th e n the A p r e f e r e n c e a > b does n o t i m p l y t h a t the p r o b a b i l i t y o f B i s g r e a t e r t h a n the p r o b a b i l i t y o f D, i t o n l y i m p l i e s f o r some re a s o n t h a t we p r e f e r a l t e r n a t i v e a t o a l t e r n a t i v e b. For example, a l t e r n a t i v e a may seem l e s s r i s k y i n the sense t h a t we have some i n f o r m a t i o n o f the l i k e l i h o o d o f B o c c u r r i n g , and no i n f o r m a t i o n o f t h e l i k e l i h o o d o f D o c c u r r i n g . That i s we may d i f f e r e n t i a t e between "known" v e r s u s "unknown" p r o -b a b i l i t i e s . Axiom IV g i v e s us a method t o determine a p a r t i a l o r d e r i n g f o r a l l a c t i o n s w h i c h would r e s u l t i n one o f two A rewards. I f a > b, w i t h a reward f u n c t i o n as g i v e n i n Axiom IV, t h e n f o r a c t i o n s e and f w i t h reward f u n c t i o n s to £ B X ( to,e) = { _ and X ( t o/f) = to £ B V 0) £ D t to £ D A the p r e f e r e n c e e > f i s i m p l i e d i f U(v) > U ( t ) . However, we can not s t a t e a p r e f e r e n c e between a and e based on our development so f a r and t h e r e f o r e o n l y a p a r t i a l o r d e r i n g can be d e t e r m i n e d . I t seems i n t u i t i v e l y c l e a r , however, t h a t i f A U(v) > U(r) and U(t) > U(s) t h e n e > a. T h i s p o i n t w i l l be c o n s i d e r e d l a t e r i n s e c t i o n 5.0, so f o r the time b e i n g i t i s s u f f i c i e n t t o c o n s i d e r t h e axiom as s t a t e d . 3.2 I m p l i c a t i o n s o f Axiom IV Axiom IV, t o g e t h e r w i t h t h e p r e v i o u s axioms, r e q u i r e s the P-measure t o have s e v e r a l p r o p e r t i e s . I n t h i s s e c t i o n we s h a l l s p e c i f y t h o s e w h i c h a r e used i n subsequent s e c t i o n s . Lemma 3.2.1. The P-measure i s monotone on 3 , i . e . , i f D C B the n P(D) < P (B) . P r o o f . By Assumption 3 t h e r e e x i s t s a,b e A such t h a t f o r any reward r , s e R w i t h U(r) > U(s) r c o e B - f r c o e D X(o),a) [ " s u e B ^ s to e D A By Axiom I l i i , a > b, and t h i s i m p l i e s by Axiom IV t h a t P(B) > P(D) . Lemma 3.2.2. The P-measure i s bounded by the i n n e r and o u t e r measure i n d u c e d by y, i . e . , f o r any D £ 6 , y*(D) < P(D) < y*(D) * where y* and y are i n n e r and o u t e r measures r e s p e c t i v e l y (see Appendix I I f o r d e f i n i t i o n , p. 235). 138 P r o o f . L e t C C D <z. E be s e t s such t h a t D e g and C , E e 0 . S i n c e P-measure i s monotone on g , P(C) ^ P(D) < P ( E ) . Thus, s i n c e C , E e 0 , we have y(C) < P(D) ^ y ( E ) , and t h e d e f i n i t i o n o f o u t e r and i n n e r measures i m p l i e s t h a t y*(D) < P(D) * y * ( D ) . Lemma 3.2.3. The P-measure agrees w i t h the l a r g e s t unique e x t e n s i o n o f y such t h a t the e x t e n s i o n i s a measure, i . e . , P(D) = y (D) i f D e S where = • S =.{D C Q | y * ( D ) • ; i * ( D ) } . P r o o f . T h i s f o l l o w s d i r e c t l y from the p r e v i o u s lemma. Lemma 3 . 2 . 4 . The P-measure i s " n e a r l y a d d i t i v e " ; t h a t i s , f o r any C e 8 , D e 9 , such t h a t D A C = 0 1) iP(D U C) - P(D) - P(C) | < y*(C) - y*(C) and f o r any C e '8 and D" e 0 , 2) |P(C O D) + P(C O D) - P(C) | < y * ( C ) - y * ( C ) . By " n e a r l y a d d i t i v e " we mean, t h e r e f o r e , t h a t g i v e n two d i s j o i n t s e t s , C and D such t h a t C e g , and D e 0 t h e n the sum P(C) + P(D) can n o t d i f f e r from P(D U C) by more t h a n t h e 139 d i f f e r e n c e o f t h e o u t e r and i n n e r measure o f C. T h e r e f o r e , i f t h e o u t e r and i n n e r measures a r e e q u a l f o r t h e s e t C, t h e n P(D) + P(C) = P ( D U C) f o r a l l s e t s D e 9 such t h a t D n c = 0. P r o o f . Two p r o p e r t i e s o f i n n e r and o u t e r measures a r e (Appendix I I , p. 2 3 5 ) : I f C A D = 0 and C e 6 , D e 9 , * * Jc t h e n y (D) + y (C) = y (D U C) and M*(D) + y * (C) = y * ( D U C) . By lemma 3.2.2, y * (D U C) < P(D VJ C) * y * (D U C) , o r , u s i n g the above, y*(D) + y * ( C ) ^  P ( D U O ^ V*(D) + y * ( C ) . * S i n c e D e 0 w h i c h a l s o i m p l i e s t h a t y*(D) = y (D) = P(D) s u b t r a c t i o n y i e l d s y * ( C ) < P ( D U C ) - P(D) < y * ( C ) . Combining t h i s i n e q u a l i t y w i t h t h e f o l l o w i n g (from lemma " 3.."2 . 2 , 140 y * ( C ) < P ( C ) * y * ( C ) , now y i e l d s | P ( D U C ) - P ( D ) - P ( C ) | < y * ( C ) - y * ( C ) . Thus, P must be " n e a r l y a d d i t i v e " f o r s e t s such t h a t D r\ C = 0 and D e 0 . T h e r e f o r e P must b e ^ a d d i t i v e • on S. ." . • - < - - -S i m i l a r l y , t h e i n n e r and o u t e r measures a l s o s a t i s f y t h e e q u a l i t i e s y * ( C ) = y * ( C n D ) + y * ( C P v D ) and y ( C ) = y ( C A D ) + y ( C O D ) . By lemma 3.2.2 we have y * ( C A D ) < P ( C A D ) < y * ( C A D ) , y * ( C A D ) ^ P ( C A t T ) < y * ( C A D ) , and hence y * ( C O D ) + y * ( C A D ) < P ( C A D ) + - P ( C A D ) < y ( C A D ) + y ( C A D ) . 141 Thus, w i t h the e q u a l i t i e s above, we have y*(C) ^ P ( C A D) + P ( C H D)4 y*(C) and, as b e f o r e , | P (C r\ D) + P(C /> D). - P(C) | < y * (C) - y * (C) ; i . e . , P(C) must be " n e a r l y " e q u a l t o P(C O D) + P(C A D). Savage's axioms i m p l y Axiom IV. Of t h o s e approaches c o n s i d e r e d i n P a r t I o f t h e t h e s i s , we r e c a l l t h a t i n the Savage approach a p r o b a b i l i t y measure was d e r i v e d on t h e s u b s e t s o f $. The method used was s u b s t a n t i a l l y d i f f e r e n t , however, and i t i s not o b v i o u s t h a t t h e P-measure s p e c i f i e d here i s c o n s i s t e n t w i t h t h a t approach. R e c a l l t h a t Savage d e f i n e s an o r d e r i n g on a l l s u b s e t s o f Q ( i . e . , 0 = 2 ^ ) by Q A B > D i f and o n l y i f a > b whenever U(r) > U ( s ) , X_,(-,a) = X_(',b) = r and Jo D X^(-ra) = X-(-,b) = s. Savage t h e n p r o v e s t h a t t h e r e e x i s t s a r e a l v a l u e d f u n c t i o n P on 0 such t h a t 0 P(C) > P(D) i f and o n l y i f B > D. That i s P(C) > P(D) i f and o n l y i f a > b, w h i c h i m p l i e s Axiom IV. 3.3 Comparison o f d i f f e r e n t approaches u s i n g E l l s b e r g ' s paradox L e t us i l l u s t r a t e some p r o p e r t i e s o f t h e e v a l u a t i o n f u n c t i o n f w h i c h a r e s t a t e d i n the f i r s t f o u r axioms. I t i s a p p r o p r i a t e t o use the E l l s b e r g paradox I I ( E l l s b e r g , 1961) f o r t h i s purpose, s i n c e e m p i r i c a l s t u d i e s u s i n g t h i s paradox have shown t h a t most p e o p l e d i f f e r e n t i a t e between r i s k and u n c e r t a i n t y . ( T h i s p o i n t i s d i s c u s s e d f u r t h e r i n s e c t i o n 8.) Suppose we a r e g i v e n an u r n c o n t a i n i n g 30 r e d b a l l s o u t o f 90 b a l l s , and t h e r e m a i n i n g 6 0 a r e an unknown m i x t u r e o f y e l l o w and black.- We s h a l l assume f o r s i m p l i c i t y t h a t t h e p r o b a b i l i t y o f a b a l l o f a g i v e n c o l o u r b e i n g drawn i s e q u a l t o the p r o p o r t i o n a l number o f b a l l s o f t h a t c o l o u r t o the t o t a l number o f b a l l s . L e t A c o n t a i n the f o l l o w i n g a l t e r n a t i v e s : a) r e c e i v i n g $100 i f a r e d b a l l i s drawn (event R) r e c e i v i n g $0 o t h e r w i s e b) r e c e i v i n g $100 i f a y e l l o w b a l l i s drawn (event Y) r e c e i v i n g $0 o t h e r w i s e c) r e c e i v i n g $100 i f a r e d o r a b l a c k b a l l i s drawn (event B U R) r e c e i v i n g $0 o t h e r w i s e d) r e c e i v i n g $100 i f a y e l l o w o r b l a c k b a l l i s drawn (event YVJB) r e c e i v i n g $0 o t h e r w i s e e) r e c e i v i n g $100 i f a b l a c k b a l l i s drawn (event B) r e c e i v i n g $0 o t h e r w i s e g) r e c e i v i n g $0 143 T h e r e f o r e by t h e st a t e m e n t o f the problem we have t h e f o l l o w i n g : A = {a,b,c,d,e,g} A q = {a,d,g} 0 = { 0,ft,{R},{BVY}} 3 = ' { 0 , n,{R},{Y},{B},{RUY},{RUB},{BUY}}. v\0) = 0 via) = l u(R) = 1/3 y(BUY) = 2/3. L e t X(«,i) denote t h e reward f u n c t i o n c o r r e s p o n d i n g t o a l t e r n a t i v e i f o r any i e A. By Axiom I I I the p r e f e r e n c e o r d e r i n g on A Q ( f o r any s t r i c t l y i n c r e a s i n g f u n c t i o n U) must A n A o . be d > u a > g s i n c e EUX(w,d) > EUX(u>,a) > EUX(u,g) and f ( d ) = EUX(aj,d), f ( a ) = EUX(w,a) and f ( g ) = EU(o>,g). A l l approaches s u g g e s t e d i n t h e i n t r o d u c t i o n agree w i t h t h e s e v a l u e s o f f . B e f o r e we c o n s i d e r t h e p r o p e r t i e s f must have 144 a c c o r d i n g t o the axioms s p e c i f i e d so f a r l e t us i l l u s t r a t e how the d i f f e r e n t approaches w h i c h were i d e n t i f i e d i n the i n t r o d u c t i o n e x t e n d t h e e v a l u a t i o n f u n c t i o n f t o A. 1) Savage approach I f we now c o n s i d e r t h e e x t e n s i o n o f y on 9 t o P on 3 such t h a t P i s a p r o b a b i l i t y (as done by Savage, 1954, o r Good, 1965) t h e n P must be i n the f o l l o w i n g form: P(0) = 0, P(fl) = 1, P(R) = 1/3, P(Y) = p, P(B) = 2/3-p. T h i s would A A t h e n i m p l y t h a t i f a > b t h e n c > d. I n p r a c t i c e many d e c i s i o n makers would d i f f e r e n t i a t e between r i s k and u n c e r -A A t a i n t y and have t h e p r e f e r e n c e a > b and d > c. T h i s approach t h e r e f o r e i m p l i e s , a p r e f e r e n c e w h i c h i s o f t e n c o n t r a d i c t e d by e m p i r i c a l s t u d i e s . 2) C o m b i n a t i o n o f e v a l u a t i o n f u n c t i o n f o r u n c e r t a i n t y and r i s k C o n s i d e r the second method d e s c r i b e d i n t h e i n t r o d u c t i o n , i . e . , a v a r i a t i o n o f t h e maximin. L e t n denote the s e t o f a l l p o s s i b l e c o m b i n a t i o n s o f b l a c k and y e l l o w b a l l s . S i n c e t h e s m a l l e s t p o s s i b l e number o f y e l l o w b a l l s i n the u r n i s z e r o , we have min EUX(co,b) = U(O) . n S i n c e U(X(« , f ) ) = U ( 0 ) , t h i s would i m p l y t h a t we a r e i n d i f f e r -e n t between a l t e r n a t i v e s b and f , which a l l s u g g e s t must c o n t r a d i c t most d e c i s i o n makers' p r e f e r e n c e . 3) E l l s b e r g approach T h i s method c o n s i d e r s the most " l i k e l y " p r o b a b i l i t y o f a y e l l o w b a l l b e i n g drawn. E l l s b e r g does not make i t c l e a r how t h i s can be d e t e r m i n e d , but assumes f o r s i m p l i c i t y t h i s means P(Y) = P(B) = 1/3. The e v a l u a t i o n f u n c t i o n would be (assuming U(0) = 0) f (a) = (1/3)U(100) f (b) = ( l - p j (1/3)U(100) f (c) = ( 1 - P 2 ) ( 1 / 3 ) U ( 1 0 0 ) + (1/3)U(100) f (d) = (2/3)U(100) f (e) = (1-P 3) d/3)U(100) f (g) = 0 where p^, 0 < p^ 4 1 , i = l , 2 , 3 , i s the "ambiguity" f a c t o r . Hence E l l s b e r g p o s t u l a t e s t h a t i t i s i m p o s s i b l e t o have a A p r e f e r e n c e o f b > a. I n E l l s b e r g ' s a pproach, i t would t h e r e -f o r e be i m p o s s i b l e t o p r e f e r u n c e r t a i n t y t o r i s k . A l t h o u g h t h i s i s p r o b a b l y t r u e i n most c a s e s , E l l s b e r g h i m s e l f has sug g e s t e d a d e c i s i o n problem where he p r e d i c t s t h a t u n c e r -t a i n t y i s p r e f e r r e d t o r i s k 1 . 1 C o n s i d e r : Urn I has 1,000 b a l l s i n i t , each b a l l i d e n t i f i e d by a number, and each number from 1 t o 1,000 r e p r e s e n t e d . The p r o b a b i l i t y o f a random draw y i e l d i n g a b a l l w i t h a g i v e n num-ber i s 1/1,000. Urn I I has 1,000 b a l l s each i d e n t i f i e d by a number from 1 t o 1,000. Thus, i n the f i r s t u r n a l l numbers from 1 t o 1,0 00 a r e r e p r e s e n t e d and i n t h e second u r n any num-ber may appear z e r o , one o r more t i m e s . The d e c i s i o n - m a k e r i s t o l d t h a t t h e number o f o c c u r r e n c e s o f any g i v e n number i s c o n s t r a i n e d o n l y by the l i m i t s o f 0 and 1,0 00 and by the f a c t t h a t t h e t o t a l number o f b a l l s i s 1,000. The d e c i s i o n - m a k e r must d e c i d e w h i c h u r n he p r e f e r s t o draw from, g i v e n t h a t he w i l l w i n i f any o f n s p e c i f i e d numbers i s drawn and w i l l l o s e n o t h i n g i f a number n o t i n the s e t o f n i s drawn. The s u b s e t o f n numbers must come from t h e s e t o f numbers from 1 t o 1,000 E l l s b e r g s u g g e s t s t h a t , i f n i s v e r y s m a l l , t h e d e c i s i o n maker w i l l p r e f e r Urn I I where t h e r e may be as many as 1,000 b a l l s 146 4) F e l l n e r ' s a pproach F e l l n e r argues t h a t a c t i o n s i n A q can n o t be compared t o a c t i o n s i n A — A Q . Hence we may compare b t o e and a,d and 9 t o each o t h e r , b u t b o r d can n o t be compared. F e l l n e r assumes th e e x p e c t e d u t i l i t y c r i t e r i o n can be used t o o r d e r t h e a l t e r n a t i v e s i n A , i . e . , F e l l n e r assumes o Axiom I I I . F o r b,d an axiom s i m i l a r t o Axiom IV i s used t o d e t e r m i n e a "P-measure" from which a p r o b a b i l i t y measure i s d e r i v e d , and a l t e r n a t i v e s b and d a r e o r d e r e d by t h e e x p e c t e d u t i l i t y o f t h e d e r i v e d p r o b a b i l i t y . However, F e l l n e r does n o t d e s c r i b e i n h i s paper how a l t e r n a t i v e s such as c a r e h a n d l e d where a m i x t u r e o f known and unknown p r o b a b i l i t i e s e x i s t . 5) P-measure approach I n the t h e o r y d e v e l o p e d h e r e we have by Axiom I I I t h a t i f A a > b t h e n P(R) > P(Y) and i f A d > c t h e n P(BUY) > P(B U R ) . I n the t h e o r y d e v e l o p e d here we do not f i n d t h i s p r e f e r -ence c o n t r a d i c t o r y . However, t h i s i n d i c a t e s t h a t P(BUY) may n o t e q u a l P(B) + P ( Y ) , o r P(BUR) may n o t e q u a l P(B) + P(Y) o r from the w i n n i n g s e t , w h i l e Urn I has e x a c t l y n w i n n i n g b a l l s . Here a m b i g u i t y i s c o n s i d e r e d t o be f a v o r a b l e . As n i n c r e a s e s , E l l s b e r g s u g g e s t s t h a t a p o i n t w i l l be r e a c h e d where th e am-b i g u i t y a s s o c i a t e d w i t h the a c t i o n "draw from Urn I I " i s con-s i d e r e d t o be u n f a v o r a b l e . (Becker and Brownson, 196 4) b o t h . The s e t f u n c t i o n P. t h e r e f o r e may n o t be a d d i t i v e and hence n o t a measure. T h i s example shows t h e r e f o r e some o f t h e d i f f i c u l t i e s o f the methods proposed f o r t h e p a r t i a l r i s k problem. I t a l s o shows t h a t the P-measure i s l e s s r e s t r i c t e d than t h e o t h e r approaches. There a r e , o f c o u r s e , some d i f f i c u l t i e s w i t h the P-measure. F o r example, i t may even be i m p o s s i b l e t o determine P(Y) i f we do not assume a •; second u r n ( U T T ) w i t h a known p r o -p o r t i o n y o f r e d b a l l s and w i t h t h e a p p r o p r i a t e e x t e n s i o n o f the a - a l g e b r a 0 . The q u a n t i t y P(Y) may be e s t i m a t e d by com-p a r i n g a l t e r n a t i v e b w i t h the f o l l o w i n g a l t e r n a t i v e : h) r e c e i v i n g $100 i f a r e d b a l l i s drawn from U.^ r e c e i v i n g $0 o t h e r w i s e I f b i s p r e f e r r e d t o h t h e n P(Y) > y and s i m i l a r l y i f h i s p r e f e r r e d t o b th e n P(Y) < y by Axiom I I I . By the t r a n s i t i v i t y a s s umption t h e r e e x i s t s a unique y such t h a t y = P ( Y ) . To determine P(Y) by t h i s method, we need t h e e x i s t e n c e o f U^ .^  w h i c h we s h a l l f o r m a l i z e as an assu m p t i o n i n s e c t i o n 3 . 3 . I n Axiom I I I we assumed t h a t f ( a ) = J UX(-,a)dy f o r a l l a e A q . T h i s d e f i n i t i o n u n i q u e l y d e f i n e s the e v a l u a t i o n f u n c t i o n f ( * ) , p r i m a r i l y because y i s a measure. One can not d i r e c t l y e x t e n d the e v a l u a t i o n f u n c t i o n f(«) t o a l l a c t i o n s i n A, by s i m p l y r e p l a c i n g t h e measure y by the P-measure P i n t h e i n t e g r a l d e f i n i t i o n s i n c e i t may n o t be u n i q u e l y d e f i n e d . I n o r d e r t o s p e c i f y a p r o p e r i n t e g r a l r e p r e s e n t a t i o n o f f ( • ) a d d i t i o n a l assumptions r e g a r d i n g the P-measure must be made. T h i s may p r e s e n t problems i n t h a t the d e s i r a b l e p r o p e r t i e s o f P, e.g., n o n - a d d i t i v i t y f o r some e v e n t s , a r e t o be r e t a i n e d . 148 F o r example,/ i f we assume t h a t h e A Q f y < 1 / 3 and U ( 0 ) = 0 then by r e p l a c i n g y by P i n the i n t e g r a l d e f i n i t i o n we o b t a i n f (h) = Y U C$100) and f (e) = P(B)U($100) . Hence the d e c i s i o n maker a c t s as though h i s p r o b a b i l i t y o f o b t a i n i n g $100 i s e q u a l t o P ( B ) , and the p r o b a b i l i t y o f o b t a i n i n g $0 i s e q u a l t o l - P ( B ) . So f o r t h e e x t e n s i o n o f t h e e x p e c t e d v a l u e u s i n g a P-measure i s v a l i d . However, i f we e x t e n d t h e number o f rewards t o t h r e e i n a l t e r n a t i v e k) we have k) r e c e i v i n g $100 i f a b l a c k b a l l i s drawn from r e c e i v i n g $100 + e i f a y e l l o w b a l l i s drawn from r e c e i v i n g $0 o t h e r w i s e . A Assume f o r s i m p l i c i t y t h a t P(Y) = P ( B ) , and a l s o t h a t a > b. By Axiom I I , p a r t i i , k > d. C a l c u l a t i n g the " e xpected v a l u e " w i t h P as a measure y i e l d s U(100)-2/3 < U(100)P(B) + U(100 + e ) P ( Y ) . T h i s i m p l i e s P(B) > — 2U(100) 3(U(100 + U(100 + e)) T a k i n g t h e l i m i t s as e tends t o z e r o , we o b t a i n P(B) > l / 3 , w h i c h would i n t u r n i m p l y b > a, c o n t r a d i c t i n g our a s s umption. I t i s o h v i o u s t h e r e f o r e t h a t P can not be used t o c a l c u l a t e t h e e x p e c t e d u t i l i t y as i f i t were a measure. 