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UBC Theses and Dissertations

Two representation theorems and their application to decision theory 1980

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TWO REPRESENTATION THEOREMS AND THEIR APPLICATION TO DECISION THEORY by i^CHEW; SOO HONG M.A., Claremont G r a d u a t e S c h o o l , 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES ( I n t e r d i s c i p l i n a r y Programme) ( M a t h e m a t i c s , Economics, Management S c i e n c e ) 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 November- 1980 0 ChewJ, Soo Hong, 1980 In presenting th is thes is in p a r t i a l fu l f i lment o f the requirements fo an advanced degree at the Un ivers i ty of B r i t i s h Columbia,r I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l ica t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of I n t e r d i s c i p l i n a r y The Univers i ty of B r i t i s h Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 Date ABSTRACT 1 1 T h i s d i s s e r t a t i o n c o n s i s t s o f two p a r t s . P a r t I c o n t a i n s t h e s t a t e - ments and p r o o f s o f two r e p r e s e n t a t i o n theorems. The f i r s t t h eorem, p r o v e d i n C h a p t e r 1, g e n e r a l i z e s t h e q u a s i l i n e a r mean o f Hardy, L i t t l e w o o d and P o l y by weakening t h e i r axiom o f q u a s i l i n e a r i t y . G i v e n two d i s t r i b u t i o n s w i t h t h e same means, q u a s i l i n e a r i t y r e q u i r e s t h a t m i x t u r e s o f t h e s e d i s t r i b u t i o n s w i t h a n o t h e r d i s t r i b u t i o n i n t h e same p r o p o r t i o n s s h a r e t h e same mean, r e g a r d l e s s o f t h e d i s t r i b u t i o n t h a t t h e y a r e m i x e d . w i t h . We weaken t h e quas l i n e a r i t y axiom by a l l o w i n g t h e p r o p o r t i o n s t h a t g i v e r i s e t o t h e same means t o be d i f f e r e n t , T h i s l e a d s t o a more g e n e r a l mean, denoted by M , w h i c h has t h e form: % ( F ) = * ~ V R < ^ d F / / R c . d F ) , where <J> i s c o n t i n u o u s and s t r i c t l y monotone, a i s c o n t i n u o u s and s t r i c t l y p o s i t i v e ( n e g a t i v e ) and F i 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 . The q u a s i l i n e a r mean, denoted by M^, r e s u l t s when t h e a f u n c t i o n i s c o n s t a n t . We showed, i n a d d i t i o n , t h a t t h e M mean has t h e i n t e r m e d i a t e v a l u e p r o p e r t y , and can be c o n s i s t e n t w i t h t h e s t o c h a s t i c dominance ( i n c l u d i n g h i g h e r degree ones) p a r t i a l o r d e r . We a l s o g e n e r a l i z e d a w e l l known i n e q u a l i t y among q u a s i l i n e a r means, v i a t h e o b s e r v a t i o n t h a t t h e mean o f a d i s t r i b u t i o n F can be w r i t t e n as t h e q u a s i l i n e a r mean o f a d i s t r i b u t i o n F , where F i s d e r i v e d f r o m F v i a a as t h e Radon-Nikodym d e r i v a t i v e o f F w i t h r e s p e c t t o F. We n o t e d t h a t the mean i n d u c e s an o r d e r i n g among p r o b a b i l i t y d i s t r i b u t i o n s v i a t h e maximand, /Ra<}>dF//.RadF, t h a t c o n t a i n s t h e ( e x p e c t e d u t i l i t y ) maximand, /^<J)dF, o f t h e q u a s i l i n e a r mean as a s p e c i a l c a s e . C h a p t e r 2 p r o v i d e s an a l t e r n a t i v e c h a r a c t e r i z a t i o n o f t h e above r e p r e s e n - t a t i o n f o r s i m p l e p r o b a b i l i t y measures on a more g e n e r a l outcome s e t where mean v a l u e s may not be d e f i n e d . In t h i s c a s e , axioms a r e s t a t e d d i r e c t l y i n terms o f p r o p e r t i e s o f t h e u n d e r l y i n g o r d e r i n g . We r e t a i n e d s e v e r a l s t a n d a r d 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 , namely weak o r d e r , s o l v a b i l i t y and m o n o t o n i c i t y b u t r e l a x e d t h e s u b s t i t u t a b i l i t y axiom o f P r a t t , R a i f f a and S c h l a i f e r , w h i c h i s e s s e n t i a l l y a r e s t a t e m e n t o f q u a s i - l i n e a r i t y i n t h e c o n t e x t o f an o r d e r i n g . P a r t I I o f t h e d i s s e r t a t i o n c o n c e r n s one s p e c i f i c a r e a o f a p p l i c a t i o n d e c i s i o n t h e o r y . I n t e r p r e t i n g t h e M ^ ( F ) mean o f C h a p t e r 1 as t h e c e r t a i n t y e q u i v a l e n t o f a monetary l o t t e r y F, t h e c o r r e s p o n d i n g i n d u c e d b i n a r y r e l a t i o n has t h e n a t u r a l i n t e r p r e t a t i o n as ' s t r i c t p r e f e r e n c e ' between l o t t e r i e s . F o r non-monetary ( f i n i t e ) l o t t e r i e s , we a p p l y t h e r e p r e s e n t a t i o n theorem o f C h a p t e r 2. The h y p o t h e s i s , t h a t a c h o i c e a g e n t ' s p r e f e r e n c e among l o t t e r i e s can be r e p r e s e n t e d by a p a i r o f a and cj) f u n c t i o n s t h r o u g h t h e i n d u c e d o r d e r i n g , i s r e f e r r e d t o as a l p h a u t i l i t y t h e o r y . T h i s i s l o g i c a l l y 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 c h o i c e agent obeys e i t h e r t h e mean v a l u e ( c e r t a i n t y e q u i v a l e n t ) axioms o r t h e axioms on h i s s t r i c t p r e f e r e n c e b i n a r y r e l a t i o n . A l p h a u t i l i t y t h e o r y i s a g e n e r a l i z a t i o n o f e x p e c t e d u t i l i t y t h e o r y i n t h e sense t h a t the e x p e c t e d u t i l i t y r e p r e s e n t a t i o n i s a s p e c i a l c a s e o f t h e a l p h a u t i l i t y r e p r e s e n t a t i o n . The m o t i v a t i o n f o r g e n e r a l i z i n g e x p e c t e d u t i l i t y comes from d i f f i c u l t i e s i t f a c e d i n t h e d e s c r i p t i o n o f c e r t a i n c h o i c e phenomena, e s p e c i a l l y t h e A l l a i s p a radox. These a re summarized i n C h a p t e r 3. C h a p t e r 4 c o n t a i n s t h e f o r m a l s t a t e m e n t s o f as s u m p t i o n s and t h e d e r i v a t i o n s o f n o r m a t i v e and d e s c r i p t i v e i m p l i c a t i o n s o f a l p h a u t i l i t y t h e o r y . We s t a t e d c o n d i t i o n s , t a k e n from C h a p t e r 1 , f o r c o n s i s t e n c y w i t h i v s t o c h a s t i c dominance and g l o b a l r i s k a v e r s i o n and d e r i v e d a g e n e r a l i z e d A r r o w - P r a t t i n d e x o f l o c a l r i s k a v e r s i o n . We a l s o d e m o n s t r a t e d how 1 a l p h a u t i l i t y t h e o r y can be c o n s i s t e n t w i t h t h o s e c h o i c e phenomena t h a t c o n t r a d i c t t h e i m p l i c a t i o n s o f 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 e i t h e r s t o c h a s t i c dominance o r l o c a l r i s k a v e r s i o n . The c h a p t e r ended w i t h a c o m p a r i s o n o f a l p h a u t i l i t y w i t h two o t h e r t h e o r i e s t h a t have a t t r a c t e d a t t e n t i o n ; namely, A l l a i s ' t h e o r y and p r o s p e c t t h e o r y . S e v e r a l o t h e r a p p l i c a t i o n s o f t h e r e p r e s e n t a t i o n theorems o f P a r t I are c o n s i d e r e d i n t h e C o n c l u s i o n o f t h i s d i s s e r t a t i o n . These i n c l u d e t h e use o f the mean as a model o f t h e e q u a l l y - d i s t r i b u t e d - e q u i v a l e n t l e v e l o f income ( A t k i n s o n , 1 970), and as a measure o f asymmetry o f a d i s t r i b u t i o n ( C a n n i n g , 1934). The a l p h a u t i l i t y r e p r e s e n t a t i o n can a l s o be u s e d t o r a n k s o c i a l s i t u a t i o n s i n t h e sense o f H a r s a n y i (1977). We ended by p o i n t i n g out an open q u e s t i o n r e g a r d i n g c o n d i t i o n s f o r c o m p a r a t i v e r i s k a v e r s i o n and s t a t e d an e x t e n s i o n o f Samuelson's (1967) c o n j e c t u r e t h a t Arrow's i m p o s s i b i l i t y theorem would h o l d i f i n d i v i d u a l s and s o c i e t y e x p r e s s t h e i r p r e f e r e n c e s by von Neumann-Morgenstern u t i l i t y f u n c t i o n s . CONTENTS v INTRODUCTION 1 PART I W REPRESENTATION THEOREMS 1 GENERALIZING THE QUASI LINEAR MEAN OF HARDY, LITTLEWOOD AND POLYA 7 1.1 INTRODUCTION 7 1.2 AXIOMS OF MEAN VALUE 8 1.3 REPRESENTATION THEOREM 14 1.4 PROPERTIES OF THE M , MEAN 29 a<j> 2 GENERALIZING THE EXPECTED U T I L I T Y REPRESENTATION THEOREM 36 2.1 INTRODUCTION 36 2.2 PRELIMINARY DEFINITIONS 36 2.3 AXIOMS 38 2.4 REPRESENTATION THEOREMS 40 PART II APPLICATION TO DECISION THEORY BACKGROUND 59 3 CRITIQUE OF EXPECTED U T I L I T Y THEORY 61 3.1 INTRODUCTION 61 3.2 SYSTEMATIC VIOLATION OF THE STRONG INDEPENDENCE PRINCIPLE 64 v i 3.3 CONCURRENCE OF RISK SEEKING AND RISK AVERTING BEHAVIOR 70 3.4 SOME PROBLEMS WITH PROBLEM REPRESENTATION 77 3.5 SUMMARY 80 4 A NEW THEORY 81 4.1 INTERPRETING MEAN VALUE AS CERTAINTY EQUIVALENT 81 4.2 REPRESENTATION OF A PREFERENCE BINARY RELATION 86 4.3 NORMATIVE IMPLICATIONS 90 4.3.1 R a t i o C o n s i s t e n c y 91 4.3.2 Assessment 97 4.3.3 S t o c h a s t i c Dominance 100 4.3.4 G l o b a l R i s k A v e r s i o n 103 4.3.5 L o c a l R i s k A v e r s i o n : The A r r o w - P r a t t Index 105 4.4 DESCRIPTIVE IMPLICATIONS 108 4.4.1 S y s t e m a t i c V i o l a t i o n o f t h e S t r o n g Independence Axiom 108 4.4.2 S t o c h a s t i c Dominance 114 4.4.3 L o c a l and G l o b a l R i s k P r o p e r t i e s : C o n c u r r e n c e o f R i s k A v e r t i n g and R i s k S e e k i n g B e h a v i o r 116 4.4.4 Some Problems w i t h Problem R e p r e s e n t a t i o n 122 4.5 CRITIQUE OF ALLAIS* THEORY AND PROSPECT THEORY 124 4.5.1 A l l a i s 1 Theory 124 4.5.2 P r o s p e c t Theory 131 4.5.3 Comparison 137 v i i CONCLUSION 5 CONCLUSION 141 5.1 SUMMARY 1 4 1 5.2 EXTENSIONS 143 v i i i T A B L E S 3.1 Summary o f E m p i r i c a l R e s u l t s R e l e v a n t t o t h e HILO L o t t e r y S t r u c t u r e 68 4.1 A l l o w a b l e C h o i c e P a t t e r n s u n d e r A l p h a U t i l i t y Theory 110 4.2 Comparison among T h e o r i e s 139 i x FIGURES 3.1 A " t y p i c a l " von Neumann-Morgenstern u t i l i t y f u n c t i o n 62 3.2 Four d e c i s i o n problems 64 3.3 The c o m p o s i t i o n o f t h e A T ( B ) l o t t e r y from A ( B n ) 65 3.4 The HILO s t r u c t u r e o f t h r e e consequence l o t t e r i e s 66 3.5 S t a n d a r d l o t t e r y c o m p a rison 70 3.6 Examples o f u c and u ^ (based on d a t a from A l l a i s ( 1 9 7 7 ) , Appendix C) 74 3.7 Examples o f u^, and u^ (based on d a t a from MacCrimmon e t . a l . (1972)) 75 3.8 G r a p h i c a l r e p r e s e n t a t i o n o f two l o t t e r i e s 77 3.9 A s e q u e n t i a l r e p r e s e n t a t i o n o f l o t t e r y B 77 4.1 R a t i o C o n s i s t e n c y i l l u s t r a t e d u s i n g b a r y c e n t r i c c o o r d i n a t e s 92 4.2 G e o m e t r i c p r o o f o f t h e R a t i o C o n s i s t e n c y p r o p e r t y 95 4.3 P r o b a b i l i t y E q u i v a l e n t method 98 4.4 T e s t o f S u b s t i t u t i o n axiom 98 4.5 P r e f e r e n c e P a t t e r n f o r ct(I) < 1 109 4.6 C o n s i s t e n c y C o n d i t i o n s f o r S t o c h a s t i c Dominance 115 4.7 C o n d i t i o n s f o r L o c a l R i s k A v e r s i o n 117 4.8 An a d m i s s i b l e a l p h a f u n c t i o n 118 4.9 A p a i r o f u £ and u ^ d e r i v e d from an a l p h a u t i l i t y d e c i s i o n maker 120 4.10 A p a i r o f u and u d e r i v e d from an a l p h a u t i l i t y d e c i s i o n maker 120 g 3/i» r 5.1 An a l p h a f u n c t i o n t h a t d i s c r i m i n a t e s a g a i n s t t h e r i c h 145 PREFACE The preface i s perhaps a s u i t a b l e place f o r an i n f o r m a l d i s c u s s i o n of ideas l e a d i n g to t h i s d i s s e r t a t i o n . In t h e i r a x i o m a t i z a t i o n of expected u t i l i t y , von Neumann and Morgenstern l i k e n e d the " u t i l i t y " , l o o s e l y speaking, of a l o t t e r y to the center o f g r a v i t y of a mass d i s t r i b u t i o n . In other words, the u t i l i t y , u(x^,...> X N 5 P ^ »•••»P n)» °f a l o t t e r y , (x^,...,x ;p^,..., Pn)» which pays x^ with p r o b a b i l i t y p^, should be the mean or average of the u t i l i t i e s a s s o c i a t e d w i t h each probable outcome. This l e d to the expected u t i l i t y e xpression: n u(x,,...,x ;p,,...,p ) = .S,p.u(x.). ^ 1 n r l r n i = l r i 1 Together with t h e i r elegant axiomatic c h a r a c t e r i z a t i o n , there seemed to be a strong case f o r the adoption of expected u t i l i t y . The A l l a i s - S a v a g e contro- versy however clouded the p i c t u r e somewhat. Some, i n c l u d i n g A l l a i s , suggested r e p l a c i n g the p r o b a b i l i t i e s , { p ^ ^ - i * with a more general set of weights, ^ i ^ i - 1 ' w*1^c'1 depend on the l o t t e r y , and sum to u n i t y . In t h i s case, n u(x.,...,x ;p,,...,p ) = .2 , 4 . (x..,... ,x ;p.,...,p ) u ( x . ) . 1 n r l r n i = l r i v 1 n r l r n I At t h i s l e v e l o f g e n e r a l i t y , the only t e s t a b l e i m p l i c a t i o n i s that the u t i l i t y o f a l o t t e r y i s intermediate i n value between the maximum and the minimum a t t a i n a b l e u t i l i t i e s . This property, which may be c a l l e d intermediate value property, i s compatible w i t h our i n t u i t i o n about the u t i l i t y of a l o t t e r y as a mean value. To impose more s t r u c t u r e , one may consider r e s t r i c t i n g the <JK weights to: w (Xj,...,x n;p 1 }...,p n) 0) i(x 1,...,x n;p 1,...,p n) = , j ? l w j C x i > • • • > x n ' P i ' • • • 'Pn-1 where {w^}^^ i s a set of p o s i t i v e l y valued weight f u n c t i o n s that depend on x i the l o t t e r y , (x^,.. . >xn»P-_> • • • >Pn) • A f u r t h e r r e s t r i c t i o n i s obtained by imposing a d e s i r a b l e property c a l l e d combination property: the u t i l i t y of a l o t t e r y remains unchanged i f we combine d i f f e r e n t p r o b a b i l i t i e s o f g e t t i n g the same outcome. This i m p l i e s t h a t the ŵ  f u n c t i o n s are of the form: w.(x.,...,x ;p,,...,p ) = p.w.(x.;x .,p . ) , 1 1 n r l r n r i 1 l - i ' - i / ' where x ^ and p ^ denote the outcome v e c t o r and p r o b a b i l i t y v e c t o r with the i t h component dele t e d . With a bo l d s l e i g h t o f hand, as yet unsubstantiated by any a p r i o r i reason, the values are assumed to be obtained from a s i n g l e f u n c t i o n a evaluated at the i t h outcome, x^. This leads to a f a i r l y t r a c t a b l e g e n e r a l i z a t i o n o f expected u t i l i t y : n p i a ( x i ) u ( x 1 , . . . f x n ; p 1 , . . . , p n ) = J^I } u ( x . ) . j ? l P j a ( x j ) Of course, a n i c e expression i s j u s t the f i r s t step. The next t h i n g i s to work back and f o r t h i n order to i d e n t i f y a minimal set of c h a r a c t e r i s t i c p r o p e r t i e s from which the a and u fun c t i o n s can be constructed. These p r o p e r t i e s , once found, would then n e c e s s a r i l y be weaker than the cor r e s - ponding ones f o r expected u t i l i t y . The f i r s t proof o f such a r e p r e s e n t a t i o n theorem i s i n the context of g e n e r a l i z i n g the q u a s i l i n e a r mean of Hardy, L i t t l e w o o d , and Polya given i n Chapter 1. The corresponding r e s u l t i n terms o f a preference ordering i s given i n Chapter 2. Although the b a s i c r e s u l t f o l l o w s from a s t r a i g h t f o r w a r d r e i n t e r p r e t a t i o n of those of Chapter 1, a s e l f - c o n t a i n e d treatment o f Chapter 2 i s presented so t h a t readers who are used t o the preference o r d e r i n g approach can s k i p Chapter 1. Bob Weber provided an elegant geometrical i n t e r p r e t a t i o n o f one of the axioms (Ratio Consistency) i n an e a r l i e r v e r s i o n of the p r e f e r - ence o r d e r i n g theorem and demonstrated that i t was redundant. This l e d to a x i i much weaker axiom i n the current proof of the mean value representation theorem. As i s usual, the organization of the d i s s e r t a t i o n assumes the normal trappings of academic wr i t i n g . Formal r e s u l t s , which are developed r i g o r - ously i n Part I, precede i n t e r p r e t a t i o n s i n the context of decision theory i n Part I I , designed to b u i l d a prima f a c i e case f o r the adoption of the more general expected u t i l i t y hypothesis on both t h e o r e t i c a l and empirical grounds. This d i v i s i o n may cause the appearance of repetitiousness, but has the advantage of making Part II self-contained. Parts of the material i n Part II are the r e s u l t of j o i n t work with Ken MacCrimmon, my research supervisor. I would also l i k e to acknowledge my debt to the other members on my guidance committee. The numerous instances when I went to Shelby Brumelle and Cindy Greenwood for help had been instrumental i n enabling me to carry through the analysis i n the proofs of the representation theorems, and also i n the understanding of some basic mathematics. I have benefited from Daniel Kahneman's research with Amos Tversky on the psychology of judgement and decision-making under uncertainty and also from the seminars i n h i s home. I have also benefited from discussions with Dave Donaldson on h i s work with Charles Blackorby on the measurement of i n e q u a l i t y and poverty. I was exposed to research i n the economics of uncertainty during Yoshitsugo Kanemoto's seminars. Thanks are also due to a non-member, John Butterworth, f o r suggesting the problem during the f i n a l examination of h i s information choice course. x i i i To my QH.a.ndmotkQA. 