3.4 E v a l u a t i o n o f P-measures So f a r , we have o n l y assumed the e x i s t e n c e o f t h e P-measure. There i s no o b v i o u s method o f d e t e r m i n i n g what P(D) i s e q u a l t o f o r an a r b i t r a r y D e g . For example, i f 0 o n l y c o n t a i n s a f i n i t e number o f s e t s , as i n E l l s b e r g ' s paradox I I , then i f a c t i o n a i s p r e f e r r e d t o a c t i o n b, 0 < P(Y) < 1/3. However P(Y) can n o t be s p e c i f i e d c o m p l e t e l y s i n c e e v e n t s f o r w h i c h the p r o b a b i l i t y o f o c c u r r e n c e i s between 0 and 1/3 do n o t e x i s t i n t h i s problem. T h e r e f o r e t o be a b l e t o s p e c i f y the P-measure f o r any e v e n t we need t o assume t h e e x i s t e n c e o f Urn I I o r t h a t i f a i s a r e a l number between z e r o and one, t h e n t h e r e e x i s t s an event C e 0 such t h a t y (C) = a , and a l s o t h a t i f a reward f u n c t i o n i s d e f i n e d by X ( to,a) = to e C to e C t h e n a e A . o T h e r e f o r e , f o r any D e 0 - 0 , t h e P-measures o f D can e a s i l y be d e t e r m i n e d by comparing a c t i o n s a t o b where ( r to E D s to E D Thus, by v a r y i n g a u n t i l we become i n d i f f e r e n t between a and b we w i l l d e t e r m i n e a s p e c i f i c a such t h a t P(.D) = a. 150 A second d i f f i c u l t y a r i s e s i n t h e development o f the t h e o r y because t h e reward s e t R i s d e f i n e d as b e i n g an a r b i -t r a r y s e t ; t h a t i s , we have n o t s p e c i f i e d i t s c a r d i n a l i t y . I t may be f i n i t e , c o u n t a b l e o r even u n c o u n t a b l e . We s h a l l assume t h a t the reward s e t R i s not f i n i t e ; t h a t i s , i t may be c o u n t a b l e o r u n c o u n t a b l e . I n t h a t way we do n o t r e s t r i c t t h e a c t i o n s e t A i n any way. These assumptions do n o t i n d i c a t e any o r d e r i n g p r e f e r e n c e among the a c t i o n s i n A - A q , t h a t i s , i t o n l y s p e c i f i e s some c o n d i t i o n s on d e c i s i o n problems under r i s k and t h e r e f o r e ought n o t t o i n f l u e n c e our p r e f e r e n c e under p a r t i a l r i s k p roblems. Assumption 4. i ) I f Y(-) i s any f u n c t i o n from ft t o R such t h a t UY(•) i s B o r e l measurable w i t h r e s p e c t t o 0 , t h e n t h e r e e x i s t s a e A such t h a t X(-,a) = Y(«). i i ) There e x i s t s an a c t i o n a e A such o t h a t Y " U ( r ) y(UX(oj,a) < y) = - r - 5  U(r°)-U(r Q) f o r each y e [ u (r ) ,U (r°)] . P a r t i ) o f t h e assu m p t i o n i m p l i e s t h a t i f Y i s a f u n c t i o n from ft t o R f o r whi c h EUY i s d e f i n e d then t h e r e e x i s t s a e A o such t h a t Y(-) = X ( - , a ) . P a r t i i ) i s more g e n e r a l than the assum p t i o n d i s c u s s e d p r e v i o u s l y , t h a t i s , f o r e v e r y a between zero and one t h e r e e x i s t s an event C e 0 such t h a t y(C) = a . The a s s u m p t i o n t h a t t h e r e e x i s t s an a c t i o n a which has a u n i f o r m d i s t r i b u t i o n on [ u (r ) ,U (r°)]| i m p l i e s t h a t i f a i s any number between ( 0 , 1 ) and C = { c o:UX ( t o,a) < y] where y = a[u(r°) - U ( r Q j ] + U ( r Q ) t h e n P(C) = a . However, the c o n v e r s e does not n e c e s s a r i l y h o l d . That i s , i f we assume f o r any a e ( 0 , 1 ) t h e r e e x i s t s a C e 0 such t h a t y (C) = a , t h e n t h i s does n o t i m p l y t h a t t h e r e e x i s t s a reward f u n c t i o n w i t h u n i f o r m d i s t r i b u t i o n . P a r t i i ) o f t h e a ssumption a l s o i m p l i e s t h a t the rewards a r e a t l e a s t c o u n t a b l e , n o t f i n i t e . 4.0 Sequences o f reward f u n c t i o n s I n s e c t i o n 3 v/e showed t h a t the P-measure i s monotone which i s , o f c o u r s e , a l s o one o f t h e p r o p e r t i e s o f a measure. I n t h i s s e c t i o n we s h a l l make t h e a d d i t i o n a l assumption t h a t P-measure i s c o n t i n u o u s from below, t h a t i s , i f B^,!^,... i s an i n c r e a s i n g sequence o f s e t s i n 8, t h e n l i m P ( B . ) = P ( l i m B . ) . 1 x T h i s h o l d s as a p r o p e r t y o f a measure a l s o . (See, f o r example Halmos, 1950, Theorem E, pp 38.) A c t u a l l y , we s h a l l make t h e assumption s l i g h t l y s t r o n g e r i n terms o f reward f u n c t i o n s r a t h e r t h a n i n terms o f t h e mea-s u r e . F o r most p r a c t i c a l purposes t h i s axiom i s not needed s i n c e i t i s u s u a l l y s u f f i c i e n t t o c o n s i d e r o n l y a f i n i t e number o f s t a t e s o f n a t u r e , o r a t most f i n i t e l y many s e t s i n 8. I t i s needed, however, f o r a c o n s i s t e n t m a t h e m a t i c a l development. I n s e c t i o n 4.1 v/e s h a l l s t a t e the axiom, and i n s e c t i o n 4.2 we s h a l l c o n s i d e r some o f i t s i m p l i c a t i o n s . 4.1 Statement o f t h e axiom B e f o r e s t a t i n g the axiom, we s h a l l need an a d d i t i o n a l d e f i n i t i o n o f what i s meant by "convergence o f a sequence o f reward f u n c t i o n s " o r "convergence o f a sequence o f a c t i o n s " . The normal way would be t o d e f i n e t o p o l o g i e s on A and R, and d e s c r i b e convergence i n terms o f t h e s e t o p o l o g i e s . F o r our purposes i t i s s u f f i c i e n t t o use t h e f o l l o w i n g : 153 D e f i n i t i o n . I f a i s a sequence o f a c t i o n s i n A , t h e n n A ' we s h a l l say a n converges t o a (or a sequence o f reward f u n c t i o n s X ( * , a n ) converges t o a reward f u n c t i o n X ( - , a ) ) i f limU(X ( u),a )) = U(X(a>,a)) f o r a l l w e f t , h n I n m a t h e m a t i c a l terms we have i n d u c e d a t o p o l o g y on A by the f u n c t i o n U (X(-,a)) from the n a t u r a l t o p o l o g y on t h e r e a l l i n e . I f a n converges t o a, we s h a l l denote t h i s by l i m a n = a o r a n -> a. S i m i l a r l y t h e convergence o f X(«,a n) t o X(-,a) i s denoted by l i m X ( - , a ) = X(-,a) o r X ( - , a ) -+X(-,a) n n L e t X ( * , a n ) be a sequence o f reward f u n c t i o n s , such t h a t UX(-,a ) converges t o a f u n c t i o n Y ( - ) . I f a e A f o r a l l - n then n n o ¥>(•). " * i s / a L ©-measurable f u n c t i o n . However, t h e r e may not e x i s t an a e A q such t h a t Y(-) = UX(«,a). F or example i f B e 0 , R i s the open i n t e r v a l (0,1) and U(x) = x th e n the sequence o f f u n c t i o n s d e f i n e d by X ( u f a ) n converges t o t h e f u n c t i o n 1-1/n a> e B 1/n ai e B 0 ) £ B U ) £ B . I t i s c l e a r t h a t t h e r e does not e x i s t an a e A such t h a t o Y(') = X ( - , a ) . T h i s w i l l l e a d t o some d i f f i c u l t i e s i n s e c t i o n 5. We s h a l l t h e r e f o r e make the f o l l o w i n g assumption: Assumption 5 . I f a n i s a sequence i n A q such t h a t a -> a th e n a e A . n o Assumption 5 t h e r e f o r e s t a t e s , t h a t i n a c e r t a i n t o p o l o g y • t h a t A q i s a c l o s e d s e t . .• v E q u i v a l e n t assumption would be t o assume R t o be a c l o s e d s e t , and s t a t e the assumption i n terms o f a reward f u n c t i o n . I f a sequence o f a c t i o n s a n t h e r e f o r e converges t o a, t h e reward X ( o o,a n) becomes c l o s e r t o t h e reward X(co,a) f o r a l l to e ft, as n tends t o i n f i n i t y and t h e two reward f u n c t i o n s become i n d i s t i n g u i s h a b l e . I t seems i n t u i t i v e l y o b v i o u s then t h a t f ( a ) s h o u l d become c l o s e r t o f ( a ) as n i n c r e a s e s , n Axiom V f o r m a l i z e s t h i s i n t u i t i o n . Axiom V. I f X ( - , a n ) n=l,2,... i s a sequence o f reward A. A f u n c t i o n s such t h a t b > a" ., ^  a-; f o r n=l,2,... and n+1 . n limX ( c o,a ) = X (o i,b) f o r a l l to e ft, t h e n f o r any a c t i o n . A A c f o r wh i c h b >c - t h e r e e x i s t s an N such t h a t a $• c f o r n " a l l n > N. As an example o f t h e i m p l i c a t i o n o f Axiom V c o n s i d e r t h e f o l l o w i n g example: L e t a be an a r b i t r a r y number from the i n t e r v a l [ 0 , 1 ] . We s h a l l say t h a t t h e e v e n t B o c c u r r e d i f n the number chosen b e l o n g s t o t h e i n t e r v a l [0,3/4-1/n) where n > 2. S i m i l a r l y we say t h e event B o c c u r r e d i f t h e number chosen b e l o n g s t o t h e i n t e r v a l [ 0 , 3 / 4 ) . C o n s i d e r t h e f o l l o w i n g a l t e r n a t i v e s : a ) r e c e i v i n g $100 i f B o c c u r s n • n r e c e i v i n g $0 o t h e r w i s e b) r e c e i v i n g $100 i f B o c c u r s r e c e i v i n g $0 o t h e r w i s e As n i n c r e a s e s , t h e s e t B becomes c l o s e r t o t h e s e t B and, n t h e r e f o r e l i m X ( • , a n ) = X ( • , b ) . T h e r e f o r e i f n i s v e r y l a r g e , Axiom V assumes t h a t we would be " n e a r l y " i n d i f f e r e n t between a and b. That i s , l e t •* n an a d d i t i o n a l a l t e r n a t i v e c be d e f i n e d by: c) r e c e i v i n g $100 i f C o c c u r s r e c e i v i n g $0 o t h e r w i s e . A I f b > c i n the s t r i c t sense ( t h a t i s , P(B) > P(C)) and i f a i s " n e a r l y " i n d i f f e r e n t t o b. Axiom V a s s e r t s t h a t a i s n 1 ' n A a l s o p r e f e r r e d t o c, i . e . , a n c. 4.2 I m p l i c a t i o n s o f the axiom The i m p l i c a t i o n s o f t h i s axiom t o g e t h e r w i t h p r e v i o u s assumptions a r e v e r y s t r o n g from a m a t h e m a t i c a l v i e w p o i n t . We s h a l l prove two o f t h e s e i m p l i c a t i o n s h e r e . The f i r s t i m p l i e s t h a t t h e P-measure i s c o n t i n u o u s , t h a t i s , l i m P ( B ^ ) = P ( l i m B ^ ) f o r any i n c r e a s i n g sequence o f s e t s i n 0 . A v A The second i m p l i c a t i o n i s t h a t i f a -> a where a > a , > a n n+1 n f o r a l l n, then f ( a n ) •> f ( a ) . T h i s i m p l i e s t h e r e f o r e t h a t i f a e- A and a e A , t h a t n o o 156 J UX ( c o,a)dy = J limUX ( c o , a n) dy = limj'ux ( c o , a n) dy . We s h a l l a l s o show t h a t i n Savage's approach t h i s axiom i s n o t s a t i s f i e d . S e v e r a l i m p l i c a t i o n s a r e now v e r i f i e d . Lemma 4.2.1. I f B n i s a sequence " o f i n c r e a s i n g s e t s i n B such t h a t l i m B = B, then l i m P ( B ) = P ( B ) . n n P r o o f . F o r any r , s e R such t h a t U(r) > U ( s ) , c o n s i d e r the reward f u n c t i o n s X(co,a ) = n (0 £ B n and X(co,b) = < co £ B n A . A Then b. -> ... r co £ B s co £ B A f A > a 3 > a 2 ^ a i hy f o r each n=l,2,... Axiom i t , and by Axiom IV(b) t h i s i m p l i e s t h a t P ( B 1 ) «: P ( B 2 ) < P ( B 3 ) . . . < P(B) That i s , P ( B n ) i s an i n c r e a s i n g sequence w i t h an upper bound. T h i s i m p l i e s t h a t l i m P ( B ^ ) = a e x i s t s . I f a < P(B) t h e n by Assumption 4, t h e r e e x i s t s a C £ 0 such t h a t a < P(C) < P ( B ) , and a l s o t h e r e e x i s t s an a c t i o n c £ A such t h a t o X(co,c) = < CO' '£ C CO £ C A A b i n c e . b > c and c > a f o r a l l n by Axiom IV and a -> b by n r, J n d e f i n i t i o n w h i c h c o n t r a d i c t s Axiom V. Hence l i m P ( B i ) = P(B) 157 Lemma- 4.2.2 . . I f X(.v,a n) n = l f 2 , ... . i s a sequence o f A -'A -reward f u n c t i o n s such t h a t a > a , , >a f o r n=l,2,, n+1 n and a- ,-«*• ';.-a-.X£/\^ th e n f (a ) -> f (a) n^- v— ...' ' n P r o o f . Assume l i m f ( a n ) = a < f ( a ) t h e n by Assumption 3 t h e r e e x i s t s c e A d e f i n e d by the reward f u n c t i o n o X ( to,c) = r° to E B r to E B where f(a)+a U ( r ) 2 x o V ( B ) = U(r°)-U(r o) Hence £ < a ) + a - u.(r ) f ( c ) = U ( r ) + [U(r°) - U ( r )] — ^ — °-U(r°)-U(r). o f ( a ) + a A A, T h e r e f o r e , a >' ~c > l i m a- c o n t r a d i c t i n g Axiom V. n Savage approach does n o t s a t i s f y Axiom V. A l t h o u g h Savage d i d n o t e x p l i c i t l y argue a g a i n s t t h i s axiom, h i s axioms c o n t r a d i c t i t . To see t h i s , l e t us assume t h a t we s t a r t w i t h a g i v e n measure space ( f t , 0 , y ) where y i s an a d d i t i v e measure. U s i n g the f i r s t s i x axioms i n h i s approach, Savage prove d t h a t t h e r e e x i s t s a f u n c t i o n U from R t o t h e r e a l l i n e such t h a t f o r any s i m p l e f u n c t i o n (.see Appendix I I f o r d e f i n i t i o n ) 158 n £(a) = EUX(-,a) = £ UX B "(•,a ) y(B i) i = l i where UXfi (*,a) i s a c o n s t a n t f o r each i = l , . . . . n . S i n c e Savage d e f i n e s 9 as a l l s u b s e t s o f ft, t h e r e does n o t e x i s t , i n g e n e r a l , a a - a d d i t i v e s e t f u n c t i o n y on 0 . Savage does not r e s t r i c t ft o r 9 i n h i s approach, but r a t h e r assumes t h a t y i s o n l y f i n i t e l y a d d i t i v e . We s h a l l show t h a t i f Axiom V were a c c e p t e d t h e n the measure y must be a - a d d i t i v e and hence t h e g e n e r a l a s s u m p t i o n o f ft and 0 can n o t be made. I f i = l , . . . i s a sequence o f d i s j o i n t s e t s , we a r e r e q u i r e d t o have £ y(B.) = y(B) where UB. = B, i = l 1 1 S i n c e i n Savage's approach t h e r e always e x i s t rewards such t h a t U (r) = 1 and U(s) = 0, t h e f o l l o w i n g reward f u n c t i o n s may be c o n s i d e r e d : UX(to,a) = GO e B a) e B and UX ( GO , a ) = n n 1 u) e Y B . i = l 1 0 o t h e r w i s e Then, a a n f o r a l l n by Axiom I I , and l i m X ( • , a n ) = X(•,a) . 159 I t i s a l s o c l e a r t h a t X(.-,a) and X(.,a. n) n = l , 2 , ... b e l o n g t o A . o The e x p e c t e d v a l u e o f X(*,a) i s e q u a l t o y ( B ) , and t h e n n e x p e c t e d v a l u e o f X ( ' , a ) i s e q u a l t o p( £ B . ) = ] T y ( B . ) , i = l i = l by f i n i t e a d d i t i v i t y . I f Axiom V h o l d s , n «i y ( B ) = l i m y ( T U ( B.)) = £ y ( B . ) , i = l i = l 1 and hence y i s a - a d d i t i v e . T h e r e f o r e i f we assume t h a t a sequence o f reward f u n c -t i o n s s a t i s f i e s Axiom V, we must a l s o have a - a d d i t i v i t y o f the measure i f f i n i t e a d d i t i v i t y i s assumed. T h i s i m p l i e s , t h e r e f o r e , t h a t t h e s e t o f 'the s t a t e s o f n a t u r e , i . e . , t h e s e t ft, must be s u f f i c i e n t l y s m a l l so t h a t a l l s u b s e t s may be made measurable i f Savage a c c e p t s Axiom V. S i n c e Savage does n o t r e s t r i c t ft, o r A f o r t h a t m a t t e r , and assumes f i n i t e a d d i t i v i t y , we can o n l y c o n c l u d e t h a t Savage r e j e c t s Axiom V. 5.0 S t o c h a s t i c dominance axiom I n p r e v i o u s s e c t i o n s o f P a r t I I we have d i s c u s s e d t h e p r o p e r t i e s o f t h e e v a l u a t i o n f u n c t i o n f ( ' ) f o r a c t i o n s which r e s u l t i n o n l y two rewards. So f a r t h e axioms g i v e o n l y a p a r t i a l o r d e r i n g on t h e s e t o f a c t i o n s . I n t h i s s e c t i o n we s h a l l s p e c i f y one a d d i t i o n a l axiom w h i c h w i l l a l l o w f ( • ) t o be s p e c i f i e d c o m p l e t e l y . I n s e c t i o n 5.1 we s h a l l s t a t e t h e axiom and i n s e c t i o n 5.2 we s h a l l c o n s i d e r some o f i t s i m p l i c a t i o n s . 5.1 Statement o f t h e axiom i Axiom VI can be c o n s i d e r e d as a g e n e r a l i z a t i o n o f Axiom IVb and w i l l , i n f a c t , r e p l a c e i t . R e c a l l t h a t Axiom IVb s t a t e s t h a t i f t h e a c t i o n s a,b e A a r e d e f i n e d by t h e reward f u n c t i o n s X (to,a) = < r s to e B to e B X(to,b) = < r s to e D to e D where U(r) > U ( s ) , t h e n a £ b i f and o n l y i f P(B) ^ P ( D ) . I f we now c o n s i d e r the case where a,b e A are d e f i n e d by r r B, X ( to,a) = < s X (to,b) = < s B. f o r B . n B. = 0, D.n D. = 0 i ^ j , t h e n i n some cases the p r e v i o u s axioms would be s u f f i c i e n t t o s p e c i f y t h e p r e f e r e n c e between a and b. For example, i f D^C B l f D 1 O  D2 <~ B l ^  B 2 and U(r) > U(.s) > U ( t ) , t h e n t h e dominance axiom (Axiom I I ) A would s p e c i f y t h e p r e f e r e n c e a > b. Axiom IVc g e n e r a l i z e d Axiom I I by n o t r e q u i r i n g t h a t D C B, but o n l y t h a t P(B) > P ( D ) . I n t h e same way we s h a l l g e n e r a l i z e Axiom V I , A u s i n g Axiom l i b w h i c h , i n t h i s c a s e , r e q u i r e s t h a t a >/ b i f and o n l y i f P ( B 1 ) > P(D X) and P ( B 1 U B 2) £ P ( D 1 0 D 2) . A n a l o g o u s l y , i n terms o f two a r b i t r a r y a c t i o n s a,b e A, t h i s A would r e q u i r e t h a t a > b i f and o n l y i f P(X ( co,a) > r) P(X(w,b) > r) f o r a l l r e R. However, i t i s more c o n v e n i e n t t o c o n s i d e r P(UX (co,a) > a ) r a t h e r t h a n P(X ( t o,a) > r) , s i n c e the f u n c t i o n F(a,a) = l-P(UX(o>,a) > a) d e f i n e s a p r o b a b i l i t y d i s t r i b u t i o n on the r e a l l i n e , as we s h a l l show i n lemma 5.2.1. Hence, A th e d e s i r e d p r o p e r t y can be r e s t a t e d more s i m p l y as a > b i f and o n l y i f F ( a , a ) ^ F ( a , b ) f o r a l l r e a l numbers a . T h i s c o n c e p t i s not new i n t h e t h e o r y o f e x p e c t e d u t i l i t y t h e o r y . However i n most cases R i s c o n s i d e r e d t o be t h e r e a l l i n e , and i n t h i s case th e d i s t r i b u t i o n f u n c t i o n F(a,a) o f t h e reward f u n c t i o n X(-,a) i s d e f i n e d i n t h e normal way, t h a t i s F ( a , a ) = P(X( • ,a) ^ a ) . I n t h i s case th e f o l l o w i n g theorem can be s t a t e d : Theorem. (L. T e s f a t s i o n , 1974). A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r EUX(-,a) > EUX(«,b), where U i s any n o n d e c r e a s i n g bounded u t i l i t y f u n c t i o n i s t h a t F ( a , a ) < F ( a , b ) f o r e v e r y r e a l number a . I n terms o f the o r d e r i n g on A t h i s s t a t e s t h e r e f o r e t h a t 3 o A o a -> b i f . F ( a , a ) < F ( a , b ) . Axiom V I extends t h i s p r o p e r t y t o a l l a c t i o n s i n A. Axiom V I . I f P(UX ( to,a) > a ) > P(UX ( to,b) > a ) f o r a l l A r e a l numbers a t h e n a ^ b and i f P(UX(u,a) > a ) > A P(UX (co,b) > a ) f o r some a t h e n a > b. 5.2 I m p l i c a t i o n s o f A x i o n V I In t h i s s e c t i o n we s h a l l prove two lemmas w h i c h g i v e t h e f o u n d a t i o n f o r s p e c i f y i n g f ( • ) c o m p l e t e l y . The most i m p o r t a n t i m p l i c a t i o n o f the assumptions so f a r i s t h a t t h e P-measure i n d u c e s a d i s t r i b u t i o n f u n c t i o n on t h e r e a l l i n e . Lemma 5.2.1. I f F ( a , a ) = 1-P (co:UX (co,a) > a ) then F ( a , a ) i s a d i s t r i b u t i o n f u n c t i o n . P r o o f . We must show t h a t F ( - , a ) s a t i s f i e s i ) N o n - n e g a t i v i t y , i . e . , F (-,a) > 0. i i ) C o n t i n u i t y from above, i . e . , i f i s a d e c r e a s i n g sequence o f r e a l numbers c o n v e r g i n g t o a Q then F ( a ^ , a ) converges t o F ( a , a ) . i i i ) I f ctj i s a d e c r e a s i n g sequence o f r e a l numbers, otj -»• - 0 0 then F(otj,a) + 0 . i v ) I f oij i s an i n c r e a s i n g sequence o f r e a l numbers oij -»- + 0 0 t h e n F(ou,a) -»• 1. v) I f a-j^ ^ a 2 / t h e n F (a-^,a) ^ F ( a 2 , a ) . S i n c e t h e P-measure i s always bounded between 0 and 1 (by Axiom IVa and Axiom I I ) t h i s i m p l i e s t h a t F ( a , a ) i s always g r e a t e r than o r e q u a l t o z e r o . To show i i ) , l e t be a d e c r e a s i n g sequence o f r e a l numbers c o n v e r g i n g t o a . D e f i n e B n = (to :U ( X ( t o , a) ) > a n ) t h e n B n i s an i n c r e a s i n g sequence o f s e t s i n 6 , such t h a t l i n u 3 n = B = {to:U (X ( t o ,a) > oc }. By-lemma 4 . 2 . 1 s e c t i o n 4 . 2 ( i . e . , by Axiom V) t h e r e f o r e l i m P ( B ) = P(B) n o r l i m F ( a ,a) = 1 - l i m P (to : U ( X ( t o , a) ) > a ) 1-P ( t o:U(X ( t o,a) ) > a ) = F (a ,a) . o P r o p e r t i e s ( i i i ) and ( i v ) now f o l l o w d i r e c t l y s i n c e U ( r ) <: U(X ( t o,a)) < U(r°); t h a t i s , i f a < U ( r Q ) t h e n P (to:U (X ( t o , a) ) > a ) = 1 and i f a > U(r°), P (to :U (X (to,a) ) > a ) = 0 , (v) f o l l o w s d i r e c t l y from lemma 3 . 2 . 1 . A s t a n d a r d r e s u l t i n p r o b a b i l i t y t h e o r y i s t h a t ^ a d F ( a , a ) = EUX(*,a) (see, f o r example, Theorem 1 . 6 . 1 2 i n Ash ( 1 9 7 2 ) ) . T h i s i m p l i e s , t h e r e f o r e , t h a t the o r d e r i n g on A Q must s a t i s f y t he n u m e r i c a l o r d e r i n g o f ^ a d F ( a , a ) f o r a l l a e A Q. We s h a l l s t a t e t h i s r e s u l t as a lemma. Ae Lemma 5 . 2 . 2 . L e t a , b e A . I f a > b, then o y"adF(a,a) >yadF(a,b) Our o b j e c t i v e now i s t o show t h a t the o r d e r i n g on A must a l s o s a t i s f y a > b^ i m p l i e s y"adF(a,a) >y"adF(a,b). 164 We s h a l l do t h i s i n two s t e p s . F i r s t , we s h a l l show t h a t i f F ( a,b) i s any d i s t r i b u t i o n w i t h b £ A , t h e r e e x i s t s an a E A q such t h a t F ( a,b) = F ( a , a ) . T h i s i s s t a t e d i n Theorem 5.2.3. Theorem 5.2.3. L e t F ( a , a ) be the d i s t r i b u t i o n f u n c t i o n i n d u c e d by X(-,a)- f o r any a £ A - A q . Then t h e r e e x i s t s a b £ A q such t h a t F ( a , a ) = F ( a,b) f o r a l l r e a l numbers a . P r o o f . To prove t h i s we s h a l l c o n s t r u c t a sequence o f measurable f u n c t i o n s w h i c h converge t o a reward f u n c t i o n w i t h the r e q u i r e d d i s t r i b u t i o n f u n c t i o n . L e t S = { Y / • • W Y ) be a s t r i c t l y i n c r e a s i n g s e t o f n ' o 'm 2 ^ numbers > i n the i n t e r v a l [U(r Q),U(r°)] such t h a t Y = U ( r ) , Y = U(r°) and 'o o 'm F ( Y ~ . . n / a ) - F( Y-#a) < ^ f o r i = l , . . . , m - l l + l I n where F ( Y , a) = l i m F(a) . a < Y a + Y S i n c e F i s r i g h t - c o n t i n u o u s such a '.sequence must always e x i s t . L e t T denote the s e t o f a l l d i s c o n t i n u i t y p o i n t s o f F(-,a) such t h a t i f y e T t h e n J n F(y,a) - F (y~,a) > K I t i s o b v i o u s t h e n t h a t T C S . n ~^ n The n e x t s t e p i n the p r o o f i s t o d e f i n e a measurable reward f u n c t i o n w i t h a d i s t r i b u t i o n f u n c t i o n which approximates F ( d , a ) . To do so, we must r e f e r back t o Assumption 4 . T h i s assumption s p e c i f i e s the e x i s t e n c e o f a reward f u n c t i o n U X ( - , c ) w i t h a u n i f o r m d i s t r i b u t i o n . D e f i n e the s e t s B Q = ( U J : U X ( W , C ) « F ( Y - L ' / H ) [ u(r°) - U ( r Q ) ] + U ( r Q ) } . B I = {(ii:F(yi,a) U(r°) - U ( r Q ) + U ( r Q ) < U X (u>,c) <c < F ( Y ~ i + 1 , a ) [u(r°) - U(r o)"] + U ( r Q ) } f o r a l l y. e S - T , 1=1,2,...,m-l. l n n' . C ± = { o o : F ( Y i , a ) [ u ( r ° ) - U ( r Q ) 3 + U ( r Q ) < U X ( t o , c ) < < F ( Y ~ i , a ) [ _ U ( r ° ) - U ( r Q ) } + U ( r Q ) } f o r a l l Y • £ T , i=0,1,2,...,m. 11 n f Then t h e s e t B Q , B ^ , . . . and C^,... are a l l p a i r w i s e d i s j o i n t , and i f y^ e S n and i ^ 1 th e n y ( B I ) = F(y~ira) - F ( Y i _ r ) a and a l s o i f y. e T then l n y (C i) = F ( Y ± ) - F ( Y ~ I ) . Assumption 3 a l s o i m p l i e s t h a t t h e r e e x i s t s U K e ft such t h a t UX ( c o i,c) e [yilYi+l/n)f)[vi,yi+^\ f o r i= 0 ,...,m-1 and a l s o to e f t such t h a t m UX(u> .c) e ( y - l / n , Y j C\ (Y I - T 1 m' 1 m 'ml 1 'm-l 'mj De f i n e a reward f u n c t i o n UX ( c o,b n) = < UX(o).,a) to e B i i = 0 , l , . . . , m -y± to e C i d T = l , 2 , ' . V . ,m' Then b e A by Assumption 4 i s i n c e UX(«,b ) i s a measurable n o n f u n c t i o n . Then f o r any a e j^ U (r ) ,U ( r ° ) ] we have: Case 1. a e T n P(UX(« ,b n) < a) = F(a,a) Case 2 . a / T n # t h e n t h e r e e x i s t s an i n t e r v a l [ Y ^ + ^ ' Y ^ ) o r ( Y M _ ] _ / Y M ] t o which a b e l o n g s , say [ Y ^ + 1 / Y ^ ) « Then F ( Y i , a ) <: P(UX(-,b n) <= a ) < F ( ^ i + i ' a ) and t h e r e f o r e |P(UX(',b ) < a) - F ( a , a ) | < F ^ Y i + 1 » a ) - F(Y i»a) 4 ^ . 167 S i n c e Y n i s s t r i c t l y i n c r e a s i n g UX(to,fc>n) converges as n i n c r e a s e s and by Assumption 4 t h e r e e x i s t s V b e A Q such t h a t b n converges t o b. T h i s completes t h e p r o o f . T h i s theorem t h e r e f o r e completes the b a s i c i d e a b e h i n d P a r t I I o f the t h e s i s . Our arguments are as f o l l o w s : Suppose we a r e i n t e r e s t e d i n comparing a c t i o n s a^ and b^ where a^,b^ e A-A . By theorem 5.2.3 t h e r e e x i s t s an a.b e A such t h a t o o F{a,a-^) and F(a,b^) have t h e same d i s t r i b u t i o n as F(a,a) and A A F(a,b) . Axiom VI then s p e c i f i e s t h a t a^ = a and b^ = b. T h e r e f o r e the p r e f e r e n c e between a-^  and b-^  can be d e t e r m i n e d by the p r e f e r e n c e between a and b, w h i c h can e a s i l y be done s i n c e t h e s e s a t i s f y t h e e x p e c t e d u t i l i t y c r i t e r i o n . I n s e c t i o n 6 we s h a l l f o r m a l i z e t h i s argument. 6.0 Summary o f assumptions and axioms and the b a s i c r e s u l t s I n t h i s s e c t i o n we s h a l l summarize the assumptions and the axioms so f a r made and we s h a l l a l s o show i n theorem 6.0.1 t h a t t h e y a r e s u f f i c i e n t t o s p e c i f y the e v a l u a t i o n f u n c t i o n f ( - ) f o r a l l a e A. The o n l y d i f f e r e n c e between the axioms here and i n p r e v i o u s s e c t i o n s i s i n Axiom IV, p a r t b, which has been r e p l a c e d by Axiom V I . The f o l l o w i n g f i v e assumptions were made: Assumption 1. For each r e R, t h e r e e x i s t s an a e A Q such t h a t X(co,a) = r f o r a l l to e ft, Assumption 2. i ) There e x i s t r Q and r° i n R, such t h a t f o r each r £ R o R R r > r > r ' o i i ) The f u n c t i o n U X ( - , a ) , a e A q i s a B o r e l measurable f u n c t i o n w i t h r e s p e c t t o 0 . The f u n c t i o n UX(«,a), a e A i s a B o r e l measurable f u n c t i o n w i t h r e s p e c t t o 3 -Assumption 3. For any D e (3 and f o r any r , s e R t h e r e e x i s t s b e A such t h a t r to e D X(to,b) = s to £ D Assumption 4. i ) I f • Y ( * ) i s any f u n c t i o n from ft t o R such t h a t .UY(•) , i s B o r e l measurable w i t h r e s p e c t t o 0 , then there, e x i s t s a £ A such t h a t X(-,a) = *Y (•) . t h a t i i ) There e x i s t s an a c t i o n a e A Q such Y " U ( r J y ( U X ( c o,a) < Y ) = -U(r°)-U(r Q) f o r each y £^U(r Q), U(r°)J. Assumption 5 . I f a i s a sequence i n A such t h a t n o a -> a then a e A . n o The f o l l o w i n g s i x axioms were made: Axiom I . There e x i s t s a r e a l - v a l u e d f u n c t i o n f ( . ) on A such t h a t A a > b i f and o n l y i f f ( a ) > f ( b ) Axiom I I . i ) I f f o r any r e R t h e r e e x i s t s a,b e A such t h a t X(cu,a) = r and X(co,b) = r f o r a l l co e ft then f (a) = f (b) . A i i ) I f X(-,a) dominates X(-,b) t h e n a > b. Axiom I I I . I f b e A Q t h e n f ( b ) = EUX(-,b). Axiom IV. There e x i s t s a s e t f u n c t i o n P d e f i n e d on 0 such t h a t i f B e 9 t h e n P(B) = y ( B ) . Axiom V. I f X ( * , a n ) n = l , 2 , . . . i s a sequence o f reward f u n c t i o n s such t h a t b f ^ ^ ^ . a n f o r n = l , 2 , . . . and limX ( o o,a ) = X ( w,b) f o r a l l w e f t then f o r any a c t i o n r . n J A A c f o r which c < b t h e r e e x i s t s an N such t h a t c < a f o r n a l l n > N. Axiom V I . I f P(UX ( t o,a) > a ) > P(UX ( w,b) > a ) f o r a l l A r e a l numbers a then a > b, i f i n a d d i t i o n P(UX(*,a) > a ) > P(UX(*,b) > a ) f o r some a t h e n a > b. Theorem 6.0.1 extends lemma 5.2.2 i n w h i c h we showed t h a t f ( a ) = ^~adF(a,a) f o r a l l a e A Q where F(x,a) = 1 - P { w:UX ( w,a) > x } . Here i t w i l l be shown t h a t t h i s i s the case f o r a l l a e A. Theorem 6.0.1. I f the Axioms I-VI and the Assumptions 1-5 s t a t e d above h o l d , then f ( a ) =JadF(a,a) f o r a l l a e A. P r o o f . I t i s s u f f i c i e n t t o show t h a t t h i s h o l d s f o r a e A-A^. I f a e A-A then by Theorem 5.2.3 t h e r e e x i s t s o o b e A such t h a t F(a,a) = F(a,b) f o r a l l r e a l numbers a. o T h i s i m p l i e s t h a t P ( t o:U(X ( t o,a) > a) > P ( w : U ( X ( w , b ) ) > a) A f o r a l l r e a l numbers a, so by Axiom VI a > b. S i m i l a r l y , P(u>:U(X(u>,b) > a ) > P (w :U (X ( w , a) ) > a ) , A A f o r a l l r e a l numbers a, hence b > a. T h e r e f o r e a = b o r f ( a ) = f ( b ) . S i n c e f ( b ) = / x d F ( x f b ) = / xdF(-,a) = f ( a ) the p r o o f i s completed. To summarize the r e s u l t s , we have shown t h a t f o r a g i v e n u t i l i t y f u n c t i o n U, and f o r any a c t i o n s a,b e A such t h a t One o f the p r o p e r t i e s o f e x p e c t e d u t i l i t y i s t h a t any a f f i n e t r a n s f o r m a t i o n o f t h e u t i l i t y f u n c t i o n would s a t i s f y the same o r d e r i n g on A^, t h a t i s , i t i s i m m a t e r i a l i f we use U(X) or aU(X)+y f o r any a > 0. I n the development here we have made use o f a s p e c i f i c u t i l i t y f u n c t i o n and f o r the t h e o r y t o be r e a s o n a b l y u s e f u l we must show t h a t f o r any a f f i n e u t i l i t y f u n c t i o n the same o r d e r i n g w i l l be o b t a i n e d . where F ( z , i ) = l-P{cocU (X (• , i ) ) > z i=a,b i m p l i e s t h a t y z d F ' ( z , a ) > J zdF' (z,b) where F ' ( z , i ) = 1-P (co : aU (X (• , i ) ) + Y > z) f o r a > 0 and y any r e a l number. S i n c e F ' ( z , i ) = F ( ( z - y ) / a , i ) i=a,b the f o l l o w i n g i d e n t i t y h o l d s : A a > b then T h e r e f o r e we must show t h a t We have t h a t i f z d F ( z , b ) , t h e n a / z d F ( z , a ) + y > a /zdF(a,b) + y 172 o r ^ z d F ' (z,a) > J~ zdF 1 (z,a) f o r any a > 0 and any y . T h e r e f o r e t h e same t r a n s f o r m a t i o n o f U would p r e s e r v e t h e o r d e r i n g on A. 6.1 R e l a t i o n o f t h e P-measure approach t o a l t e r n a t i v e approaches I n the i n t r o d u c t i o n we s p e c i f i e d s e v e r a l approaches t o the p a r t i a l r i s k problem. We s h a l l c o n s i d e r some o f t h e s e i n r e l a t i o n t o the approach d e v e l o p e d h e r e . The P-measure as a p r o b a b i l i t y w i l l be c o n s i d e r e d i n s e c t i o n 8.0. I n two o f t h e approaches, a s e t IT o f p o s s i b l e p r o b a b i l i t y measures on 3 must be s p e c i f i e d . Hence e x i s t e n c e o f the p r o -b a b i l i t y measure i s t a k e n f o r g r a n t e d . T h i s i s not an o b v i o u s a s s u m p t i o n f o r a t h e o r e t i c a l development, but f o r p r a c t i c a l problems t h i s i s not a s e r i o u s drawback. One o f the c r i t e r i a i s then s p e c i f i e d by c h o o s i n g the a l t e r n a t i v e a e A f o r w h i c h minEU(X(•,a)) i s maximum. T h i s n approach t h e r e f o r e i g n o r e s t h e p o s s i b i l i t y t h a t one o f the d i s t r i b u t i o n s i n H may be more l i k e l y t han a n o t h e r . For example, i n E l l s b e r g ' s paradox, we may be t o l d t h a t t h e r e i s more th a n a .50 chance t h a t t h e r e are more b l a c k than y e l l o w b a l l s . T h i s type o f i n f o r m a t i o n can n o t be used i n t h i s a pproach. I n o r d e r t o r e l a t e t h i s t o E l l s b e r g ' s paradox I I a g a i n , c o n s i d e r the f o l l o w i n g a l t e r n a t i v e s : a) r e c e i v i n g $100 i f a b l a c k r e c e i v i n g $0 o t h e r w i s e Then, s i n c e t h e r e may be ze r o b a l l i s drawn b l a c k b a l l s minEU(X(- ,a) = U(0) . n T h i s i s e q u i v a l e n t t o s e t t i n g the P-measure e q u a l t o y * { b l a c k b a l l i s drawn}, the i n n e r measure o f t h e event i n q u e s t i o n . I n the more g e n e r a l c a s e , i f U ( r ^ ) > U ( r 2 ) > ... >' U ( and a l t e r n a t i v e b i s d e f i n e d by b) r e c e i v i n g r ^ i f o c c u r s r e c e i v i n g r 2 i f B 2 o c c u r s • • • • • • r e c e i v i n g r i f B o c c u r s ^ n n where B i O B^ = 0, i ^ j . Then the P-measures would be d e f i n e d as P ( B 1 ) = U.*(B 1) P ( B 2 ) = y * ( B 2 ) 'n-1 P ( B n ) = 1 - 2 P(B. ) 1=1 Hence the approach which uses minEU (X (• ,b)) as a d e c i s i o n c r i t e r i o n i s a s p e c i a l case o f t h e P-measure. However, i t seems u n l i k e l y t h a t anyone would have such an extreme P-measure. 174 In E l l s b e r g ' s approach we must d e t e r m i n e the "most l i k e l y " l i n e a r c o m b i n a t i o n s o f d i s t r i b u t i o n s i n II. T h i s c o m b i n a t i o n i s denoted by P and i t i s n o t o b v i o u s how i t can be de t e r m i n e d . L e t us assume t h a t t h i s can be d e t e r m i n e d and then the e v a l u -a t i o n f u n c t i o n i s s p e c i f i e d by where p i s an " a m b i g u i t y f a c t o r " . I t i s o b v i o u s t h a t p must be a f u n c t i o n o f t h e " a m b i g u i t y " o f r e c e i v i n g c e r t a i n r e wards, r a t h e r t h a n a f u n c t i o n o f a m b i g u i t y o f the p r o b a b i l i t i e s o f c e r t a i n e v e n t s . The " a m b i g u i t y f a c t o r " t h u s becomes a f u n c t i o n on t h e a c t i o n s , i . e . , each a c t i o n may have a d i f f e r e n t " a m b i g u i t y f a c t o r " s p e c i f i e d by t h e d e c i s i o n maker. T h e r e f o r e t h i s method i s no d i f f e r e n t from t h e Hurwicz a - c r i t e r i o n , where a i s the " a m b i g u i t y f a c t o r " , which s i m p l y d e f i n e s f o r each a c t i o n where a depends on the a c t i o n . A l t h o u g h t h i s e v a l u a t i o n f u n c -t i o n can c e r t a i n l y be used t o s p e c i f y an o r d e r i n g on A g i v e n Axiom I and Assumption 1, i t g i v e s l i t t l e u n d e r s t a n d i n g o f how t o make d e c i s i o n s . In o r d e r f o r E l l s b e r g ' s c r i t e r i o n t o be u s e f u l , some method has t o be found t o de t e r m i n e the " a m b i g u i t y f a c t o r " p , i n d e p e n d e n t l y o f the a c t i o n . Becker and Brownson (1964) d i d t h i s f o r a v a r i a t i o n on the E l l s b e r g paradox I I . They con-f ( a ) = p/UX(-,a)dP + (1 - p )minEUX(•,a) J n f (a) = aq.(,r ) + (l-a)U(r°) 175 s i d e r e d problems i n the form o f E l l s b e r g ' s paradox I I w i t h the a d d i t i o n a l i n f o r m a t i o n t h a t t h e r e are a t l e a s t x y e l l o w b a l l s and a t l e a s t y b l a c k b a l l s . Hence the o u t e r and i n n e r measure o f t h e e vent {a y e l l o w b a l l drawn}, {a b l a c k b a l l * drawn} were v a r i e d . T h e r e f o r e i f y (D) > y A ( D ) t h e y suggested t h a t P ( D ) = y * ( D ) - ^ * ( D ) and the " a m b i g u i t y f a c t o r " i s a f u n c t i o n o f t h e magnitude o f * the d i f f e r e n c e y (D) - y ^ ( D ) . I t i s n o t o b v i o u s how t h i s r e s u l t can be g e n e r a l i z e d . F o r example, assume t h a t t h e r e e x i s t s an u r n w i t h the f o l l o w i n g c o n t e n t s : 90 b a l l s are a m i x t u r e o f y e l l o w , b l a c k , and orange b a l l s , 50 b a l l s a r e a m i x t u r e o f orange and green b a l l s . We a r e a l s o t o l d t h a t t h e r e a r e e x a c t l y 40 orange and y e l l o w b a l l s . There i s no o b v i o u s way t o determine the "most l i k e l y " number o f orange b a l l s (event (0)) and n e i t h e r i s the e s t i m a t e y * ( O ) - y * ( O ) P ( O ) = -an o b v i o u s c o n c l u s i o n . F e l l n e r ' s (1961) approach i s s i m i l a r t o the approach d e v e l o p e d here i n the sense t h a t t h e P-measure may not n e c e s s a r i l y be a d d i t i v e . Thus, the d i f f i c u l t y a r i s e s as t o how P-measure can be used t o d e f i n e an e x p e c t e d v a l u e . F e l l n e r 176 d i d n o t s p e c i f y any p r o p e r t i e s he would e x p e c t t h e P-measure t o have e x c e p t t h a t i t must be t r a n s f o r m a b l e i n t o a p r o b a b i l i t y measure. For example, i f P-^  and P 2 a r e the P-measures o f e v e n t s B and B, we form the c o r r e c t e d p r o b a b i l i t i e s P-L = P ] / ^ ! +  F2 ) a n d P 2 = P 2 / ( P l + P 2 ) * Hence P^ and P 2 a r e a d d i t i v e f o r the e v e n t s B and B. However l e t C e 0 such t h a t y (C) = P-^B), and c o n s i d e r the a l t e r -n a t i v e s : X(co,a) = ^$100 co e B j $100 co £ C X(co,b) = < $0 co e B I $0 co e C I f U($0) = 0 then U(100) y (C) = UdOOJP, (B) . Now i f we use the P^, we must m o d i f y t h e u t i l i t y f u n c t i o n so t h a t U(100)y (C) = u 2 ( 1 0 0 ) P 1 t h a t i s , U 2(100) = U ( 1 0 0 ) P 1 / P 1 -F e l l n e r t h e r e f o r e argues t h a t U 2(•) c o n t a i n s n o t o n l y the amount o f money we may r e c e i v e b u t a l s o o ur " d i s l i k e " o r " l i k e " f o r t he " a m b i g u i t y " o f the event B. In t h i s approach t h e r e f o r e we must f i r s t d e r i v e the P-measures such t h a t they can be t r a n s f o r m e d t o p r o b a b i l i t i e s and s e c o n d l y we must d e r i v e u t i l i t y f u n c t i o n f o r each a c t i o n . T h e r e f o r e r a t h e r than h a v i n g an " a m b i g u i t y f a c t o r " depending on the a c t i o n as i E l l s b e r g ' s c a s e , F e l l n e r s u g g e s t e d a d i f f e r e n t u t i l i t y f u n c t i o n f o r each a c t i o n . 178 7.0 D e r i v a t i o n o f P-measure I n s e c t i o n 6 we summarized our s e t o f assumptions and axioms f o r a t h e o r y o f d e c i s i o n making under p a r t i a l r i s k . A l t h o u g h t h o s e c o n d i t i o n s a r e s u f f i c i e n t f o r t h e "expected u t i l i t y " (Theorem 6.0.1) t o h o l d , some d i f f i c u l t i e s a r i s e when we attempt t o d e r i v e the P-measure f o r a r b i t r a r y s e t s i n 3 , u n l e s s a d d i t i o n a l assumptions a r e made. T h i s i s i n c o n t r a s t t o Savage's approach. F o r example, i f we d e r i v e the s u b j e c t i v e p r o b a b i l i t i e s u s i n g Axiom IV (or Savage Axiom S4) f o r a c l a s s o f p a i r w i s e d i s j o i n t s e t s B^,...,B n t h e n we know the s u b j e c t i v e p r o b a b i l i t y o f a l l s e t s which can be formed by t a k i n g u n i o n s , i n t e r s e c t i o n s o r complements o f the s e t s B,,...,B . However, s i n c e t h e P-measure i s not n e c e s s a r i l y I n •* a d d i t i v e , t h a t i s , P ( B ^ O B 2) may not e q u a l P ( B 1 ) + P ( B 2 ) , t h e s e p r o b a b i l i t i e s a r e g e n e r a l l y n o t known. Hence t o d e r i v e the P-measure f o r a l l p o s s i b l e e v e n t s , we must c o n s i d e r e v e r y p o s s i b l e c o m b i n a t i o n o f u n i o n , i n t e r s e c t i o n and complement. T h i s would be v e r y time consuming i f n o t i m p o s s i b l e when the number o f ev e n t s i s v e r y l a r g e . However, i f t h e r e e x i s t some ev e n t s f o r w h i c h t h e P-measure would have t h e s t a n d a r d p r o -p e r t i e s o f a measure, i . e . , a d d i t i v i t y , t h e d e r i v a t i o n would be s i m p l i f i e d s u b s t a n t i a l l y . I n s e c t i o n 7.1 we s h a l l suggest those s e t s f o r w h i c h i t seems most l i k e l y t h a t the P-measure i s a d d i t i v e . I n s e c t i o n 7.2 we s h a l l show t h a t t h e r e always e x i s t s a P-measure h a v i n g the p r o p e r t i e s we assumed. F i n a l l y , i n s e c t i o n 7.3 we s h a l l g i v e some p o s s i b l e P-measures when 3 c o n t a i n s o n l y f i n i t e l y many s e t s . 