1 INTRODUCTION Two w e l l known r e p r e s e n t a t i o n theorems p r o v i d e t h e s t a r t i n g p o i n t f o r t h i s d i s s e r t a t i o n . The f i r s t , due t o Hardy, L i t t l e w o o d and P o l y a ( 1 9 3 4 ) , i s an a x i o m a t i c c h a r a c t e r i z a t i o n o f a r a t h e r g e n e r a l c l a s s o f mean v a l u e s c a l l e d t h e q u a s i l i n e a r mean: ^ ( F ) = C J T V R cj>dF). Mjj(F) denotes t h e q u a s i l i n e a r mean a s s o c i a t e d w i t h a p r o b a b i l i t y d i s t r i b u t i o n F and c h a r a c t e r i z e d by a s t r i c t l y monotone f u n c t i o n <J>. Ha r d y , L i t t l e w o o d and P o l y a p r o v e d t h e i r c h a r a c t e r i z a t i o n f o r s i m p 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 d e f i n e d on a compact i n t e r v a l , . Examples o f q u a s i l i n e a r means i n c l u d e t h e w i d e l y u s e d a r i t h m e t i c mean (<b i s l i n e a r ) , t he g e o m e t r i c mean (<j> i s l o g a r i t h m i c ) , t he harmonic mean (cj> i s o f t h e form ^) and t h e r t h moment mean, a l s o known as t h e g e n e r a l mean o f o r d e r r (cj> i s o f t h e form x r ) . The second r e p r e s e n t a t i o n theorem has i t s g e n e s i s i n t h e S t . P e t e r s b u r g ' s paradox -- an i n d i v i d u a l i s not w i l l i n g t o s t a k e a l l t h a t he p o s s e s s e s t o t a k e p a r t i n a l o t t e r y t h a t pays 2 1 d o l l a r s w i t h ^ chance; t h u s , d e m o n s t r a t i n g t h e l i m i t a t i o n o f u s i n g m a t h e m a t i c a l e x p e c t a t i o n o f p a y o f f s as a g e n e r a l r u l e f o r t h e o r d e r i n g o f r i s k y p r o s p e c t s . T h i s l e d B e r n o u l l i t o propose, i n 1738, t h e e x p e c t a t i o n o f a 'moral w o r t h ' f u n c t i o n , u, o f w e a l t h as an a l t e r n a t i v e . I n p a r t i c u l a r . , he used t h e l o g a r i t h m i c f u n c t i o n d e r i v e d by assuming t h a t an i n f i n i t e - s i m a l i n c r e a s e i n t h e wo r t h o f w e a l t h i s p r o p o r t i o n a l t o an i n f i n i t e - s i m a l i n c r e a s e i n w e a l t h b u t i n v e r s e l y p r o p o r t i o n a l t o w e a l t h i t s e l f . The monetary w o r t h o r c e r t a i n t y e q u i v a l e n t M(F) c o r r e s p o n d - 2 i n g t o a l o t t e r y F i s t h e n g i v e n by, u(M(F)) = / R u d F ; o r a l t e r n a t i v e l y , M(F) = u _ : L ( r R u d F ) . Note t h a t t h i s e x p r e s s i o n i s t h e same as t h a t d e f i n i n g t h e q u a s i - l i n e a r mean. The f i r s t a x i o m a t i c t r e a t m e n t l e a d i n g t o t h e e x p e c t a t i o n o f a f u n c t i o n o f p a y o f f s as a r u l e f o r t h e o r d e r i n g o f l o t t e r i e s i s g i v e n by Ramsey (1926) i n h i s " F o u n d a t i o n o f M a t h e m a t i c s " , von Neumann and M o r g e n s t e r n (1947) p r o v i d e d an a l t e r n a t i v e a x i o m a t i z a t i o n i n d e p e n d e n t l y i n t h e i r "Theory o f Games and Economic B e h a v i o r " and i n i t i a t e d t h e use o f t h e term " u t i l i t y " . They p r o v e d t h e e x i s t e n c e o f an o r d e r - p r e s e r v i n g map on a m i x t u r e s e t (e.g. t h e space 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 ) s u b j e c t t o a m i n i m a l s e t o f p o s t u l a t e s such t h a t t h e o r d e r - p r e s e r v i n g map i s t h e e x p e c t a t i o n o f a u t i l i t y f u n c t i o n . T h e i r r e s u l t i s now commonly r e f e r r e d t o a s . t h e e x p e c t e d u t i l i t y theorem. The u s e f u l n e s s o f q u a s i l i n e a r means needs no e l a b o r a 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 . e . , t h e a p p l i c 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 theorem t o d e c i s i o n - m a k i n g by i n t e r p r e t i n g t h e b i n a r y r e l a t i o n as a p r e f e r e n c e r e l a t i o n and the m i x t u r e s e t as a s e t o f r i s k y a l t e r n a t i v e s F o r a s u r v e y o f t h e use o f t h e r t h moment mean i n s t a t i s t i c s , see N o r r i s ( 1 9 7 6 ) , B l a c k o r b y § Donaldson (1978a) c o n t a i n s examples o f t h e use o f q u a s i l i n e a r mean i n t h e measurement o f income i n e q u a l i t y . Weerahandi § Z i d e k (1979) p r o v i d e s an a l t e r n a t i v e c h a r a c t e r i z a t i o n o f the r t h moment mean f o r p r o b a b i l i t y d i s t r i b u t i o n s d e f i n e d on the p o s i t i v e h a l f - l i n e . Ben-Tal (1977) showed t h a t q u a s i l i n e a r means ar e o r d i n a r y a r i t h m e t i c means d e f i n e d on l i n e a r spaces w i t h s u i t - a b l y chosen o p e r a t i o n s o f a d d i t i o n and m u l t i p l i c a t i o n . 3 or e q u i v a l e n t l y , t h e i n t e r p r e t a t i o n o f t h e q u a s i l i n e a r mean as a c e r t a i n t y e q u i v a l e n t , has a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n s i n c e i t s i n c e p t i o n . A l t e r n a t i v e a x i o m a t i z a t i o n s were g i v e n by Marschak ( 1 9 5 0 ) , Samuelson ( 1 9 5 2 ) , H e r s t e i n and M i l n o r ( 1 9 5 3 ) , Savage ( 1 9 5 4 ) , Anscombe and Aumann ( 1 9 6 3 ) , P r a t t , R a i f f a and S c h l a i f e r ( 1 9 6 4 ) , J e n s e n ( 1 9 6 7 ) , DeGroot ( 1 9 7 0 ) , F i s h b u r n ( 1 9 7 0 ) , Arrow (1971) and o t h e r s . Savage ( 1 9 5 4 ) , B l a c k w e l l and G i r s h i c k (1961) and DeGroot (1970) a p p l i e d e x p e c t e d u t i l i t y t h e o r y t o s t a t i s t i c a l d e c i s i o n s . I n a d d i t i o n , i t s e r v e d as the f o u n d a t i o n f o r Arrow (1971) and Marschak and Radner (1972) i n t h e i r i n v e s t i g a t i o n o f t h e economics o f u n c e r t a i n t y , and f o r Howard (1964) and Keeney and R a i f f a (1976) i n t h e i r work on d e c i s i o n a n a l y s i s . E x p e c t e d u t i l i t y t h e o r y , t h o ugh, has been l e s s s u c c e s s f u l i n d e s c r i b i n g and e x p l a i n i n g a c t u a l c h o i c e s (Edwards, 1961; S l o v i c , F i s c h h o f f and L i c h t e n s t e i n , 1977). Even von Neumann and M o r g e n s t e r n r e a l i z e d a t t h e o u t s e t t h a t e x p e c t e d u t i l i t y r u l e s out c o m p l e m e n t a r i t y among m u t u a l l y e x c l u s i v e consequences, a u t i l i t y f o r g a m b l i n g p e r s e , and o t h e r b e h a v i o r s t h a t seem r e l a t i v e l y common (von Neumann and M o r g e n s t e r n , 1947, Ap p e n d i x A, Sec. 3 ) . S u b s e q u e n t l y , v a r i o u s c h a l l e n g e s , b e g i n n i n g w i t h t h e A l l a i s p a radox ( A l l a i s , 1953), have c a l l e d i n t o q u e s t i o n t h e e m p i r i c a l v a l i d i t y o f a key p r o p e r t y o f e x p e c t e d u t i l i t y t h e o r y , t h e s t r o n g independence p r i n c i p l e (Marschak, 1950; Samuelson, 1952). The s t r o n g independence p r i n c i p l e r e q u i r e s t h a t r a n k i n g among l o t t e r i e s r e m a i n s u n a l t e r e d when each l o t t e r y i s composed w i t h an i d e n t i c a l l o t t e r y u s i n g t h e same p r o b a b i l i t y . T h i s i s c l o s e l y l i n k e d t o t h e axiom o f q u a s i l i n e a r i t y o f Hardy, L i t t l e w o o d and P o l y a ( 1 9 3 4 ) } 4 t h e " s u b s t i t u t i o n o f l o t t e r i e s " o f P r a t t , R a i f f a and S c h l a i f e r (1964) and t h e " s u r e - t h i n g " p r i n c i p l e o f Savage (1954). P a r t I o f t h i s d i s s e r t a t i o n c o n t a i n s t h e s t a t e m e n t s and p r o o f s o f two r e p r e s e n t a t i o n theorems. The f i r s t t h e o r e m , p r o v e d i n C h a p t e r 1 g e n e r a l i z e s t h e q u a s i l i n e a r mean o f Hardy, L i t t l e w o o d and P o l y a by weakening t h e i r axiom o f q u a s i l i n e a r i t y . G i v e n two d i s t r i b u t i o n s w i t h t h e same means, q u a s i l i n e a r i t y r e q u i r e s t h a t m i x t u r e s o f t h e s e d i s t r i b u t i o n s w i t h a n o t h e r d i s t r i b u t i o n i n t h e same p r o p o r t i o n s s h a r e t h e same mean, r e g a r d l e s s o f t h e d i s t r i b u t i o n t h a t t h e y mixed w i t h . We weaken t h e q u a s i l i n e a r i t y axiom by a l l o w i n g t h e p r o p o r t i o n s t h a t g i v e r i s e t o t h e same means t o be d i f f e r e n t . T h i s g i v e s r i s e t o a more g e n e r a l mean, M^, t h a t i s s p e c i f i e d by a c o n t i n u o u s and s t r i c t l y p o s i t i v e ( n e g a t i v e ) f u n c t i o n , a , and a c o n t i n o u s and s t r i c t l y monotone f u n c t i o n , <J>. The q u a s i l i n e a r mean r e s u l t s when t h e a f u n c t i o n i s c o n s t a n t . I n a d d i t i o n , we show t h a t t h e mean has t h e I n t e r m e d i a t e V a l u e P r o p e r t y , and p r o v i d e 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 s f o r c o n s i s t e n c y w i t h t h e s t o c h a s t i c dominance ( i n c l u d i n g h i g h e r d e g r e e ones) p a r t i a l o r d e r . We a l s o g e n e r a l i z e a w e l l known i n e q u a l i t y among q u a s i l i n e a r means, by t h e o b s e r v a t i o n t h a t t h e mean o f a d i s t r i b u t i o n F can be w r i t t e n as t h e q u a s i l i n e a r mean o f a d i s t r i b u t i o n F where F i s d e r i v e d f r o m F v i a a a s t h e Radon-Nikodym ct d e r i v a t i v e o f F w i t h r e s p e c t t o F. As was n o t e d e a r l i e r , t h e mean i n d u c e s an o r d e r i n g among d i s t r i b u t i o n s v i a t h e ( e x p e c t e d u t i l i t y ) maximand, / R * d F . C o r r e s p o n d i n g l y , t h e M ^ mean i n d u c e s a more g e n e r a l o r d e r i n g v i a t h e 5 maximand, / Ra<f>dF// RadF. We p r o v e , i n C h a p t e r 2, an a l t e r n a t i v e c h a r a c t e r i z a t i o n o f t h e above r e p r e s e n t a t i o n f o r s i m p l e p r o b a b i l i t y measures on a more g e n e r a l outcome s e t where mean v a l u e s may n o t be d e f i n e d . I n t h i s c a s e , axioms are s t a t e d d i r e c t l y i n t e r m s o f p r o p e r t i e s o f t h e u n d e r l y i n g o r d e r i n g . We r e t a i n s e v e r a l s t a n d a r d 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 , namely weak o r d e r , s o l v a b i l i t y and m o n o t o n i c i t y but r e l a x t h e s u b s t i t u t i o n p r i n c i p l e o f P r a t t , R a i f f a and S c h l a i f e r , w h i c h i s e s s e n t i a l l y a r e s t a t e m e n t o f q u a s i l i n e a r i t y i n t h e c o n t e x t o f an o r d e r i n g . The m o t i v a t i o n f o r t h e r e s e a r c h c o n t a i n e d i n P a r t I comes from p a r a d o x e s i n t h e f i e l d o f d e c i s i o n t h e o r y . Hence, t h e f o r m u l a t i o n o f a new t h e o r y o f c h o i c e ( c a l l e d a l p h a u t i l i t y t h e o r y ) t h a t g e n e r a l i z e s e x p e c t e d u t i l i t y t h e o r y c o n s t i t u t e s P a r t I I ( C h a p t e r s 3 and 4) o f t h i s d i s s e r t a t i o n . C h a p t e r 3 g a t h e r s t o g e t h e r t h e c r i t i c i s m s and e m p i r i c a l f i n d i n g s which c o n t r a d i c t t h e i m p l i c a t i o n s o f e x p e c t e d u t i l i t y t h e o r y t o pave t h e way f o r t h e development o f a l p h a u t i l i t y i n t h e f i r s t two s e c t i o n s o f C h a p t e r 4. S e c t i o n s 3 and 4 c o n t a i n r e s p e c t i v e l y , t h e d e r i v a t i o n o f t h e n o r m a t i v e i m p l i c a t i o n s ( e . g . s t o c h a s t i c dominance, g l o b a l and l o c a l r i s k a v e r s i o n ) and d e s c r i p t i v e i m p l i c a t i o n s ( i n p a r t i c u l a r , r e l a t i n g t o t h e d e s c r i p t i v e i n a d e q u a c y o f e x p e c t e d u t i l i t y t h e o r y ) o f a l p h a u t i l i t y t h e o r y . The c h a p t e r ends w i t h a c o m p a r i s o n i n S e c t i o n 5 o f a l p h a u t i l i t y w i t h two a l t e r n a t i v e t h e o r i e s t h a t have a t t r a c t e d s i g n i f i c a n t i n t e r e s t . PART I REPRESENTATION THEOREMS 1 7 GENERALIZING THE QUASILINEAR MEAN OF HARDY/ LITTLEWOOD AND POLVA 1.1 INTRODUCTION What i s mean v a l u e ? C o n v e n t i o n a l wisdom t e l l s us t h a t i t r e p r e - s e n t s , t y p i f i e s o r i n some way measures t h e c e n t r a l t e n d e n c y o f a d i s - t r i b u t i o n . We a r e r e s c u e d from t h e a m b i g u i t y , as i n e l e m e n t a r y s t a t i s t i c s t e x t s , by examples. Some f a m i l i a r n o t i o n s o f mean v a l u e i n c l u d e median, mode, a r i t h m e t i c mean, g e o m e t r i c mean, harmonic mean and root-mean-square o r more g e n e r a l l y t h e r t h r o o t o f t h e r t h moment o f a p o s i t i v e random v a r i a b l e (known a l s o as t h e g e n e r a l mean o f o r d e r r ) . Of t h e s e , t h e a r i t h m e t i c mean i s t h e most w i d e l y used. There a r e however s i t u a t i o n s f o r w h i c h t h e a r i t h m e t i c mean may n o t be t h e most a p p r o p r i a t e ' t y p i c a l ' v a l u e : n o t a b l y , t h e d i s c r e p a n c y between p e r c a p i t a income and t h e ' t y p i c a l ' income f o r a s o c i e t y , t h e b u l k o f whose w e a l t h i s i n t h e hands o f a few. In t h e i r p i o n e e r i n g work o f 1934, Hardy, L i t t l e w o o d and P o l y a showed how a g e n e r a l c l a s s o f mean v a l u e s , c a l l e d t h e q u a s i l i n e a r mean, can be t a i l o r e d t o o u r needs s u b j e c t t o a n e c e s s a r y and s u f f i c i e n t s e t o f axioms. T h i s c l a s s i n c l u d e s as s p e c i a l c a s e s a l l t h e examples o f mean v a l u e s m entioned above e x c e p t f o r median and mode wh i c h do n o t s a t i s f y t h e i r axioms. We g e n e r a l i z e , i n t h i s c h a p t e r , t h e q u a s i l i n e a r mean v i a a weaker s e t o f axioms s t a t e d i n s e c t i o n 2. They a r e shown i n s e c t i o n 3 t o be 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 r e p r e s e n t a t i o n o f a c l a s s o f mean v a l u e s t h a t g e n e r a l i z e s the q u a s i l i n e a r mean. F i n a l l y , we d e r i v e some p r o p e r t i e s o f o u r mean 8 v a l u e i n s e c t i o n 4. A p p l i c a t i o n s w i l l be d i s c u s s e d i n P a r t I I and t h e c o n c l u s i o n o f t h i s d i s s e r t a t i o n . 1.2 AXIOMS OF MEAN VALUE 2 L e t Dj denote t h e space 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 w i t h a l l t h e i r mass c o n c e n t r a t e d i n some i n t e r v a l J o f t h e r e a l l i n e R ( J need n o t be bounded). We c o n s i d e r a f u n c t i o n a l M whose domain i s Dj. What p r o p e r t i e s s h o u l d M p o s s e s s i n o r d e r t o be a mean v a l u e ? A n a t u r a l c a n d i d a t e , m o t i v a t e d by t h e mean-value theorems o f e l e m e n t a r y c a l c u l u s , i s : P r o p e r t y 1: I n t e r m e d i a t e V a l u e P r o p e r t y JM(F) e conv S u p p ( F ) , V F e pj . The s u p p o r t o f a d i s t r i b u t i o n F, S u p p ( F ) , c o n s i s t s o f each p o i n t x such t h a t e v e r y open s e t c o n t a i n i n g x has p o s i t i v e mass, Conv Supp(F) i s t h e s m a l l e s t i n t e r v a l c o n t a i n i n g S u p p ( F ) . The i n t e r m e d i a t e v a l u e p r o p e r t y r e q u i r e s t h a t the mean o f a d i s t r i b u t i o n be n e i t h e r g r e a t e r t h a n t h e maximum a t t a i n a b l e v a l u e n o r l e s s t h a n t h e minimum a t t a i n a b l e v a l u e . Axiom 1. below i s a consequence o f t h e i n t e r m e d i a t e v a l u e p r o p e r t y . A* 1 0" 1 1 ' C o n s i s t e n c y w i t h C e r t a i n t y M ( 5 ) - X , V J C e J , The d i s t r i b u t i o n 6^ r e f e r s t o t h e s t e p f u n c t i o n a t x w h i c h , i n terms o f p r o b a b i l i t y , i n d i c a t e s o b t a i n i n g x w i t h p r o b a b i l i t y 1. A n o t h e r p r o p e r t y t h a t seems r e a s o n a b l e i s g i v e n by Axiom 2. _ I n t h i s s e c t i o n , t h e terms axiom and p r o p e r t y a r e used i n t e r c h a n g e a b l y . P r o p e r t i e s c a r r y t h e "axiom" l a b e l i f t h e y appear i n t h e f i n a l r e p r e s e n t - a t i o n theorem as a c h a r a c t e r i s t i c p r o p e r t y . 9 Axiom 2: Betweenness V F , G e D j , i f M ( F ) < M (G ) t h e n V B e (0,1), M ( 3 F+(l-B )G)e ( M ( F ) , M ( G ) ) . I t i s s t r a i g h t f o r w a r d t o check t h a t Betweenness i s e q u i v a l e n t t o P r o p e r t y 2 s t a t e d below: P r o p e r t y 2: M i x t u r e - m o n o t o n i c i t y V F , G e D j , i f M ( F ) < M ( G ) , the n M ( B F+(1- B ) G ) < M ( Y F + ( 1 - Y )G ) i f 1>B>Y*0. Lemma 1.1: Axiom 2 <=» P r o p e r t y 2. P r o o f : O m i t t e d . A d i s t r i b u t i o n G i s s a i d t o s t o c h a s t i c a l l y dominate a n o t h e r d i s t r i b u t i o n F i n t h e f i r s t d e g r e e , denoted by G > F, i f G i s always n o t g r e a t e r t h a n F p o i n t w i s e . I f i n a d d i t i o n , G i s s t r i c t l y l e s s t h a n F at some p o i n t , t h e n G s t o c h a s t i c a l l y dominates F s t r i c t l y i n t h e f i r s t l d e g r e e , denoted by G > F. S t o c h a s t i c dominance o f t h e f i r s t d egree i s an a p p e a l i n g p a r t i a l o r d e r . C o n s i s t e n c y o f mean v a l u e w i t h t h i s p a r t i a l o r d e r i s s t a t e d as P r o p e r t y 3: P r o p e r t y 3: M o n o t o n i c i t y V F , G e D j , G > F => M (G ) > M ( F ) . The n e x t axiom d e a l s w i t h t h e e f f e c t on mean v a l u e o f c e r t a i n changes i n t h e c o m p o s i t i o n o f t h e u n d e r l y i n g d i s t r i b u t i o n . Axiom 3: Weak S u b s t i t u t i o n V F , G £ D j , i f M ( F ) = M (G ) t h e n V B e ( 0,l ) 3 y e (0,1) 9 V H E DJ, M ( B F+(1-6 ) H ) = M ( Y G + ( 1 - Y )H ) . 