7.1 A d d i t i v i t y o f the P-measure B e f o r e s t a t i n g t h e axiom, we s h a l l c o n s i d e r some o f t h e d i f f e r e n c e s between d e c i s i o n making under p a r t i a l r i s k and r i s k . E l l s b e r g (1961) r e p o r t e d a v a r i e t y o f c h o i c e s among h y p o t h e t i c a l l o t t e r i e s , i m p l y i n g t h a t " a m b i g u i t y " a s s o c i a t e d w i t h the p r o b a b i l i t i e s o f some e v e n t s i n f l u e n c e s the c h o i c e s . By " a m b i g u i t y " he meant t h a t t h e p r o b a b i l i t i e s were not s p e c i f i e d p r e c i s e l y , d e f i n i n g i t as "...a q u a l i t y depending on the amount, t y p e , r e l i a b i l i t y and ' u n a n i m i t y ' o f i n f o r m a t i o n and g i v i n g r i s e t o one's degree o f ' c o n f i d e n c e ' i n an e s t i m a t e o f r e l a t i v e l i k e l i h o o d " . (p.657) Be c k e r and Brownson (1964) m o d i f i e d t h i s d e f i n i t i o n t o s t a t e " . . . a m b i g u i t y i s d e f i n e d by any d i s t r i b u t i o n o f p r o -b a b i l i t i e s o t h e r t h a n p o i n t e s t i m a t e s " . (p.64) I m p l i c i t i n b o t h d e f i n i t i o n s i s t h e e x i s t e n c e o f a "second o r d e r " p r o b a b i l i t y d i s t r i b u t i o n . However, i f i t were r e a s o n a b l e t o assume a g i v e n second o r d e r p r o b a b i l i t y d i s t r i -b u t i o n , then Marschak's Axiom M2 (see Appendix I) ought t o be used t o c a l c u l a t e the e x p e c t e d u t i l i t y . I n our development the e x i s t e n c e o f a p r o b a b i l i t y measure i s n o t n e c e s s a r y . N e i t h e r o f t h e s e d e f i n i t i o n s i s , t h e r e f o r e , a p p r o p r i a t e h e r e . Note, however, t h a t i n b o t h d e f i n i t i o n s we may speak o f the "a m b i g u i t y o f t h e event C" i f t h e p r o b a b i l i t y o f C i s not pre c i s e l y s p e c i f i e d o r d e r i v a b l e i n the sense t h a t the o u t e r measure o f C i s s t r i c t l y g r e a t e r t h a n the i n n e r measure, i . e . * y (C) > y * ( C ) . 180 A l s o , i t seems o b v i o u s t h a t some ev e n t s may have "more" a m b i g u i t y t h a n o t h e r s . F o r example, i f we know t h a t an u r n c o n t a i n s 100 b a l l s w h i c h a r e a m i x t u r e o f r e d and b l a c k , t h e n the e v e n t o f drawing a r e d b a l l has "more" a m b i g u i t y i n t h i s case than i f we knew t h a t t h e r e were between 49 and 51 r e d b a l l s i n the u r n . We can, t h e r e f o r e , a l s o speak o f t h e "degree o f a m b i g u i t y " . We s h a l l f o r m a l i z e b o t h o f t h e s e i d e a s i n the f o l l o w i n g d e f i n i t i o n . D e f i n i t i o n 7.1.1. The degree o f a m b i g u i t y o f an event C i s d e f i n e d t o be the d i f f e r e n c e between i t s o u t e r and i n n e r measures, i . e . , y*(C) - y * ( C ) . We speak, t h e r e f o r e , o n l y o f "the a m b i g u i t y o f a s e t C" i f i t s "degree o f a m b i g u i t y " i s s t r i c t l y g r e a t e r t h a n z e r o . T h i s d e f i n i t i o n has s e v e r a l i m p l i c a t i o n s c o n c e r n i n g the degree o f a m b i g u i t y o f t h e e v e n t s . We s h a l l summarize some o f t h e s e h e r e . We s h a l l s t a t e t h e s e i m p l i c a t i o n s as lemmas a l t h o u g h the p r o o f s a r e t r i v i a l and f o l l o w d i r e c t l y from the p r o p e r t i e s o f i n n e r and o u t e r measure. The f i r s t p r o p e r t y i s t h a t an e v e n t and i t s complement must have t h e same degree o f a m b i g u i t y . Lemma 7.1.1. F o r any C y*(C) - y*(C) = y*(C) - y*(C) . P r o o f . From Appendix I I we have 181 y CC) = 1—y* (C) f o r any event C. T h e r e f o r e y*(C) - y*(C) = [ l - y * ( C " ) ] - [ 1 - y * ( C ) ] = y*(C) - y * ( C ) . The o t h e r p r o p e r t i e s we s h a l l c o n s i d e r a r e concerned w i t h "the degree o f a m b i g u i t y " o f u n i o n o f e v e n t s . I f D i s any ev e n t f o r which the p r o b a b i l i t y i s known, and C i s any event d i s j o i n t w i t h D, th e n i t seems r e a s o n a b l e t h a t t h e event D 0 C has the same degree o f a m b i g u i t y as C. T h i s p r o p e r t y i s t h e n e x t lemma. Lemma 7.1.2. I f D e 0 , C e 6 and D A C = 0 t h e n y*(C) - y*(C) = y * ( D U O - y * (D U C) . P r o o f . By the p r o p e r t i e s o f i n n e r and o u t e r measure we have y * ( C U D) = y*(C) + y*(D) y * ( C U D ) = y * (C) + y * (D) . Hence y * ( D U O - y*(D O C) = i = y (D.) + y CO - y * CD) ^ y * (C) = y * ( C ) - y * (C) * s i n c e y (D) = y *(D) . The second a d d i t i v i t y p r o p e r t y i s concerned w i t h the o c c a s i o n when b o t h s e t s have a degree o f a m b i g u i t y . C o n s i d e r , f o r example, the u n i o n o f an e v e n t C and i t s complement. I n t h i s case t h e degree o f a m b i g u i t y must c a n c e l o u t s i n c e t h e e v e n t C \J C must have t h e degree o f a m b i g u i t y o f z e r o . I n g e n e r a l n o t h i n g can be s a i d o f the u n i o n o-f e v e n t s where each has a p o s i t i v e degree o f a m b i g u i t y . Lemma 7.1.3. I f C,D e 3 and t h e r e e x i s t two s e t s E,F E 0 such t h a t C C E, D C F and E r\ F = 0 t h e n y * ( C U D ) - y * ( C O D ) = ( y*(C) - y * ( O ) + ( y*(D) - y * (D) ) P r o o f . The two s e t s E and F g i v e t h e c o n d i t i o n f o r t h e degree o f a m b i g u i t y o f the two s e t s C and D t o be s e p a r a t e d and i n t h i s case the p r o o f f o l l o w s d i r e c t l y from the e q u a t i o n s y * ( C U D) = y*(C) + y*(D) and y * (C ( J D) = (C) + y * (D) . 183 Here we s h a l l assume t h a t the P-measure i s dependent on the degree o f a m b i g u i t y . We s h a l l i l l u s t r a t e what we mean by t h i s i n the f o l l o w i n g examples. F i r s t we c o n s i d e r some o f the a d d i t i v i t y p r o p e r t i e s u s i n g E l l s b e r g 1 s paradox I I , and a m o d i f i c a t i o n t h e r e o f . C o n s i d e r t h e a l t e r n a t i v e s : b) r e c e i v i n g $100 i s a y e l l o w b a l l i s drawn r e c e i v i n g $0 o t h e r w i s e c) r e c e i v i n g $100 i s a r e d o r a b l a c k b a l l i s drawn r e c e i v i n g $0 o t h e r w i s e . I f we assume the d e c i s i o n maker has i n d i c a t e d h i s b e l i e f s a r e such t h a t P(B) = P(Y) t h e n i t i s i m p l i e d t h a t he b e l i e v e s t h e b e s t e s t i m a t o r f o r t h e p r o b a b i l i t y o f drawing a b l a c k b a l l i s e q u a l t o 1/3. Due t o t h e degree o f a m b i g u i t y , however, he " d i s c o u n t s " t h a t p r o b a b i l i t y t o P(Y) (where P(Y) may a c t u a l l y be g r e a t e r than 1/3). The amount t h a t he d i s c o u n t s the event i s e q u a l t o 1/3 - P (B). T h e r e f o r e , s i n c e t h e event R U B o f a l t e r n a t i v e c must have t h e same degree o f a m b i g u i t y by lemma 7.1.2, he would " d i s c o u n t " the event by the same amount. That i s , P ( R U B ) = 2/3 - (1/3 - P(Y)) = 1/3 + P(Y) = 1/3 + P ( B ) . T h i s i m p l i e s t h e r e f o r e t h a t the P-measure i s a d d i t i v e f o r 184 the e v e n t R U B , T h i s would a l s o i m p l y t h a t i f P(R) > P ( Y ) , P(Y) = PCB) th e n P(.R \JB) 4 P & O B) w h i c h would e x p l a i n E l l s b e r g ' s Paradox I I . We s h a l l a l s o c o n s i d e r t h e case where n e i t h e r the s e t s C o r D b e l o n g t o 0 ; t h i s d e c i s i o n problem i s a v a r i a t i o n o f E l l s b e r g ' s paradox. M o d i f i e d E l l s b e r g ' s paradox I I . C o n s i d e r an u r n which c o n t a i n s 100 b a l l s , 50 o f wh i c h a r e a m i x t u r e o f r e d and y e l l o w b a l l s and 50 o f wh i c h a r e a m i x t u r e o f green and orange b a l l s . L e t us assume t h a t the e v e n t {R \J Y} = {a r e d o r y e l l o w b a l l drawn} has a p r o b a b i l i t y o f 1/2 o f o c c u r r i n g . S i m i l a r l y the p r o -b a b i l i t y o f t h e e v e n t { G O O } = {a green o r an orange b a l l drawn} i s 1/2. The "degree o f a m b i g u i t y " f o r the e v e n t {R} = {a r e d b a l l i s d r a w n } i s , t h e r e f o r e , e q u a l t o 1/2 s i n c e * u {a r e d b a l l i s drawn} = 1/2 and y*{a r e d b a l l i s drawn} = 0. S i m i l a r l y the degree o f a m b i g u i t y f o r t h e e v e n t {G} = {a green b a l l i s drawn} i s e q u a l t o 1/2. I f v/e now c o n s i d e r t h e e v e n t {R O G} = {a r e d o r a green b a l l i s drawn}, we note t h a t t h e degree o f a m b i g u i t y i s e q u a l t o 1, i . e . , t h e sum o f t h e "degree o f a m b i g u i t y " o f the e v e n t s . T h e r e f o r e i f P(R) = P(Y) and P(G) = P(O) t h a t i s , t h e e v e n t {R} has been " d i s c o u n t e d " by 1/4 - P ( Y ) , and s i m i l a r l y {0} has been " d i s c o u n t e d " by 1/4 - P ( G ) . S i n c e the degree o f a m b i g u i t y f o r t h e e v e n t {R U 0} i s the sum o f t h e degrees o f a m b i g u i t y o f b o t h e v e n t s , i t seems r e a s o n a b l e t o s u b t r a c t the d i s c o u n t f a c t o r s from b o t h o f t h e s e . That i s , P ( R U O) = 1/2 - (1/4 - P(R)) - (1/4 - P ( 0 ) ) = P (R) + P (O) . Hence the p r o b a b i l i t y f o r t h e e v e n t {R \J, 0} i s a d d i t i v e f o r the e v e n t {R} and {0}. We summarize t h e s e i d e a s now as Axiom V I I . Axiom V I I . For each C e (3 and D e 0 we have P (C) = P (C H D) + P (C O D) . I f i n E l l s b e r g ' s paradox I I we l e t C = (R 0 B) and (D) = (R) t h e n Axiom V I I i m p l i e s t h a t P ( R U B ) = P(R) + P(B) i f R r\ B = 0 which we i l l u s t r a t e d i n t h e b e g i n n i n g o f t h i s s e c t i o n , l a r l y , i f i n t h e m o d i f i c a t i o n we l e t C = (R U 0) and D then P (R U 0) = P (R) + P (0) , as i l l u s t r a t e d i n the, l a s t example. Axiom V I I t h e r e f o r e s p e c i f i e the s e t s f o r w h i c h t h e P-measure ought t o be a d d i t i v e . 7.2 E x i s t e n c e o f P-measure I n s e c t i o n 3.2 we noted t h e P-measure was " n e a r l y " a d d i -t i v e f o r c e r t a i n s e t s ; i n Axiom V I I we assumed a d d i t i v i t y f o r e x a c t l y t h o s e s e t s . Hence a t l e a s t we a r e c o n s i s t e n t i n the f o r m u l a t i o n s o f t h e axioms. S i m i -= (ROY) 186 One o f the d i f f i c u l t i e s w h i c h a r i s e s when s p e c i f y i n g a s e t o f axioms we b e l i e v e the d e c i s i o n maker ought t o have, i s t h a t t h e r e may not e x i s t a P-measure w i t h t h o s e p r o p e r t i e s . For example, we have shown t h a t t h e Savage axioms c o n t r a d i c t Axiom V. I n s e c t i o n 8 we s h a l l see t h a t i f we r e q u i r e t h e P-measure t o be a p r o b a b i l i t y , a d d i t i o n a l r e s t r i c t i o n s must be imposed on B. That i s , B can n o t be an a r b i t r a r y s e t o f e v e n t s and hence A can not be an a r b i t r a r y s e t o f a c t i o n s . The q u e s t i o n now a r i s e s as t o whether the axioms we have assumed would r e s t r i c t B i n any way. I f we t r a n s l a t e t h e axioms i n t o p r o p e r t i e s the P-measure must s a t i s f y , t hen we can summ-a r i z e t h e problem as f o l l o w s : G i v e n an a r b i t r a r y p r o b a b i l i t y space ( f t , 0,y) and an a r b i -t r a r y 0 - a l g e b r a B c o n t a i n i n g 0 , does t h e r e e x i s t a s e t f u n c t i o n P such t h a t 1) P(C) i s d e f i n e d f o r a l l C e B 2) P(C) = y(C) f o r a l l C e 0 3) P(C) < P(D) f o r a l l C C D and C,D e B 4) I f B. C B., n and B. e B f o r a l l i , and l i m B . = B, l l + l l l t h e n l i m P ^ ) = P (B) 5) P(C) = P(C A D) + P(C H D) f o r a l l C e B and D e 0 . * F o r t u n a t e l y t h i s problem i s e a s i l y s o l v e d s i n c e y has t h e s e p r o p e r t i e s . However 4) i s n o t always t r u e f o r an a r b i -t r a r y o u t e r measure, b u t o n l y a r e g u l a r o u t e r measure (see Appendix I I ) . T h i s does not c o n c e r n our development h e r e , however, s i n c e the o u t e r measure we a r e c o n s i d e r i n g i s always i n d u c e d by t h e measure y and hence t h e r e f o r e always r e g u l a r . 187 7 . 3 Some p o s s i b l e P-measures I n t h i s s e c t i o n we s h a l l c o n s i d e r some p l a u s i b l e P-meas-ur e s when 8 c o n t a i n s o n l y a f i n i t e number o f s e t s . T h e r e f o r e we a r e o n l y i n t e r e s t e d i n P-measures w h i c h s a t i s f y p r o p e r t i e s 1 , 2 , 3 and 5 i n s e c t i o n 6 . 2 . F o r s i m p l i c i t y we s h a l l assume t h a t t h e r e e x i s t f i n i t e l y many d i s j o i n t s u b s e t s o f ft denoted by D^,...,D which g e n e r a t e the a - a l g e b r a 9 . L e t C be any s e t such t h a t C i 9 , ; • and l e t 8 be t h e a - a l g e b r a g e n e r a t e d by C,D^,...,D. We a l s o assume t h a t a measure y i s d e f i n e d on 0 . Our o b j e c t i s the n t o d e f i n e a P on 8 . * We note t h a t b o t h P(-) = y ( • ) o r P(«) = y * ( 0 s a t i s f y a l l * p r o p e r t i e s 1 t o 5 , i n t h i s c a s e . I f we a c c e p t y as our P-measure t h e n , from a d e c i s i o n maker's p o i n t o f v i e w t h i s w ould i m p l y t h a t i n t h e E l l s b e r g paradox, P(B) = 2 / 3 and P(Y) = 2 / 3 . I f we a c c e p t y ^ as our P-measure, P(B) = 0 and P(Y) = 0 . Both seem v e r y u n l i k e l y t o be a c c e p t e d and e m p i r i -c a l s t u d i e s i n d i c a t e t h a t t h e y a r e not a c c e p t e d . * A more l i k e l y c a n d i d a t e would be a w e i g h t e d average o f y * and y A , i . e . , P ^ C ) = a y (C) + ( 1 - a ) y * (C) f o r a l l C e 8 , where a e ( 0 , 1 ) . I t i s o b v i o u s t h a t P^ s a t i s f i e s t h e f i r s t o f the p r o p e r t i e s on p . 1 8 1 . and s i m i l a r l y t h e second f o l l o w s s i n c e * y (C) = y*(C) = y(C) f o r a l l C e 0 . The t h i r d p r o p e r t y f o l l o w s s i n c e y ' a n d y ^ ".are monotone The f'ourth f o l l o w s o b-v i o u s l y , and, hence we need on-ly show "that t h e f i f t h p r o p e r t y h o l d s . L e t D be an a r b i t r a r y s e t b e l o n g i n g t o 0 and C e 8• Then we have 1 8 8 \* (C) ~ y * ( C A D) + y * (C A 5 ) and y*(C) = y * ( C A D) + y ^ ( C A D) (see Appendix I I ) T h e r e f o r e P 1 ( C ) = a y * ( C ) + ( l - a ) y ^ ( C ) = a [ y ( C A D ) + y (C A D) ] + + ( 1 - a ) [ y * (C A D) + y * (C A D) ] = [ a y * ( C f \ D ) + ( 1 - a ) y * (C A D) ] + + [ a y * ( C A D ) + ( 1 - a ) y ^ ( C A D ) ] = P (C A D) + P (C A D) , and hence t h e f i f t h p r o p e r t y i s s a t i s f i e d by P ^ ( • ) • I n terms o f t h e E l l s b e r g paradox I I , t h i s would i m p l y t h a t P(B) = P(Y) and may be any number between 0 and 2 / 3 i n c l u s i v e and P (B)+P (Y) = P ( B O Y ) i f and o n l y i f P(B) = 1 / 3 . T h e r e f o r e , u s i n g t h i s measure, E l l s b e r g ' s paradox i s e a s i l y e x p l a i n e d . A p r e f e r e n c e o f a b e t on (R) o v e r {B} i n d i c a t e s P(R) = 1 / 3 > P(B) and s i n c e P^(B) = P 1 ( Y ) we have P , ( B U Y ) = 2 / 3 P r ( R O XI = .P1(_R). + P - L O O by Axiom V I I = 1/3 + P 1 ( Y ) < 2/3 s i n c e P-^Y) < 1/3. Th e r e f o r e a p r e f e r e n c e o f a b e t on {R} ove r {B} i m p l i e s a p r e f e r e n c e o f a b e t on {B V J Y} o v e r {R O Y}. S i m i l a r l y a p r e f e r e n c e o f {B} ove r {R} would i m p l y a p r e f e r e n c e o f {R U Y} ov e r {B V j Y}, and i n d i f f e r e n c e between {R} and {B} would i n d i c a t e an i n d i f f e r e n c e between {B V j Y} and ( R ( J Y}. T h i s d e f i n i t i o n o f P-measure can be g e n e r a l i z e d t o t h e case where B c o n t a i n s a f i n i t e c o l l e c t i o n o f non-measurable s e t s . That i s we assume t h a t t h e r e e x i s t s a c o l l e c t i o n o f d i s j o i n t s u b s e t s o f ft, denoted by Dn,...,D and l e t 0 be th e J J 1 m a - a l g e b r a g e n e r a t e d by the s e t s ,i=l,...,m. L e t C^,...,C be s u b s e t s o f ft such t h a t C\ / 0 i = l , . . . , n , and l e t B be the a - a l g e b r a g e n e r a t e d by C^,...,C , D^,...,D and a g a i n we assume t h a t a measure y i s d e f i n e d on 0 . As b e f o r e we s h a l l e x t e n d y t o 0 . One way o f d o i n g so would be t o d e f i n e E.. = D. A C., i= l , . . . , m , j = l , . . . , n , and l j I j l e t E. . ^ 0, j e T 1, E. . = 0, o t h e r w i s e . That„ i s , , , T 1 l j ' l j i s an i n d e x s e t f o r w h i c h t h e s e t E ^ j i s n o t e q u a l t o 0 f o r a f i x e d i . * D e f i n e P~ (E . .) = a . y (E . .) + (1-cO y . (E . .) where a:., i s 2. l j 1 l j 1 * l j l a r e a l - v a l u e d f u n c t i o n on the c a r d i n a l i t y o f T 1 t o the i n t e r -v a l [ 0 , 1 ] . F i r s t c o n s i d e r t h e u n i o n \J'^±y, Where S C T""" j£.S 190 D e f i n e P 2 ( U E ) = P 2 ( D ± ) T C S T P 9 ( E . . ) o t h e r w i s e , F i n a l l y , i f F i s an a r b i t r a r y s e t i n 8 l e t G be t h e l a r g e s t s u b s e t o f F, such t h a t G e 0 , we d e f i n e m P 2 ( F ) = P 2(G) + 21 P 2 ( D i ^• (F-G) ) • i = l C l e a r l y , the f i r s t f o u r p r o p e r t i e s h o l d f o r the same r e a s o n as f o r P^. To show t h a t t h e f i f t h p r o p e r t y h o l d s , l e t F and B be a r b i t r a r y s e t s b e l o n g i n g t o 6 and 0 r e s p e c -t i v e l y . Then we are r e q u i r e d t o show t h a t P 2 (F) = P 2 (F n B) + P 2 (F C\ B) . L e t G be the l a r g e s t s u b s e t o f F such t h a t G e 0 , and t h e n , c l e a r l y , B A, G must be t h e l a r g e s t s u b s e t o f F A, B such t h a t B A , G e 0 . S i m i l a r l y , B A G must be t h e l a r g e s t s u b s e t o f F A B . m P 2 ( F ) = P 2(G) + 21 P 2 (Di'A (F-G) ) i = l m = P 2 ( G A B) + P 2 ( G A B) + £ P 2(D. r\ (F-G) ) i = l 1 S i n c e D. e i t h e r i s c o n t a i n e d i n B o r B, P 2 (D-L r\ ( F - G ) ) = P 2 ( D I H ( ( F P\ B ) - ( G C\ B ) ) + P 2 (D i A ( (F C\ B) - (G A B) ) . Hence by recombindihg terms P 2 ( F ) = P 2 ( F A B ) + P 2 ( F A B ) . A p p l y i n g t h i s P-measure t o t h e E l l s b e r g paradox I I , we would o b t a i n e x a c t l y the same measure as d i s c u s s e d b e f o r e . P 2(W) + P 2 ( 0 ) and no r e l a t i o n would n e c e s s a r i l y h o l d between P 2 ( B ) t o P 2 ( G ) . The p r e f e r e n c e s between {B} and {G} may be e i t h e r way w i t h o u t c o n t r a d i c t i n g the t h e o r y . Note t h a t the way we c o n s t r u c t e d the l a s t P-measure works f o r the case where 6 i s f i n i t e . That i s , f i r s t d e f i n e P f o r the " s m a l l e s t " u n i t E.., n e x t d e f i n e the P-measure f o r s e t s i j E,^ C D'; and f i n a l l y impose the a d d i t i v i t y c o n d i t i o n , However, f o r the m o d i f i e d paradox we would have P 2 ( G U W) = P 2 ( G U 0) = P 2(W U 0) = P 2(G) + P 2(W) = P 2 ( G ) + P 2 ( 0 ) = P(F) = P(G) 192 I n d o i n g s o , we would always s a t i s f y t h e f i v e p r o p e r t i e s on p. 186. The P-measure f o r an a r b i t r a r y s e t E^ .. can p r o b a b l y be approximated i n g e n e r a l as k (y (E ) - y * ( E . . ) ) P(E..) = (a.y (E..) + (1-a) y* (E. . J ) e J J where ou = 1 / c a r d i n a l i t y o f T 1 and k„ any r e a l number. To e x p l a i n t h i s f o r m u l a we note t h a t f o r the E l l s b e r g * paradox I I a^y ( E ^ j ) + ( 1 - a ^ ) y * ( E ^ ^ ) f o r t h e e v e n t {B} and {Y} would be 1/2(2/3) + 1/2(0) = 1/3. F o r the m o d i f i e d paradox and f o r the e v e n t {B} and {Y}, we would have 1/2(1/3) + 1/2(0) = 1/6 and f o r t h e e v e n t {G}, {W}, and {0} we would have 1/3(1/2) + 1/3(0) = 1/6. Hence i f k. = 0, we would have t h e P-measure as a measure I ( i . e . , a d d i t i v e ) . I f k^ < 0, t h i s i m p l i e s we would d i s c o u n t t h i s measure based on the d i f f e r e n c e between the o u t e r and i n n e r measure. I f k^ > 0, t h i s would i n d i c a t e a p r e f e r e n c e o f u n c e r t a i n t y w h i c h i n c r e a s e s as t h e d i f f e r e n c e between o u t e r and i n n e r measures i n c r e a s e s . None o f t h e P-measures mentioned here a r e , o f c o u r s e , n e c e s s a r y f o r t h e t h e o r y t o h o l d . We have suggested some o f t h e s e as they seem t o have e m p i r i c a l s u p p o r t , and would e x p l a i n t h e d i f f i c u l t i e s i n the paradoxes c o n c e r n i n g u n c e r t a i n t y v s . r i s k . The advantage o f a p p r o x i m a t i n g the P-measure i s , o f c o u r s e , t h a t we o n l y need t o d e r i v e one c o n s t a n t k^ and from t h i s we can d e r i v e t h e P-measure f o r a l l s e t s . 