10 Hardy, L i t t l e w o o d and P o l y a (1934) u s e d a s p e c i a l case o f Axiom 3 (They c a l l e d i t q u a s i l i n e a r i t y . ) s t a t e d as P r o p e r t y 4 below f o r t h e i r q u a s i - l i n e a r mean. P r o p e r t y 4: S u b s t i t u t i o n ( Q u a s i l i n e a r i t y ) V F, G, H e f j j , i f M(F)=M(G), t h e n V 6 e (0,1) M(BF+(1-B)H)=M(BG+(1-B)H) . S t a r t i n g w i t h two d i s t r i b u t i o n s w i t h t h e same mean v a l u e , q u a s i l i n e a r i t y o r t h e s u b s t i t u t i o n p r o p e r t y r e q u i r e s t h a t m i x t u r e s o f t h e s e d i s t r i b u t i o n s w i t h a n o t h e r d i s t r i b u t i o n i n t h e same p r o p o r t i o n s s h a r e t h e same mean r e g a r d l e s s o f t h e d i s t r i b u t i o n t h a t t h e y mixed w i t h . The weak s u b s t i t u t i o n axiom a l l o w s m i x t u r e p r o p o r t i o n s t h a t g i v e r i s e t o t h e same mean v a l u e t o be d i f f e r e n t . The f o l l o w i n g p r o p e r t y c a l l e d R a t i o C o n s i s t e n c y i s a consequence o f Axioms 2 and 3 ( I n an e a r l i e r p a p e r (Chew, 1979), t h e R a t i o C o n s i s t e n c y p r o p e r t y was i m p l i c i t i n a s t r o n g e r s t a t e m e n t o f Axiom 3. The weaker v e r s i o n i s due t o s u g g e s t i o n s o f Weber, Myerson, and M i l g r o m . F o r a d i s c u s s i o n o f an i n t e r e s t i n g g e o m e t r i c a l i n t e r p r e t a t i o n o f R a t i o C o n s i s t e n c y due t o Weber, see C h a p t e r 4, s e c t i o n 3.1): P r o p e r t y 5: R a t i o C o n s i s t e n c y Suppose F, G, H e D j and B^, B 2 > Y^> Y 2 e (0»1) 3 M(F) = M(G) t M(H), and M ( B i F + ( l - B i ) H ) = M ( Y i G + ( l - Y i ) H ) f o r i = l , 2 . B j / l - B j B 2 / l - B 2 ^ V 1 " * ! = V 1 _ Y 2 • Lemma 1.2: Axioms 2 and 3 i m p l y P r o p e r t y 5. P r o o f : Suppose t h a t M(H) < M(F) = M(G) w i t h o u t l o s s o f g e n e r a l i t y . 11 Axiom 3 = > 3 f : (0,1) -»• (0,1) g V B e ( 0 , 1 ) , M(BF+(1-B)H) = M(£(B)G+(1-£(B))H) ; Lemma 1.1 =* f i s a 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 . T h i s t o g e t h e r w i t h Axioms 3 and 4 i m p l i e s t h a t f " 1 e x i s t s . T h e r e f o r e , t h e y a r e c o n t i n u o u s f u n c t i o n s , and hence d i f f e r e n t i a b l e a.e, D e f i n e x : (0,1) - R + by T(3) = f ^ g / i l g ( 3 ) » (1.1) Note t h a t T i s c o n t i n u o u s and d i f f e r e n t i a b l e a.e.. We show below t h a t T i s a c o n s t a n t t o complete t h e p r o o f . C o n s i d e r 0 < B <B + <5 < 1. I t f o l l o w s a f t e r s u b s t i t u t i n g G f o r F u s i n g Axiom 3: M ( ( B . « ) F * ( I - ( ^ ) ) H ) . M ( [. ts^; i,:i;:; ). ( B t, iG. ( l l t, ) i;^_;;i. ( B T T )H) But (B+6)F+(1-(B+5))H = BF + (1-g) 1- 6 r l-B-6 .. F + , „ H 1-B (1.2) T h e r e f o r e L.H.S. o f (1.2) = M(BF+(1-B) & _ l - B - 6  - - T B F + " W H ) (1.3) M B M B ) G + Bx(B)+l-B BT(B)+1-B 4 F + 1 # H l - B (1.4) a f t e r s u b s t i t u t i n g G f o r F u s i n g Axiom 3. A p p l y i n g t h e same argument f o r t h e r e m a i n i n g F-component i n e x p r e s s i o n ( 1 . 4 ) , we o b t a i n : 1 2 R.H.S. =M 6-r(6/e-r(e)+i-e)+eT(e) 6 x ( S / 3 x ( B ) + l - 3 ) + 3 x ( B ) + l - ( B + 6 ) + 1 -(3+6) \ 6x ( 6 / B x ( B ) + l - 3 ) + B x ( B ) + l - ( 3 + 6 ) HJ ' C1"5-1 Comparing e x p r e s s i o n s ( 1 . 2 ) and e x p r e s s i o n ( 1 . 5 ) , i t f o l l o w s t h a t ( B + 6 )T ( B + 6 ) = Bx(B) + 6x(6/Bx(B) + 1-6) . ( 1 . 6 ) Suppose w i t h o u t l o s s o f g e n e r a l i t y t h a t x i s d i f f e r e n t i a b l e at 6. Then Lim ( 3 + 6 ) T ( B ^ - B T ( B ) = t ( 8 ) + B t , ( b ) = Lim T C 5 / ). o+U o o+U (1 .7) T h e r e f o r e , t h e r i g h t hand l i m i t o f x a t 0 denoted by x ( 0 + ) e x i s t s , A p p l y i n g t h e same argument f o r o t h e r B's f o r w h i c h x i s d i f - f e r - e n t i a b l e , we o b t a i n fg [ e x(B)] = x ( 0 + ) ' a.e. . ( 1 . 8 ) T h e r e f o r e x ( 3) = x ( 0 + ) a.e., and hence x(B) = T ( 0 + ) by c o n t i n u i t y . Q.E.D. F i n a l l y , we r e q u i r e o ur mean v a l u e t o be a c o n t i n u o u s f u n c t i o n a l i n t h e sense o f Axioms 4 and 5 . Axiom 4 : C o n t i n u i t y I f {F } ,C D T converges i n d i s t r i b u t i o n t o — n n=l J 6 F e D j and F has compact s u p p o r t , t h e n M(F) = Lim M(F ). n - H » n 13 Convergence i n d i s t r i b u t i o n has the f o l l o w i n g c h a r a c t e r i z a t i o n which i s used sometimes as i t s d e f i n i t i o n . F converges i n d i s t r i b u t i o n t o F e D T i f and o n l y i f n J / T f d F c o n v e r g e s t o / , f d F V f e C ( J ) , where C ( J ) i s the space J n u o f a l l bounded c o n t i n u o u s f u n c t i o n s on J . Note t h a t when J i s unbounded, t h e a r i t h m e t i c mean o f F does not n e c e s s a r i l y n c o n v e r g e t o t h e a r i t h m e t i c mean o f F s i n c e t h e f u n c t i o n x does n o t b e l o n g t o C ( J ) . We impose the c o n d i t i o n o f compact s u p p o r t i n Axiom 4 i n o r d e r n o t t o e x c l u d e t h e a r i t h m e t i c mean fr o m our c l a s s o f mean v a l u e s . The r e q u i r e m e n t o f C o n t i n u i t y i s u s e f u l because i t t e l l s us t h a t the mean o f a d i s t r i b u t i o n may be a p p r o x i m a t e d by t h e mean o f a d i f f e r e n t d i s t r i b u t i o n t h a t i s c l o s e t o i t . When J i s a compact i n t e r v a l , Axiom 4 i s e q u i v a l e n t t o c o n t i n u i t y o f t h e mean v a l u e , M, w i t h r e s p e c t t o t h e I?'-norm. When J i s unbounded, t h e f o l l o w i n g c o n d i t i o n t e l l s us how t o e s t i m a t e t h e mean v a l u e o f a d i s t r i b u t i o n F w i t h o u t compact s u p p o r t ( i f i t e x i s t s ) by i t s r e s t r i c t i o n t o a compact i n t e r v a l , K, denoted by F„. K Axiom 5: E x t e n s i o n Let {^n^™! D e a n i n c r e a s i n g f a m i l y o f compact i n t e r v a l s such t h a t L im .= J , t h e n M(F) = L i m M ( F K ) , V F e D j . The mean v a l u e f o r a d i s t r i b u t i o n F w i t h o u t compact s u p p o r t i s g i v e n by t h e l i m i t o f t h e mean v a l u e s o f t h e sequence o f t r u n c a t e d d i s t r i b u t i o n s , "•FK ^ = 1' f o r a n y i n c r e a s i n g f a m i l y , {K }°° whose l i m i t i s J.' S i n c e n n ~ the sequence o f mean v a l u e s , {M(F )} does n o t always c o n v e r g e , t h e i i n=0 mean v a l u e f o r a d i s t r i b u t i o n w i t h o u t compact s u p p o r t need n o t e x i s t . A good example i s t h e a r i t h m e t i c mean. 14 1.3 PvEPRESENTATION THEOREM We b e g i n w i t h a s t a t e m e n t o f t h e q u a s i l i n e a r mean r e p r e s e n t a t i o n theorem. D°[A,B] denotes t h e r e s t r i c t i o n o f D[A , B ] t o s i m p l e d i s t r i b u - t i o n s w i t h a f i n i t e number o f d i s c o n t i n u i t i e s . Theorem 1.1: (Hardy, L i t t l e w o o d S, P o l y a ) Suppose 3M : D° [ A , B ] R. Then M s a t i s f i e s Axiom 1, P r o p e r t y 3 and P r o p e r t y 4, i f and o n l y i f 3 cj> : [A,B]->-R, c o n t i n u o u s , s t r i c t l y monotone such t h a t M(F) = <J> (J $ dF) , V F E D [ A , B ] . (1.9) A M oreover, i f 3 <f>* : [ A , B ] •+ R such t h a t M(F) = <{>* 1 ( f <j)* dF) , V F e D [ A , B ] A t h e n V x e [ A , B ] , <J>*(X) = a<j>(x) + b> f ° r some a,b w i t h a i 0. (1.10) P r o o f : ( O m i t t e d s i n c e i t i s a s p e c i a l case o f Theorem 1.2). I n o t h e r words, t h e most g e n e r a l certainty consistent, monotone and quasilinear f u n c t i o n a l o f F i s t h a t d e f i n e d by ( 1 . 9 ) . We s h a l l c a l l i t quasilinear mean. S i n c e t h e q u a s i l i n e a r mean M(F) i s c o m p l e t e l y s p e c i f i e d by a c o n t i n u o u s , 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 <(>, up t o an a f f i n e t r a n s f o r m a t i o n , ( 1 . 1 0 ) , i t i s c o n v e n i e n t t o w r i t e i t as M ^ ( F ) . The f o l l o w i n g theorem g e n e r a l i z e s t h e theorem o f H ardy, L i t t l e w o o d and P o l y a and e x t e n d s t h e i r a n a l y s i s f r o m D ° [ A , B ] t o D [ A , B ]. A f u r t h e r e x t e n s i o n t o f J T f o r t h e a r b i t r a r y J f o l l o w s l a t e r . 15 Theorem 1.2: Suppose 3 M : D[A,B] -> R. Then M s a t i s f i e s Axiom 1, Axiom 2, Axiom 3 and Axiom 4 i f and o n l y i f 3cp : [A,B] ->• R, c o n t i n u o u s , s t r i c t l y monotone, and a : [A,B] -»• R +, c o n t i n u o u s , s t r i c t l y p o s i t i v e , B B such t h a t M(F) = <j> _ 1(f ctcp dF/ J a dF) , V F e D[A,B]. (1.11) A A Mo r e o v e r , i f 3<j>* : [A,B] -> R, and a* : [A,B] -»- R*, i , B B such t h a t M(F) = <J>* (/ a*cf>*dF/J a* dF) , V F e D[A,B] A A t h e n V x e [ A , B ] = l ^ ^ ^ r l H U ^ d - " ) k(<|>(x)-<f>(A)) + (<J>(B)-<J)(x)) and a * ( x ) = c a (x) {k (<j> (x) -<|> (A)) + (<J> (B) -<|> ( x ) ) } , (1.13) f o r some a,b,c,k w i t h a,c i 0, k > 0 . P r o o f : ( N e c e s s i t y ) Axiom 1 f o l l o w s i m m e d i a t e l y . Axiom 2 f o l l o w s from t h e o b s e r v a t i o n t h a t , 3 ( / , B a dF)<J>fM(F)) + (1-6) (//a dG)4>(M(G)) <|>(M(BF+(1-B)-G))= n \ •e(/A« dF) + ( 1 - B ) ( / A a dG) i n c r e a s e s s t r i c t l y i n 6 when M(F)>M(G). C o n s i d e r F,G,H e D[A,B], Suppose M(F) = M(G). Then, *(M(BF + (1-B)H))= ^ l * d F ) K M ( F ) ) + ( 1 - g ) ( j f r dH)<j)(M(H)) B ( / A B a dF) + ( l - B ) ( J B a dH) <KM( YG + ( l - Y ) H ) ) W i t h = ^ d ¥ ) n U a dG) ' V 3 £ [A'B]' 16 Hence, Axiom 3. Le t IF > ^ converge t o F. S i n c e a,<f>,c)> a r e c o n t i n u o u s on a compact i n t e r v a l [ A , B ] , B B B B Ua d F n ^ A a d F a n d d F n d F " B B B B - lA^dFn/JAa d F n - /Aa<|> dF /J^ dF . -1 f B f B -1 rB ,B =» M ( F n ) = cj> (/ a* d F n / J A a d F n ) •+ <j, (J^acj) d F / / A a dF) = M(F) . Axiom 4 f o l l o w s . ( S u f f i c i e n c y ) D e f i n e ^ : [ 0 , l ] + [A,B] as f o l l o w s . K p ) = M ( S p ) , V P e [ 0 , 1 ] , where S p = pSg + ( l - p ) 6 A . (1.14) Axiom 1 =* ̂ CO) = A and 4^(1) = B. Axiom 2 =* IJJ i s s t r i c t l y i n c r e a s i n g . L e t ( P n ^ - i c o n v e r g e t o P- Then Sp^ converges i n d i s t r i b u t i o n t o S . Axiom 4 =» K p ) = M ( S p ) = Lim M ( S p n ) = Lim * ( p n ) . I t f o l l o w s t h a t i> i s c o n t i n u o u s and s t r i c t l y i n c r e a s i n g and t h e r e - f o r e has an i n v e r s e a) : [A.B] -»- [ 0 , l ] w h i c h i s c o n t i n u o u s and s t r i c t l y i n c r e a s i n g . I f x = »Kp) > t h e n p = <j>(x) and M ( 6 x ) = x = K p ) = M(S ) = M ( S ~ ( x ) ) . (1.15) Lemma 1.2 i m p l i e s t h e e x i s t e n c e o f a s t r i c t l y p o s i t i v e c o n s t a n t t h a t depends on x such t h a t , V H e D[ A , B ] , V 6 e [ ' 0 , l ] , M ( 3 6 x + ( l - B ) H ) = M C g ^ . g j S~ ( x ) + H) . (1.16) 17 C o n s t r u c t a: (A,B) -> (0,°°) by a s s i g n i n g a ( x ) = x , V x e ( A , B ) . The f o l l o w i n g argument e s t a b l i s h e s t h e c o n t i n u i t y o f a on (A,B) and t h e n e x t e n d s i t s domain t o i n c l u d e t h e e n d - p o i n t s . C o n s i d e r g ( x ) = H(h& +h&J = M( "C>0 S2r , + ? , i A ) x A J ^a(x)+l <|>(x) a ( x ) + l A J M(S { a ( x ) K x ) / 5 ( x ) + l } ) = K a ( x ) K x ) / a ( x ) + l ) L e t t x n } "^ converge t o x e (A , B ) . Then ^ ^ x n + ^ A c o n v e r g e s i - n d i s t r i - b u t i o n t o %& + % 6 A . Axiom 4 i m p l i e s t h a t , g(x) = m&x + % 6 A ) = Lim M ( % 6 X n + h&p) = Lim g (x n) . T h e r e f o r e , g i s c o n t i n u o u s i n ( A , B ) . I t f o l l o w s t h a t a i s c o n t i n u o u s i n ( A , B ) . L e t ( x n } "^ co n v e r g e t o B from below, t h e n , Uh) = MCS^) = Lim H(h6Xn + h&A) = Lim g(x n) = Lim * ( $ ( x n ) / ( 1 + 1/5(x n))) . => Lim a ( x ) = 1, s i n c e i ( B ) = 1. S i m i l a r l y , we can show t h a t Lim a ( * n ) = 1 as x^ converges t o A from above. We e x t e n d a t o [A,B] c o n t i n u o u s l y by a s s i g n i n g 1 t o a a t the e n d - p o i n t s . Now, we a r e r e a d y t o show t h a t , V F e D [ A , B ] , t h e f u n c t i o n s a and <f> s a t i s f y c o n d i t i o n ( 1 . 1 1 ) . n ° r L e t { x i ^ _ i be t h e s u p p o r t o f a d i s t r i b u t i o n F i n D [ A , B ] , and n r e p r e s e n t F i n t h e f o r m , F = £ ei<5 x. » w n e r e 6. = F ( x ^ ) - F ( x . ) . i = l 1 n M(F) = M( I 6.6 x ) i = l i 6l°(xi) " 9 j = M ( o i a ( x 1 ) + z e . s * ( X l ) + i £ 2 e 1 5 C x 1 ) + E e j 5 x i ) • a f t e r s u b s t i t u t i n g S ^ x ^ f o r 6 ^ u s i n g e x p r e s s i o n (.1.16). R e p e a t i n g (n-1) t i m e s on t h e r e m a i n i n g 6 X^, i= 2 , . . . , n , y i e l d s , ~ - l ,B . ,-B = ^(Ee.a(x )<j>(x ) / E 6 . a ( x . ) ) = <j, A ( / acJ>dF// adF) . : 3 A A F i n a l l y , we exten d o u r c o n s t r u c t i o n t o F e D [ A , B ] . Suppose F E D[A,B] - D°[A,B]. C o n s t r u c t t h e f o l l o w i n g sequence {F } °° i n D°[A,B], n n=l V FCAJ«A . fiFCA* F(A. I i ^ l „ 5 A + i ( B . A ) / 2 n By c o n s t r u c t i o n , F n ( x ) F ( x ) , V x e { A + ^ ~ I : i , n e I + , i < 2 n} whi c h i s dense i n [ A , B ] . T h e r e f o r e {F^} °° co n v e r g e s t o F. n=l Axiom 4 =» M(F) = Lim M(F ) ~ 1 B B = Lim <j> (/ 54 dF // 5 dF ) n-x=° Y V J A Y n J A n y — 1 B B = $ (/ S$ dF// a d F ) , A A s i n c e c]>,<f> and a a r e c o n t i n u o u s on [A , B ] , 19 ( U n i q u e n e s s ) Suppose 3 a : [A.B-] -»• R + a n d cf> : [A,B] -»- R, t h a t s a t i s f y c o n d i t i o n (1.11) Then x = M ( 6 x ) = M ( S - ( x ) ) = - 1 r a ( B ) c ^ ( B ) | ( x ) + a ( A ) ^ > ( A ) ( l - $ ( x ) ) " * 1 a ( B ) $ ( x ) + a ( A ) ( l - $ ( x ) ) J ' U ' 1 / J and V B e ( 0 , 1 ) , - 1 B a ( x ) ( K x ) + (l-B)a(A)<|>(A) , * L Ba(x) + ( l - B ) a ( A ) J = M ( B 6 x + ( 1 - B ) 6 A ) = M r + Cl - B ) , U l B a ( x ) + ( l - B ) <Kx) Ba(x) + (1-B) ° A -» - 1 B5(x)»(x)a(B)(l>(B) + (Ba(x)(l-»(X))-f(l-g))a(A)(|,(A) * 1 B 5 ( x ) $ ( x ) a ( B ) + (B5(x)(l-<|)(x)) + ( l - B ) ) a ( A ) J ' (1.18) a f t e r a p p l y i n g a and $ t o t h e e q u a l i t i e s (1.15) and ( 1 . 1 6 ) . L e t a = <j>(B) - <j>(A), b = <KA), c = a (A) , and k = c t(B ) /c t(A). I t i s s t r a i g h t f o r w a r d t o check t h a t , V x e [A,B ] , <j>(x) = a { k $ ( x ) / ( k * ( x ) + ( l - $ ( x ) ) ) } + b, and a ( x ) = c a ( x ) { k $ ( x ) + ( l - $ ( x ) ) } . Suppose a* and cj>* a r e a n o t h e r p a i r o f f u n c t i o n s t h a t s a t i s f y c o n d i - t i o n ( 1 . 1 1 ) . Then (J>*(x) = - a * { k * * ( . x ) / ( k * $ (x) + Q-* ( x ) ) ) } + b* , 20 and a * ( x ) = c * o ( x H k * $ ( x ) + ( l - $ ( x ) ) } , w i t h a* = t}.*(B) - <)>*(A), b* = <f>*(A), c* = a * ( A ) , and k* = a * ( B ) / a * ( A ) . F i n a l l y , we check t h a t , V x e [ A , B ] , 4>*(x) = a ' { k - ( H x ) - K A ) ) / ( k - ( < K x ) - K A ) M K B ) - K x ) ) ) } + b ' , a * ( x ) = c ' a ( x ) { k ' ( < K x ) - < K A ) ) + (<£ (B) -<J> ( x ) ) } , Our g e n e r a l i z a t i o n o f t h e q u a s i l i n e a r mean, d e f i n e d by (1.1-1), i s c o m p l e t e l y s p e c i f i e d b y a p a i r o f f u n c t i o n s (ct,£>) . A c c o r d i n g t o Theorem 1.2, t h i s i s t h e most g e n e r a l mean f o r d i s t r i b u t i o n s d e f i n e d on a compact i n t e r v a l t h a t s a t i s f i e s Consistency with Certainty, Betweenness, Weak Substitution and Continuity. I n k e e p i n g w i t h p r e c e d e n t , we denote o u r g e n e r a l i z e d mean by M^. The p a i r o f f u n c t i o n s (a,<j)) denotes a p a r t i c u l a r member o f t h e c l a s s , {a,<)>},, o f f u n c t i o n s t h a t y i e l d t h e same mean on D T. When J i s a compact i n t e r v a l , such as [A,B] i n Theorem 1.2, we can form t h e f o l l o w - i n g s u b c l a s s o f {a,<j)} T, c a l l e d k-ratio subclass, f o r k > 0. f o r a' = (cf>*(B)-<}>*(A))/a = a*/a, b' = <j>*(A) = b*. c' = a*(A)/{a(A)(<HB)-<j>(A))} = c * / c a , and k' = { a * ( B ) a ( A ) / a * ( A ) a ( B ) } = k * / k . Q.E.D. [A,B] = { (a,<j>) e { a , tj)} [A,B] : a ( B ) / a ( A ) = k ) . 2 1 k ,k k r k , We denote by (a ,<f> ) a g e n e r i c element and (ci , cj> ) [A,B] t h e oanoniaal element o f {a,<J>} [A,B] t h a t s a t i s f i e s : \ ( A ) = 0 , $ ( B ) = 1 , a ( A ) = l and a K ( B ) = k . I t can be shown, u s i n g e x p r e s s i o n s ( 1 . 1 2 ) and ( 1 . 1 3 ) , t h a t t h e elements o f a k - r a t i o s u b c l a s s are r e l a t e d t o each o t h e r v i a an a f f i n e t r a n s f o r m a t i o n k k f o r t h e (j> component and a s c a l a r t r a n s f o r m a t i o n f o r t h e a component. The c l a s s {ct.cbjj-^ can be o b t a i n e d from i t s k - r a t i o s u b c l a s s e s by t a k i n g t h e i r u n i o n o v e r a l l p o s i t i v e k's. The f o l l o w i n g c o r o l l a r i e s o f Theorem 1 . 2 a r e needed t o e x t e n d our r e s u l t s t o i n c l u d e noncompact i n t e r v a l s . The r e s t r i c t i o n o f {a,<f>}r. D -i L A i , t o t h e i n t e r v a l [ A n , B n ] i s denoted by {a,4)} r. „ -. r D . U U L A 1 > Bl - I L A Q ' o C o r o l l a r y 1 . 1 : L e t A j < A Q < B Q < B j . Then ( a , aS} [ A J . B J ] U, {a, aS}. [ A 0 , B 0 ] = K E C ! ? 0 1 ' H 0 1 > K . B j ' o*"o- where 0 1 a | ( B n ) ^(Bp) â lAoT FfAoT ( 1 . 1 9 ) and • 0 1 q ^ B p ) ( 1 - $ * ( B n ) ) ^ ( A Q ) ( 1 - $ K A 0 ) ) ( 1 . 2 0 ) P r o o f : Denote by ( o ^ 1 , ^ 1 ) an element o f { c t ^ } . - ^ 1 n -, L A i , v>\J k i k i Observe t h a t ( a i 1 , ^ ! 1 ) [ A 0 , B 0 ] E { a ^ } [ A ° , B 0 ] where k _ " l | ( B o ) _ &i 1 (Bp) ( M 1 1 (Bp ) * ( 1 - j i . 1 ( B n ) ) ) ( , 0 " ̂ P ( A 0 ) " a 1 l ( A 0 ) ( k 1 $ 1 l ( A 0 ) + ( 1 - $ 1 1 ( A 0 ) ) ) ' U - ' i J which i s a c o n t i n u o u s , s t r i c t l y i n c r e a s i n g and o n t o f u n c t i o n o f k 22 w i t h domain (0,°°) and range (h ,h ) "01 01 C o r o l l a r y 1.2: L e t A 2 < A i < A 0 < B 0 < Bi < B 2 1 I 1- Q.E.D. Denote by ( c L 1 , ^ 1 ) t h e c a n o n i c a l element o f t h e u n i t a r y s u b c l a s s - f o r t h e i n t e r v a l [ A . . B.1, i = l , 2 . i i Then where and b o 2 < h 0 1 and h 0 2 > h 0 1 , 0 1 a . ^ A o ) ^ ( A Q ) ' h 0 i = g J ^ B o H l - ^iHBn)) d . U A 0 ) ( l - $ . U A 0 ) ) ' (1.22) f o r i = 1,2, P r o o f : C o n s t r u c t t h e f u n c t i o n s £. . : (0,°°) -> ( h . . , h . . ) , f o r i < j , 3 >1 - i j i j v i a k. = K. • (k.) i ] . i 3 a.HB.JCk.^.lCB.) + (1-ijL 1 (B.))) a^CA.JCk.^.UA.) + (l-^.^A.))) Note t h a t , by c o n s t r u c t i o n , (1.23) k. { a ' * } [ A . , B . ] 3 3 [ A . , B . ] [ A . , B . ] Note a l s o t h a t i s c o n t i n u o u s , s t r i c t l y i n c r e a s i n g , and onto from (0,°°) t o (h. . ,h. .) . -1J i r Suppose h 0 2 > h 0 i . P i c k kj_ e [h 1 2,°°). Then C i o ^ i ) < h 0 1 < h 0 2 . =* 3 k 2 e (0,°°) such t h a t t ; 2 0 ( k 2 ) = £,l0{k1) . 23 B u t { a ' * } [ A 2 , B 2 ] [ A 0 , B 0 ] [ A i . B i ] [ A i . B i ] [ A 0 , B 0 ] [ A 0 , B 0 ] A s i m i l a r argument e s t a b l i s h e s h 0 1 < h 0 2 . Q.E.D. According to C o r o l l a r y 1.1, we have to r e s t r i c t the k Q - r a t i o correspond- ing t o the [ A O , B Q ] i n t e r v a l t o w i t h i n a range of values i f we want (co 0'*!) 0) t o a g r e e with ( c x i 1 , ^ 1 ) r e s t r i c t e d to [ A Q . B Q ] f o r a l a r g e r i n t e r v a l [ A X , B I ] . C o r o l l a r y 1.2 t e l l s us that the range of p e r m i s s i b l e k g - r a t i o ' s gets squeezed as we go from [ A i , B i ] t o a l a r g e r i n t e r v a l [ A 2 , B 2 ] . NOW we extend Theorem 1.2 t o the case of a r b i t r a r y i n t e r v a l J . Theorem 1.3: Suppose 3 M : D R. Then M s a t i s f i e s Axiom 1, Axiom 2 , Axiom 3 , Axiom 4 and Axiom 5 i f and only i f 3 <J> : J -»• R, continuous, s t r i c t l y monotone, and a : J -*• R +, continuous, s t r i c t l y p o s i t i v e , such t h a t 3 M ( F ) = f 1 (jja<|>dF/JpdF) , V F e D j . (1.24) Moreover, i f (a*,<j>*) i s another p a i r of f u n c t i o n s that s a t i s f i e s 3 The r a t i o fj a^dF/fj adF f o r F without compact support i s defined by expression (1.25) . 24 c o n d i t i o n ( 1 . 2 4 ) , t h e n V i n t e r v a l [ A,B] C J , 3 a , b, c, k w i t h a, c * 0 and k > 0 a V x e [A,B] , Av r f x ! = a k(d)(x) - 4>(A)) * 1 j d k ( K x ) - cj) (A)) + (cf,(B) - <j>(x)) b ' a*(x) = ca(x){k(<})(x) - cf>(A)) + (<j> (B) - <|>(x))}. P r o o f : ( N e c e s s i t y ) V e r i f i c a t i o n o f Axiom 1, Axiom 2 , Axiom 3 , and Axiom 4 i s the same as i n p r o o f o f Theorem 1.2. Axiom 5 f o l l o w s t r i v i - a l l y f rom t h e d e f i n i t i o n ( e x p r e s s i o n (1.25)) o f M(F) f o r F w i t h o u t compact s u p p o r t . ( S u f f i c i e n c y ) I f J i s compact, t h e n we a r e done. O t h e r w i s e , l e t OO j - _ CO {K } = {[A ,B J} be a sequence o f i n t e r v a l s such t h a t n n=0 n n n=0 00 CO { A n } n _ Q '-'lBn n̂-o-' :""s a s t r i - c t l y d e c r e a s i n g ( i n c r e a s i n g ) sequence, and Lim{K n}= J . C o r o l l a r y 1.1 says t h a t , { a i ^ i } t A i 5 B . ] | [ A o , B 0 ] = k / l h o i , h o . ) { a O ^ O > [ ; 0 ) B o ] f a . ^ B o ) ( l - ^ ^ C B o ) ) a ^ C B o ) $ ^ ( 6 0 ) where (h . ,h .) = ( , ) • ' 0 1 0 1 a . ^ A o ) ( l - ^ . ' f A o ) ) a . ^ A o ) $ i l ( A 0 ) C o r o l l a r y 1.2 s a y s t h a t , { h Q ^ } 0 0 i s s t r i c t l y i n c r e a s i n g and { h 0 . } °° i s s t r i c t l y i = l i = l d e c r e a s i n g . 25 Let C . = [ ( h .+h . J / 2 , (h .+h . J / 2 ] , 1 1 -01 -0 i + l Oi 0 l + l and D. = ( h 0 . + 1 , h 0 . Then D i ^ C . ^ ( h o i > n o i ) f o r i = 1,2,3,... . Observe t h a t ( h ^ j h ^ J , C^,D^ a r e s t r i c t l y d e c r e a s i n g sequences o f s e t s by i n c l u s i o n . S i n c e C \ i s compact f o r each i , t h e r e f o r e , Lim C . = C t <t> . S i n c e Di C. C . 5 (h . ,h J V i i t f o l l o w s t h a t , 1 i l ^ - o i 01 Lim D. c C C Lim ( h . . , h . J . But Lim D. = Lim (h„.,h„J. Hence Lim (h . ,h\.) = C . - O i O i 0 0 To c o n s t r u c t (a,<j>) d e f i n e d on J t h a t s a t i s f i e s c o n d i t i o n ( 1 . 2 4 ) , p i c k k 0 e C M . D e f i n e ( a ( x ) , < K x ) ) = (aj°(x) , <J»J°(x)) f o r x e [ A 0 , B 0 ] , = ( a ^ f x ) , ^ ( x ) ) f o r x e [ A ^ B j J - [ A 0 , B „ ] , C a ^ C x ) , ^ ( x ) ) f o r x e [A. ,B.] - [ A . ^ . B . such t h a t ^ i O ^ i - 1 = k o = 5 0 ° ( B o ) > (see e x p r e s s i o n ( 1 . 2 3 ) ) , 4 i ( A 0 ) = l = a 5 ° ( A 0 ) , ^ i ( A 0 ) = 0 = $ 5 ° ( A 0 ) , 26 <f>*i(B0) = 1 = $ ^ 0 ( B o ) Observe t h a t ( a 1 ^ 1 ! 1 , d ^ i t 1 ) v l + l ' v i + l -* agree a t A 0 and a t B 0. k • k • [A.,B.] E ' • " i 1 ' ^ i 1 ^ s i n c e t h e y G i v e n any d i s t r i b u t i o n F w i t h compact s u p p o r t , p i c k [A^,B^] such t h a t Supp(F) C [A , B i ] . Then M(F) = «j>ki 1 ( J ® 1 ^ 1 ^ 1 d F / J ^ a ^ d F ) , i i = f\jja<i> d F / / j C t dF) . Fo r any d i s t r i b u t i o n F e D j , i f Supp(F) i s not compact, t h e n we o b t a i n M(F) , i f i t e x i s t s , from Axiom 5 as f o l l o w s . CO L e t { K n ) n _ Q be an i n c r e a s i n g sequence o f compact i n t e r v a l s whose l i m i t i s e q u a l t o J . Denote by F j ^ t h e r e s t r i c t i o n o f F t o K^, Then Axiom 5 =* M(F) = Lim M ( F K ) = Lim <J. _ 1(/ a* d F j ^ / J a d F ^ ) . (1.25) When t h e l i m i t (1.25) e x i s t s and does n o t depend on t h e c h o i c e o f t h e sequence, I K i , i t i s denoted by tj> (/Ta<J) d F / / T a dF) . However, n n = 0 <J J t h e above l i m i t does n o t always e x i s t . An example i s g i v e n by t h e a r i t h m e t i c mean f o r a cauchy d i s t r i b u t i o n . ( U n i q u e n e s s ) T h i s f o l l o w s d i r e c t l y from a p p l y i n g Theorem 1.2 t o a r b i t r a r y i n t e r v a l s [A,BJ i n J . Q.E.D. 27 We have c h a r a c t e r i z e d t h e c l a s s o f mean v a l u e s f o r d i s t r i b u t i o n s on t h e r e a l l i n e h a v i n g t h e p r o p e r t i e s o f Consistency with Certainty3 Betweenness, Weak Substitution, Continuity and Extension w i t h a p a i r o f f u n c t i o n s (a,<J>) . F o r d i s t r i b u t i o n s w i t h o u t compact s u p p o r t s , t h e i r c o r r e s p o n d i n g means do n o t n e c e s s a r i l y e x i s t (see ( 1 . 2 5 ) ) . 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 t h a t e n s u r e s e x i s t e n c e i s g i v e n b e l o w . C o r o l l a r y 1.5: ^a(j,CF) e x i s t s V F e rjj i f and o n l y i f e i t h e r <J> i s bounded o r a»<|> i s bounded. P r o o f : The s u f f i c i e n c y p a r t o f t h e p r o o f i s s t r a i g h t f o r w a r d . To p r o v e n e c e s s i t y , suppose f o r t h e p a i r (a,tj>), n e i t h e r <j> n o r a-cj) i s bounded. We may assume, w i t h o u t l o s i n g g e n e r a l i t y , t h a t <j> i s n o t bounded from above. There a r e two c a s e s . As x t e n d s t o +«, e i t h e r i ) a ( x ) i s bounded from above, o r i i ) a ( x ) t e n d s t o +«>. Case i ) : C o n s i d e r a sequence {x. 9 a(x^)<j>(xp=2 • Lim — — - does n o t c o n v e r g e . m-**> 2 : 1 a ( x . ) / 2 i = l ^ \ J oo 1 Case i i ) : C o n s i d e r a sequence L > ^ } ^ _ J 3 <K X^) = 2 . v m 1 i = l a ' - X i ' ' Then M . ( 2 . , —r^5 0 = Lim r- does n o t c o n v e r g e . otcj)v 1=1 „i x±J m ,„i 2 y m-x» 2, . a ( x . ) / 2 Then M ( 2 T —r-5 ) = acfr 1=1 2 i X j / 28 A s i m i l a r argument e s t a b l i s h e s t h e r e s u l t f o r t h e c a s e when 4 i s unbounded from below. Q.E.D. The above c o r o l l a r y i s u s e f u l i n C h a p t e r 4 when we i n t e r p r e t mean v a l u e as t h e c e r t a i n t y e q u i v a l e n t o f a l o t t e r y and i n s i s t t h a t a c e r t a i n t y e q u i v a l e n t s h o u l d always be f i n i t e . 2 9 1.4 PROPERTIES OF THE M ± MEAN ad) Of p o s s i b l e p r o p e r t i e s f o r mean v a l u e , t h e I n t e r m e d i a t e V a l u e p r o p e r t y ( P r o p e r t y 1) e n j o y s a r a t h e r s p e c i a l s t a t u s , somewhat l i k e a d e f i n i n g p r o p e r t y . A f t e r a l l , even measures such as median and mode, w h i c h a r e r e j e c t s o f t h e q u a s i l i n e a r mean, e x h i b i t t h i s p r o p e r t y . The c o n c l u s i o n t h a t M has t h e i n t e r m e d i a t e v a l u e p r o p e r t y f o l l o w s from t h e o b s e r v a t i o n t h a t , M r n = c *• Mx)(<K*) - 4>(c))dF(x) _ a<r r j ° A x ( x ) d F ( x ) " U Hence, C o r o l l a r y 1.4: M s a t i s f i e s P r o p e r t y 1 ( I n t e r m e d i a t e V a l u e p r o p e r t y ) l C o n s i s t e n c y w i t h s t r i c t s t o c h a s t i c dominance ' > ' ( P r o p e r t y 3: M o n o t o n i c i t y ) i s deemed d e s i r a b l e f o r many a p p l i c a t i o n s o f mean v a l u e . The c o r o l l a r y b e l o w g i v e s t h e c o n d i t i o n u n d e r w h i c h M , i s c o n s i s t e n t . o<J» . 4 w i t h s t o c h a s t i c dominance ( n o n s t r i c t ) 1 > 1. C o r o l l a r y 1.5: Suppose a and <f> a r e b o t h bounded on J . 1 Then V F,G e D j , F > G M a < j , ( F ) I M a ( j > ( G ^ i f and o n l y i f V s e J , a( x ) (<|>(x)-<f>(s)) (1-26) i s a n o n d e c r e a s i n g f u n c t i o n ( n o n i n c r e a s i n g f u n c t i o n ) . P r o o f : We s h a l l assume w i t h o u t l o s s o f g e n e r a l i t y t h a t d) i s s t r i c t l y i n c r e a s i n g . The p a r t i a l o r d e r ' > 1 i s d e f i n e d by G > F i f G(x) < F ( x ) , V x e J . The s t r o n g e r p a r t i a l o r d e r ' > ' d e f i n e d e a r l i e r ( P r o p e r t y 3) i s t h e above w i t h s t r i c t i n e q u a l i t y f o r some x. 30 1 1 ( S u f f i c i e n c y ) Suppose G > F . Then F Q ' > Fe whenever 9' > 6 , where F Q = (l - e ) F+9G, V 9 e ( 0 , 1 ) . (1.27) D e f i n e £(x ; F ) = ( a ( x ) //jCidF}{d>(x)-Q (F ) } , (fi as i n p. 22) (1.28) where fi(F) = fj acf>dF//ja dF. T h e n , ^ ( F Q ) = / ^ ( x ^ ^ G C x ) - F ( x ) ) , (1.29) = / J ( G ( x ) - F ( x ) d C ( x ; F Q ) > 0. S i n c e the i n t e g r a n d i s n o n n e g a t i v e and t, i s n o n d e c r e a s i n g V F e pjj, n(G ) - n(F) = / J { / J ( G ( x ) - F ( x ) ) d ? ( x ; F e ) } d e > o M .(G) > M .(F). ( N e c e s s i t y ) Suppose a ( x ) (<j)(x),-<f>(s*)) i s s t r i c t l y d e c r e a s i n g a t some x* f o r some s* e i n t J . S i n c e a ( x ) (<}>(x)-c|>(s)) i s a c o n t i n u o u s f u n c t i o n , i t i s s t r i c t l y d e c r e a s i n g f o r some open n e i g h b o u r h o o d (x*-£,x*+£). Assume w i t h o u t l o s s o f g e n e r a l i t y t h a t s* > x*. P i c k any y* > s* and compute p* such t h a t , s* = M ( p * 6 y * + ( l - p * ) 6 x " ) = M a < ) )(F*), where x" = x*-hE,. C o n s i d e r G* = p*<5y* + ( 1 - p * ) 6 ^ , f o r some x' e [x*,x*+£). Compute, / j ( G * ( x ) - F * ( x ) ) d < ; ( x ; F * ) = (1 - p * ) (5 (x» ; F * ) - c ( x " ; F * ) ) ' < 0. But M a^((l - e)F*+9G*) = M (F* Q) i s n o n d e c r e a s i n g i n 9. ^ - ^ ( F * E ) = / J ( F * ( x ) - G * ( x ) ) d ? ( x ; F * e ) > 0. S i n c e t h e R.H.S. i s c o n t i n u o u s , i t s l i m i t as 0 approaches 0 from above i s n o n n e g a t i v e , w h i c h i s a c o n t r a d i c t i o n . The e x t e n s i o n t o p o s s i b l e e n d - p o i n t s o f J f o l l o w s f r o m t h e c o n t i n u i t y and bounded- ness o f a and <j>. Q.E.D. 31 . The f u n c t i o n ? ( x ; F ) can be us e d t o g e n e r a t e t h e f o l l o w i n g l i n e a r f u n c t i o n a l , 5 * F 0 ) = / j C ( - ; F ) d ( . ) . Observe t h a t e x p r e s s i o n (1.29) i s t h e Gateaux d i f f e r e n t i a l o f Q, a t F Q i n t h e d i r e c t i o n G-F, wh i c h may be w r i t t e n a s : dVfi(Fe) = C G - p ) - e The f u n c t i o n a l C*p(*) and t h e f u n c t i o n r,(«;F) a r e b o t h r e f e r r e d t o as t h e Gateaux d e r i v a t i v e o f 5. a t F. We now i n t e r p r e t c o n d i t i o n (1.26) as f o l l o w s . The Gateaux d e r i v a t i v e o f 0, a t F, c,(_•;¥), i s n o n d e c r e a s i n g f o r e v e r y F i n DT. T h i s g e n e r a l i z e s t h e c o r r e s p o n d i n g c o n d i t i o n f o r q u a s i l i n e a r mean i f we o b s e r v e t h a t t h e Gateaux d e r i v a t i v e o f (<|>oM ) a t F i s s i m p l y 4> w h i c h i s s t r i c t l y i n c r e a s i n g i r r e s p e c t i v e o f F. An o t h e r u s e f u l p a r t i a l o r d e r i s second degree s t o c h a s t i c dominance ' 1 ', d e f i n e d by, 2 G > F i f / x ( G ( y ) - F ( y ) ) d y < 0, V x e J and / T ( G ( y ) - F ( y ) ) d y = 0 where J X = { y e J : y < x }. (1.30) The above says t h a t . G dominates F i n t h e second degree i f t h e y have t h e same a r i t h m e t i c mean ( i f t h e y e x i s t ) and t h e a r i t h m e t i c mean x x o f G t r u n c a t e d by J i s n o t l e s s t h a n t h a t o f F t r u n c a t e d by J f o r e v e r y x i n J . T h i s i s e q u i v a l e n t ' t o t h e n o t i o n o f mean preserving spread ( R o t h s c h i l d § S t i g l i t z , 1970) i n u n c e r t a i n t y e c o n o m i c s , and t h e principle of transfer ( D a l t o n , 1920) w h i c h s t a t e s t h a t a s o c i e t y ' s w e l f a r e i s n o t d i m i n i s h e d by a t r a n s f e r o f w e a l t h from t h e r i c h t o 32 t h e p o o r . Q u a s i l i n e a r mean M, i s known t o be c o n s i s t e n t w i t h second degree s t o c h a s t i c dominance when cj> i s i n c r e a s i n g and concave o r d e c r e a s i n g and convex. H a v i n g n o t e d t h e s i m i l a r i t y between <j> and ? ( * ; F ) i n d e r i v i n g c o n s i s t e n c y c o n d i t i o n s f o r f i r s t degree s t o c h a s t i c dominance, we e n t e r t a i n t h e c o n j e c t u r e t h a t t h e c o r r e s p o n d i n g second degree c o n d i t i o n f o r M i s t h a t c(-;F) i s concave (convex) i f $ i s i n c r e a s i n g ( d e c r e a s i n g ) f o r e v e r y F i n D j . The v e r i f i c a t i o n o f t h i s c o n j e c t u r e i s c o n t a i n e d as a s p e c i a l case o f a more g e n e r a l r e s u l t d e v e l o p e d i n t h e n e x t p a r a g r a p h . We b e g i n w i t h the f o l l o w i n g d e f i n i t i o n o f kth degree stochastic dominance, G > F i f / J { / J Z n _ 1 ( - . - { / j Z 3 { / j Z 2 ( G ( z 1 ) - F ( z 1 ) ) d z 1 } d z 2 } d z 3 } - - - } d z n . 2 } d z n - i ) = 0 * f o r n = 2,. . . ,k, and / J Z ] < { / J Z ] c _ 1 { « • •{ as above } v • } d z k _ 2 ) d z k _ l } < 0, V z k c J . When t h e n t h moment about t h e o r i g i n e x i s t s f o r d i s t r i b u t i o n s F and G f o r n = l , . . . , k , t h e n G dominates F i n t h e k t h degree i f t h e i r n t h moments agree f o r n = l , . . . k , and t h e k t h moment about t h e o r i g i n o f G t r u n c a t e d by J Z K i s n o t l e s s ( g r e a t e r ) t h a n t h a t o f F t r u n c a t e d a t J Z K i f k i s odd (even) f o r e v e r y z^ i n J . The f o l l o w i n g c o r o l l a r y g i v e s c o n d i t i o n s on a and d> f o r c o n s i s t e n c y o f M , w i t h k t h degree s t o c h a s t i c dominance. C o r o l l a r y 1.6: Suppose a , a ' , a " , - " , a ^ " 1 ^ , and <J>, dp' , <(>",•••, <f> ^ 1 ' a r e c o n t i n u o u s and bounded on J . 33 Then V F,G e D T, G > F =* M . ( G ) > M . ( F ) i f and o n l y i f V F e D _, r, ̂ k - 1 ^ ( x ; F ) i s a n o n d e c r e a s i n g ( n o n i n c r e a s i n g ) f u n c t i o n i f <f> i s i n c r e a s i n g ( d e c r e a s i n g ) when k i s odd, o r i s a n o n i n c r e a s i n g ( n o n d e c r e a s i n g ) f u n c t i o n i f <f> i s i n c r e a s i n g ( d e c r e a s i n g ) when k i s even. P r o o f : Assume w i t h o u t l o s s o f g e n e r a l i t y t h a t <£ i s i n c r e a s i n g and k i s even. k k ( S u f f i c i e n c y ) Suppose G > F . Then F Q , > F Q whenever 6' > 6, where F E (1-9)F+9G, f o r 0 e ( 0 , 1 ) . T h e n H e " f i C F Q ) = / J ? ( x ; F 6 ) d ( G ( x ) - F ( x ) ) = ( - l ) k / j { / J X { as i n p. 3 2 } d z k _ 1 } d c ( x ; F E ) = ( - l ) k / J I k ( x ) d c ( x ; F e ) > 0, where i s t h e k-time i t e r a t e d i n t e g r a l o f ( G ( x ) - F ( x ) ) on t h e i n t e r v a l J (see e x p r e s s i o n 1.30), (k-1) s i n c e i s n o n p o s i t i v e and c, i s n o n d e c r e a s i n g ( n o n i n - c r e a s i n g ) f o r k odd (even) V F e D T. u I t f o l l o w s t h a t , fi(G)-fl(F) = / J { / J T ^ ( x ) d c ( x ; F e ) }d0 > 0 =»• M . ( G ) > M . ( F ) . ( N e c e s s i t y ) T h i s f o l l o w s from an argument t h a t i s e s s e n t i a l l y t h e same as t h e one used i n t h e n e c e s s i t y p r o o f o f C o r o l l a r y 1.5. Q.E. D. We end t h i s s e c t i o n by o f f e r i n g a l i n k between M . and M. t h a t l e a d s t o a u s e f u l c o n d i t i o n under w h i c h c e r t a i n known i n e q u a l i t i e s 34 f o r M, can be ext e n d e d t o M ,. We d e r i v e from a d i s t r i b u t i o n F, a t h r o u g h t h e f u n c t i o n a, a n o t h e r d i s t r i b u t i o n F , F a ( x ) = / adF//, adF, f o r e v e r y x e J , (1.31) j X u i f t h e de n o m i n a t o r e x i s t s . I n t h i s case, M , and M, a r e r e l a t e d i n a<j> cj) t h e f o l l o w i n g manner: M ,(F) = M f F a ) . acp cp T h i s l e a d s i m m e d i a t e l y t o : Lemma 1.3: Suppose M X ( F ) > M, (F) V F e V C D T . ct Then i f F e V whenever F does, t h e n M (F) > M (F) V F e V • art) = ctip One use o f t h e above i s t h e e x t e n s i o n o f t h e r e s u l t , I f r > s then M (F) > M (F) f o r e v e r v F e D, ,, * = s (o, 0 0) where M (F) = M _ ( F ) = {/°x rdF} ^. r J x r J 0 , c» -p 00 ^1/ t o M (F) = / a ( x ) x d F ( x ) / / a ( x ) d F ( x ) } / r . The f u n c t i o n a has t h e a, r 0 0 s t a n d a r d m e a s u r e - t h e o r e t i c i n t e r p r e t a t i o n as a Radon-Nikodym deriva- tzve o f F w i t h r e s p e c t t o F. We may, on t h e o t h e r hand, c o n s i d e r ct F as an ' i n t e g r a l ' o f F t h r o u g h t h e f u n c t i o n OL. Can we d e f i n e F a even when / TcxdF does n o t e x i s t ? Our d e f i n i t i o n o f M (F) when F J ad) v does n o t have compact s u p p o r t ( e x p r e s s i o n (1.25)) s u g g e s t s t h e f o l l o w i n g . _ 00 L e t (K } be an i n c r e a s i n g f a m i l y o f compact i n t e r v a l s whose n n=l l i m i t i s J . Then - f j f d F = Lim ^K^^ d F / J j ^ a dF, f o r e v e r y f e C°(J) where C°(J) denotes the space o f c o n t i n u o u s f u n c t i o n s on J . 