8.0 The P-measure as a p r o b a b i l i t y measure I n d e c i s i o n t h e o r y one f r e q u e n t l y comes i n c o n t a c t w i t h o t h e r names used f o r a p r o b a b i l i t y measure. The most common ones, i n a l p h a b e t i c o r d e r , a r e : I n B a y e s i a n s t a t i s t i c s we a l s o have t h e a d d i t i o n a l terms o f p r i o r p r o b a b i l i t y and p o s t e r i o r p r o b a b i l i t y . Most o f t h e s e s a t i s f y Kolmogorov's axiom (see Appendix I I ) ; o t h e r s d i f f e r o n l y by assuming f i n i t e a d d i t i v i t y r a t h e r t h a n a - a d d i t i v i t y . For our purpose we s h a l l d i v i d e them i n t o two c a t e g o r i e s . The f i r s t c a t e g o r y assumes a - a d d i t i v i t y , and w i l l be c a l l e d m a t h e m a t i c a l p r o b a b i l i t i e s ; t h e second c a t e g o r y assumes o n l y f i n i t e a d d i t i v i t y and w i l l be c a l l e d Savage p r o b a b i l i t i e s . We a r e concerned o n l y w i t h the p r o p e r t i e s o f p r o b a b i l i t i e s S p e c i f i c a l l y , our i n t e r e s t i s i n t h e i m p l i c a t i o n s o f assuming the P-measure t o be f i n i t e l y a d d i t i v e o r e q u i v a l e n t l y .' a - a d d i t i v e . I n s e c t i o n 2.0 we assumed t h a t t h e r e e x i s t s a p r o -b a b i l i t y space ( f t , 9 , y ) . We a l s o assumed t h a t the reward f u n c t i o n s X ( - , a ) , a e A a r e not a l l m easurable, t h a t i s 9 i s too s m a l l . We had t h e r e f o r e t o e x t e n d t h e measure y t o the s m a l l e s t a - a l g e b r a f o r w h i c h a l l f u n c t i o n s X(*,a) a r e measur-a b l e . I t would, t h e r e f o r e , be o f i n t e r e s t t o d e t e r m i n e the l a r g e s t a - a l g e b r a f o r w h i c h y can be extended as a p r o b a b i l i t y E m p i r i c a l p r o b a b i l i t y Geometric p r o b a b i l i t y I m p e r sonal p r o b a b i l i t y I n d u c t i v e p r o b a b i l i t y I n t u i t i v e p r o b a b i l i t y Judgment p r o b a b i l i t y L o g i c a l p r o b a b i l i t y Degree o f c o n f i r m a t i o n Degree o f c o n v i c t i o n Degree o f r a t i o n a l b e l i e f M a t h e m a t i c a l p r o b a b i l i t y O b j e c t i v e p r o b a b i l i t y P e r s o n a l p r o b a b i l i t y P h y s i c a l p r o b a b i l i t y P s y c h o l o g i c a l p r o b a b i l i t y Random chance R e l a t i v e f r e q u e n c y S t a t i s t i c a l p r o b a b i l i t y S u b j e c t i v e p r o b a b i l i t y ( F i s h b u r n , 1964, pp.132) 194 measure. I n s e c t i o n 8,1, v/e s h a l l c o n s i d e r t h e s e arguments f o r Savage's p r o b a b i l i t i e s . I n s e c t i o n 8.2 v/e s h a l l c o n s i d e r the P-measure as a a - a d d i t i v e measure and d i s c u s s the p o s s i b l e e x t e n s i o n o f p. I n s e c t i o n 8.3 we s h a l l c o n s i d e r an a l t e r -n a t i v e t o d e c i s i o n making under p a r t i a l r i s k . The r e a s o n f o r d o i n g so here i s t h a t t h i s a l t e r n a t i v e a l s o i n d u c e s a p r o -b a b i l i t y measure. 8.1 Savage e x t e n s i o n s I t i s w e l l known, and prove n i n most b a s i c t e x t b o o k s on measure t h e o r y , t h a t i f (ft,G,y) i s d e f i n e d such t h a t ft i s t h e r e a l l i n e , 0 a l l B o r e l s e t s , and y t h e Lesbeque measure, i t i s i m p o s s i b l e t o e x t e n d y t o a l l s u b s e t s o f t h e r e a l - l i n e i f i t i s a l s o r e q u i r e d t h a t y be a - a d d i t i v e . I t i s a l s o w e l l known t h a t i f we o n l y r e q u i r e a f i n i t e l y a d d i t i v e measure the e x t e n s i o n e x i s t s (see Royden, 1968, p.53). Most m a t h e m a t i c i a n s today assume a - a d d i t i v i t y o f t h e measure a l t h o u g h some r e s e a r c h i s s t i l l t a k i n g p l a c e c o n c e r n i n g f i n i t e l y a d d i t i v e measures. From a d e c i s i o n maker's v i e w p o i n t the i m p l i c a t i o n o f the measure n o t b e i n g d e f i n e d on a l l s u b s e t s can be i l l u s -t r a t e d by t h e f o l l o w i n g s i m p l e s i t u a t i o n . There e x i s t s a s e t D C ft such t h a t i f v/e a r e o f f e r e d a l o t t e r y t i c k e t d e f i n e d by X ( to,a) = I $100 to e D $0 to e D i t would be i m p o s s i b l e t o determine i t s e q u i v a l e n t v a l u e , i . e . , how much i t i s w o r t h . Savage, f o r one, d i s l i k e d t h i s i m p l i c a t i o n , and was t h e r e -195 f o r e f o r c e d t o assume o n l y a d d i t i v e measures. I n the Savage approach a p r e f e r e n c e o r d e r i n g was d e t e r m i n e d on the s u b s e t s o f ft. T h i s can be done by a v a r i a t i o n o f Axiom I I I . One method d e t e r m i n e s a p r e f e r e n c e o r d e r i n g on 2^ by l e t t i n g B c ft, D c. ft, U(r) > U(s) and oo e B J r <o E D X(uo,b) = | to e B J s w e D. A 2^ Then a b i m p l i e s B > D. S e v e r a l a u t h o r s (De F i n e t t i , 1937; Savage, 1954) have s t u d i e d t h e problem o f d e t e r m i n i n g c o n d i t i o n s on the o r d e r i n g on 2^ under w h i c h t h e r e e x i s t s a r e a l - v a l u e d o r d e r p r e s e r v i n g f u n c t i o n P on 2^ t h a t can be i n t e r p r e t e d as an a d d i t i v e p r o -b a b i l i t y measure. The e x t e n s i o n t o a - a d d i t i v i t y was made by K r a f t , P r a t t and S e i d e n b e r g (1959). I n s e c t i o n 6.2 o f P a r t I we l i s t e d the c o n d i t i o n s on t h i s p r e f e r e n c e o r d e r i n g f o r t h e 2 f t e x i s t e n c e o f a p r o b a b i l i t y measure such t h a t i f A ^ B t h e n P(A) ^ P ( B ) . A l t h o u g h a P-measure i s n o t n e c e s s a r i l y a p r o -b a b i l i t y measure, i t would be o f i n t e r e s t t o determine what c o n d i t i o n s on the p r e f e r e n c e o r d e r i n g we s a t i s f i e d u s i n g o u r P-measure. We s h a l l r e p e a t f o r easy r e f e r e n c e , t h e axiom s t a t e d i n s e c t i o n 6.2 o f P a r t I . Axiom o f o r d e r i n g 2ft 2ft 1. I f C e 0 , B e 0 , t h e n e i t h e r C ^ B o r B ^ C. 2 f t 2. F o r any s e t C e 0 , C < C. 2 f t 2 f t 2^ 3. I f C B and B ^ D t h e n C ^ D. 2ft 2^ 2^ 4. 0 < ft and f o r any e v e n t C,0 £ C < ft. A l l t h e s e axioms can e a s i l y be d e r i v e d from Axioms I and I I . Axiom o f m o n o t o n i c i t y 2ft 2 9-1. I f B n B 2 = 0, C 1 4 B and C 2 < B 2 , 2^ t h e n C±{J C 2 ^ B 1 U B 2-2 f t 2. I f B 2 = 0, C 1 4 B 1 and C 2 < B 2 , 2ft t h e n C1U C2 < B± Vj B 2 . The axioms o f m o n o t o n i c i t y may n o t be s a t i s f i e d by t h e P-measure. C o n s i d e r , f o r example, E l l s b e r g ' s paradox I I . C^ = {a y e l l o w b a l l drawn} B^ = {a r e d b a l l drawn} C 2 = {a b l a c k b a l l drawn} B 2 = {a y e l l o w b a l l drawn} Then we m i g h t have t h e p r e f e r e n c e C\ B 2 = 0, C-^ 4 B^, C 2 .< but 2 C x O C 2 B 1 v j B 2 . Axiom Of monotone sequence For e v e r y monotone sequence ' i n c r e a s i n g e v e n t s such .that C .* -C and an e v e n t B such t h a t n : C n « B, f o r a l l n, t h e n C 4 B. T h i s axiom f o l l o w s from Axiom V and Axiom I I I . To see t h i s , d e f i n e a sequence o f reward f u n c t i o n s as f o l l o w s : X ( o),a n) = 03 e C n 0) £ C n where U(r) >.. ::.u:(s) and C e $. By Assumption 3, a £ A f o r n ' n a l l n. S i n c e C n i s a sequence o f i n c r e a s i n g s e t s c o n v e r g i n g t o C, a n converges t o a £ A where a i s d e f i n e d by the reward f u n c t i o n X(ai,a) = 0) £ C 0) £ C I f b e A, and b i s d e f i n e d by the reward f u n c t i o n X ( o),b) = r OJ £ B s 0) £ B where B £ 6. By lemma 3.2.1 P ( C 1 ) p ( c 2 ) <  p ( > c 3 ) 4 and l i m P ( C ) = P(C).. T h e r e f o r e i f P (C n) < P (B) f o r a l l n, t h e n P(C) < P(B) a l s o . Axiom o f p a r t i t i o n o f event E v e r y e v e n t can be p a r t i t i o n e d i n t o two e q u a l l y p r o b a b l e e v e n t s . T h i s assumption i s needed t o s p e c i f y the p r o b a b i l i t y measure. V i l l e g a s (1964) showed t h a t t h i s axiom, under c e r -t a i n a s s u m p t i o n s , i s e q u i v a l e n t t o the e x i s t e n c e o f a random v a r i a b l e w i t h u n i f o r m d i s t r i b u t i o n (Assumption 3 ) . T h e r e f o r e , 198 the o n l y axiom w h i c h may not h o l d i n the P-measure approach i s the axiom o f m o n o t o n i c i t y . 8.2 M a t h e m a t i c a l p r o b a b i l i t i e s I n t h i s s e c t i o n we s h a l l a l s o assume t h e r e e x i s t s a pro-b a b i l i t y space ( f t , 0 , y ) . Our aim i s t o s t u d y t h e p o s s i b l e e x t e n s i o n o f 0 t o a a - a l g e b r a c o n t a i n i n g 0 . The Savage p r o -b a b i l i t y s a t i s f i e s : 1) y(C) e x i s t s f o r a l l C C ft. 2) y (C) = y (E) C e 0 . 3) I f C,,...,C are d i s j o i n t s e t s t h e n I n J n n V ( U C. ) = Z>y(C ) . i = l i = l I n t h i s s e c t i o n we s h a l l r e p l a c e 1) by: 1) y(C) e x i s t s f o r a l l C e 6 where 8 i s a v e r y s p e c i a l a - a l g e b r a , and r e p l a c e 3) by: 3) I f C^ i = l , . . . i s a d i s j o i n t sequence C^ e (3, then ( U C. ) = Z ] y ( C . ) i = l 1 i - 1 1 The f i r s t e x t e n s i o n r e s u l t we s h a l l s t a t e i s t h e s t a n d a r d t e x t b o o k case w h i c h can be j u s t i f i e d i n the f o l l o w i n g way. Assume t h a t f o r a g i v e n s e t B we know the p r o b a b i l i t y o f o c c u r r e n c e i s e q u a l t o o n e - h a l f . L e t D be a s u b s e t o f B which does n o t b e l o n g t o 0 . I f we w i s h t o i n c l u d e D i n the a - a l g e b r a the d i f f i c u l t y i s t o d e t e r m i n e w h i c h measure t o a s s i g n D such t h a t we do not c o n t r a d i c t any o f t h e axioms o f the measure. There i s one case, however, when t h i s problem does not a r i s e ; t h a t i s , i f y ( B ) = 0 . Then, f o r any subset the only p o s s i b l e measure we c o u l d a s s i g n would be zero. When t h i s i s done f o r a l l subsets o f the measure zero, i t i s c a l l e d the completion of the p r o b a b i l i t y space. D e f i n i t i o n . A measure y on a a - a l g e b r a 8 i s s a i d to be complete i f and only i f whenever D C B and y ( B ) = 0 , B e 0 then D e 6 . The completion of a p r o b a b i l i t y space ( f t , 0 , y ) i s d e f i n e d as f o l l o w s . L e t 8 be the a - a l g e b r a generated by the s e t { G U D} where G e 0 and D C B where y ( B ) = 0 f o r some B e © . We extend y to y by y ( G U D) = y ( G ) . The i m p l i c a t i o n i s t h a t i f a d e c i s i o n maker r e f u s e s to pay anything f o r a l o t t e r y t i c k e t which wins any amount i f B o c c u r s , he would a l s o r e f u s e to pay anything f o r a l o t t e r y t i c k e t which wins the same amount i f any subset of B occurs. T h i s i s an obvious r e s u l t from a d e c i s i o n maker's p o i n t o f view and hence we can always assume t h a t the p r o b a b i l i t y space we are working w i t h i s complete. There a l s o e x i s t s a second case when t h i s problem i s e a s i l y s o l v e d . I f we again assume D C B , then by monotonicity of the measure we know t h a t y(D) must be l e s s than y ( B ) . I f we a l s o know t h a t there e x i s t s a s e t N c o n t a i n e d i n D(N C D) such t h a t N e Q and y(N) = y ( B ) , then c l e a r l y y (D) = y ( B ) . T h i s method was used by Lebesque ( 1 9 0 1 ) i n d e f i n i n g a measurable s e t f o r which the i n n e r and outer measures are e q u a l . I t i s a l s o knownthat the a - a l g e b r a generated by a l l s e t s such t h a t i n n e r and outer measures are equal i s the l a r g e s t 200 g - a l g e b r a f o r which y can be extended u n i q u e l y , we s h a l l denote t h i s a - a l g e b r a by S. The o u t e r measure w i l l be denoted as * b e f o r e by y and the i n n e r measure as b e f o r e by y^. I f we now remove the c o n d i t i o n t h a t y be u n i q u e l y extended, can we s t i l l e x t e n d S? L e t Z be any s e t such t h a t Z £ S, then we denote t h e s m a l l e s t a - a l g e b r a g e n e r a t e d by the c o l l e c t i o n o f s e t s Z and a l l s e t s i n S by ( S , Z ) . We s h a l l n e x t s t a t e how y can be extended t o t h i s a - a l g e b r a . Theorem 8.2.1. Los and Marczewski (194 9) L e t y be a p r o b a b i l i t y on a a - a l g e b r a S on ft and Z e f t . D e f i n e y and y f o r each s e t E e (S,Z) by * _ y(E) = y* (E H Z) + y (E f l Z) y (E) = y* (E r\ Z) + y* (E A Z) Then y_ and y are e x t e n s i o n s o f y t o (S,Z) and y(Z) = y*(Z) _ * and y(Z) = y ( Z ) . C o r o l l a r y 8.2.2. The f u n c t i o n m(E) = ( l - a ) y ( E ) + ay(E) a e (0,1) i s a l s o an e x t e n s i o n o f y t o (S, Z ) . * _ We note t h a t i f y (Z) = y*(Z) t h e n y(E) = y(E) f o r a l l _ * E e (S,Z) so the o n l y i n t e r e s t i n g case i s i f y (Z) > y * ( Z ) . We a l s o note t h a t f o r any s e t E c (S,Z) we have y * ( E ) < y ( E ) < y (E) y * ( E ) y ( E ) < y * ( E ) y * (E) < m(E) < y * (E) . We can o b v i o u s l y r e p e a t t h i s p r o c e s s by c o n s i d e r i n g the i n n e r and o u t e r measure g e n e r a t e d by y_, y o r m, and hence we can e x t e n d t o the l a r g e r a - a l g e b r a g e n e r a t e d by a d d i n g a f i n i t e number o f s e t s . Los and M arczewski (1949) showed, however, t h a t we may not be a b l e t o add a c o u n t a b l e number o f s e t s . That i s , t h e r e may s t i l l e x i s t s e t s w h i c h a r e s t i l l n o t measurable, and t h e r e f o r e , we s t i l l have the d i f f i c u l t y o f e x p l a i n i n g t h e e x i s t e n c e o f a s i m p l e d e c i s i o n problem i n s e c t i o n 8.2. 2 0 2 8 . 3 P r o b a b i l i t y measures on Fu z z y s e t s The t h e o r y o f d e c i s i o n making i n a f u z z y environment (see Zadeh ( 1 9 6 5 ) , ( 1 9 6 8 ) , ( 1 9 6 9 ) ) i s r e l a t e d t o t h e d e c i s i o n making under p a r t i a l r i s k . The d i f f e r e n c e however i s t h a t e v e n t s i n the f u z z y environment a r e not c l e a r l y d e f i n e d . Examples o f f u z z y s e t s a r e : "X i s a p p r o x i m a t e l y e q u a l t o 5 " ; o r " I n twenty t o s s e s o f a c o i n t h e r e a r e s e v e r a l more heads than t a i l s " . Because o f t h e vagueness o f t h e d e s c r i p t i o n o f eve n t s d i f f i c u l t i e s i n s p e c i f y i n g the p r o b a b i l i t i e s f o r the s e e v e n t s can be a n t i c i p a t e d , as w i t h d e c i s i o n making under p a r t i a l r i s k . I n t h i s s e c t i o n we s h a l l summarize how t h i s p r oblem i s h a n d l e d f o r f u z z y s e t s , and r e l a t e the i d e a t o d e c i s i o n making under p a r t i a l r i s k . To do so we must d e f i n e f u z z y s e t s i n more m a t h e m a t i c a l terms. D e f i n i t i o n . (Zadeh, 1 9 6 5 ) . L e t ft = {w} be a c o l l e c t i o n o f o b j e c t s ( s t a t e s ) . A f u z z y s e t B i n ft i s a s e t o f o r d e r p a i r s B = (w, J B (w) ) w e f t where Jg(w) i s a r e a l v a l u e d f u n c t i o n from ft t o M = [ 0 , 1 ] . I f M = { 0 , 1 } t h e n t h i s i s e q u i v a l e n t t o the normal d e f i n i t i o n o f a s u b s e t o f ft. The f u n c t i o n Jg(w) can be thought o f as "the degree o f c o n f i d e n c e " we have o f w b e l o n g i n g t o B. The f u n c t i o n i s e q u a l t o 1 i f we a r e c e r t a i n o f w b e l o n g i n g t o B and 0 i f we a r e c e r t a i n t h a t t h i s i s n o t t h e c a s e . To i l l u s t r a t e t h e f u n c t i o n J^(yr) r l e t ft be a s e t o f 60 b a l l s , numbered from 1 t o 60. A l l the b a l l s a r e e i t h e r y e l l o w o r b l a c k and we know f o r c e r t a i n t h a t t h e f i r s t 10 b a l l s a r e y e l l o w and the l a s t 10 a r e b l a c k . Somewhere between 11 and 51 t h e r e e x i s t s a number y such t h a t a l l numbers below i n c l u d i n g y a r e y e l l o w , and t h o s e above a r e b l a c k . The f u n c -t i o n Jg(w) where B i s t h e number o f b l a c k b a l l s can be i l l u s -t r a t e d g r a p h i c a l l y as i n F i g . 7.4. No. o f ,black b a l l s w' F i g . 7.4. I l l u s t r a t i n g the degree o f c o n f i d e n c e i n r e l a t i o n t o the p o s s i b l e number o f b l a c k b a l l s i n an u r n w i t h p a r t i a l knowledge. F o r example the graph i l l u s t r a t e s t h a t we a r e 50% c o n f i -d ent t h a t t h e r e a r e a t l e a s t 15 b l a c k b a l l s . A l l s t a n d a r d s e t o p e r a t i o n s can be d e f i n e d f o r f u z z y s e t s as f o l l o w s : The i n t e r s e c t i o n o f two s e t s C and B i s denoted by C. f\ B and d e f i n e d by t h e membership f u n c t i o n JC r N B ( ' ) = m i n ( J c ( •) , J B ( •) ) • The complement C o f t h e s e t C i s d e f i n e d by t h e membership f u n c t i o n = 1 - J C ( . ) . The u n i o n C U B , o f t h e s e t C and B i s d e f i n e d as J C U B ( , ) = ™x(Jc>JB]' F o r example t h e complement s t a t e s t h a t i f we a r e 50% c o n f i d e n t t h a t t h e r e a r e a t l e a s t 15 b l a c k b a l l s we a r e a l s o 50% c o n f i -d e n t t h e r e a r e a t most 45 y e l l o w b a l l s . I n d e f i n i n g p r o b a b i l i t i e s on f u z z y s e t s we assume f i r s t t h e r e e x i s t s a p r o b a b i l i t y space ( f t , 0 , y ) where 0 does not c o n t a i n f u z z y s e t s . where J (w) i s the i n d i c a t o r f u n c t i o n f o r t h e s e t B. L e t B Hence B denote the a - a l g e b r a o f f u z z y s e t s t h e n i f C e 6 t h e p r o b a b i l i t y o f C i s d e f i n e d by I t can be shown t h a t y s a t i s f i e s the s t a n d a r d p r o p e r t i e s o f a p r o b a b i l i t y measure. Hence t h i s approach i s n o t d i f f e r e n t from any o t h e r approach w h i c h d e v e l o p s p r o b a b i l i t i e s on t h e s e t g . I t i s v e r y l i k e l y t h i s a p p r o a c h : m a y b e u s e d t o d e v e l o p s i m i l a r r e s u l t s a s P - m e a s u r e s . H o w e v e r , i n t h i s c a s e t h e s t a n d a r d d e f i n i t i o n o f i n t e r s e c t i o n , u n i o n a n d c o m p l e m e n t w o u l d h a v e t o b e r e - d e f i n e d . 206 9.0 E m p i r i c a l s t u d i e s r e l a t e d t o P-measure The r e a s o n f o r c o n s i d e r i n g a t h e o r y w h i c h d i f f e r e n t i a t e s between r i s k and u n c e r t a i n t y i s t h a t t h i s seems t o be a com-monly made d i s t i n c t i o n . I n t h i s s e c t i o n we s h a l l s u b s t a n t i a t e t h i s c l a i m by summarizing some e m p i r i c a l s t u d i e s made c o n c e r n -i n g t h i s p o i n t . We s h a l l a l s o show t h a t t h e s e s t u d i e s i n d i c a t e a c orrespondence between the degree o f a m b i g u i t y and P-measure. Our f i r s t o b j e c t i v e w i l l be t o show t h a t i f Axiom IVb i s a c c e p t e d , the P-measure d e f i n e d t h e r e b y i s not n e c e s s a r i l y a p r o b a b i l i t y measure. T h i s can be done by u s i n g E l l s b e r g 1 s paradox I . R e c a l l t h a t i n E l l s b e r g ' s paradox I we c o n s i d e r e d two u r n s , one w i t h 50 r e d and 50 b l a c k b a l l s , and t h e second where the c o m p o s i t i o n of r e d and b l a c k b a l l s was unknown. A l t e r n a t i v e s a and c r e f e r r e d t o a c t i o n s d e f i n e d on b a l l s drawn from th e u r n w i t h t h e known c o m p o s i t i o n and a l t e r n a t i v e s b and d on the o t h e r . I f the P-measure P i s a p r o b a b i l i t y then a p r e f e r e n c e o f a o v er b i m p l i e s d must be p r e f e r r e d t o c. S i m i l a r l y i f b i s p r e f e r r e d t o a, then c must be p r e f e r r e d t o d. For a d e t a i l e d s t u d y o f the E l l s b e r g paradox I , see s e c t i o n 6.4, P a r t I , and r e f e r e n c e s t o o t h e r r e l a t e d s t u d i e s . L e t us now summarize some r e s u l t s c o n c e r n i n g t h i s p a r a -dox from the s t u d y by MacCrimmon and L a r s s o n (1975). F i f t e e n o u t o f 19 s u b j e c t s chose a P-measure wh i c h i s not a p r o b a b i l -i t y measure. Another s u b j e c t r a nked t h r e e o f t h e b e t s e q u a l l y w i t h a l t e r n a t i v e b l o w e r . Two s u b j e c t s r a nked a l l f o u r c h o i c e s e q u a l l y and the r e m a i n i n g s u b j e c t r a n k e d the c h o i c e s i n a c c o r d a n c e w i t h a p r o b a b i l i t y measure. 207 Hence 15 o u t o f 19 d i d n o t choose a l t e r n a t i v e s as though the P-measure were a p r o b a b i l i t y measure and o n l y t h r e e d i d choose as though i t were. A l t h o u g h o t h e r s t u d i e s have n o t shown such a l a r g e r e j e c t i o n o f P-measure as a p r o b a b i l i t y , the p e r c e n t a g e o f r e j e c t i o n has been s u b s t a n t i a l . T h i s i n d i -c a t e s , t h e r e f o r e , t h a t i t i s r e a s o n a b l e t o assume t h a t t h e r e a r e d i f f e r e n c e s between r i s k and u n c e r t a i n t y . A s t u d y made by Becker and Brownson (1964) shows s u p p o r t f o r Axiom V I I and shows t h a t a m b i g u i t y can be measured as the d i f f e r e n c e between the o u t e r and i n n e r measure. T h e i r s t u d y a l s o i n d i c a t e s t h a t t h e P-measure i s n o t always a p r o b a b i l i t y measure. T h e i r s t u d y was based on a v a r i a t i o n o f E l l s b e r g paradox I . They c o n s i d e r e d f i v e urns w i t h the maximum and minimum number o f r e d and b l a c k b a l l s as shown below: Red B a l l s B l a c k B a l l s Minimum Maximum Minimum Maximum Number Number Number Number Urn I 0 100 0 100 Urn I I 50 50 50 50 Urn I I I 15 85 15 85 Urn IV 25 75 25 75 Urn V 40 60 40 60 The o b j e c t o f t h e i r s t u d y was t o f i n d s u p p o r t f o r t h e f o l -l o w i n g h y p o t h e s e s : H y p o t h e s i s I . - I n d i v i d u a l s a r e w i l l i n g t o pay money t o a v o i d a c t i o n s i n v o l v i n g a m b i g u i t y . The e x p e r i m e n t sought t o v e r i f y E l l s b e r g ' s f i n d i n g s 208 under a c t u a l p a y o f f s i t u a t i o n s , (Evidence s u p p o r t i n g t h i s h y p o t h e s i s does no in.ore t h a n c o n f i r m t h a t a t h i r d v a r i a b l e appears t o be a f f e c t i n g d e c i s i o n b e h a v i o r o f some p e o p l e ; i t does n o t l e n d d i f f e r e n t i a l s u p p o r t t o any p a r t i c u l a r d e f i n i t i o n o f t h e t h i r d v a r i a b l e . ) H y p o t h e s i s I I . - Some pe o p l e behave as i f t h e y a s s o c i a t e a m b i g u i t y w i t h t h e d i s t r i b u t i o n on each p r o b a b i l i t y . (Becker and Brownson, p.65) The r e s u l t o f t h e i r s t u d y was s p e c i f i e d i n two s e t s . F i r s t , i t was d e t e r m i n e d whether t h e s u b j e c t s had a p r e f e r e n c e as E l l s b e r g p r e d i c t e d by E l l s b e r g paradox I . The r e s u l t was as f o l l o w s : Responses t o E l l s b e r g q u e s t i o n s Number o f S u b j e c t s 50 - 50 p r e f e r r e d 16 0 - 100 p r e f e r r e d 1 I n d i f f e r e n t 5 C o l o r p r e f e r e n c e 12 T o t a l 34 Hence t h e y found a l a r g e p e r c e n t a g e h a v i n g a c o l o u r p r e f e r e n c e as w e l l as a h i g h p e r c e n t a g e r e j e c t i n g the P-measure as a p r o b a b i l i t y measure. F i f t e e n out o f t h e s i x t e e n (one was o m i t t e d due t o o v e r s i g h t on t h e p a r t o f t h e e x p e r i m e n t e r ) were s e l e c t e d f o r f u r t h e r s t u d y . Note t h a t the i n n e r measure o f drawing a r e d b a l l i s e q u a l t o t h e minimum number o f r e d b a l l s and the o u t e r mea-su r e e q u a l t o t h e maximum number o f r e d b a l l s , and s i m i l a r l y f o r b l a c k b a l l s . T h e r e f o r e , f o r example, th e degree o f 209 a m b i g u i t y i s g r e a t e r f o r t h e event o f dravzing a r e d b a l l from Urn I th a n f o r t h e ev e n t o f drawing a r e d b a l l from Urn I I I . I n t he second s t u d y t h e s u b j e c t s were asked t h e amount th e y were w i l l i n g t o pay t o have t h e b a l l drawn from t h e i r c h o i c e o f u r n f o r any two c o m b i n a t i o n s i f t h e i r w i n n i n g s were $1. A c t u a l money was used f o r t h i s e x p e r i m e n t . F o r the e x a c t f i n d i n g s o f t h e i r r e s u l t s , we r e f e r t o Ta b l e 3 and T a b l e 5 i n t h e i r paper. However, t h e i r s t u d y c l e a r l y i n d i c a t e s t h a t t h e i r s u b j e c t s p r e f e r t h e a l t e r n a t i v e w i t h a s m a l l e r degree o f a m b i g u i t y and th e y c o n c l u d e t h a t : "Evidence p r e s e n t e d i n T a b l e s 3 and 5 c o n f i r m s the f i r s t h y p o t h e s i s t h a t some pe o p l e w i l l pay t o a v o i d an ambiguous c o u r s e o f a c t i o n when t h a t a c t i o n has an ex p e c t e d v a l u e e q u a l : t o an a l t e r n a t i v e unambiguous c o u r s e o f a c t i o n . F o u r t e e n o f t h e f i f t e e n s u b j e c t s , a l l o f whom i n d i c a t e d under non - p a y o f f c o n d i t i o n s t h a t they had an a v e r s i o n t o a m b i g u i t y , were w i l l i n g t o pay a sum o f money t o have t h e o p p o r t u n i t y t o s e l e c t t h e i r p r e f e r r e d c o u r s e of a c t i o n . When c h o o s i n g between t h e 50-50 u r n and t h e 0-100 u r n , t h e s u b j e c t s o f f e r e d t o pay an average of $0.36 (the average o f t h e amounts i n the f i r s t column o f T a b l e s 3 and 5) i n o r d e r t o a v o i d an ambiguous c o u r s e o f a c t i o n whose e x p e c t e d v a l u e was $0.50. (One must wonder whether t h e s e s u b j e c t s , i n r e t r o s p e c t , would c o n s i d e r t h e d i s c o m f o r t a v o i d e d w o r t h the p r i c e t hey p a i d . ) The second h y p o t h e s i s , t h a t a m b i g u i t y i s a s s o c i a t e d w i t h the d i s t r i b u t i o n on each p r o b a b i l i t y , i s a l s o s u p p o r t e d by t h e d a t a . I n a l l c a s e s the p r e f e r r e d u r n , f o r w h i c h t h e s u b j e c t would pay a premium, was t h a t u r n whic h had t h e s m a l l e r range around E ( P r ) . " (Becker and Brownson, p.70) T h e i r s t u d y was o f c o u r s e o n l y d e s i g n e d f o r E l l s b e r g paradox I and does n ot i n g e n e r a l i n d i c a t e t h a t most p e o p l e p r e f e r a c t i o n s w i t h no a m b i g u i t y , o n l y i n t h i s v e r y s p e c i a l example. A l s o note t h a t i n t h e i r s t atement "...an ambiguous c o u r s e o f a c t i o n when t h a t a c t i o n has an e x p e c t e d v a l u e e q u a l t o an a l t e r n a t i v e unambiguous c o u r s e of. a c t i o n . . . " . (Becker and Brownson, p.70) 210 They i m p l y t h e r e e x i s t s , a, second o r d e r d i s t r i b u t i o n but t h i s may n o t be t h e c a s e . The s t u d i e s so f a r i n d i c a t e t h a t t h e r e a r e t h o s e w h i c h do d i f f e r e n t i a t e between r i s k and u n c e r t a i n t y and t h a t t h e degree o f a m b i g u i t y may be one f a c t o r w h i c h d e t e r m i n e s t h e P-measure. However t h i s does n ot i m p l y t h a t t h e y a c c e p t t h e e x p e c t e d u t i l i t y c r i t e r i o n b e i n g a c c e p t e d i n the f i r s t place,' t h a t i s , t h e r e j e c t i o n may be by tho s e who r e j e c t the e x p e c t e d u t i l i t y c r i t e r i o n . • A s t u d y made by Y a t e s and Zukowski (1975) i n d i c a t e s t h a t many s u b j e c t s who a r e c o n s i s t e n t w i t h the e x p e c t e d u t i l i t y c r i t e r i o n f o r d e c i s i o n making under r i s k a l s o d i f f e r e n t i a t e between r i s k and u n c e r t a i n t y . They c o n s i d e r e d t h e f o l l o w i n g a l t e r n a t i v e s : A l t e r n a t i v e a. Step 1: D e s i g n a t i o n o f " v a l u a b l e " c h i p . A c o l o r o f poker c h i p s , r e d o r b l u e , i s d e s i g n a t e d by the p l a y e r as " v a l u a b l e " . Step 2: Bookbag c o m p o s i t i o n . F i v e v a l u a b l e and f i v e non-v a l u a b l e c h i p s a r e p l a c e d i n a bookbag by the p l a y e r . a Step 3: Drawing. The p l a y e r draws one c h i p a t random from the bookbag. Step 4: P a y o f f . The p l a y e r becomes e n t i t l e d t o r e c e i v e $1. i f a v a l u a b l e c h i p i s drawn i n St e p 3 o r n o t h i n g i f a n o n - v a l u a b l e c h i p i s drawn. A l t e r n a t i v e b. Step 1: D e s i g n a t i o n o f " v a l u a b l e " c h i p . A c o l o r o f poker c h i p s , r e d o r b l u e , i s d e s i g n a t e d by the p l a y e r as " v a l u a b l e " . S tep 2: F i r s t bookbag c o m p o s i t i o n . E l e v e n w h i t e c h i p s , marked 0 t h r o u g h 10, a r e p l a c e d by the p l a y e r i n the f i r s t o f two bookbags. Step 3: F i r s t d r a w i n g . The p l a y e r draws one c h i p a t random 211 from t h e f i r s t bookbag. Step 4: Second bookbag c o m p o s i t i o n . The number o f v a l u a b l e c h i p s c o r r e s p o n d i n g t o t h e number drawn i n Step 3 a r e p l a c e d by t h e p l a y e r i n the second bookbag. Non-v a l u a b l e c h i p s are added t o make a t o t a l o f 10 v a l -u a b l e and n o n - v a l u a b l e c h i p s i n the second bookbag. The c h i p s a r e t h e n mixed w e l l . Step 5: Second dr a w i n g . The p l a y e r draws one c h i p a t random from th e second bookbag. Step 6: P a y o f f . The p l a y e r becomes e n t i t l e d t o r e c e i v e $1. i f a v a l u a b l e c h i p i s drawn i n Step 5 o r n o t h i n g i f a n o n - v a l u a b l e c h i p i s drawn. A l t e r n a t i v e c. Step 1: Bookbag c o m p o s i t i o n . The p l a y e r i s i n f o r m e d t h a t t h e r e a r e t e n poker c h i p s i n a bookbag. Each poker c h i p i s e i t h e r r e d o r b l u e . The number o f c h i p s o f e i t h e r c o l o r can be any number from z e r o t h r o u g h 10, w i t h the t o t a l number o f c h i p s b e i n g 10. (The p l a y e r i s n o t i n f o r m e d o f t h e number o f c h i p s o f each c o l o r . ) S t e p 2: D e s i g n a t i o n o f " v a l u a b l e " c h i p s . A c o l o r o f poker c h i p s , r e d o r b l u e , i s d e s i g n a t e d by the p l a y e r as " v a l u a b l e " . Step 3: Drawing. The p l a y e r draws one c h i p a t random from the bookbag. Step 4: P a y o f f . The p l a y e r becomes e n t i t l e d t o r e c e i v e $1. i f a v a l u a b l e c h i p i s drawn i n Step 3 o r n o t h i n g i f a n o n - v a l u a b l e c h i p i s drawn. We note t h a t a l t e r n a t i v e s a and b a r e d e c i s i o n making under r i s k , and a l t e r n a t i v e c d e c i s i o n making under p a r t i a l r i s k . I n E l l s b e r g ' s paradox I , i t has sometimes been s u g g e s t e d t h a t i f t h e s u b j e c t i s asked from which u r n he wants the r e d b a l l drawn, he may b e l i e v e t h a t t h e u n c e r t a i n t y a l t e r n a t i v e has been " r i g g e d " a g a i n s t him, and t h e r e f o r e he may choose th e one w i t h g i v e n p r o b a b i l i t i e s . Y a t e s and Zukowski (1975) have e l i m i n a t e d t h i s f a c t o r i n t h e i r e x p e r i m e n t by l e t t i n g the p l a y e r choose the c o l o u r . I f t h e s u b j e c t s do not have a c o l o u r 212 p r e f e r e n c e then a l t e r n a t i v e s a and b ought t o have the same p r e f e r e n c e i f t h e e x p e c t e d u t i l i t y c r i t e r i o n i s used. I n t h e i r s t u d y t h e y used 108 s t u d e n t s and each s t u d e n t was c l a s s i f i e d i n one o f t h e f o l l o w i n g c l a s s e s : a-b ( 2 0 ) ; b-a ( 1 4 ) ; a-c ( 1 9 ) ; c-a (12); b-c ( 1 6 ) ; c-b ( 2 7 ) ; i . e . , 20 s t u d e n t s had t h e c h o i c e between a l t e r n a t i v e a o r a l t e r n a t i v e b ( i n d i f f e r e n c e n ot p e r m i t t e d ) where a l t e r n a t i v e a was l i s t e d f i r s t ; 14 s t u d e n t s a l s o had t h e c h o i c e between a l t e r n a t i v e a o r a l t e r n a t i v e b, b u t i n t h i s case a l t e r n a t i v e b was l i s t e d f i r s t . A c t u a l p a y o f f was used. Each s t u d e n t was r e q u i r e d t o choose w h i c h a l t e r n a t i v e he p r e f e r r e d i n t h e c l a s s t o w h i c h he b e l o n g e d . I n d i f f e r e n c e was n o t p e r m i t t e d . T h e i r r e s u l t was as f o l l o w s : A l t e r n a t i v e A l t e r n a t i v e a b e a 16 7 b 1 8 - 1 4 c 24 29 -where th e column a l t e r n a t i v e was chosen o v e r the row a l t e r n a t i v e . T h e i r s t u d y t h e r e f o r e shows t h a t l i t t l e d i f f e r e n c e e x i s t s between a l t e r n a t i v e a o r a l t e r n a t i v e b, b u t a l t e r n a t i v e c i s d e f i n i t e l y not p r e f e r r e d t o e i t h e r a c r b. F o r the e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d t h e e x p e c t e d f r e q u e n c y would be: 213 A l t e r n a t i v e a A l t e r n a t i v e . . b. . . c a 17 21 1/2 b 17 15 1/2 c 15 1/2 21 1/2 T h e i r r e s u l t then i n d i c a t e s a s u b s t a n t i a l d i f f e r e n c e between a l t e r n a t i v e s c and a and a l s o , t o a s m a l l e r e x t e n t , between c and ;b. T h i s i n d i c a t e s a d i s t i n c t i o n between r i s k and u n c e r t a i n t y . These r e s u l t s c a n, t h e r e f o r e , be summarized as f o l l o w s : 1) The P-measure d e r i v e d based on Axiom IV may n o t be a p r o -b a b i l i t y . F o r E l l s b e r g ' s paradoxes t h e r e i s a s u b s t a n t i a l p e r c e n t a g e o f r e j e c t i o n o f t h e P-measure as a p r o b a b i l i t y . 2) A m b i g u i t y can be measured i n terms o f t h e d i f f e r e n c e between o u t e r and i n n e r measure. 3) Even i f t h e e x p e c t e d u t i l i t y c r i t e r i o n i s a c c e p t e d f o r d e c i s i o n making under r i s k , t h e r e i s a l a r g e p e r c e n t a g e o f peo-p l e who d i f f e r e n t i a t e between r i s k and u n c e r t a i n t y . 214 10.0 Summary o f P a r t I I I n the second p a r t o f the t h e s i s , we have suggested a c r i t e r i o n f o r d e c i s i o n making under p a r t i a l r i s k . The d e f i n i -t i o n o f p a r t i a l r i s k i n c l u d e s b o t h d e c i s i o n making under r i s k and d e c i s i o n making under u n c e r t a i n t y . T h e r e f o r e any c r i t e r i o n s u ggested f o r p a r t i a l r i s k problems must a l s o be a c c e p t e d f o r r i s k and u n c e r t a i n t y problems. Here we assumed t h a t t h e ex p e c t e d u t i l i t y c r i t e r i o n i s a c c e p t e d f o r r i s k problems. T e c h n i c a l l y , t h e r e f o r e we have extended the e x p e c t e d u t i l i t y c r i t e r i o n t o d e c i s i o n making under p a r t i a l r i s k . The i d e a of t h i s e x t e n s i o n i s not new; b o t h Savage (1954) and F e l l n e r (1961) d i d s i m i l a r e x t e n s i o n s . We d i f f e r from Savage s i n c e h i s approach does n ot d i f f e r e n t i a t e between r i s k and u n c e r t a i n -t y . We a l s o d i f f e r from F e l l n e r ' s s i n c e h i s approach assumes a d i f f e r e n t u t i l i t y f u n c t i o n f o r each a c t i o n . What we have i n common w i t h each o f t h e s e approaches i s t h a t f o r any e x t e n s i o n o f the e x p e c t e d u t i l i t y c r i t e r i o n a p r o b a b i l i t y measure must be d e r i v e d on t h e s e t o f p o s s i b l e rewards. I t i s s u f f i c i e n t t h a t a monotonic c o n t i n u o u s non-n e g a t i v e s e t f u n c t i o n ( c a l l e d P-measure) i s s p e c i f i e d on the s t a t e s o f n a t u r e . S i n c e e v e r y p r o b a b i l i t y measure i s a monotonic, c o n t i n u o u s and n o n - n e g a t i v e s e t f u n c t i o n , a d d i t i o n a l assumptions on t h e P-measure must be made f o r the P-measure t o be a p r o b a b i l i t y measure. T h e r e f o r e , d e c i s i o n t h e o r i s t s who b e l i e v e t h a t t h e e x p e c t e d u t i l i t y c r i t e r i o n ought t o be used f o r d e c i s i o n making under u n c e r t a i n t y would a l s o a c c e p t the approach d e v e l o p e d h e r e . They would perhaps i n s i s t t h a t a d d i t i o n a l assumptions be made on t h e P-measure, f o r c i n g i t t o 215 . be a d d i t i v e . We do n o t e x p e c t a l l d e c i s i o n t h e o r i s t s t o s u p p o r t t h e assumptions d e v e l o p e d h e r e . We do, however, b e l i e v e t h a t i f Axiom I I i s a c c e p t e d , i . e . , t h e e x p e c t e d u t i l i t y c r i t e r i o n f o r d e c i s i o n making under r i s k , t hen the a d d i t i o n a l assumptions made here f o l l o w i n the s p i r i t o f Axiom I I and would be g e n e r a l l y a c c e p t e d . I n c o n s i d e r i n g the axioms i n more d e t a i l we e x p e c t t o f i n d t h a t the most c o n t r o v e r s i a l axiom i n P a r t I I i s Axiom V, w h i c h i s where th e sequence o f reward f u n c t i o n s i s d i s c u s s e d . Here we have chosen t o d e s c r i b e t h e axiom i n terms o f i n c r e a s i n g reward f u n c t i o n s . An e q u i v a l e n t r e s u l t would be t o d e s c r i b e the axiom i n terms o f d e c r e a s i n g reward f u n c t i o n s . However i f we s t a t e the axiom i n terms of an a r b i t r a r y c o n v e r g e n t sequence o f reward f u n c t i o n s , the e x i s t e n c e o f P-measure i s not o b v i o u s (and has not y e t been prov e n as f a r as we know) . In terms o f measure t h e o r y , P a r t I I can be d e s c r i b e d as a mix-t u r e o f s t a n d a r d measure t h e o r y and Savage's approach. One o f t h e o b j e c t i v e s o f measure t h e o r y can be summarized by the f o l l o w i n g q u o t e : "The l e n g t h 1 ( 1 ) o f an i n t e r v a l I i s d e f i n e d , as u s u a l , t o be the d i f f e r e n c e o f t h e e n d p o i n t s o f the i n t e r v a l . . . . I n the case o f l e n g t h the domain i s the c o l l e c t i o n o f a l l i n t e r v a l s . We s h o u l d l i k e t o e x t e n d the n o t i o n o f l e n g t h t o more c o m p l i c a t e d s e t s than i n t e r v a l s . . . . we would l i k e t o c o n s t r u c t a s e t f u n c t i o n m which a s s i g n s t o each s e t E i n some c o l l e c t i o n $ o f s e t s o f r e a l numbers a n o n n e g a t i v e extended r e a l number mE c a l l e d the measure o f E. I d e a l l y , we s h o u l d l i k e m t o have the f o l l o w i n g p r o p e r t i e s : i . mE i s d e f i n e d f o r each s e t E o f r e a l numbers; t h a t i s , <£> = T (R) ; i i . f o r an i n t e r v a l I , ml = 1 ( 1 ) ; i i i . i f E i s a sequence o f d i s j o i n t s e t s ( f o r which 1 m i s d e f i n e d ) , m( {J E ) - m(E ) ; n n n 216 i v . m i s t r a n s l a t i o n i n v a r i a n t ; t h a t i s , i f E i s a s e t f o r w h i c h m i s d