35 Cfc We have d e f i n e d F so t h a t t h e e q u a l i t y , h o l d s even when F does n o t have compact s u p p o r t . I t i s s t r a i g h t - f o r w a r d t o check t h a t Lemma 1.3 h o l d s f o r t h e e x t e n d e d d e f i n i t i o n 2 36 GENERALIZING THE EXPECTED UT IL ITY REPRESENTATION THEOREM 2.1 INTRODUCTION The p r e c e d i n g c h a p t e r g e n e r a l i z e d the q u a s i l i n e a r mean by weaken ing the ax iom o f q u a s i l i n e a r i t y . As we n o t e d i n t h e i n t r o d u c t i o n , t h e q u a s i l i n e a r mean, M^, r e p r e s e n t s a n o t h e r way t o a x i o m a t i z e e x p e c t e d u t i l i t y , v i a t h e maximand, /R<t>dF. C o r r e s p o n d i n g l y , o u r g e n e r a l i z e d mean, M ^ , i n d u c e s a more g e n e r a l o r d e r i n g v i a the maximand, /Da<f>dF/.L a d F . K K T h i s c h a p t e r t r e a t s the p r o b l e m o f e x t e n d i n g t h e above r e p r e s e n t a t i o n f o r the c a s e o f s i m p l e p r o b a b i l i t y measures on a more g e n e r a l outcome s p a c e t h a n the r e a l l i n e . S i n c e the n o t i o n o f mean v a l u e may not be d e f i n e d f o r a more g e n e r a l outcome s p a c e ( c o n s i d e r , e . g . , t h e outcome s p a c e c o n s i s t i n g o f g e t t i n g a p r o m o t i o n , s t a t u s quo and b e i n g f i r e d ) , we need t o s t a t e axioms d i r e c t l y i n terms o f p r o p e r t i e s o f the u n d e r l y i n g o r d e r i n g . The deve lopments o f r e s u l t s p a r a l l e l t h o s e o f C h a p t e r 1. C o n s e q u e n t l y , the p r o o f s h e r e a re s t r a i g h t f o r w a r d a d a p t a t i o n s o f t h e c o r r e s p o n d i n g ones i n C h a p t e r 1. They a re n o n e t h e l e s s i n c l u d e d s o t h a t C h a p t e r 2 may be r e a d i n d e p e n d e n t l y o f C h a p t e r 1. Unlike C h a p t e r 1, most d e f i n i t i o n s u s e d h e r e a r e g i v e n e x p l i c i t l y b e c a u s e t h e y a r e r e l a t i v e l y u n f a m i l i a r . 2.2 PRELIMINARY DEFINITIONS D e f i n i t i o n 2.1: A s i m p l e p r o b a b i l i t y measure P on a s e t X i s a r e a l - v a l u e d f u n c t i o n d e f i n e d on t h e s e t o f a l l s u b s e t s o f X s u c h t h a t : 1) . P(A) > 0, V A C X ; 2) P(X) = l ; 3) P(AuB) = P(A)+P(B) when A, B C X and A n B = <j>; 4) P(A) = 1 f o r some f i n i t e A c X . 37 A s i m p l e p r o b a b i l i t y measure P on a s e t X has t h e p r o p e r t y P ( { x } ) = 0 f o r a l l b u t a f i n i t e number o f x e X and f o r a l l A c X, P(A) = I P ( x ) where P(.{x}) i s w r i t t e n as P ( x ) . xSA D e f i n i t i o n 2.2: A p o i n t mass, 6 , a t x i s t h e spm w i t h P(x) = 1. X D e f i n i t i o n 2.3: F o r $ G ( 0 , 1 ) , t h e g - m i x t u r e o f a spm P w i t h a n o t h e r spm Q, gP + ( l - g ) Q , i s t h e r e a l - v a l u e d f u n c t i o n t h a t a s s i g n s BP(A) + ( l - g ) Q ( A ) f o r e v e r y A C X. I t i s c l e a r t h a t gP + ( l - g ) Q i s a spm when P, Q a r e spm's. In n g e n e r a l , £ 6 - p - i s a s P m i f P- i s a spm f o r i = 1, 2,..., n and i = l 1 1 1 n I 6 I = 1 w i t h & > 0 f o r i = 1, 2, .. . , n. F o r a spm P on a s e t X, i = l l e t { x . } n c X be t h e s e t o f p o i n t s f o r w h i c h P ( x . ) > 0 f o r 1 i = l n i = 1, 2, n. I t i s easy t o check t h a t P = £ p.6 , where 1=1 1 P i = P ( x i ) f o r i = 1, 2, ...,n. D e f i n i t i o n 2.4: The e x p e c t a t i o n , E ( f , P ) , o f a r e a l - v a l u e d f u n c t i o n f d e f i n e d on X r e l a t i v e t o a spm on X i s d e f i n e d by E ( f , P ) = I f ( x ) P ( x ) x s X n n For P H I p fi , E ( f , P ) = I p . f ( x ) . i = l I i = l D e f i n i t i o n 2.5: A b i n a r y r e l a t i o n -< on a s e t Y i s a weak o r d e r i f - < i s asymmetric ( i . e . Vx, y e Y, x -< y =* n o t (y ^ : x ) ) and n e g a t i v e l y t r a n s i t i v e ( i . e . Vx, y, z e Y, n o t (x •< y) n n o t (y ^ z) =*• not ( x - < z ) ) . 38 We summarize some p r o p e r t i e s o f a weak order,-<,- v i a t h e f o l l o w i n g . Lemma 2.1: Suppose -< i s a weak o r d e r on Y . D e f i n e b i n a r y r e l a t i o n s ~ , =< on Y by x ~ y ** n o t (x •< y) n n o t (y -< x) , Vx, y e Y and x =< y *» (x -c y) u (x ~ y) , Vx, y e Y . Then, i ) ~ i s an e q u i v a l e n c e r e l a t i o n i i ) =$ i s t r a n s i t i v e , i i i ) =S i s c o n n e c t e d ( i . e . Vx, y € Y , (x y) u (y x ) ) . i v ) (x -< y) n (y ~ z) =* x -< z, and (x ~ y) n (y -< z) =• x -< z, Vx, y, z e Y . P r o o f : ( O m i t t e d ) . In a p r e f e r e n c e c o n t e x t , -< i s c a l l e d ' s t r i c t p r e f e r e n c e ' and x-< y i s r e a d as 'y i s s t r i c t l y p r e f e r r e d t o x'; = 5 i s c a l l e d 'weak p r e f e r e n c e ' and x y i s r e a d as 'x i s not p r e f e r r e d t o y'; ~ i s c a l l e d ' i n d i f f e r e n c e ' and x ~ y i s r e a d as 'x i s i n d i f f e r e n t t o y'. 2.3 AXIOMS The f o l l o w i n g a r e c o n d i t i o n s on a b i n a r y r e l a t i o n s on |_̂ , t h e s e t o f spm's d e f i n e d on a s e t X . Axiom 1 : O r d e r i n g -c i s a weak o r d e r . Axiom 2: S o l v a b i l i t y VP, Q, R e |_ P-< Q and Q -< R =* 33 e ( 0 , 1 ) 3 3 P + ( l - g ) R ~ Q. Axiom 3: M o n o t o n i c i t y VP, Q £ L ^ , P -< Q =• BP + ( l - B ) Q -< yP + ( l - Y ) Q f o r 0 < Y < g < 1 . 39 Axiom 4: Weak Independence VP, Q 6 P ~ Q =» VB e ( o , i ) a Y G ( o , i ) 3 V R £ Lx» BP + ( l - B ) R ~ YQ + ( 1 - Y ) R . Axioms 1, 2, and 3 a r e s t a n d a r d p r o p e r t i e s o f a b i n a r y r e l a t i o n t h a t can be r e p r e s e n t e d by t h e e x p e c t a t i o n o f a u t i l i t y f u n c t i o n . Axiom 4 i s our o n l y d e p a r t u r e . I f we i n s i s t t h a t B and Y be i d e n t i c a l , t h e n A x i o m 4 r e d u c e s t o t h e s u b s t i t u t i o n p r i n c i p l e , w hich i s a n o t h e r p r o p e r t y o f e x p e c t e d u t i l i t y . The f o l l o w i n g p r o p e r t y i s a r e s t a t e m e n t o f P r o p e r t y 5 o f C h a p t e r 1 i n t h e c o n t e x t o f a weak o r d e r , -< , on D e f i n i t i o n 2.6: ( R a t i o C o n s i s t e n c y ) I f HP, Q, R € ^ and 8 i , B 2 , Y i , Y 2 e (0,1) 3 P ~ Q and B.P + ( l - B . ) R ~ Y.Q + ( l - Y . ) R i i i i f o r i = 1, 2, t h e n Y i / 1-Yl Y2 / 1-Y2 B i / 1-Bi B 2 / 1-B 2 Lemma 2.2: Axioms 1, 3 and 4 => R a t i o C o n s i s t e n c y . P r o o f : The p r o o f o f Lemma 2.2 i s e s s e n t i a l l y i d e n t i c a l t o t h a t o f Lemma 1.2 i n C h a p t e r 1. The i n t e r p r e t a t i o n o f our axioms and t h e R a t i o C o n s i s t e n c y p r o p e r t y i n t h e c o n t e x t o f c h o i c e w i l l be d e f e r r e d u n t i l C h a p t e r 4 where we a p p l y t h e r e p r e s e n t a t i o n theorems o f t h i s c h a p t e r and t h a t o f Ch a p t e r 1 t o d e c i s i o n t h e o r y . 40 2.4 REPRESENTATION THEOREMS To f a c i l i t a t e t h e s t a t e m e n t o f our r e p r e s e n t a t i o n theorem, we have D e f i n i t i o n 2.7: L e t •< be a b i n a r y r e l a t i o n on L ^ , t h e s e t o f s i m p l e p r o b a b i l i t y measures on a s e t X . The i n d u c e d b i n a r y r e l a t i o n -< on X i s d e f i n e d by, Vx. y e X , x < y<* ii < 5 . ' J ' J x y I f -< i s a weak o r d e r , t h e n -< i s a l s o a weak o r d e r . We d e r i v e t h e b i n a r y r e l a t i o n s :< and ̂  from -< as i n Lemma 2.1. D e f i n i t i o n 2.8: L e t -< be a weak o r d e r on a s e t Y , an element ye. Y i s a maximal ( m i n i m a l ) element i f Vx 6 Y , x =§ y (y ̂  x ) . Theorem 2.1: L e t be t h e s e t o f s i m p l e p r o b a b i l i t y measures d e f i n e d on a s e t X . Suppose -< i s a b i n a r y r e l a t i o n on w i t h t h e i n d u c e d b i n a r y r e l a t i o n on X denoted by -< . Then t h e r e e x i s t f u n c t i o n s a : X ->- R + and v : X ->• R such t h a t v i s non c o n s t a n t and a t t a i n s i t s supremum and infimum o v e r X and VP, Q e |_̂ , r < 0 ~ E(av,P) < E ( g v , Q ) 4 E(ct.P) E(a,Q) • ^ ' 1 J i f and o n l y i f -< s a t i s f i e s Axioms 1-4 and X c o n t a i n s a maximal element x and a m i n i m a l element x such t h a t x -< x. Moreover, i f a , v and a*, v* s a t i s f y t h e c o n d i t i o n o f t h i s theorem, th e n 3 a, b, c, k w i t h a, c, k > 0 such t h a t Vx £ X , 41 a * ( x ) = c a ( x ) { k [ v ( x ) - v ( x ) ] + [ v ( x ) - v ( x ) ] } and k [ v ( x ) - v ( x ) ] V W = a k [ v ( x ) - v ( x ) ] + [v(X) - v ( x ) ] + b- P r o o f : N e c e s s i t y : L e t x, x e X 3 v ( x ) = i n f v and v ( x ) = sup v X€X X € X Vx e X o b s e r v e t h a t v ( x ) < v ( x ) < v ( x ) 6 =$6 =S 6- X X X x =< x =5 x =*• x, x a r e m i n i m a l and maximal elements o f X r e s p e c t i v e l y . F u r t h e r m o r e , v i s n o n c o n s t a n t =* I n f v < Sup v =* 6 -< 6-x x ** x -< X . Axiom 1 f o l l o w s i m m e d i a t e l y . Axiom 2 f o l l o w s from t h e o b s e r v a t i o n t h a t VP, Q S E(av,6P + ( l - B ) Q ) . . D E ( a > * ( l - B ) Q ) 1 5 c o n t i n u o u s i n B- VP, Q e L x , P -< Q =* E ( a v » p ) ^ E ( a v , Q J E(a,P) E(a,Q) _ E(av,gP + ( l - B ) Q ) , . • . . • « , 0 E(a,3P + ( l - B ) Q ) d e c r e a s e s s t r i c t l y m B. (2.3) =* Axiom 3. 42 Suppose P, Q € L , and P ~ Q ~ f f ^ i = £ £ ™ z ? l E(a,P) E(a,Q) I t f o l l o w s t h a t VR e 1^ and VB G (0,1) E(gy,BP + ( l - B ) R ) = E(av,yQ + ( l - y ) R ) E(a,BP + ( l - B ) R ) E(a,yQ + ( l - y ) R ) w h e r e v'u-y) = E c « » p ) 6/(1-6) E(a,Q) • Hence, Axiom 4. S u f f i c i e n c y : L e t x, x be m i n i m a l and maximal elements o f X , r e s p e c t i v e l y . D e f i n e VP e [ 0 , 1 ] , S = p6- + (1 - p)6 . p * x ^ x By h y p o t h e s i s 6^ -< 6-. I t f o l l o w s from Axiom 3 t h a t S -< S * » 0 < p < q < l . (2.4) p q Vx S X 3 6 -< 6 and 6 -< 6-, i t f o l l o w s from Axiom 2 t h a t X X X X 3q e (0,1) 3 6 - S 4 I t i s c l e a r from (2.4) t h a t q i s u n i q u e . We c o n s t r u c t a r e a l - v a l u e d f u n c t i o n v : X [0,1] i n t h e f o l l o w i n g manner. v ( x ) = 0 , v ( x ) = 1, Vx e X - {x,x} , v ( x ) = q 3 6 ~ S . ' n x q 43 From c o n s t r u c t i o n , V x e X - { x , x } , 6 x ~ S v ( x ) (2.5) Lemma 2.2 => 3 x x > 0 3 VR G |^ and B E (0,1) B 6 x + ^ R ~ BxTiTa S(x)+ frrfif R ^ A X C o n s t r u c t a p o s i t i v e r e a l - v a l u e d f u n c t i o n a on X i n t h e f o l l o w i n g manner. a(x) = a(x) = 1 V x e X - {x,x} , a ( x ) G i v e n a spm P E J p.6 . 1 i x . i = l l A p p l y i n g (2.6) t o x^, x 2 , x n s e q u e n t i a l l y , i t f o l l o w s t h a t f n P,a(x ) n p I p 6 ~ S v ( x ) + I ' S 6 x ' 1=1 i p a(x , ) + z p. v l x i j i=2 p.a(x.) + z P . i 1 1 j=2 J 1 1 J = 2 J P 1 a(x 1) -n s. P^Cxj+p-aCxj + z p, ^ ( x i } j=3 ^ P 2a(x 2) P 1a(x 1)+p 2a(x 2) +_.i 3 P j LV n P. + I 3= _ 6 x i=3 p a(x )+p 7a(x 9) + z p. x i 1 1 2 z i=3 J 44 n p.acx.) I — 1— S n I iL=i ? p a ( x ) *C*i> " [ . i j P W ^ ' l p.acx.)] Hence, VP € L v , P ~ S E ( a v , P ) E(a,P) • I t f o l l o w s from Lemma 2.1 ( i v ) t h a t VP, Q 6 L ^ , P -< Q ~ s E ( a v , P ) S E ( a v , Q ) E(a,P) * E(S,Q) E(av,P) < E(5v,Q) E(a,P) E(a,Q) Un i q u e n e s s : Suppose H a : x ^ R + and v : X R such t h a t v i s n o n c o n s t a n t and a t t a i n s i t s infimum and supremum o v e r X a t v, y, r e s p e c t i v e l y , and VP, Q e L x , r i Q ^ E(av,P) < E(gy,Q) ' ^ E(a,P) E(a,Q) - U " / J C l e a r l y , y, y a r e m i n i m a l and maximal elements o f X - (y E x) U (6 ~ 6 ) and (y = x) U (6- ~ 6-). - - v y x v / ^ v y x =*• v ( x ) = v ( y ) and v ( x ) = v ( y ) . By construction, Vx € X - { x , x l , 6 ~ S ~ . (see r e l a t i o n ( 2 . 5 ) ) (2.8) x v ( x ) ,. r Y l _ v ( x ) a ( x ) v ( x ) + (1 - v ( x ) ) a ( x ) v ( x ) v(.xj - = =_ C2 a-) v ( x ) a ( x ) + (1 - v ( x ) ) a ( x ) ' l / ' y j 45 after applying (2.7) to (2.8). Also V f 6 ( 0 , 1 ) , BS(x) S + (1-3)6 3 6 x + Cl-B)«x g ~ ( g + ( i - B ) - C2.10) Bct(x)v(x) + (1-B)a(x)v(x) Ba(x) + (l-B)a(x) (2.11) _ Ba(x)v(x)a(x)v(x) + [Bct(x) (l-v(x)) + (1-B)]a(x)v(x) B5(x)v(x)a(x) + [Bct(x) (l-v(x)) + (1-B)]a(x) after applying (2.7) to (2.10) Let a = v(x) - v(x), b = v(x), c = a(x), a(x) . It i s easy to check that (2.9) and (2.11) become r -i kv(x) v ( x ) = a i ~ r s — + b kv(x) + (1 - v(x)) and a(x) = cS(x)[kv(x) + (1 - v(x)) ] . (2.12) Suppose a*, v* are another pa i r of functions that s a t i s f y the hypotheses of the theorem. Then V*W = a* i * ~ r r!CM ~ ( ^ + b * k*v(x) + (l-v(x)) and a*(x) = c*fi(x)[k*v(x) + (1 - v ( x ) ) ] , (2.13) 4 6 where a* = v * ( x ) - v * ( x ) , b* = v * ( x ) , c* = «*(x) , and k* = ^ [ . F i n a l l y , i t i s s t r a i g h t f o r w a r d t o check t h a t , k ' [ v ( x ) - y ( x ) ] v * f x 1 = a' = : + D w k ' [ v ( x ) - v ( x ) ] + v ( x ) - v ( x ) and a * ( x ) = c ' a ( x ) { k 1 [ v ( x ) - v ( x ) ] + v ( x ) - v ( x ) > (2.14) f o r a' = ( v * ( x ) - v * ( x ) )/a = a*/a b' = v * ( x ) = b* c' = ̂ SE ( 1 £* . 1 a M v ( x ) - v ( x ) J C c a =  a * ( x ) a ( x ) = k^ a ( x ) a*(x) k Q.E.D. We showed t h a t any b i n a r y r e l a t i o n on (_̂ , t h a t s a t i s f i e s t h e o r d e r i n g , m o n o t o n i c i t y , s o l v a b i l i t y and weak independence axioms, i s c h a r a c t e r i z e d by a p a i r o f f u n c t i o n s (a,v) d e f i n e d on X. When a i s c o n s t a n t , o u r r e p r e s e n t - a t i o n , E ( o c v , P ) / E ( a , P ) , r e d u c e s t o t h e e x p e c t e d u t i l i t y r e p r e s e n t a t i o n Ct E ( v , P ) . I f we d e f i n e a s i m p l e p r o b a b i l i t y measure P d e r i v e d from P i n t h e f o l l o w i n g manner, P a ( A ) = E ( c d A , P ) / E ( a , P ) , V A C X, where 1^ denotes t h e i n d i c a t o r f u n c t i o n o f A, t h e n we can s t a t e our r e p r e s e n t a t i o n i n t h e a l t e r n a t i v e f a s h i o n b elow; ECv.P"). Our r e p r e s e n t a t i o n i s t h e n s i m p l y t h e e x p e c t a t i o n o f t h e v - f u n c t i o n 47 ct w i t h r e s p e c t t o t h e measure P d e r i v e d from P v i a t h e a - f u n c t i o n w h i c h i s t h e Radon-Nikodym d e r i v a t i v e o f P a w i t h r e s p e c t t o P. We r e n d e r t h e r o l e o f a t r a n s p a r e n t by c o n s i d e r i n g a u n i f o r m measure N P = £ N ' ^ e c o r r e s P o n d i n g r e p r e s e n t a t i o n o f P i s g i v e n b y, i = l i N ot(x.) E(ctv,P)/E(ct,P) = £ ~N v ( x . ) . 1 = 1 .Z.a(x.) J = l 3 N M T h i s i s a w e i g h t e d a v e r a g e o f ( v ( x ^ ) } ^ _ ^ w i t h w e i g h t s , ( c t C x ^ ) } / ^ . The s t a t e m e n t o f Theorem 2.1 r e q u i r e s t h e s e t X t o be bounded by a maximal and a m i n i m a l element. The r e m a i n d e r o f t h i s s e c t i o n d e a l s w i t h t h e e x t e n s i o n o'f Theorem 2.1 t o t h e case where X has n e i t h e r a maximal n o r a m i n i m a l element. T h i s p a r a l l e l s t h e development towards t h e p r o o f o f Theorem 1.3 i n C h a p t e r 1. D e f i n i t i o n 2.9: L e t -< be a weak o r d e r on a s e t X- P ° r s, t £ X, an i n t e r v a l [ s , t ] C X i s d e f i n e d by [ s , t ] = {x £ X : s < x , x = s t } . When X c o n t a i n s b o t h a maximal element x and a m i n i m a l element x r e l a t i v e t o a weak o r d e r -<, t h e n X = [ x , x ] . When s^, s 2 , t ^ , t 2 e X, such t h a t s 2 •< Sj -< t j -< t 2 , t h e n [ S j . t j ] ^ [ s 2 , t 2 ] . D e f i n i t i o n 2.10: A p a i r o f f u n c t i o n s (a,v) i s s a i d t o r e p r e s e n t a weak o r d e r - < on L Y i f (<*,v) s a t i s f i e s c o n d i t i o n ( 2 . 1 ) . 