e f i n e d and i f E + y i s t h e s e t {x + y:x e E}/ o b t a i n e d by r e p l a c i n g each p o i n t x i n E by the p o i n t x + y, then m(E + y) = mE. U n f o r t u n a t e l y , i t i s i m p o s s i b l e t o c o n s t r u c t a s e t f u n c t i o n h a v i n g a l l f o u r o f t h e s e p r o p e r t i e s , and i t i s not known whether t h e r e i s a s e t f u n c t i o n s a t i s f y i n g the f i r s t t h r e e p r o p e r t i e s . C o n s e q u e n t l y , one o f t h e s e p r o p e r t i e s must be weakened..." (Royden, pp.52-53) I t i s n o t o b v i o u s which one s h o u l d be weakened. Mathe-m a t i c i a n s d e c i d e d on the f i r s t o f t h e s e , and r e s t r i c t e d the t o t a l number o f s e t s . The i m p l i c a t i o n f o r d e c i s i o n t h e o r y i s t h a t t h e r e e x i s t s a s u b s e t D o f Q such t h a t no p r i c e can be s e t on t h e l o t t e r y t i c k e t w h i c h g i v e s r $100 to e D X(w,a) = < $0 03 £ D I . e . , t h e r e e x i s t s i m p l e d e c i s i o n problems f o r which the e x p e c t e d u t i l i t y c r i t e r i o n can n o t be used. T h i s i s an u n d e s i r a b l e c o n c l u s i o n . Savage (1954) r e j e c t e d the t h i r d a l t e r n a t i v e , b u t o n l y f o r a c o u n t a b l e number of s e t s and as a consequence, he r e j e c t e d Axiom V o f P a r t I I . We b e l i e v e t h i s t o be an e q u a l l y u n d e s i r -a b l e c o n c l u s i o n , and we a l s o r e j e c t t h e t h i r d a l t e r n a t i v e , a l t h o u g h we were more s e l e c t i v e i n d e t e r m i n i n g on which s e t s t h e measure might be a d d i t i v e . T h i s would, o f c o u r s e , g i v e the d e s i r a b l e f l e x i b i l i t y o f d i s t i n g u i s h i n g between r i s k and un-c e r t a i n t y . Summary I n P a r t I o f t h e t h e s i s we. d i s c u s s e d a s e t o f axioms n e c e s s a r y f o r t h e e x p e c t e d u t i l i t y c r i t e r i o n t o h o l d . Empir c a l s t u d i e s showed t h a t by f a r t h e most c o n t r o v e r s i a l axiom was the a d d i t i v i t y axiom (Axiom I I , P a r t I) w h i c h s p e c i f i e d t h e e x i s t e n c e o f a r e a l - v a l u e d f u n c t i o n h, such t h a t i f B. i = l , . . . i s a d i s j o i n t sequence o f s u b s e t s o f ft, and each one b e l o n g s t o 3 then f ( a ) = £ h ( B . , a ) . i T h i s a ssumption o b v i o u s l y must be s p e c i f i e d s i n c e t h e i n t e g r a l i s a - a d d i t i v e o v e r d i s j o i n t s e t s . I t a l s o p a r t l y i m p l i e s t h e e x i s t e n c e o f a measure W on 8 which r e f l e c t s the d e c i s i o n maker's b e l i e f o f t h e l i k e l i h o o d o f a g i v e n e v e n t o c c u r r i n g . That i s , i f i t i s b e l i e v e d t h a t B i s more l i k e l y t o o c c u r t h a n C, t h e n W(B) > W(C). A second axiom w h i c h g i v e s r i s e t o some c o n c e r n i s Axiom IV. I n t h i s case a a - a l g e b r a B c o n t a i n i n g 0 was assumed t o e x i s t . A p r o b a b i l i t y measure y was d e f i n e d on 9 , and Axiom IV s p e c i f i e d t h e r e l a t i o n between y and W. S e v e r a l o f t h e paradoxes d i s c u s s e d were shown t o c o n t r a d i e t e i t h e r (or both) o f t h e s e . That i s a measure W, on 8 c o u l d n o t be d e f i n e d such t h a t Axiom I I would a l s o be s a t i s f i e d . I n A l l a i s ' paradox i t was shown by MacCrimmon and L a r s s o n (1975) t h a t t h i s c o n t r a d i c t i o n t o t h e axiom o n l y o c c u r r e d when the amount o f reward money was s u b s t a n t i a l l y above what most p e o p l e a r e used t o h a n d l i n g . Hence, A l l a i s ' paradox does not c o n t r a d i c t Axiom I I when i t p e r t a i n s t o d e c i s i o n s c o n c e r n i n g amounts o f money w h i c h most p e o p l e a r e accustomed t o h a n d l i n g . The same argument i s n o t t r u e , however, f o r E l l s b e r g ' s paradox, s i n c e i n t h a t s i t u a t i o n c o n t r a d i c t i o n s o c c u r r e d f o r even s m a l l amounts o f money. A l t h o u g h b o t h E l l s b e r g ' s paradox and A l l a i s ' paradox c o n t r a d i c t Axiom I I , i t seems r e a s o n a b l e , however, t o assume t h a t E l l s b e r g ' s paradox d i f f e r e n t i a t e s between known v e r s u s unknown p r o b a b i l i t i e s , r a t h e r than v i o l a t i n g t h e a d d i t i v i t y a s s umptions. These assumptions a r e , o f c o u r s e , n o t m u t u a l l y e x c l u s i v e s i n c e a r e j e c t i o n o f the e x i s t e n c e o f p r o b a b i l i t i e s on unknown e v e n t s would a l s o l e a d t o t h e r e j e c t i o n o f t h e a d d i t i v i t y assumption. In P a r t I I o f the t h e s i s , we a t t e m p t e d t o d i f f e r e n t i a t e between unknown and known p r o b a b i l i t i e s i n an a l t e r n a t i v e manner. T h i s was done by d e f i n i n g a l e s s r e s t r i c t e d s e t f u n c t i o n on a l l e v e n t s such t h a t the s e t f u n c t i o n i s e q u a l t o the p r o b a b i l i t y o f known e v e n t s but does n o t n e c e s s a r i l y s a t i s f y a l l the laws of p r o b a b i l i t y f o r unknown e v e n t s . That i s , the s e t f u n c t i o n i s l e s s s t r u c t u r e d i n t h e l a t t e r c a s e . A l t h o u g h o t h e r approaches e x i s t which d i f f e r e n t i a t e between p a r t i a l r i s k problems and r i s k o r u n c e r t a i n t y problems, t h e i r major emphasis has n o t .focussed on the unknown e v e n t s t h e m s e l v e s . R a t h e r , a form o f c o r r e c t i o n f a c t o r i s assumed t o a p p l y a f t e r the e x p e c t e d u t i l i t y c r i t e r i o n has been c a l -c u l a t e d a c c o r d i n g t o p r o b a b i l i t y laws. In a l l o t h e r approaches t h e r e f o r e t h e e x i s t e n c e o f p r o b a b i l i t i e s f o r a l l e v e n t s has been t a k e n f o r g r a n t e d . The method used i n P a r t I I was t o s p e c i f y a s e t o f axioms which a d e c i s i o n maker would be w i l l i n g t o f o l l o w i f 219 he a c c e p t s the e x p e c t e d u t i l i t y c r i t e r i o n f o r d e c i s i o n problems under r i s k . Most o f the axioms a r e p r o p e r t i e s o f t h e e x p e c t e d u t i l i t y c r i t e r i o n . The axioms i n P a r t I I , t h e r e f o r e , reduce t o the ex p e c t e d u t i l i t y c r i t e r i o n i f a d d i t i o n a l assumptions a r e made on the P-measure. 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"Fuzzy s e t s " , I n f o r m a t i o n and C o n t r o l 8^:338-353 (1965) . Zadeh, L. A. " P r o b a b i l i t y measures o f f u z z y e v e n t s " , J . Math. A n a l , and A p p l . 23:421-427 (1968) . Zadeh, L. A. "Toward a t h e o r y o f f u z z y systems", ERL Report No. 69-2, E l e c t r o n i c s Research L a b o r a t o r i e s , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , June (1969) . 226 Appendix I Some o f t h e main theorems which guarantee the e x i s t e n c e o f the e v a l u a t i o n f u n c t i o n f are summarized i n t h i s a ppendix. As was s t a t e d i n s e c t i o n 3, the c o n d i t i o n s f o r the e x i s t e n c e o f f a r e o f two t y p e s — one s e t o f c o n d i t i o n s on t h e o r d e r r e l a t i o n and the second on the t o p o l o g y on A. Because o f t h e i r i m p o r t a n c e we s h a l l g i v e a b r i e f summary o f t h e t y p e s o f r e l a t i o n s t o be c o n s i d e r e d . D e f i n i t i o n . L e t S be a s e t . A b i n a r y r e l a t i o n Y on S i s a s u b s e t o f the C a r t e s i a n p r o d u c t S x S. S y m b o l i c a l l y , Y C S x S. By t h i s d e f i n i t i o n the empty s e t 0 i s a r e l a t i o n . T h i s i s the " s m a l l e s t " r e l a t i o n s i n c e i t i s a s u b s e t o f a l l o t h e r r e l a t i o n s . S i m i l a r l y , we have t h e " l a r g e s t " r e l a t i o n L = SxS whi c h c o n t a i n s a l l r e l a t i o n s on S x S. Other r e l a t i o n s w h i c h o c c u r o f t e n enough t o deserve mention a r e t h e d i a g o n a l  r e l a t i o n , V , o f a s e t S, d e f i n e d by V = {(x,y) e SxS|x = y } , and the d u a l r e l a t i o n Y 1 o f a r e l a t i o n Y, d e f i n e d by Y' = { ( x , y ) | ( y , x ) e Y}. The d u a l i s , t h e r e f o r e , the m i r r o r image a c r o s s the d i a g o n a l . The complement r e l a t i o n , Y o f a r e l a t i o n Y, i s the complement o f the s e t Y, i . e . , Y = L-Y, t h e u s u a l complement used i n s e t t h e o r y . M u l t i p l i c a t i o n o f two r e l a t i o n s can a l s o be d e f i n e d . L e t C and D be two r e l a t i o n s on the same s e t S. The p r o d u c t denoted by C • D, i s d e f i n e d by C • D = { ( x , y ) | ( x , z ) e C,(z,y) e D, f o r some z e S}. 227 We s h a l l n o t d i s c u s s the p r o p e r t i e s o f t h i s p r o d u c t , b u t we s h a l l use i t i n some d e f i n i t i o n s . Some r e l a t i o n s have p r o -p e r t i e s o c c u r r i n g so o f t e n t h a t names have been g i v e n t o t h e s e p r o p e r t i e s . We s h a l l summarize t h e s e here and use t h e de-f i n i t i o n s by Chipman (1960) f o r the more common ones. D e f i n i t i o n . L e t t i n g M be any r e l a t i o n on a s e t S we d e f i n e t h e terms: 1. T r a n s i t i v e : M • M C M 2. N e g a t i v e T r a n s i t i v e : M • M C M 3. R e f l e x i v e : V C M 4. I r r e f l e x i v e : V C M 5. C o m p a r a b i l i t y : L C M O M' 6. Symmetric: M' C M 7. Asymmetric: M'C M 8. A n t i s y m m e t r i c : M C\ M1 C V D e f i n i t i o n . A r e l a t i o n M i s a p a r t i a l o r d e r i n g on a s e t S p r o v i d e d M i s r e l f e x i v e , a n t i s y m m e t r i c and t r a n s i t i v e . S S I t i s common t o w r i t e x < y (or y > x) r a t h e r t h a n (x,y) e M. We s h a l l adopt t h e same n o t a t i o n h e r e . As a r u l e we s h a l l denote any o r d e r i n g by < (or >) i f i t i s o b v i o u s on which s e t t h e o r d e r i n g i s d e f i n e d . Where doubt may o c c u r we A i n d i c a t e t h e s e t a l o n g w i t h t h e o r d e r i n g , i . e . , < i s an o r d e r r e l a t i o n on A. S A s s o c i a t e d w i t h any p a r t i a l o r d e r i n g < we can d e f i n e a S S S S r e l a t i o n < by: x < y i f and o n l y i f x ^ y and not y 4 x. S C l e a r l y < i s t r a n s i t i v e and i r r e f l e x i v e , and i s sometimes used as a d e f i n i t i o n o f a p a r t i a l o r d e r i n g . Here we w i l l c a l l i t 228 a s t r i c t p a r t i a S 1 o r d e r . We can a l s o d e f i n e a r e l a t i o n = by: S S S x = y i f and o n l y i f x U and y ^ x. The second t y p e o f o r d e r i n g we s h a l l c o n s i d e r i s a weak o r d e r i n g . S S D e f i n i t i o n . A r e l a t i o n < i s a weak o r d e r i n g p r o v i d e d < i s asymmetric and n e g a t i v e t r a n s i t i v e . S i n c e an asymmetric and n e g a t i v e t r a n s i t i v e b i n a r y r e l a t i o n i s i r r e f l e x i v e and t r a n s i t i v e a s t r i c t p a r t i a l o r d e r i n g i s c o n t a i n e d i n a weak o r d e r i n g . I f M i s e i t h e r a s t r i c t p a r t i a l o r d e r i n g o r a weak o r d e r i n g , t h e n we d e f i n e M = by: M M M M x = y i f n o t x < y and n o t y < x and d e f i n e ^ by: M M M x < y i f x < y o r x = y. In d o i n g s o , we g a i n the c o m p a r a b i l i t y p r o p e r t y , i . e . , M M g i v e n any x,y e M th e n e i t h e r x y o r y ^ x. Thus, we have reached t h e o r d e r i n g t h a t r e a l numbers s a t i s f y e x c e p t f o r perhaps t h e an t i s y m m e t r y p r o p e r t y . M De f i n i t i o n . A r e l a t i o n < i s a complete o r d e r i n g i f i t i s comparable and t r a n s i t i v e . I t w i l l , t h e r e f o r e , not come as a s u r p r i s e i f the o r d e r i n g we need f o r t h e e x i s t e n c e o f f i s e i t h e r a weak o r d e r i n g o r a p a r t i a l o r d e r i n g . 229 T o p o l o g i c a l n e c e s s i t i e s . I f we o n l y assume t h a t t h e o r d e r i n g g i v e n i s e i t h e r a weak o r d e r i n g o r a s t r i c t p a r t i a l o r d e r i n g on a s e t A, then i t i s not s u f f i c i e n t t o s p e c i f y an o r d e r p r e s e r v i n g r e a l - v a l u e d f u n c t i o n . We have t o make a d d i t i o n a l assumptions on the c a r d i n a l i t y o f t h e s e t A. A Theorem 1. I f < i s a weak o r d e r i n g on A and t h e c a r d i n a l i t y o f A i s c o u n t a b l e t h e n t h e r e i s a r e a l -v a l u e d f u n c t i o n f on A such t h a t A a < b i m p l i e s f ( a ) < f ( b ) f o r a l l a,b e A. The r e s u l t would s t i l l h o l d i f we r e p l a c e d t h e weak o r d e r i n g by a s t r i c t p a r t i a l o r d e r i n g . We s h a l l n o t prove t h i s theorem here as the p r o o f i s easy and can be found i n most books on u t i l i t y t h e o r y . I t was f i r s t proved i n t h i s form by Debreu (1954). I t i s n o t d i f f i c u l t however t o see t h a t t h e c a r d i n -a l i t y assumption i n Theorem 1 c o u l d be weakened. We would o n l y need t o have t h e s e t o f e q u i v a l e n c e c l a s s e s i n d u c e d by the weak o r d e r i n g t o be c o u n t a b l e . In t h i s c a s e , we i n d u c e an A o r d e r i n g on t h e e q u i v a l e n c e c l a s s e s by the o r d e r i n g <. I f we now remove the c o n d i t i o n t h a t A must be c o u n t a b l e (or t h a t t h e s e t o f e q u i v a l e n c e c l a s s e s must be c o u n t a b l e ) , we must s t i l l keep some c o n t r o l o v e r the c a r d i n a l i t y o f A. The c o n d i t i o n r e q u i r e d i s the e x i s t e n c e o f a c o u n t a b l e , dense s u b s e t o f A. T h i s was used by F i s h b u r n (1970) . P e l e g (1970) c a l l e d t h i s c o n d i t i o n a s e p a r a b i l i t y c o n d i t i o n . Here, we s h a l l use F i s h b u r n ' s t e r m i n o l o g y s i n c e s e p a r a b i l i t y as used i n 230 t o p o l o g y means something q u i t e d i f f e r e n t . A D e f i n i t i o n . L e t < be a b i n a r y r e l a t i o n on A, and l e t Z be a s u b s e t o f A. Then Z i s o r d e r - d e n s e i n A i f and o n l y A A i f whenever a ^ b t h e r e i s a z e Z such t h a t a ^ z and A z < b. L e t A be the s e t of a l l r e a l numbers and Z t h e s e t o f a l l r a t i o n a l numbers, and l e t t h e o r d e r i n g d e f i n e d by t h e n a t u r a l o r d e r i n g o f numbers on b o t h A and Z. Then Z i s o r d e r - d e n s e s i n c e between e v e r y two d i s t i n c t numbers t h e r e e x i s t s a r a t i o n a l number. T h i s c o n d i t i o n c o n t r o l s the c a r d i n a l i t y o f the s e t A i f the c a r d i n a l i t y o f Z i s s p e c i f i e d and i s s u f f i c i e n t f o r p r o v i n g the second theorem ( F i s h b u r n , 1970, p . 2 7 ) . Theorem 2. I f A i s weakly o r d e r e d and t h e r e i s a c o u n t a b l e s u b s e t Z C A which i s o r d e r - d e n s e i n A, t h e n t h e r e i s a r e a l - v a l u e d f u n c t i o n f on A such t h a t A a < b i m p l i e s f ( a ) < f ( b ) f o r a l l a,b e A. We would now l i k e t o r e l a t e the concept o f o r d e r - d e n s e s e t s t o more i n t u i t i v e c o n c e p t s i n t o p o l o g y . To do so we must i n t r o d u c e some o f t h e i d e a s i n t o p o l o g y . One o f t h e fundamental c o n c e p t s o f t o p o l o g y i s t h a t o f " n e a r n e s s " . One n a t u r a l way o f d e f i n i n g what we mean by " a c t i o n a i s near a c t i o n b" f o r some a,b e A, i s by i n d u c i n g a d i s t a n c e measure on A. L e t us f i r s t assume t h a t t h e r e e x i s t s a r e a l - v a l u e d f u n c t i o n f on A . A weak t o p o l o g y may then be d e f i n e d on A , be assuming t h a t f o r any open s e t o f r e a l numbers N c o n t a i n i n g f ( a ) , the s e t {b e A | f ( b ) e N} i s d e f i n e d t o be open f o r a l l a e A . I n d o i n g so, f would always be c o n t i n u o u s and the c o n c e p t o f "nearness" i s i n h e r i t e d from "nearness" on t h e r e a l l i n e . T h i s d e f i n i t i o n i s e q u i v a l e n t t o s a y i n g t h a t t h e s e t s {z e A | f ( z ) < f ( a ) } and {z £ A | f ( a ) < f ( z ) } a r e open f o r a l l a e A . I t a l s o i m p l i e s t h a t i f a ^ , i = l , 2 , . . . i s a sequence o f a c t i o n s , {a^} converges t o a i f and o n l y i f l i m f (a.) = f ( a ) . I f the f u n c t i o n f i s not g i v e n , b u t a weak o r d e r i n g A A e x i s t s on A such t h a t a,b £ A i m p l i e s e i t h e r a < b o r b < a, t h e n a t o p o l o g y T may be d e f i n e d on A by l e t t i n g the s e t s A A {z £ A | a < z l a n d {z e A | Z < a} f o r a l l a £ A form a subbase f o r the t o p o l o g y T. We s h a l l denote th e c l o s u r e o f t h e s e t i A . A , {z £ A | a < z} by {z £ A | a ^ z } . We now show how t h i s can be r e l a t e d back t o o r d e r dense s e t s . The c o n d i t i o n n e c e s s a r y f o r t h i s i s : i f A i s s e p a r a b l e and c o n n e c t e d ( i n a t o p o l o g i c a l s e n s e ) , t h e n t h e r e e x i s t s a c o u n t a b l e s u b s e t Z o f A which i s o r d e r dense i n A . A A To see t h i s , l e t a < b. Then the s e t s {z £ A | z < a} and {z e A | b ^ z} a r e d i s j o i n t , c l o s e d and non-empty and t h e r e f o r e , A , A {z £ A | z < b} f\ {z £ A | a < z} 232 i s open and non-empty. S i n c e A i s s e p a r a b l e i t must c o n t a i n a c o u n t a b l e dense s u b s e t Z such t h a t t h e r e e x i s t s a z e Z A A s a t i s f y i n g a < z and z < b. T h e r e f o r e i f A i s weakly o r d e r e d , and i s s e p a r a b l e and c o n n e c t e d , t h i s i m p l i e s t h a t t h e r e e x i s t s a c o u n t a b l e s u b s e t Z C A w h i c h i s o r d e r dense i n A. A l s o note t h a t t h e e x i s t e n c e o f such a s e t Z i s a l l t h a t i s needed f o r the e x i s -t e n c e o f a r e a l - v a l u e d f u n c t i o n f on A such t h a t A a < b i m p l i e s f ( a ) < f ( b ) f o r a l l a,b e A by Theorem 2. T h i s g i v e s us the f o l l o w i n g theorem (Debreu, 1954) . Theorem 3. I f A i s w e a k l y - o r d e r e d , s e p a r a b l e and c o n n e c t e d , w i t h the n a t u r a l t o p o l o g y T, t h e n t h e r e e x i s t s a r e a l - v a l u e d f u n c t i o n f on A, such t h a t : A i f a < b then f ( a ) < f ( b ) f o r a l l a,b e A. S i m i l a r l y , P e l e g (1970) has p r o v e d the same theorem f o r s t r i c t p a r t i a l o r d e r i n g : Theorem 4. I f A i s a s t r i c t p a r t i a l o r d e r i n g on a connected, s e p a r a b l e s e t A w i t h the n a t u r a l t o p o l o g y T, then t h e r e e x i s t s a r e a l - v a l u e d f u n c t i o n f on A such A t h a t : i f a < b then f ( a ) < f ( b ) f o r a l l a,b e A. A summary o f c o n d i t i o n s n e c e s s a r y f o r the e x i s t e n c e o f f now f o l l o w s : 1) O r d e r i n g a s s u m p t i o n , i . e . , t r a n s i t i v i t y , p a r t i a l , weak 233 o r complete. 2) Topology a s s u m p t i o n , i . e . , the c a r d i n a l i t y assumption on A. From a d e c i s i o n maker's p o i n t o f view we would t h e r e f o r e be most i n t e r e s t e d i n the t r a n s i t i v i t y and c o m p l e t e n e s s , s i n c e t h e t o p o l o g i c a l a ssumption does not a f f e c t p r e f e r e n c e s among a l t e r n a t i v e s . 234 Appendix I I The purpose o f t h i s appendix i s t o summarize some o f t h e b a s i c d e f i n i t i o n s and r e s u l t s c o n c e r n i n g measure and f u n c t i o n t h e o r y . 1. Measures and t h e e x t e n s i o n theorem The c l a s s 0 o f s u b s e t s o f ft whi c h has the f o l l o w i n g p r o p e r t i e s : a) ft e 0 b) i f B e 0 t h e n B e 0 , where B = ft - B CO c) i f B 1,B 2,... t h e n U B e 0 i = l i s c a l l e d a a - f i e l d o r a - a l g e b r a . A measurable space i s a s e t ft, and a a - a l g e b r a 0 o f s u b s e t s o f ft. A measure on a a - f i e l d 0 i s a n o n - n e g a t i v e , extended r e a l - v a l u e d f u n c t i o n y such t h a t whenever B ^  ,B 2,. . . form a f i n i t e o r c o u n t a b l y i n f i n i t e c o l l e c t i o n o f d i s j o i n t s e t s i n 0 we have u ( U B n ) = Z vte >. n n I f y ( f t ) = 1, y i s c a l l e d a p r o b a b i l i t y measure. A measure space i s a t r i p l e ( f t , 0 , y ) where ft i s a s e t , 0 i s a a - a l g e b r a o f s u b s e t s o f ft, and y i s a measure on 0 . I f y i s a p r o b a b i l i t y measure ( f t , 0 , y ) i s c a l l e d a p r o b a b i l i t y space. The d e f i n i t i o n o f a p r o b a b i l i t y measure i m p l i e s t h e f o l l o w i n g consequences where a l l s e t s a r e members o f 0 . 1. y ( 4 > ) = 0. 2. y ( E ) = 1 - y ( E ) . 235 3. p ( E U F ) + y ( E f\ F ) = y ( E ) + y ( F ) . 4. E C F i m p l i e s t h a t y ( E ) = y ( F ) - ( F ~ E ) < y ( F ) . 5. Monotone p r o p e r t y . I f E N i s an i n c r e a s i n g sequence o f s e t o r a d e c r e a s i n g sequence o f s e t c o n v e r g i n g t o E , then y ( E N ) -> y ( E ) . 6. B o o l e ' s i n e q u a l i t y . y ( U E . ) 4 JE. U (E.j_) . i i On each s u b s e t D o f ft, we can d e f i n e an o u t e r measure by y*(D) = i n f y(M) M e 6 MOD and an i n n e r measure by y*(D) = sup y(M) M e 9 MCD The f o l l o w i n g p r o p e r t i e s o f the i n n e r and o u t e r measure a r e e a s i l y v e r i f i e d (or can be found i n Halmos, 1950, Lebesque, 1901, o r Royden, 1968). * i ) V * ( D ) -4 Vi (D) f o r a l l D e f t i i ) i f C U D £ 9 , C A D M then y* (C) + y* (D) = y (CO D) * i i i ) i f C A D = 0, t h e n y* (C (J D) ^ V* (C) + y (D) < ^ y*(C U D) i v ) y*(D) = y*(D) = y(D) i f D e 9 . F o r D e 0 and D P\ C = 0 the f o l l o w i n g s i x p r o p e r t i e s h o l d : if ^ £ v) y (D O C) = y (D) + y (C) v i ) y ^ ( D O c ) =y*(D) + y * ( C ) v i i ) y* (C) = y* (C 0 D) + y* (C ( J D) v i i i ) \i* (C) = y* (C U D) + y* (C U D) i x ) y*(C) = 1 - y*(C) 236 x) i f D C F and C C E where F C\ E = 0 and F,E e 0 t h e n y * ( C O D ) = y * ( C ) + y*(D) y * ( c O D) = y * ( C ) + u*(D) . * A more g e n e r a l d e f i n i t i o n o f an o u t e r measure y i s sometimes used. I t i s an extended r e a l - v a l u e d s e t f u n c t i o n d e f i n e d on a l l s u b s e t s o f a space ft and h a v i n g the f o l l o w i n g p r o p e r t i e s : i ) y * ( 0 ) = 0 i i ) i f A C B then y (A) £ y (B) oo oo ^ i i i ) i f E C U E. th e n y (E) < 2EIU (E. ) . i = l 1 i = l 1 The second p r o p e r t y i s c a l l e d m o n o t o n i c i t y and t h e t h i r d c o u n t a b l e s u b a d d i t i v i t y . I n view o f ( i ) f i n i t e s u b a d d i t i v i t y f o l l o w s from ( i i i ) . From an o u t e r measure a new measure y ^ can be d e f i n e d . I n t h i s case t h e d i f f i c u l t y a r i s e s i n making s u r e from w h i c h s e t s i n ft the a d d i t i v i t y p r o p e r t i e s h o l d . T h i s i s u s u a l l y done by s p e c i f y i n g a c l a s s o f measurable s e t s i n the f o l l o w i n g * way: A s e t E i s measurable w i t h r e s p e c t t o y i f f o r e v e r y s e t A we have u*(A) = y * ( A H E ) + y * (A A E) . * T h i s guarantees t h a t y i s a d d i t i v e o u t o f t h i s c l a s s o f s e t s , and a l s o t h a t t h e c l a s s o f measurable s e t s i s a a - a l g e b r a (Royden, 1968, Theorem 1, p.251). 237 * Theorem A I : The c l a s s f o f y -measurable s e t s i s a a - a l g e b r a . * I f we s t a r t w i t h an o u t e r measure y , and d e f i n e a measure y ^ * _ from t h e y , and t h e n i n d u c e an o u t e r measure y ^ from y ^ , and * _ * i f i t so happens t h a t y = y ^ we say y i s a r e g u l a r o u t e r measure. * A r e g u l a r o u t e r measure y has t h e p r o p e r t y t h a t i f i s a sequence o f i n c r e a s i n g s e t s such t h a t U B ^ = B then l i m y ( B i ) = y ( B ) . T h i s i s t h e p r o p e r t y needed i n Axiom V ( P a r t I I o f t h e t h e s i s ) . 2. Measurable f u n c t i o n s G i v e n two spaces ft, R, and a mapping X ( c o ) : f t •+ R, the i n v e r s e image o f a s e t B C R i s d e f i n e d as X _ 1 B = {co e ft;X(co) e B } . Denote t h i s by { X e B > . The t a k i n g o f i n v e r s e images p r e s e r v e s a l l s e t o p e r a t i o n s ; t h a t i s { X e U B X ) = vJ { X e B X > , X X { X e O M = e B i > ' x A X { X e B } = { X e B } 238 Given two measurable spaces ( f t , 0 ) , {R,V), a f u n c t i o n X:ft + R i s c a l l e d measurable i f the i n v e r s e o f e v e r y s e t i n ¥ i s i n 0 . T h e r e f o r e i f a measure i s d e f i n e d on 0 , an i n d u c e d  measure P may be d e f i n e d on ¥ by the measurable mapping X as f o l l o w s : By the p r o p e r t i e s o f the i n v e r s e image P i s g u a r a n t e e d t o be a measure. I f a sequence o f measurable f u n c t i o n s c o nverges t o a f u n c t i o n , then t h i s f u n c t i o n i s a l s o measurable (Royden, 1968, Theorem 6, p.223). Theorem A l l . The c l a s s o f ©-measurable f u n c t i o n s i s c l o s e d under p o i n t w i s e convergence. That i s , i f X (•) a r e ©-measurable f o r each n, and l i m X (co) n n n e x i s t s f o r e v e r y co, th e n X(co) = l i m X (co) i s J ' n n ©-measurable. 3. I n t e g r a t i o n s L e t X(«) be a f u n c t i o n from ft t o R, th e n X(') i s a s i m p l e f u n c t i o n i f t h e r e e x i s t s a f i n i t e number o f s u b s e t s o f ft, denoted by B n,...,B such t h a t 1 n P (B) = y (X e B) f o r a l l B B . O B . = i 3 i and X_, (•) = r . f o r a l l i = l , . . . , n . 239 To d e f i n e t h e i n t e g r a l we f i r s t d e f i n e t h e i n t e g r a l f o r s i m p l e f u n c t i o n s , t h e n n o n - n e g a t i v e f u n c t i o n s and l a s t l y f o r a r b i t r a r y f u n c t i o n s . The i n t e g r a l . Take ( f t , 9 , y ) t o be a measure space, and l e t R be the r e a l l i n e , t h e n / Xdy o f the n o n - n e g a t i v e n measurable s i m p l e f u n c t i o n i s d e f i n e d by 5" r.y ( B . ) . L e t i = l 1 1 X(co) ^ 0 be a n o n - n e g a t i v e 0-measurable f u n c t i o n . To d e f i n e t h e i n t e g r a l o f X l e t X n 0 be s i m p l e f u n c t i o n s such t h a t X i n c r e a s e s t o X. n Fo r X i n c r e a s i n g t o X, i t i s easy t o show t h a t n 3 ; X L,dy ^ / X dy > 0. n+1 J n D e f i n e J'xdy as l i n ^ y x ^ y . F u r t h e r m o r e , the v a l u e o f t h i s l i m i t i s the same f o r a l l sequences o f n o n - n e g a t i v e s i m p l e f u n c t i o n s c o n v e r g i n g up t o X (Halmos, p.101). I f J\x\d\i < °°, d e f i n e ^ X d y = fx+d\i - y"x~dy; where X ( - ) + = max{X (• , 0)} and X ( - ) ~ = max{ - X ( - ) , 0 } . The e l e m e n t a r y p r o p e r t i e s o f the i n t e g r a l a r e : I f t h e i n t e g r a l s o f X and Y e x i s t , i ) X > Y i m p l i e s ^ d y > J"Ydy, i i ) J{aX + 8 Y ) d y = ajXdy + 8 ^ Y d y , i i i ) A ,B £ G,A O B = 0 i m p l i e s ^ X d y = + J*d]i • A « - > B A B Some nonelementary p r o p e r t i e s a r e : Monotone Convergence Theorem A I I I . (Halmos, 1950, Theorem B , p.112). F o r X n > 0 n o n - n e g a t i v e i n c r e a s i n g 240 0 - m e a s u r a b l e f u n c t i o n s , w h i c h converge t o X, then l i m dn = ^ X d y . Theorem AIV. (Halmos, 1950, Theorem I , p . 9 8 ) . L e t X(-) be a B o r e l measurable f u n c t i o n such t h a t J Xdy e x i s t s n I f B. i s a s e t o f d i s j o i n t s u b s e t s o f Q t h e n x ^ X d y = Z yXdy. B i T h i s summarizes the b a s i c d e f i n i t i o n s and r e s u l t s needed i n P a r t I and P a r t I I o f the t h e s i s . 241 Appendix I I I I n t h i s a ppendix we s h a l l summarize some o f t h e d i f f e r e n t approaches t o the e x p e c t e d u t i l i t y t h e o r y . B e f o r e s t a t i n g t h e axiom we s h a l l summarize the n o t a t i o n s used c o n c e r n i n g the reward f u n c t i o n s . For each a e A, a f u n c t i o n X(«,a) from ft t o R i s c a l l e d a reward f u n c t i o n . I f Y(') i s a f u n c t i o n from B t o R where B CI ft, th e n Y i s c a l l e d a r e s t r i c t i o n o f X(-,a) i f X(',a) = Y(«) f o r a l l (i) e B. The f u n c t i o n Y(-) w i l l , i n t h i s c a s e , be denoted by X B ( • , a ) . L e t X_,(*,a) and X-(*,c) be two r e s t r i c t i o n s o f the reward f u n c t i o n s X(-,a) and X(-,c) i f t h e r e e x i s t s a f u n c t i o n Y(-) from B U C t o R such t h a t Y(«) = X_,(',a) and Y(-) = X ^ ( * , c ) . I t w i l l be c o n v e n i e n t a t some c a s e s t o denote t h e f u n c t i o n Y(') by X_ .. _ , ( - , a c ) , e s p e c i a l l y so i n t he Luce and K r a n t z approach. Approaches t o e x p e c t e d u t i l i t y t h e o r y . The d i f f e r e n t approaches we have c o n s i d e r e d were d e v e l o p e d by: 1) vonNeumann-Morganstern (1947) 2) Marschak (1959) 3) Savage (1954) 4) Arrow (1971) 5) Luce and K r a n t z (1971) 242 We s h a l l summarize t h e axioms assumed by each o f t h e s e . vontTeumann-Morgenstern Axioms. The vonNeumann-Morgenstern (1947) approach does n ot d i r e c t l y make any ass u m p t i o n s on t h e u n d e r l y i n g p r o b a b i l i t y space (Q,. Q , y) o r on r . I t does assume t h a t n, t h e s e t o f a l l i n d u c e d p r o b a b i l i t y measures on R, i s e q u a l t o t h e s e t o f a l l d i s c r e t e p r o b a b i l i t y measures on R. They i n d u c e an o r d e r i n g on n from an o r d e r i n g on R. To p r e s e n t t h e i r a p proach, we need some new d e f i n i t i o n s . L e t n " = (0,1) x R x R, and l e t F be a f u n c t i o n from n" t o R. The s e t n" can be tho u g h t o f as the s e t o f t h o s e i n d u c e d p r o -b a b i l i t y measures on R which are non-zero f o r e x a c t l y two elements o f R. The v a l u e F ( a , r ^ , r 2 ) = can be tho u g h t o f as the reward r ^ e R wh i c h would make us i n d i f f e r e n t between the gambles. 1. r e c e i v i n g r ^ w i t h the p r o b a b i l i t y a r e c e i v i n g r 2 w i t h the p r o b a b i l i t y 1-a 2. r e c e i v i n g r ^ w i t h t h e p r o b a b i l i t y 1 In what f o l l o w s , i t i s assumed t h a t r ^ ^ ^ r ^ e R and t h a t a, 8 and y a r e r e a l numbers on ( 0 , 1 ) . R Axiom NMl: < i s a complete, h e r e d i t a r y o r d e r i n g . R R Axiom NM2: r ^ < r 2 i m p l i e s r ^ < F ( a r ^ , r 2 ) and R F ( c x , r ^ , r 2 ) < r 2 f o r a l l a e (0,1) R R Axiom NM3: r ^ ^  r ^ < r 2 i m p l i e s the e x i s t e n c e o f an R a e (0,1) and a y e (0,1) w i t h F ( a , r ^ , r 2 ) < r ^ and 243 R r 3 < F ( Y , r 1 , r 2 ) . R Axiom NM4: F ( a , r ^ , r 2 ) = F (1-a ,r>, ,r^) and R F ( a , F ( y , r 1 , r 2 ) , r 2 ) = F f a y , ^ , ^ ) f o r a l l a,y e ( 0 , 1 ) . These assumptions a r e s u f f i c i e n t t o prove t h a t a r e a l v a l u e d f u n c t i o n U e x i s t s such t h a t R r.^ < r 2 i m p l i e s U(r^) < U ( r 2 ) and U ( R a , r l f r 2 ) ) = a U ( r 1 ) + ( l - a ) U ( r 2 ) . Marschak Axioms. Marschak (1950) was t h e f i r s t t o adopt an approach o f e s t a b l i s h i n g an o r d e r i n g on t h e p r o b a b i l i t y measures. Samuelson (1952), H e r s t e i n and M i l n o r (1953) and o t h e r a u t h o r s have a l s o adopted t h i s f o r m u l a t i o n . The axioms we s h a l l g i v e h e r e a r e e s s e n t i a l l y t h e same as Jensen's (1964) axioms, and he has shown them t o i m p l y Marschak's axioms. I n t h i s approach a l s o , we i g n o r e the u n d e r l y i n g p r o b a b i l -i t y space s i n c e a l l assumptions a r e based on t h e i n d u c e d p r o b a b i l i t y measures. R i s assumed t o be a f i n i t e s e t . We denote II, as b e f o r e , t o be t h e s e t o f a l l p r o b a b i l i t y measures on R. n Axiom M l : < i s a complete, h e r e d i t a r y o r d e r i n g . n Axiom M2: I f P^,P 2,P 3 e II and P 1 < P 2 , t h e n f o r any r e a l number a e ( 0 , 1 ) , 244 n a P 1 + ( l - a ) P 3 < a P 2 + ( l - a ) P 3 n n Axiom M3: I f P^ < P 2 < P 3 t h e n t h e r e e x i s t s r e a l n numbers a, 3 e (0,1) such t h a t aP^ + ( l - a ) P 3 < P 2 and n P 2 < &?1 + ( 1 - 8 ) P 3 . These t h r e e a s sumptions a r e s u f f i c i e n t t o d e r i v e a u t i l i t y f u n c t i o n . Savage Axioms. Savage (1954) s t a r t s w i t h an u n d e r l y i n g p r o b a b i l i t y space ( f t , 0 , y ) . The t h e o r y does n o t h o l d , however, f o r an a r b i t r a r y p r o b a b i l i t y space. There a r e r e s t r i c t i o n s on t h e c a r d i n a l i t y o f A (Axiom S6) and 0 must c o n t a i n a l l s u b s e t s o f ti. The p r o b a b i l i t y y i s not e x p l i c i t l y d e f i n e d b u t i s d e r i v e d from p r e f e r e n c e r e l a t i o n s on s u b s e t s o f 0,. r Axiom S I ; < i s a complete h e r e d i t a r y o r d e r i n g . L e t B,B e 8 and ^ T.' r2 e R a n d a / b , c , d e A -Axiom S2: I f Xg(-,a) = X B ( * , b ) , X B(«,c) = X B(«,d), X B ( - , a ) = X - ( . , c ) , X B ( - , b ) = X-(-,d) r and X(•,a) ^ X(•,b) r t h e n X(•,c) * X(•,d) Axiom S3: L e t X(-,a) = r-j and X(-,b) = r 2 I f X B ( - , c ) = X B ( - , a ) , X B(-,d) = X B ( - , b ) 245 and X-( • ,c) = X-(•,d) r t h e n X(-,a) £ X(-,b) i f and o n l y i f X(-,c) < X(-,d) f o r a l l B such t h a t B i s n o t n u l l 2 Axiom S4: I f B,C C ft a,b,c,d e A, and X B ( - , a ) = r x x c ( - , b ) = ^ X B ( - , c ) = r 3 X c(-,d) = r 3 and X ( • ,a) th e n X(•,c) r i e R 1=1,2,3,4 , 3 ^ > r 2 , r 3 > r 4 X-(-,a) = r 2 X ? ( - , b ) = r 2 X B ( - , c ) = r 4 X^(-fd) = r 4 •S X(-,b) r * X ( • , d ) . Axiom S5: There i s a t l e a s t one p a i r o f rewards r l ' r 2 e ^ such t h a t i f a,b e A a r e d e f i n e d by X(-,a) = r ^ and X(*,b) 5 r 2 , t h e n r X(-,a) = r± < X(-,b) = r 2 . r Axiom S6; I f X(-,a) < X(-,b) and r i s any r e w a r d i n R, t h e n t h e r e e x i s t s a p a r t i t i o n P o f ft, such t h a t f o r any B e P 2A s e t B e 0 i s n u l l i f y(B) = 0, o r i n Savage t e r m i n o l o g y , B r i s n u l l i f X D ( - , a ) = X n ( - , c ) f o r a l l a,c e A. 246 X B ( - , c ) = r , X-(-,c) = X-(-,a) i m p l i e s r X( • ,c) < X(•,b) and X B ( - , d ) = r , X B(-,d) = X-(-,b) i m p l i e s r X(-,a) < X(-,d) r Axiom S7: I f X B U , a ) < X ( t o,b) f o r a l l u e B, t h e n X B ( - , a ) < X B ( - , b ) The f i r s t s i x axioms a r e s u f f i c i e n t t o g u a r a n t e e a prob-a b i l i t y y and a f u n c t i o n U such t h a t t h e e x p e c t e d u t i l i t y p r e s e r v e s t h e o r d e r i n g when the reward s e t i s f i n i t e . To ex t e n d t h e r e s u l t t o an i n f i n i t e reward s e t , Axiom S7 i s a l s o needed. Arrow's Axioms. Arrow (1971) does n o t make t h e r e s -t r i c t i o n s on t h e p r o b a b i l i t y space ( f t , 0 , y ) t h a t Savage does but h i s o v e r a l l approach i s s i m i l a r t o Savage's. A Axiom A l : < i s a complete, h e r e d i t a r y o r d e r i n g . A Axiom A2: G i v e n a,b e A where b < a, a reward r e R, and {E"""} a sequence o f s e t s i n 0 such t h a t E 1 +"'"c E 1 w i t h 0 E. = 0. D e f i n e a c t i o n s a 1 e A, b 1 e A by X E i ( ' ' a i ) = X E i ( - ' a ) ' X E i ( - ' a i ) = C -X I i ( - , b 1 ) = X ^ i ( - , b ) , X E i ( - , b 1 ) = c. 247 Then f o r a l l i s u f f i c i n e t l y l a r g e A i i A b < a and b < a. A Axiom A3: For any g i v e n event E, < s a t i s f i e s Axiom A2 such t h a t any two a c t i o n s a,b e A where X„(',a) = X„(-,b) hi hi w i l l be i n d i f f e r e n t g i v e n E. T h i s i s denoted as a = b|E. Weak p r e f e r e n c e i s A A denoted as a < b|E and s t r i c t p r e f e r e n c e as a < b|E. Axiom A4: L e t P be a p a r t i t i o n . G i v e n two a c t i o n s A . A a,b e A, i f f o r e v e r y E e P, b < a|E, the n b < a. I f i n a d d i t i o n , t h e r e i s a c o l l e c t i o n P' o f e v e n t s i n P, A . whose u n i o n i s n o n - n u l l , such t h a t b < a|E, E e P', A then b < a. Axiom A5: I f a,b,c,d e A and X ( t o 1,a) = X ( u ) 2 , b ) , X ( c o 1,c) = X ( t o 2 , d ) , A , . A , c < a | t o ^ i m p l i e s d < b | c o 2 . Axiom A6: The p r o b a b i l i t y d i s t r i b u t i o n o f s t a t e s o f the w o r l d i s a t o m l e s s . I f the p r o b a b i l i t y d i s t r i b u t i o n o f consequences i s the same f o r two a c t i o n s , t h e y a r e i n d i f f e r e n t . T h i s s e t o f axioms i s a l s o s u f f i c i e n t t o p r o v e t h e e x i s t e n c e o f a r e a l - v a l u e d u t i l i t y f u n c t i o n U on R such t h a t the o r d e r i n g on A s a t i s f i e s t h e e x p e c t e d u t i l i t y c r i t e r i o n . 248 Luce and K r a n t z Axioms. One argument c r i t i c i z i n g Savage's axioms has been t h a t a l l the random v a r i a b l e s have been d e f i n e d on the same s t a t e space. F or example, i n b e t t i n g t h a t heads w i l l o c c u r when a c o i n i s f l i p p e d , o r when c o n s i d -e r i n g i n v e s t i n g i n a p a r t i c u l a r s t o c k , c l e a r l y t h e s t a t e o f the w o r l d i s q u i t e d i f f e r e n t . Luce and K r a n t z (1971) dev e l o p e d an axiom system t o handle t h i s c a s e , as f o l l o w s : L e t $ denote a s e t o f f u n c t i o n s from n o n - n u l l e v e n t s and 0 a g i v e n a l g e b r a and U t h e n u l l e v e n t s i n 0 . U s i n g t h i s n o t a t i o n t h e i r axioms can be s t a t e d a s : Assume t h a t C,B e 0 . Axiom LK1: i ) I f D A B = 0 t h e n „ _, (• ,ac) z $ i i ) I f B C D, X D ( - , a ) e $ i m p l i e s X B ( - , a ) e * . > i s a weak o r d e r i n g . I f D C\ B = 0 and X ( - , a ) = x_(«,b) t h e n X D U B ( , ' a b ) = V-' a )' Axiom LK4: I f D A B = 0 t h e n X D ( ' , a ) > X D(-,b) i f and o n l y i f X D U B ( ' ' a G ) * X D U B ( * ' b c ) -Axiom LK2: Axiom LK3: 249 Axiom LK5: I f D n B = 0, X ^ i • , i ) = X g (•,i) i=a,b,c,d X D U B ( ' ' a g ) = X D U B ( ' ' b h ) a n d X D U B ( * ' k a ) = X D U B ( , ' l b ) t h 6 n X D l ) B ( - , c g ) >, X D U B ( - , d h ) i f and o n l y i f X D U B ( * ' k c ) * X D U B ( " ' l d ) Axiom LK6: I f D n B = 0, and f o r any sequence o f a c t i o n i e M, X B ( - , a i ) ? X N ( - , a 2 ) and ^ s ' - ' V l ' = X D O B ( - ' a i + l a 2 ) f ° r a 1 1 1 t h e n e i t h e r M i s f i n i t e or { X D ( - , a i ) | i e M} i s unbounded, Axiom LK7: i ) I f B e II and D C B, t h e n D e 11. i i ) B e 11 i f and o n l y i f f o r a l l X D O B ( - ' a b ) £ * ' X D U B ( , ' a b ) = V ' a ) ' Axiom LK8: 0 - 1 1 c o n t a i n s a t l e a s t t h r e e p a i r w i s e d i s -j o i n t e l e m e n t s . $ c o n t a i n s a t l e a s t two a c t i o n s such t h a t X D ( - , a ) £ X B ( - , b ) . Axiom LK9: i ) I f C and X_(-,a) a r e g i v e n t h e n t h e r e — — — ^ — — jj $ e x i s t s X^i',&) e $ f o r w h i c h X_,(-,a) = X ^ ( * , d ) . i i ) I f D (\ B = 0 and X D U B ( - ,ac) * x D u B ( -> x m » o ( * , b c ) f t h e n t h e r e e x i s t s X_(*,e) such t h a t X D U B ( - ' d ) * X D U B ( * ' e c ) -T h i s completes t h e d i f f e r e n t axioms f o r t h e d i f f e r e n t approaches c o n s i d e r e d h e r e . 

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