48 D e f i n i t i o n 2.11: The u n i q u e n e s s c l a s s r e p r e s e n t i n g a weak o r d e r -< on l_x> {ct>v}x> c o n s i s t s o f a l l p a i r s (a,v) t h a t r e p r e s e n t -< on L^- D e f i n i t i o n 2.12: L e t s, t £ X 3 s -< t , w e denote by { a , v ) r , , t h e [ s , t j k - r a t i o s u b c l a s s o f t h e u n i q u e n e s s c l a s s { a , v ) r , r e p r e s e n t i n g , t J -< on L|-s ^ - j , c o n s i s t i n g o f t h o s e p a i r s (a,v) t h a t s a t i s f y aft") k k k r \ = k. A g e n e r i c element o f { a , v } r i s denoted by (a ,v ) . D e f i n i t i o n 2.13: L e t s, t e X 9 s-< t , t h e p a i r (ct*,v*) i s s a i d t o be an (a,b,c,k) t r a n s f o r m a t i o n o f (a,v) on [ s , t ] i f 3 a, b, c, k such t h a t a, c, k > 0 and Vx € [ s , t ] a * ( x ) = c a ( x ) { k ( v ( x ) - v ( s ) ) + v ( t ) - v ( x ) } and v * ( x ) = a — k K * ) ' ( , + b k [ v ( x ) - v ( s ) ] + v ( t ) - v ( x ) We denote such a t r a n s f o r m a t i o n by (a*,v*) = T a j b j C ) k (<x,v) on [ s , t ] C X- II k I t i s c l e a r t h a t {ct,v} r , = ^ + { a , v } r ^, . L e t (a,v) be an [ s , t ] kSR [ s , t ] element o f { a , v } r ,, we can t h e n g e n e r a t e a l l o t h e r elements [ s , t j v i a t h e (a,b,c,k) t r a n s f o r m a t i o n . Note t h a t T . . . i s an a, D , l , i a f f i n e t r a n s f o r m a t i o n on v and T, . . i s a p o s i t i v e s c a l a r 1 , 1 , c , l m u l t i p l e o f a; and t h a t we can use a u n i t a r y r a t i o p a i r (a*,v*) s { a , v ) r t o g e n e r a t e a l l elements o f {a,v> r , L s , t j [ s , t j 49 s i n c e T a j b ) C ) k ( a l . v l ) e { a ' v } ^ s t ] ' W e d e n o t e by ( S k . v k ) , t h e k k k c a n o n i c a l member o f { a , v } f , wh i c h s a t i s f i e s v (s) = 0, v ( t ) = l , [ s , t J a k ( s ) = 1, and & k ( t ) = k. I t i s c l e a r t h a t ( a k , v k ) i s u n i q u e f o r each k. I n g e n e r a l , a member a, v o f {ct,v}r , i s u n i q u e l y L S j t J s p e c i f i e d by t h e v a l u e s o f a, v a t . s and t . L e t a = v ( t ) - v ( s ) , b = v ( s ) , c = a ( s ) , k = a ( t ) / a ( s ) , t h e n Vx € [ s , t ] . a ( x ) = ca1Cx) [k0 1(x) + (1 - v ^ x ) ) ] and v ( x ) = a kv 1(x) k\)1(x) + 1 - vJCx) + b C o r o l l a r y ' 2 .1 : L e t s Q , t Q , ^ e X 3 S j A; s Q -k t -k t , then { a , v} - U r ^ k \s t 1 ~ kefjz, ? ^ tot,v} r , , where L S Q , t j k f c t * 0 1 / 0 1 J 0 0 oi a , , a , , a ( s o } v ( s o } (2.15) a\tQ) 1 - v 1 ^ ) -01 .1 , i a ( s Q ) 1 - v ( s Q ) (2.16) P r o o f : [ s 0 , t 0 ] J [ s 1 > t l ] - L[SQ,tQ] / H s ^ t ^ ' T h e r e f o r e , i f a , v d e f i n e d on [s , t ] r e p r e s e n t s -<on L . t h e n a,v L ^ ^ j t ^ J 50 r e p r e s e n t s ^ o n L - { a . v K , f , c { a , v } F s , Observe t h a t {a,v} [ S j . t j ] ko f s t i = { a ' v } r s t v w h e r e L O'V ^ ' V o — I i 1 ( 2 - 1 7 ) al(sQ) k^'CSg) + (1 - v ^ S g ) ) Note t h a t £ i s a c o n t i n u o u s , s t r i c t l y i n c r e a s i n g and ont o f u n c t i o n from (0,oo) t o U 0 1 > £ 0 1 ) - Hence {a, v} k l rs t 1 = ^ ( o i . v K , [ s 0 , t Q ] I! ( ^ U { a ' v } r s t 1 = U _ {a»v } r ° , • 0 l - 0 1 * 0 1 J u C o r o l l a r y 2.1 t e l l s us i t i s p o s s i b l e t o e x t e n d our r e p r e s e n t a t i o n o f a weak o r d e r on L r ^ t o a weak o r d e r on L r ^ i f t h e [ s Q , t 0 ] [ S l , t l ] o r d e r i n g has n o t changed on L r -i • I t a l s o g i v e s c o n d i t i o n s L V 0 J . on ( a n , v n ) e { a , v } r 1 so t h a t ( a n , v n ) can be extended t o a U U [ S ^ t p J U U 51 member (ct̂ vj 6 { a » v } [ s t ] S U C h t h a t C V V = (a1,v1) i . e . t h e extended p a i r ( a ^ , v j d e f i n e d on [ s ^ , t ^ ] a g r e e s w i t h ( a 0 J v 0 ) on [ s 0 , t Q ] . These c o n d i t i o n s ((2.15) and (2,16)) a r e g i v e n i n terms o f a 1 ( t Q ) , a 1 ( s 0 ) , v 1 ( t 0 ) , v 1 ( s 0 ) ; and can be o b t a i n e d v i a r e l a t i o n s (2.5) and (2.6) i n t h e c o n s t r u c t i v e p r o o f o f Theorem 2.1. C o r o l l a r y 2.2: L e t sQ, s ^ s ^ t Q , t j , t 2 G X 3 s 2 -k s 1 4; s Q ^ t -< t j - < t 2 , and l e t (a*>^*) denote t h e c a n o n i c a l u n i t a r y a,v on [ s i , t i ] , f o r i = 1,2. Then, l02 < l0l a n d *02 > *01 (2.18) where £; 0 1 'i('o) and a ? ( t 0 ) i - « | ( t 0 ) f o r i = 1,2. P r o o f : From C o r o l l a r y 2.1, {a,v} [ s 2 , t 2 ] rs t i = U - { a ' v } r s t l ' { a , v } r U (a.v}r° [ S ° ' t o ] W r V ^ J [ S ° , t o 1 ' 52 and {a,v} [ s 2 , t 2 ] k l • U - { a ' v } r s t l where '12 .1 a , , a 2 ( s 1 J v 2 ( s 1 ) and a^tp l - v^ctp a 2 ( s 1 ) 1 - v 2 ( s 1 ) C o n s t r u c t t h e f u n c t i o n £. . : (0,°°) -»- f£..,£..), f o r i < i , v i a J , i - i j i j k. = C. .(k.) a^Ct.) k . v ^ t . ) + I - v^(t.) J i 3 3 i 3 i a*(s.) k.v*(s.) + 1 - v*(s.) 3 i 3 3 i 3 i (2.19) Note t h a t , by c o n s t r u c t i o n , k. {a,v} 3 [s. ,t.] 3 3 [ s . , t . ] l i J {a,v} ^ 3 [s.,t.] Note a l s o t h a t £. . i s c o n t i n u o u s , s t r i c t l y i n c r e a s i n g , and onto from 3 J i (0,oo) t o ( £ ^ , 1 ^ 3 . Suppose lQ2 >lQ1. P i c k k1 e [l12,co) t h e n ? 1 0 ( k l ) < l Q 1 < lQ2. 3 k 2 e ( 0 . - ) 3 ? 2 0 ( k 2 ) = e10(kl) 53 i . e . {ct,v} [ s 2 , t 2 ] [ s 0 , t Q ] {a,v} ? i o ( V But {ct,v} [ s 2 , t 2 ] { a , v} [ s 2 , t 2 ] [ S p t ^ {a,v} 5 2 1 ( k 2 ) 5 2 1Ck 2) = k x B u t 5 2 1Ck- 2) e (A , A 1 2 ) A s i m i l a r argument e s t a b l i s h e s £ Q 1 < £ Q 2 Thus f a r , we have c o n s i d e r e d e x t e n d i n g from an i n t e r v a l o f X t o a l a r g e r i n t e r v a l . P resumably x i s n o t bounded, o t h e r w i s e , we would have c o n s t r u c t e d (ct,v) on X w i t h Theorem 2.1 once and f o r a l l . The n e x t i n t e r e s t i n g case t h e n i s when X c o n t a i n s n e i t h e r a maximal n o r a m i n i m a l e l e m e n t , f o r example, t h e r e a l l i n e . W i t h a s t r u c t u r a l c o n d i - t i o n on X , we show i n Theorem 2.2 t h a t even i n t h i s c a s e , a a-v r e p r e - s e n t a t i o n e x i s t s on L,. D e f i n i t i o n 2.14: L e t •< be a weak o r d e r on a s e t Y- A sequence 00 l y . } C Y i s c o f i n a l ( c o i n i t i a l ) i f Vx € Y , x =$ y . ( y . =< x) f o r i = l i w i some p o s i t i v e i n t e g e r i . 54 Theorem 2.2: L e t be t h e s e t o f spm d e f i n e d on a s e t X> an<3 -< i s a b i n a r y r e l a t i o n on l_x w i t h t h e i n d u c e d b i n a r y r e l a t i o n on x denoted by -< . There e x i s t f u n c t i o n s a : X~*"R+ a n d v : X R such t h a t ( i ) v(x) c o n t a i n s a s t r i c t l y i n c r e a s i n g c o f i n a l sequence and a s t r i c t l y d e c r e a s i n g c o i n i t i a l sequence, and ( i i ) VP, Q e L x W E(a,P) E(a,Q) ' i f and o n l y i f , -< s a t i s f i e s Axioms 1-5 and X o r d e r e d by -< c o n t a i n s a s t r i c t l y i n c r e a s i n g c o f i n a l sequence and a s t r i c t l y d e c r e a s i n g c o i n i t i a l sequence. Moreover, i f a, v and a*, v* b o t h s a t i s f y ( i ) and ( i i ) , t h e n Vs, t e X 3 s t , 3 a , b, c, k w i t h a, c, k > 0 3 Vx e [ s , t ] , a*(x) = c a ( x ) { k [ v ( x ) - v ( s ) ] + [ v ( t ) - v ( x ) ] } , and v * f x ) = a k [ y ( x ) - y ( s ) ] and v (x) a k [ v ( x ) _ v ( s ) J + [ v ( t ) _ v ( x ) ] b. P r o o f : N e c e s s i t y : L e t {d.}°° and {e.}°° c v(X) D e a s t r i c t l y d e c r e a s i n g c o i n i t i a l 1 i=0 1 i=0 sequence and a s t r i c t l y i n c r e a s i n g c o f i n a l sequence, r e s p e c t i v e l y . P i c k s., t . e X 3 v ( s . ) = d . and v ( t . ) = e. f o r i = 0,1,2, ... i i i i ^ i i 55 I t f o l l o w s t h a t {s.} ( { t . } ) i s a s t r i c t l y d e c r e a s i n g 1 i = l 1 i = l c o i n i t i a l sequence ( s t r i c t l y i n c r e a s i n g c o f i n a l sequence) o f X- V e r i f i c a t i o n o f Axioms 1-4 i s s t r a i g h t f o r w a r d (see N e c e s s i t y P r o o f o f Theorem 2.1). S u f f i c i e n c y : oo 00 L e t {s.} ( { t . } ) be a s t r i c t l y d e c r e a s i n g c o i n i t i a l sequence 1 i=0 1 i=0 ( s t r i c t l y i n c r e a s i n g c o f i n a l sequence) o f X- Suppose w i t h o u t l o s s o f g e n e r a l i t y t h a t s^ -< t ^ . I t i s easy t o check t h a t oo X= U [ S j . t ] . i=0 Let (ex., v.) r e p r e s e n t -< on | r f o r i = 0,1,2, ... . I t can 1 i ' i always be done because o f Theorem 2.1. C o r o l l a r y 2.1 =* = U _ <vV ko [ s . , t . ] a\{t ) 1 - v * ( t ) a1 it ) v ^ t ) where (* o i,£ o i) = ( . , - f - ^ - • - i _ 2 - ) a (s ) 1 - v ( s 0 ) a ( s 0 ) v (s ) C o r o l l a r y 2.2 => "1 0 0 - 00 {A n-} i s s t r i c t l y i n c r e a s i n g and {£ n.} i s U 1 i = l 0 1 i = i s t r i c t l y d e c r e a s i n g . L e t A. - f 0 i + ; ° > i + 1 , £ Q i + + , and B. = ( £ _ . . , £ „ . J l - 0 , i + l 0,1+1 56 Then B. <~ A. c ttQi,iQi) f o r i = 1,2,3, .... Observe t h a t t Z Q ^ , l Q i ) , A , B i a r e s t r i c t l y d e c r e a s i n g sequences by i n c l u s i o n . S i n c e A^ i s compact f o r each i , n e s t e d i n t e r v a l theorem =* l i m A. = A^ i <j). i - > o o S i n c e B C A C (o j . ) V i 1 ^ l jt - O i , Oi => l i m B. C A C l i m (£.. ,JL.) . l oo . - O i Oi But l i m B. = l i m (£..,JL.) l . - O i O i Hence l i m (£..,£„.) = A -Oi Oi ° To c o n s t r u c t (ct,v) d e f i n e d on x t h a t r e p r e s e n t s -< on L ^ , p i c k k n e A . 0 0 0 ko ko D e f i n e ( a ( x j , v ( x j ) = ( a Q (x) , v Q ( x ) ) f o r x € [ s Q , t ] . k l k l = ( a 1 ( x ) , v 1 ( x ) ) f o r x e t ^ ' 1 ^ ] _ [ s 0 » t Q ] , k. k. ( a . 1 ( x ) , v . 1 ( x ) ) f o r x G [ s . , t . ] - [ s i _ 1 , t . s uch t h a t ? i 0 ( V = ko and k. = 1 = k. = 0 = k. v i ^ V = 1 = ao <so3 ~ko vo <so> vo <V 57 Observe t h a t ( a . .. , v. , ) l + l ' l + l 1 agree a t s^ and a t t Q . k. . k. , k. k. i + l i + U i _ , I i . . „, = ( a . , v. J s i n c e t h e y [ s . , t . ] l I VP, Q e L x , p i c k [ s . , t . ] B ? > Q € L [ s . , t . ] ' i i k. k. k. k. E ( a 1 v 1 , P) E f a . 1 v . 1 , Q) t h e n P •< Q ** < L _ J : E f a . 1 , P) E C a . 1 , Q) E(av,P) < EQv,Q) E(a,P) E(a,Q) by c o n s t r u c t i o n . To complete t h e s u f f i c i e n c y p r o o f , o b s e r v e t h a t ( v ( s . ) } 1 i = 0 ( { v ( t . ) } ) i s a s t r i c t l y d e c r e a s i n g c o i n i t i a l sequence 1 1 = 0 ( s t r i c t l y i n c r e a s i n g c o f i n a l sequence) o f v ( x ) - U n i q u e n e s s : T h i s f o l l o w s d i r e c t l y from a p p l y i n g Theorem 2.1 t o a r b i t r a r y i n t e r v a l s [ s , t ] i n X- Q.E.D. 58 PART 11 APPLICATION TO DECISION THEORY BACKGROUND A c h o i c e s i t u a t i o n e x i s t s when more tha n one c o u r s e o f a c t i o n i s a v a i l a b l e t o a d e c i s i o n maker. A t h e o r y o f c h o i c e s p e c i f i e s , f o r each s e t o f a v a i l a b l e a l t e r n a t i v e s , t h e one t h a t w i l l be chosen. We have a v a l i d d e s c r i p t i v e t h e o r y i f , f o r t h e r e l e v a n t domain o f c h o i c e s i t u a t i o n s , t h e t h e o r y c a n be c o m p a t i b l e w i t h t h e a c t u a l c h o i c e s . The t h e o r y i s n o r m a t i v e l y c o m p e l l i n g i f t h e u n d e r l y i n g p o s t u l a t e s a r e o f s u f f i c i e n t a p p e a l so t h a t a d e c i s i o n maker i s w i l l i n g t o change h i s c h o i c e t o conform t o t h e t h e o r y ' s s p e c i f i c a t i o n s . E x p e c t e d u t i l i t y has been c o n s i d e r e d an example o f such a t h e o r y because f o r many r e s e a r c h e r s ( e . g . Savage ( 1 9 5 4 ) , MacCrimmon (1965) and R a i f f a ( 1 9 6 8 ) ) , i t s a t i s f i e s t h e l a t t e r r e q u i r e m e n t s . Y e t , t h e r e i s enough e m p i r i c a l e v i d e n c e ( c f . C h a p t e r 3) t o suggest t h a t i t i s n o t a v e r y good d e s c r i p - t i v e t h e o r y . P e o p l e t e n d t o s y s t e m a t i c a l l y v i o l a t e t h e i m p l i c a t i o n s o f a k e y p r o p e r t y o f e x p e c t e d u t i l i t y c a l l e d t h e s t r o n g independence p r i n c i p l e o r t h e s u b s t i t u t i o n axiom. Many o f them would n o t change t h e i r c h o i c e s a f t e r b e i n g t o l d o f t h e i r v i o l a t i o n s (MacCrimmon, 1968; S l o v i c $ T v e r s k y , 1975). Due t o t h e s u c c e s s o f e x p e c t e d u t i l i t y i n t h e m o d e l i n g o f phenomena i n t h e economics o f u n c e r t a i n t y and i t s a p p l i c a t i o n t o s t a t i s t i c a l d e c i s i o n t h e o r y , i t has been f a s h i o n a b l e t o d i s c o u n t v i o l a t i o n s as m i s t a k e s n e e d i n g c o r r e c t i o n . A d e p a r t u r e f r o m t h i s t r e n d i s e v i d e n t i n t h e appearance o f s e v e r a l r e c e n t p a p e r s ( M e g i n n i s s , 1977; Handa, 1977; K armarkar, 1978; Kahneman § T v e r s k y , 1979; M a c h i n a , 1980) p r o p o s i n g a l t e r n a t i v e t h e o r i e s o f c h o i c e t o a c c o u n t f o r A l l a i s ' 'paradox' and 60 o t h e r e m p i r i c a l f i n d i n g s t h a t c o n t r a d i c t t h e i m p l i c a t i o n s o f e x p e c t e d u t i l i t y . We d e v e l o p i n C h a p t e r 4 a new t h e o r y o f c h o i c e c a l l e d a l p h a u t i l i t y t h e o r y which g e n e r a l i z e s e x p e c t e d u t i l i t y v i a . a n e c e s s a r y and s u f f i c i e n t s e t o f axioms t h a t weaken t h e c o r r e s p o n d i n g ones f o r e x p e c t e d u t i l i t y . S p e c i f i c a l l y , t h e s u b s t i t u t i o n axiom i s r e p l a c e d by a weaker axiom c a l l e d Weak Independence. G i v e n two l o t t e r i e s t h a t a r e i n d i f f e r e n t t o each o t h e r , Weak Independence a l l o w s f o r d i f f e r e n t p r o b a b i l i t i e s i n com- p o s i n g each o f t h e s e l o t t e r i e s w i t h a t h i r d l o t t e r y t o p r e s e r v e i n d i f f e r e n c e . However, t h e s e m i x t u r e - p r o b a b i l i t i e s once d e t e r m i n e d must be independent o f t h e t h i r d l o t t e r y . The axioms i m p l y t h a t t h e r a t i o o f t h e m i x t u r e ( p r o b a b i l i t y ) odds i s c o n s t a n t . We c a l l t h i s t h e Ratio Consistency p r o p e r t y . E x p e c t e d u t i l i t y r e s u l t s when t h i s r a t i o i s i d e n t i c a l l y u n i t y . Our t h e o r y has d e s c r i p t i v e r e l e v a n c e i n t h a t i t can r e p r e s e n t t h e u s u a l r e s p o n s e s g i v e n t o t h e A l l a i s p a radox and i s c o m p a t i b l e w i t h o t h e r r e p o r t e d e m p i r i c a l f i n d i n g s c o n t r a d i c t i n g t h e i m p l i c a t i o n s o f e x p e c t e d u t i l i t y . Y e t , i t can be c o n s i s t e n t w i t h such n o r m a t i v e l y a p p e a l i n g p a r t i a l o r d e r s as s t o c h a s t i c dominance and g l o b a l r i s k a v e r s i o n . As w i t h e x p e c t e d u t i l i t y , t h e c o n s t r u c t i v e p r o o f o f our r e p r e s e n t a t i o n theorem f u r n i s h e s a p r o c e d u r e f o r t h e assessment o f the a l p h a u t i l i t y f u n c t i o n s , from w h i c h e m p i r i c a l l y t e s t a b l e p r e d i c t i o n s can be d e r i v e d . CRITIQUE OF EXPECTED UTIL ITY THEORY 3.1 INTRODUCTION E x p e c t e d u t i l i t y t h e o r y has a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n s i n c e i t s r e v i v a l by von Neumann and M o r g e n s t e r n i n t h e i r "Theory o f Games and Economic B e h a v i o r " . I t s e r v e s as t h e f o u n d a t i o n f o r t h e economics o f u n c e r t a i n t y (Arrow, 1971; Marschak and Radner, 1972; Diamond and R o t h s c h i l d , 1978), s t a t i s t i c a l d e c i s i o n t h e o r y (Savage, 1954; B l a c k w e l l and G i r s h i c k , 1961; R a i f f a and S c h l a i f e r , 1961; DeGroot, 1970) and d e c i s i o n a n a l y s i s (Howard, 1964; Keeney and R a i f f a , 1976) . E x p e c t e d u t i l i t y has been l e s s s u c c e s s f u l though i n e x p l a i n - i n g and d e s c r i b i n g a c t u a l c h o i c e s (Edwards, 1954, 1961; MacCrimmon, 1965; Kahneman and T v e r s k y , 1979); t h u s , p r o v i d i n g a s t r o n g impetus f o r f u r t h e r t h e o r e t i c a l development. In t h i s c r i t i q u e , we r e v i e w b r i e f l y t h e e m p i r i c a l f i n d i n g s t h a t pose d i f f i c u l t y f o r e x p e c t e d u t i l i t y . F i r s t , we p r o v i d e a summary, based on a r e c e n t p a p e r (Chew and MacCrimmon, 1979b), o f t h e s y s t e m a t i c v i o l a t i o n s o f t h e s t r o n g independence p r i n c i p l e . The f i r s t example o f such a v i o l a t i o n i s p r o v i d e d by t h e A l l a i s p a r a d o x , w h i c h i n s p i r e d e x t e n s i v e f o l l o w - u p s t u d i e s and m o d i f i c a t i o n s . The n e x t s e c t i o n , d i s c u s s e s t h e c o n c u r r e n c e o f r i s k - a v e r t i n g and r i s k - s e e k i n g b e h a v i o r e v i d e n t i n t h e p r e v a l e n c e o f t h e p u r c h a s e of i n s u r a n c e and g a m b l i n g (Friedman and Savage, 1948). F o r i n s t a n c e , M a r k o w i t z (1952) n o t e d p r e v a l e n t r i s k - p r o n e n e s s f o r l o t t e r i e s i n v o l v - i n g l o s s e s among h i s s u b j e c t s . T h i s o b s e r v a t i o n i s a l s o n o t e d by / 62 Kahneman and T v e r s k y ( 1 9 7 9 ) , p a r t i c u l a r l y t h e i r p r o b a b i l i s t i c i n s u r a n c e example. F i g u r e 3.1 d i s p l a y s a t y p i c a l von Neumann-Morgenstern u t i l i t y f u n c t i o n w h i c h i s used t o account f o r t h e j o i n t r i s k a v e r t i n g / s e e k i n g b e h a v i o r d i s c u s s e d above. The "convex" ("concave") r e g i o n s c o r r e s p o n d t o r i s k - p r o n e n e s s ( r i s k - a v e r s i o n ) . F i g . 3.1: A " t y p i c a l " von Neumann-Morgenstern u t i l i t y f u n c t i o n u ( x ) 63 S i n c e t h e von Neumann-Morgenstern u t i l i t y f u n c t i o n cannot be convex and concave at t h e same t i m e , e x p e c t e d u t i l i t y r u l e s out c o n c u r r e n c e o f r i s k p r o n e n e s s and r i s k a v e r s i o n w i t h i n t h e same r e g i o n o f w e a l t h l e v e l s . Whether t h i s i s a c t u a l l y t h e case i s an e m p i r i c a l q u e s t i o n t h a t has y e t t o be f u l l y i n v e s t i g a t e d , There i s however i n d i r e c t e v i d e n c e (MacCrimmon, e t . a l . 1972; A l l a i s , 1977) t o t h e c o n t r a r y . D i f f e r e n t t h e o r e t i c a l l y e q u i v a l e n t p r o c e d u r e s f o r t h e e l i c i t a t i o n o f von Neumann- M o r g e n s t e r n u t i l i t y , e.g., t h e c e r t a i n t y e q u i v a l e n t , the g a i n e q u i v a l e n t , and t h e c h a i n i n g method, t e n d t o y i e l d d i f f e r e n t c u r v e s w i t h o p p o s i n g r i s k - p r o p e n s i t i e s . A d i f f i c u l t y w i t h e x p e c t e d u t i l i t y , one t h a t t o u c h e s on t h e l a r g e l y u n e x p l o r e d a r e a o f pro b l e m r e p r e s e n t a t i o n and i t s e f f e c t on t h e d e c i s i o n maker's p r e f e r e n c e , i s , t h e c o n t r o v e r s y o v e r t h e domain on w h i c h a u t i l i t y f u n c t i o n i s d e f i n e d . S h o u l d i t be f i n a l w e a l t h l e v e l s , t h e n o r m a t i v e l y c o m p e l l i n g p o s i t i o n as i n Friedman and Savage ( 1 9 4 8 ) , P r a t t (1964) and p r a c t i c a l l y a l l t h e l i t e r a t u r e on t h e economics o f u n c e r t a i n t y , o r changes i n a s s e t p o s i t i o n r e l a t i v e t o some "customary" w e a l t h l e v e l ? M a r k o w i t z (1952) and o t h e r s have o b s e r v e d t h a t p r e f e r - ences a re r e l a t i v e l y i n d e p e n d e n t o f the c u r r e n t w e a l t h l e v e l s . A n o t h e r d i f f i c u l t y i s r e l a t e d t o t h e f i n d i n g (Kahneman and T v e r s k y , 1979) that, p r e f e r e n c e s among tw o - s t a g e l o t t e r i e s may depend on whether th e d e c i s i o n maker r e p r e s e n t s t h e s e l o t t e r i e s i n t h e i r s i m p l e e q u i v a - l e n t forms. T h i s c h a p t e r expands on t h e i s s u e s i n t r o d u c e d above w i t h o u t d u p l i c a t i n g u n d u l y t h e c o n t e n t s o f o t h e r c r i t i q u e s a l r e a d y c i t e d . 64 We s h a l l e x p l o r e t h e d e s c r i p t i v e i m p l i c a t i o n s o f o u r g e n e r a l i z a t i o n o f e x p e c t e d u t i l i t y t h e o r y i n t h e n e x t c h a p t e r i n l i g h t o f t h e examples c o n s i d e r e d h e r e . 3.2 SYSTEMATIC VIOLATION OF THE STRONG INDEPENDENCE PRINCIPLE As a l e a d - i n t o a more g e n e r a l s t r u c t u r e o f l o t t e r i e s , c o n s i d e r t h e f o u r d e c i s i o n p roblems g i v e n i n F i g u r e 3.2. A Q: $ 1 , 0 0 0 , 0 0 0 f o r s u r e BQ: 1 0 / 1 1 chance o f $ 5 , 0 0 0 , 0 0 0 1/11 chance o f $0 , A L : 1 1 / 1 0 0 chance o f $ 1 , 0 0 0 , 0 0 0 8 9 / 1 0 0 chance o f $0 B L : 1 0 / 1 0 0 chance o f $ 5 , 0 0 0 , 0 0 0 9 0 / 1 0 0 chance o f $0 V 8 9 / 1 0 0 chance o f $ 5 , 0 0 0 , 0 0 0 1 1 / 1 0 0 chance o f $ 1 , 0 0 0 , 0 0 0 B H : 9 9 / 1 0 0 chance o f $ 5 , 0 0 0 , 0 0 0 1/100 chance o f $0 V $i ,000,000 f o r s u r e V 1 0 / 1 0 0 chance o f $ 5 , 0 0 0 , 0 0 0 8 9 / 1 0 0 chance o f $ 1 , 0 0 0 , 0 0 0 1/100 chance o f $0 F i g u r e 3.2: Four d e c i s i o n problems Under 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 , t h e o n l y p e r m i s s i b l e p a t t e r n s o f c h o i c e s a r e e i t h e r A^, A T , A, , A n o r B U , B T , B T , B N . I f 0 'HJ 'o- y o u r c h o i c e s a r e l i k e most p e o p l e ' s , you w i l l have chosen A u , A , B. H I L and A n , wh i c h i s not c o n s i s t e n t w i t h t h e i m p l i c a t i o n s o f e x p e c t e d u t i l i t y . The c h o i c e o f A j and B^ c o n s t i t u t e s t h e w e l l known A l l a i s 65 paradox. A l e s s e r known p a r a d o x , t h e A l l a i s r a t i o p a r a d o x , i s g i v e n by t h e c h o i c e o f B. and A,.. The v i o l a t i n g c h o i c e o f A u and B T L U n , L has n o t been s t u d i e d . The i n s i g h t one g a i n s from t h e s t r u c - t u r e i n F i g u r e 3 . 2 , r a t h e r t h a n s i m p l y c o n s i d e r i n g s e p a r a t e b i n a r y l o t t e r i e s , i s t h a t t h e v i o l a t i n g p a i r s ( A j , B L ) , (B^, A ) and (A^, B^) are a l l d e r i v a t i v e s o f t h e b a s i c v i o l a t i o n , A U , A T , B , , A „ v e r s u s ' n 1 L U V A r V V S e v e r a l f e a t u r e s o f the s t r u c t u r e i n F i g u r e 3 .2 a r e w o r t h n o t i n g . I t i s b a s e d on t h r e e consequences $ 0 , ' $ 1 , 0 0 0 , 0 0 0 and $ 5 , 0 0 0 , 0 0 0 , denoted by L, I , and H r e s p e c t i v e l y . The A (B ) a l t e r n a t i v e , where x X X s t a n d s f o r one o f t h e consequences, L, I , H, i s o b t a i n e d from t h e AQ(BQ) a l t e r n a t i v e by c o m p o s i t i o n w i t h consequence x a t p r o b a b i l i t y 8 9 / 1 0 0 . T h i s i s i l l u s t r a t e d i n F i g u r e 3 . 3 f o r t h e case x = L. -• I F i g u r e 3 . 3 : The c o m p o s i t i o n o f t h e A T (B ) Li L l o t t e r y from A Q(B Q ) 66 S i n c e A (B ) i n F i g u r e 3 . 3 has t h e same f i n a l outcomes and p r o b a b i l i t i e s as A ^ ( B j J i n F i g u r e 3 . 2 , t h e s e l o t t e r i e s a r e e q u i v a l e n t . A l t h o u g h f o r i l l u s t r a t i v e p u r p o s e s , we have o n l y c o n s i d e r e d t h e conse- quences $0, $1,000,000 and $5,000,000 and t h e c o m p o s i t i o n p r o b a b i l i t y .89, i t seems r e a s o n a b l e t o e x p e c t v i o l a t i o n s o f e x p e c t e d u t i l i t y f o r o t h e r con- sequence v a l u e s and o t h e r p r o b a b i l i t y l e v e l s . T h i s l e a d s t o a more g e n e r a l s t r u c t u r e o f d e c i s i o n p r o b l e m s , i l l u s t r a t e d i n F i g . 3 . 4 . A^ i s a s u r e p r o - s p e c t o f t h e i n t e r m e d i a t e consequence I . B^ o f f e r s a q chance a t t h e most p r e f e r r e d outcome H and a l - q chance a t t h e l e a s t p r e f e r r e d outcome L. The A^(B^) a l t e r n a t i v e i s o b t a i n e d from t h e A ^ ( B Q ) a l t e r n a t i v e l y by composing w i t h t h e x-consequence a t p r o b a b i l i t y g. F i g u r e 3 . 4 : The H I L O s t r u c t u r e o f t h r e e consequence l o t t e r i e s 67 Note t h a t t h e H ( f o r " h i g h " ) , I ( f o r " i n t e r m e d i a t e " ) and L ( f o r " l o w " ) , g i v e n i n t h e boxes o f F i g u r e 3.4, exhaust t h e p o s s i b l e c o m p o s i t i o n s from t h e b a s i c p r o b l e m (denoted as "0") . For ease o f r e f e r e n c e t h i s w i l l be c a l l e d t h e "HILO" l o t t e r y s t r u c t u r e . E x p e c t e d u t i l i t y t h e o r y imposes some s e v e r e r e s t r i c t i o n s on t h e c h o i c e s i n t h i s l o t t e r y s t r u c t u r e . The s t r o n g independence p r i n c i p l e r e q u i r e s t h a t p r e f e r e n c e between two a l t e r n a t i v e s be p r e s e r v e d when each a l t e r n a t i v e i s composed w i t h a common a l t e r n a t i v e a t t h e same g g p r o b a b i l i t y . S i n c e t h i s i s how t h e and a l t e r n a t i v e s a r e gener- a t e d , i t i m p l i e s t h a t t h e c h o i c e o f a l t e r n a t i v e A^ e n t a i l s t h e c h o i c e g g o f A w h i l e t h e c h o i c e o f a l t e r n a t i v e B_ e n t a i l s t h e c h o i c e o f B . x O x ' f o r a l l v a l u e s o f x and B. Hence, t h e c h o i c e o f an A a l t e r n a t i v e i n one o f t h e ca s e s o f t h e HILO s t r u c t u r e and a c h o i c e o f a B a l t e r n a t i v e i n a n o t h e r c a s e (such as t h e s t a n d a r d A l l a i s c h o i c e o f A , B ) v i o - X LJ l a t e s e x p e c t e d u t i l i t y . I n a d d i t i o n t o t h e s e v i o l a t i o n s a c r o s s B l p r o b l e m s , v i o l a t i o n s may o c c u r w i t h i n each p r o b l e m s i n c e A^ may be B 2 chosen f o r some p a r t i c u l a r l e v e l 8^ w h i l e B^ may be chosen f o r some d i f f e r e n t l e v e l B 2- A l t h o u g h e m p i r i c a l r e s u l t s a r e n o t a v a i l a b l e f o r a l l t h e a c r o s s - case and w i t h i n - c a s e c o m b i n a t i o n s ( s i n c e t h i s l o t t e r y s t r u c t u r e has not appeared i n t h e l i t e r a t u r e ) , t h e r e a r e r e s u l t s f o r some o f t h e p a r t i c u l a r c a s e s . T a b l e 3.1 summarizes t h e c h o i c e s from t h e main e m p i r i c a l s t u d i e s i n terms o f t h e HILO l o t t e r y s t r u c t u r e . From t h i s t a b l e i t s h o u l d be ap p a r e n t t h a t most o f t h e e f f o r t has been d e v o t e d t o s t u d y i n g v a r i o u s v e r s i o n s o f t h e s t a n d a r d A l l a i s p a r a d o x , H-H C a s e : 06 . 8 ' S H , A H (8 > 8 ' ) 0-H C a s e : B „ . A „ L-L C a s e : A ^ , (6 > 8 ' ) 0 - L C a s e : A Q , I - L C a s e : Aj, S t a n d a r d A l l a i s P a r a d o x p+ o r> I O 3 3 i 3 3 1— at 2 1 7T o» 2 I o> o -S o */» -» (/» -*• § 1 o - 3 ui a> . 2 * Q» —< "1 -*. SS 1 -n O CO 03 o» a» m 3 3 0 T 1 T T T T o> ni 1 T T l o. (T> p tc y  p tc y  u rn  —« o 3 -J — T a* ro —* U l U l CO U l o o T 1 —• O O O O U 1 tJl r o —• —' ro . . O U l O U l U l o *»• -C* U l UJ ui ui co r o <n U N M ro —« —• ^ ifl U3 3 U l 3 11 CO II —' S< —' 8 9 69 t h e I-L c a s e t Note t h a t w h i l e t h e f r e q u e n c y o f c h o o s i n g t h e v i o l a t i n g 6 6 c h o i c e s , A j , B^, v a r i e s a c r o s s s t u d i e s , t h e v i o l a t i o n seems r o b u s t o v e r q u i t e d i f f e r e n t l e v e l s o f consequences and p r o b a b i l i t i e s . R e c e i v - i n g i n c r e a s i n g a t t e n t i o n r e c e n t l y has been t h e L-L p a t t e r n , A^ , B ^ ( i n c l u d i n g t h e s p e c i a l c a s e 0-L, o f c h o i c e s A , B 2 ) . U Li The o n l y s t u d i e s which have c o n s i d e r e d s e v e r a l cases s i m u l t a n - e o u s l y a r e t h o s e o f MacCrimmon and L a r s s o n (1975) and Kahneman and T v e r s k y (1979) . The f o r m e r s t u d y i n t r o d u c e d t h e 0-H and the H-H cases i n t h e c o n t e x t o f n e g a t i v e outcomes and i s t h e o n l y a t t e m p t t o map p a t t e r n s o f p r e f e r e n c e s between A^ and B^ f o r v a r i o u s l e v e l s o f 3 ( i n c l u d i n g the s p e c i a l "0" case o f 6 = 1.0) and v a r i o u s l e v e l s o f t h e i n t e r m e d i a t e outcome, I . I t seems c l e a r from T a b l e 3.1 t h a t o u r u n d e r s t a n d i n g o f a c t u a l c h o i c e s f o r t h e d e c i s i o n problems i n F i g u r e 3.4 i s i n c o m p l e t e . S t u d i e s t h u s f a r c o n d u c t e d have c o v e r e d t h e I - L , L-0, and L-L c a s e s , f o r g a i n s , and t h e H-0 and H-H c a s e s , f o r l o s s e s . W i t h t h e HILO l o t t e r y s t r u c t u r e , t h e r e a r e p o t e n t i a l l y 6 d i s t i n c t b i n a r y v i o l a t i o n p a t t e r n s , " I - L " , " H - I " , "1-0", "L-0", "H-0" and "H-L", a c r o s s problems and t h r e e b i n a r y v i o l a t i o n p a t t e r n s , "H-H", " I - I " and " L - L " w i t h i n p r o b l e m s . The c a s e s " H - I " , "1-0", " I - I " and "H-L" re m a i n u n e x p l o r e d . To o b t a i n a more c o m p l e t e p i c t u r e , a s y s t e m a t i c s t u d y o f c h o i c e s r e l a t e d t o F i g u r e 3.4 i s needed. The t e r m i n o l o g y " I - L c a s e " r e f e r s t o t h e p a i r o f l o t t e r i e s c o m p r i s i n g t h e I case b e i n g p r e s e n t e d i n c o n j u n c t i o n w i t h t h e l o t t e r i e s o f t h e L c a s e . 70 3.3 CONCURRENCE OF RISK SEEKING AND RISK AVERTING BEHAVIOR We s h a l l n o t d i s c u s s h e r e t h e e x i s t e n c e o f n o n - o v e r l a p p i n g r i s k - s e e k i n g and r i s k - a v e r t i n g r e g i o n s o f a u t i l i t y f u n c t i o n (see e.g. F i g . 3 . 1 ) c o r r e s p o n d i n g t o t h e p u r c h a s e o f l o t t e r y t i c k e t s , p u r c h a s e o f i n s u r - ance and g r e a t e r r i s k - s e e k i n g p r o p e n s i t y f o r l o t t e r i e s i n v o l v i n g l o s s e s . T h i s has been g i v e n adequate coverage e l s e w h e r e (Friedman and Savage, 1948; M a r k o w i t z , 1952; Kahneman and T v e r s k y , 1979). I n s t e a d , we f o c u s o u r a t t e n t i o n on t h e p o s s i b l e c o n c u r r e n c e o f r i s k p r o n e n e s s and r i s k a v e r s i o n w i t h i n t he same r e g i o n ; t h u s n e g a t i n g any e x p l a n a t i o n based on m o d i f i c a t i o n s o f t h e von Neumann-Morgenstern u t i l i t y f u n c t i o n . S e v e r a l measurement p r o c e d u r e s t o e l i c i t a d e c i s i o n maker's von' Neumann-Morgenstern u t i l i t y f u n c t i o n a r e based on l o t t e r y c o m p a r i - son o f t h e s o r t g i v e n i n F i g u r e 3.5. F i g u r e 3.5: S t a n d a r d l o t t e r y c o m p a r i s o n L o t t e r y A i s a s u r e consequence o f X , an amount g r e a t e r t h a n X^ but l e s s t h a n X . L o t t e r y B i s a p chance o f g e t t i n g X and a 1-p chance o f g e t t i n g X . I f we f i x t h e amounts i n l o t t e r y B a t X® and X^ 71 r e s p e c t i v e l y b u t a l l o w X £ and p t o v a r y , t h e n t h e p a i r s o f numbers ( X c , p ) such t h a t l o t t e r y A i s i n d i f f e r e n t t o l o t t e r y B d e f i n e a f u n c t i o n w h i c h we denote by u c ( X ) , i . e . l o t t e r y A i s i n d i f f e r e n t t o l o t t e r y B whenever p = u (X ) . We denote an a f f i n e t r a n s f o r m a t i o n o f u (X) by u c ( X ) , i . e . u c ( X ) = a u c ( X ) + b, f o r some a , b w i t h a > 0. Suppose the d e c i s i o n maker i s an e x p e c t e d u t i l i t y m a x i m i z e r w i t h von Neumann- M o r g e n s t e r n u t i l i t y f u n c t i o n u ( X ) . Then A i n d i f f e r e n t t o B i m p l i e s t h a t u ( X c ) = G c(X c)u(X°) + ( l - u c ( X c ) ) u ( x J ) , o r a l t e r n a t i v e l y u ( X c ) = [u (x° ) -uCxJ) ]a c CX c D + u(X°J , (3.1) whic h i s an a f f i n e t r a n s f o r m a t i o n o f u (X ). c c T h e r e f o r e , t h e f u n c t i o n u c ( X ) i s a von Neumann-Morgenstern u t i l i t y f u n c t i o n . The above measurement p r o c e d u r e i s u s u a l l y known as t h e Certainty Equivalent method. We can a l t e r n a t i v e l y f i x t h e s u r e amount i n A a t X^ and t h e l o s s c amount i n B a t X^ and o b t a i n t h e p a i r s (X , p (X ) ) such t h a t A remains i n d i f f e r e n t t o B. A p p l y i n g e x p e c t e d u t i l i t y a g a i n , we o b t a i n t h e f o l l o w i n g r e l a t i o n . u(X°) = p (X ) u ( X ) + (1-p (XJ)u(X°J , o r a l t e r n a t i v e l y L 6 6 6 6 6 *> u ( X ? ) - u(X°) S i n c e u(X) i s r e l a t e d t o p * ( x ) t h r o u g h an a f f i n e t r a n s f o r m a t i o n , p^ *(X) i s a l s o a v o n Neumann-Morgenstern u t i l i t y f u n c t i o n ; and t h i s 72 e l i c i t a t i o n p r o c e d u r e i s c a l l e d t h e Gain Equivalent method. S i m i l a r l y , we can d e t e r m i n e t h e von Neumann-Morgenstern u t i l i t y u s i n g the Loss Equivalent method by f i x i n g and X^ t o o b t a i n the p a i r s (X ,p (X ) ) such t h a t a p (X ) chance o f X v e r s u s g e t t i n g X o t h e r w i s e i s i n d i f f e r e n t t o g e t t i n g X^ f o r s u r e . We a p p l y e x p e c t e d u t i l i t y a g a i n and o b t a i n : . u(X°) - u(X°) T h e r e f o r e , -=—^jvT i s a v o n Neumann-Morgenstern u t i l i t y f u n c t i o n . 1 _ V X J The c e r t a i n t y , g a i n and l o s s e q u i v a l e n t methods a r e o b t a i n e d from F i g . 3 . 5 by h o l d i n g c o n s t a n t the p a i r s (X , X ) , (X , X ) and (X , X ) X. g £ C C g r e s p e c t i v e l y . I f i n s t e a d we h o l d p^ c o n s t a n t , t h e n t h e r e a r e t h r e e r e m a i n i n g c a n d i d a t e s f o r a d d i t i o n a l methods known c o l l e c t i v e l y as t h e Chaining Methods which a r e o b t a i n e d by h o l d i n g c o n s t a n t t h e p a i r s ( P ^ , X j ) , (P^,X^) and (p^,X-^) r e s p e c t i v e l y . Of t h e t h r e e c a s e s , we x. C g d e s c r i b e o n l y t h e f i r s t i . e . f i x i n g ( p ^ , X ^ ) , w h i c h i s more o f t e n used i n p r a c t i c e . B e g i n n i n g w i t h an amount X^ l a r g e r t h a n X^, d e t e r m i n e an amount X 2 such t h a t a s u r e consequence o f X 2 i - n A would be i n d i f f - e r e n t t o p^ chance o f g e t t i n g X^ and o b t a i n i n g X^ o t h e r w i s e i n B. Determine a t h i r d v a l u e X^ by r e p l a c i n g X^ by X 2 and r e p e a t i n g t h e p r o c e s s . Thus, we o b t a i n a d e c r e a s i n g sequence ( X ^ , X 2 , X . j , . ..) w i t h t h e f o l l o w i n g p r o p e r t y , u ( X . + 1 ) = p°u(X.) + (l-p°)u(X°). ( 3 . 4 ) A s s i g n i n g a r b i t r a r y v a l u e s t o u ( ^ ) a n d a g r e a t e r v a l u e t o u ( X ^ ) , 73 we o b s e r v e t h a t e x p r e s s i o n (3.4) d e t e r m i n e s t h e von Neumann-Morgenstern u t i l i t y on t h e d e c r e a s i n g s e t o f p o i n t s (X^,X2,X2> . . . ) . We s h a l l denote t h e u t i l i t y f u n c t i o n o b t a i n e d u s i n g t h e c h a i n i n g method w i t h f i x e d p r o b a b i l i t y p° and f i x e d l o s s amount X^ by u 0 ( X ) . a p As we have n o t e d e a r l i e r , i f t h e d e c i s i o n maker i s a ' t r u e ' e x p e c t e d u t i l i t y m a x i m i z e r , t h e n t h e u t i l i t y f u n c t i o n s u , u , u C g J6 and Up Q would be a f f i n e t r a n s f o r m a t i o n s o f each o t h e r so t h a t any one o f them s u f f i c e s . A l l a i s (1979, A p p e n d i x C ) , i n an e x p e r i m e n t c o n d u c t e d i n 1952 found t h a t t h e u and u, c u r v e s o b t a i n e d f r o m t h e C 2 same s u b j e c t s a r e g e n e r a l l y v e r y d i f f e r e n t . A f a i r l y t y p i c a l p l o t i s g i v e n i n F i g u r e 3.6. Note t h a t under t h e e x p e c t e d u t i l i t y hypo- t h e s i s , t h e convex r e g i o n n e a r X=0 o f u^ i s not c o m p a t i b l e w i t h t h e g l o - b a l c o n c a v i t y o f Uj . B e h a v i o r a l l y , t h e r i s k s e e k i n g r e g i o n o f u ~"2 C c o r r e s p o n d s t o o u r i n t u i t i o n about t h e p s y c h o l o g y o f l o t t e r y p u r c h a s e -- p e o p l e t e n d t o f o r g o a s m a l l c e r t a i n amount i n f a v o u r o f a s m a l l chance o f a l a r g e g a i n ; w h i l e t h e c o n c a v i t y o f ux r e a f f i r m s t h e r e l u c - 2 t a n c e o f i n d i v i d u a l s t o engage i n symmetric b e t s .   76 In an o n g o i n g s t u d y on the r i s k a t t i t u d e s o f t o p - l e v e l b u s i n e s s managers c a r r i e d out by MacCrimmon and o t h e r s [ 1 9 7 2 ) , c h a i n i n g and g a i n e q u i v a l e n t s were among t h e methods used t o a s s e s s von Neumann- M o r g e n s t e r n u t i l i t y f u n c t i o n s . F i g . 3.7 d i s p l a y s a t y p i c a l p a i r o f c u r v e s u and u o b t a i n e d from a s u b j e c t u s i n g t h e c h a i n i n g method and g a i n e q u i v a l e n t method r e s p e c t i v e l y . Note a g a i n t h a t t h e c o n v e x i t y o f u n e a r X=0 i s i n c o n s i s t e n t , under 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 , w i t h the c o n c a v i t y o f u a t t h e same r e g i o n . The same i n c o n s i s t e n c y a l s o a p p l i e s , though i n t h e o p p o s i t e d i r e c t i o n , t o t h e convex r e g i o n o f Ug v e r s u s t h e c o n c a v i t y o f u ^ beyond i t s i n i t i a l convex r e g i o n . Even though t h e e m p i r i c a l e v i d e n c e on t h e c o n c u r r e n c e o f r i s k p r o n e n e s s and r i s k a v e r s i o n w i t h i n t h e same range o f w e a l t h l e v e l s i s s c a n t and f r a g m e n t a r y , what we a l r e a d y know about t h e a c t u a l a p p l i c a - t i o n o f d i f f e r e n t methods t o e l i c i t u t i l i t y f u n c t i o n s s u g g e s t s t h a t e x p e c t e d u t i l i t y does n o t account f o r t h e r e s u l t s . T h i s s u g g e s t s t h a t f u r t h e r i n v e s t i g a t i o n o f t h e c o n c u r r e n c e o f r i s k p r o n e n e s s and r i s k a v e r s i o n and e s p e c i a l l y m u t u a l i n c o m p a t i b i l i t y o f t h e d i f f e r e n t measurement p r o c e d u r e s t o o b t a i n von Neumann-Morgenstern u t i l i t y f u n c t i o n s i s w a r r a n t e d . 77 3.4 SOME PROBLEMS WITH PROBLEM REPRESENTATION Pro b l e m r e p r e s e n t a t i o n and i t s i n f l u e n c e on p r e f e r e n c e s i s a r e l a t i v e l y u n touched a r e a o f r e s e a r c h on d e c i s i o n - m a k i n g . N o r m a t i v e l y , a d e c i s i o n maker's p r e f e r e n c e s h o u l d n o t depend on t h e way a l t e r n a t i v e s a r e p e r c e i v e d o r r e p r e s e n t e d as l o n g as i t does n o t a f f e c t t h e d e s i r - a b i l i t y o f t h e u n d e r l y i n g consequences o f h i s a l t e r n a t i v e s . That t h i s may n o t be t h e case i s d e m o n s t r a t e d by Kahneman and T v e r s k y (1979) t h r o u g h a c l a s s o f phenomena termed I s o l a t i o n E f f e c t s . C o n s i d e r the c h o i c e between A and B i n F i g u r e 3.8. $0 $4000 F i g . 3.8: G r a p h i c a l r e p r e s e n t a t i o n o f two l o t t e r i e s $4000 $0 F i g . 3.9: A s e q u e n t i a l r e p r e s e n t a t i o n o f l o t t e r y B 78 I f you a r e an e x p e c t e d u t i l i t y d e c i s i o n maker, the n y o u r p r e f e r e n c e does n o t depend on how t h e p r o b a b i l i t i e s o f f i n a l outcomes a r e o b t a i n e d , so t h a t l o t t e r y C i n F i g . 3.9 i s e q u i v a l e n t t o l o t t e r y B i n F i g . 3.8, i . e . p r e f e r e n c e between A and B s h o u l d be i n t h e same d i r e c t i o n as p r e - f e r e n c e between A and C. Kahneman and T v e r s k y found f o r one group o f s u b j e c t s ( n = 95 ) , 65% p r e f e r l o t t e r y B t o l o t t e r y A. However, t h e modal p r e f e r e n c e p a t t e r n between l o t t e r y A and l o t t e r y C f o r a n o t h e r group o f s u b j e c t s i s found t o be t h e o p p o s i t e (78% p r e f e r C t o A; w i t h n=141). The p r o b l e m d e s c r i p t i o n t h a t e l i c i t e d t h e modal p r e f e r e n c e f o r l o t t e r y C v e r s u s l o t t e r y A i s r e p r o d u c e d below. (4000,.80) r e f e r s t o a l o t t e r y t h a t pays $4000 w i t h .8 p r o b a b i l i t y and $0 w i t h .2 p r o b a - b i l i t y , and (3000) denotes t h e s u r e consequence o f $3000. C o n s i d e r t h e f o l l o w i n g t w o-stage game. In t h e f i r s t s t a g e , t h e r e i s a p r o b a b i l i t y o f .75 t o end t h e game w i t h o u t w i n n i n g a n y t h i n g , and a p r o b a b i l i t y o f .25 t o move i n t o t h e second s t a g e . I f you r e a c h t h e second s t a g e you have a c h o i c e between (4000,.80) and (3000) Your c h o i c e must be made b e f o r e t h e game s t a r t s , i . e . , b e f o r e t h e outcome o f t h e f i r s t s t a g e i s known. The p r o b l e m d e s c r i p t i o n above f o c u s e s t h e s u b j e c t s ' a t t e n t i o n on the c h o i c e between (4000,.80) and (3000) r a t h e r t h a n t h e common outcome $0 w i t h the' same p r o b a b i l i t y .75. Kahneman and T v e r s k y c o n j e c t u r e d t h a t most s u b j e c t s t h e n i g n o r e t h e common outcome - p r o b a b i l i t y component under t h e above p r o b l e m r e p r e s e n t a t i o n so t h a t t h e i r c h o i c e becomes i d e n t i c a l t o t h a t between (4000,.80) and (3 0 0 0 ) . 79 A n o t h e r k i n d o f i s o l a t i o n e f f e c t c o n s i d e r e d by Kahneman and T v e r s k y i s r e l a t e d t o M a r k o w i t z (1952)'s o b s e r v a t i o n t h a t p r e f e r e n c e s are r e l a t i v e l y i n d e p e n d e n t o f c u r r e n t w e a l t h l e v e l s . They p r e s e n t e d t h e f o l l o w i n g p roblems t o two d i f f e r e n t groups o f s u b j e c t s . P roblem 1. In a d d i t i o n t o whatever you own, you have been g i v e n 1000. You a r e now asked t o choose between A: (1000,.50) and B: (500) P r o b l e m 2. In a d d i t i o n t o whatever you own, you have been g i v e n 2000. You a r e now asked t o choose between C: (-1000,.50) and D: (-500) The m a j o r i t y o f s u b j e c t s chose A i n P r o b l e m 1 and B i n P r o b l e m 2. Note> however, t h a t i n terms o f f i n a l outcomes, t h e two c h o i c e problems a r e e q u i v a l e n t i . e . you a r e e i t h e r $1500 r i c h e r i f you choose B o r D, o r you have even chance o f e n d i n g up w i t h $1000 o r $2000 more i f you choose A o r C. P e o p l e , however, seem t o p e r c e i v e P r o b l e m 1 as a c h o i c e between (1000,.50) and ( 5 0 0 ) , and Problem 2 as a c h o i c e between (-1000,.50) and (-500), w i t h t h e lump sums o f $1000 i n P r o b l e m 1 and $2000 i n P r o b l e m 2 s a f e l y i n t e g r a t e d i n t o t h e i r c u r r e n t w e a l t h l e v e l s . As an a l t e r n a t i v e t o t h e f i n a l outcome p o s i t i o n n o r m a l l y a s s o c i a t e d w i t h e x p e c t e d u t i l i t y , Kahneman and T v e r s k y p r o p o s e d t h a t p e o p l e p e r c e i v e outcomes as g a i n s and l o s s e s r e l a t i v e t o some n e u t r a l r e f e r e n c e p o i n t . A l l o w i n g t h e r e f e r e n c e p o i n t t o be d e t e r m i n e d by t h e d e c i s i o n maker i n t h e c o n t e x t o f t h e c h o i c e s i t u a t i o n he f a c e s , e x p e c t e d u t i l i t y , w i t h a u t i l i t y f u n c t i o n d e f i n e d -on changes i n a s s e t p o s i t i o n r e l a t i v e t o t h e r e f e r e n c e p o i n t , i s c o m p a t i b l e w i t h t h e modal p r e f e r e n c e s i n 80 Problems 1 and 2. The r e f e r e n c e p o i n t need n o t be t h e s t a t u s quo e s p e c i a l l y when t h e c h o i c e s i t u a t i o n i n v o l v e s ' a s u r e g a i n o f $1000. 3.5 SUMMARY We have r e v i e w e d b r i e f l y some o f t h e l i t e r a t u r e on e m p i r i c a l e v i d e n c e t h a t c o n t r a d i c t s t h e i m p l i c a t i o n s o f e x p e c t e d u t i l i t y . I t has been c l a s s i f i e d under t h e headings:' s y s t e m a t i c v i o l a t i o n s o f t h e s t r o n g independence p r i n c i p l e ( s e c t i o n 3.2), c o n c u r r e n c e o f r i s k p r o n e n e s s and r i s k a v e r s i o n ( s e c t i o n 3 . 3 ), and some problems w i t h p r o b l e m r e p r e s e n t a t i o n ( s e c t i o n 3.4). The phenomena c o n s i d e r e d i n c l u d e t h e A l l a i s p a r a d o x and i t s v a r i o u s m o d i f i c a t i o n s , i n c o m p a t i b i l i t y among d i f f e r e n t methods o f m e a s u r i n g a von Neumann-Morgenstern u t i l i t y f u n c t i o n , and Kahneman and T v e r s k y ' s i s o l a t i o n e f f e c t s . In t h e n e x t c h a p t e r , we g e n e r a l i z e e x p e c t e d u t i l i t y t h e o r y by a p p l y i n g t h e r e p r e s e n t a t i o n theorems o f P a r t I as a t h e o r y o f c h o i c e . We t h e n e x p l o r e t h e d e s c r i p t i v e i m p l i c a t i o n s o f our g e n e r a l i z a t i o n w i t h r e s p e c t t o t h e phenomena c o n s i d e r e d i n t h i s c r i t i q u e . 4 A NEW THEORY 81 We d e v e l o p i n t h i s c h a p t e r a new t h e o r y o f c h o i c e c a l l e d a l p h a u t i l i t y t h e o r y w h i c h g e n e r a l i z e s e x p e c t e d u t i l i t y b y i n t e r p r e t i n g , i n s e c t i o n s 1 and 2, t h e r e p r e s e n t a t i o n theorems o f C h a p t e r s 1 and 2 i n terms o f c h o i c e among l o t t e r i e s . We e x p l o r e i n s e c t i o n 3 some n o r m a t i v e i m p l i c a t i o n s o f our t h e o r y i n c l u d i n g c o n s i s t e n c y c o n d i t i o n s w i t h s t o c h a s t i c dominance and l o c a l and g l o b a l r i s k a v e r s i o n . The q u e s t i o n o f d e s c r i p t i v e v a l i d i t y i s c o n s i d e r e d i n s e c t i o n 4 where we show t h a t a l p h a u t i l i t y i s c o m p a t i b l e w i t h t h e phenomena r e v i e w e d i n t h e c r i t i q u e o f e x p e c t e d u t i l i t y ( C h a p t e r 3 ) . We end t h e c h a p t e r w i t h a comparison o f our t h e o r y w i t h two o t h e r a l t e r n a t i v e t h e o r i e s o f c h o i c e . 4.1 INTERPRETING MEAN VALUE AS CERTAINTY EQUIVALENT In C h a p t e r 1, we p r o v e d a r e p r e s e n t a t i o n theorem o f a mean v a l u e f u n c t i o n a l , M, f o r p r o b a b i l i t y d i s t r i b u t i o n s s u b j e c t t o a n e c e s s a r y and s u f f i c i e n t s e t o f axioms. The p r e s e n t s e c t i o n e x p l o r e s t h e i m p l i - c a t i o n s o f o u r r e p r e s e n t a t i o n theorem f o r c h o i c e among l o t t e r i e s by 2 i n t e r p r e t i n g mean v a l u e s as c e r t a i n t y e q u i v a l e n t s . I n t h e e n s u i n g d i s c u s s i o n , we assume t h a t l o t t e r i e s , d e f i n e d on some s i n g l e - a t t r i b u t e consequence set, e.g., monetary g a i n s , can be r e p r e s e n t e d by p r o b a b i l i t y d i s t r i b u t i o n s d e f i n e d on t h e r e a l l i n e . (The case o f more g e n e r a l con- sequence space i s c o n s i d e r e d i n s e c t i o n 4.2) The d e c i s i o n maker i s assumed t o have c o m p l e t e and t r a n s i t i v e p r e f e r e n c e o v e r l o t t e r i e s . The f o l l o w i n g axiom a s s e r t s t h a t he i s a b l e t o a s s i g n , c o r r e s p o n d i n g t o any l o t t e r y F, a c e r t a i n t y e q u i v a l e n t 2 T h i s was d i s c u s s e d i n Chew (1979). 82 M(F) w h i c h i s an amount such t h a t t h e d e c i s i o n maker i s i n d i f f e r e n t between g e t t i n g i t f o r s u r e and t a k i n g t h e l o t t e r y F. D denotes t h e space 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 ( l o t t e r i e s ) d e f i n e d on t h e r e a l l i n e . Axiom MO: E x i s t e n c e V F e D, M(F) e x i s t s . Axiom MO r u l e s out i n f i n i t e c e r t a i n t y e q u i v a l e n t s t h u s p r e - e m p t i n g any p o s s i b i l i t y o f a S t . P e t e r s b u r g t y p e p a r a d o x . The f o l l o w i n g f i v e axioms M1-M5 are t a k e n d i r e c t l y from Chapter 1. We d i d n o t s t a t e e x i s t e n c e (MO) as an axiom i n C h a p t e r 1 because i t i s n o t an i n t r i n s i c p r o p e r t y o f mean v a l u e s . F o r example, t h e a r i t h m e t i c mean does n o t always e x i s t . Axiom M l : C e r t a i n t y C o n s i s t e n c y M(6 ) = x V x £ R. X I t i s d i f f i c u l t t o t a k e i s s u e w i t h Axiom Ml wh i c h r e q u i r e s t h a t t h e c e r t a i n t y e q u i v a l e n t o f ̂  i . e . g e t t i n g x f o r s u r e , i s x. A n o t h e r n o r m a t i v e l y a p p e a l i n g axiom i s t h e f o l l o w i n g : Axiom M2: Betweenness V F, G e D, i f M(F) < M(G), t h e n V B € ( 0 , 1 ) , M(BF+(1-B)G) €E (M(F),M(G)). Axiom M2 r e q u i r e s t h a t t h e c e r t a i n t y e q u i v a l e n t o f a m i x t u r e o f two l o t t e r i e s be i n t e r m e d i a t e i n v a l u e between t h e c e r t a i n t y e q u i v a l e n t s o f t h e r e s p e c t i v e l o t t e r i e s . The n e x t axiom weakens t h e axiom o f q u a s i l i n e a r i t y o f Hardy, L i t t l e w o o d and P o l y a o r t h e " s u b s t i t u t i o n o f l o t t e r i e s " p r i n c i p l e o f P r a t t , R a i f f a and S c h a i f e r . 83 Axiom M3: Weak S u b s t i t u t i o n V F, G e D, i f M(F)' = M(G), t h e n V 3 e (0,1) 3 y £ (0,1) 3 V H e D, M(3F+(1-3)H) = M ( y G + ( l - Y ) H ) . Suppose F and G a r e two l o t t e r i e s w i t h t h e same c e r t a i n t y e q u i v a l e n t s i . e . i n d i f f e r e n t t o each o t h e r . C o n s i d e r t h e m i x t u r e s o r compound l o t t e r i e s 3 F + (1-3)H and y G + ( l - y ) H o f t h e r e s p e c t i v e l o t t e r i e s , w i t h a ; t h i r d l o t t e r y H a t p r o b a b i l i t i e s 6 and y. Weak s u b s t i t u t i o n weakens th e s u b s t i t u t i o n p r i n c i p l e i n t h e sense t h a t t h e m i x t u r e p r o b a b i l i t i e s 3 and y t h a t p r e s e r v e e q u a l i t y o f c e r t a i n t y e q u i v a l e n t s o r i n d i f f e r e n c e need not be the same. However, i t r e q u i r e s t h a t t h e s e m i x t u r e p r o b a b i l i - t i e s once d e t e r m i n e d be i n dependent o f t h e t h i r d l o t t e r y H. The n e x t two axioms a r e t e c h n i c a l a s s u m ptions which a r e i n t r o d u c e d t o e nsure t h a t our n o t i o n o f c e r t a i n t y e q u i v a l e n t i s w e l l behaved r e l a - t i v e t o c e r t a i n l i m i t i n g o p e r a t i o n s on p r o b a b i l i t y d i s t r i b u t i o n s . Axiom M4 p e r m i t s t h e a p p r o x i m a t i o n o f l o t t e r i e s w i t h numerous d i s c r e t e outcomes by a c o n t i n u o u s d i s t r i b u t i o n e.g. u n i f o r m d i s t r i b u t i o n . Axiom M4: C o n t i n u i t y I f ^ F n ^ _ ^ converges i n d i s t r i b u t i o n t o F and F has compact s u p p o r t , t h e n M(F) = Lim M(F ) . F o r a more t e c h n i c a l d i s c u s s i o n o f Axiom M4 see C h a p t e r 1. Our l a s t axiom s t a t e d below a s s e r t s t h a t t h e c e r t a i n t y e q u i v a l e n t o f a l o t t e r y F , t r u n c a t e d by t h e i n t e r v a l K, i s ' c l o s e ' t o t h a t o f t h e o r i g i n a l l o t t e r y F i f t h e i n t e r v a l o f t r u n c a t i o n K i s s u f f i c i e n t l y l a r g e . 84 o o Axiom M5: E x t e n s i o n L e t {K } . b e an i n c r e a s i n g f a m i l y o f — n n = 0 J We a r e now r e ady t o r e s t a t e Theorem 1.3 o f C h a p t e r 1 i n terms o f c e r t a i n t y e q u i v a l e n t s . Suppose a d e c i s i o n maker a s s i g n s a c e r t a i n t y e q u i v a l e n t M(F) c o r r e s p o n d i n g t o a l o t t e r y F, t h e n t h e assignment o f M(F) s a t i s f i e s Axioms M1-M5 i f and o n l y i f t h e r e e x i s t a s t r i c t l y p o s i t i v e l y v a l u e d f u n c t i o n a and a 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 v such t h a t V F G D, compact i n t e r v a l s such t h a t Lim K = R. n-*=° n Then M(F) = Lim Y[{Jc ) , V F e D. n M(F) = v -1 (/ ctvdF// adF) . (4.1) Moreover, i f (ct*,v*) i s a n o t h e r p a i r o f f u n c t i o n s t h a t s a t i s f i e s c o n d i t i o n ( 4 . 1 ) , t h e n , f o r e v e r y i n t e r v a l [A,B] C R, t h e r e e x i s t c o n s t a n t s a,b,c,k w i t h a,c,k > 0 such t h a t V x S [ A , B ] , v * ( x ) a k [ v ( x ) - v ( A ) ] + v ( B ) - v ( x ) k [ v ( x j - v ( A ) ] + b , and a * ( x ) = c a ( x ) [ k ( v ( x ) - v ( A ) ) ] + v(B) - v ( x ) ] . (4.2) 85 Note t h a t t h e e x p r e s s i o n f o r c e r t a i n t y e q u i v a l e n t (4.1) i s more g e n e r a l t h a n t h e c o r r e s p o n d i n g e x p r e s s i o n f o r e x p e c t e d u t i l i t y , M(F) = v _ 1 ( / R v d F ) (4.3) which r e s u l t s when a i s c o n s t a n t . The u n i q u e n e s s r e l a t i o n (4.2) sub- sumes as s p e c i a l c a s e s , an a f f i n e t r a n s f o r m a t i o n f o r v and a p o s i t i v e s c a l a r t r a n s f o r m a t i o n f o r a. To a v o i d t h e p o s s i b i l i t y o f any S t . P e t e r s b u r g t y p e p a r a d o x , we have a s s e r t e d t h r o u g h Axiom MO t h a t t h e c e r t a i n t y e q u i v a l e n t M(F) must be f i n i t e f o r any l o t t e r y F. We can t h e n a p p l y C o r o l l a r y 1.3 i n C h a p t e r 1 t o c o n c l u d e t h a t e i t h e r v i s bounded o r a.v i s bounded. Thus, we have o b t a i n e d a g e n e r a l i z a t i o n o f e x p e c t e d u t i l i t y , c a l l e d a l p h a u t i l i t y , i n the sense t h a t t h e d e c i s i o n maker's p r e f e r e n c e o v e r l o t t e r i e s i s r e p r e s e n t e d by a more g e n e r a l e x p r e s s i o n f o r h i s c e r t a i n t y e q u i v a l e n t c h a r a c t e r i z e d by an a d d i t i o n a l a l p h a - f u n c t i o n . We s h a l l d e f e r d i s c u s s i o n o f t h e a c t u a l shapes o f a and v f u n c t i o n s t i l l t h e s e c t i o n s a f t e r n e x t . The n e x t s e c t i o n c o n t a i n s a p a r a l l e l development o f the above t h e o r y u s i n g t h e r e p r e s e n t a t i o n theorems o f C h a p t e r 2. 86 4.2 REPRESENTATION OF A PREFERENCE BINARY RELATION In t h e p r e c e d i n g s e c t i o n , we d e v e l o p e d a l p h a u t i l i t y t h e o r y f o r l o t t e r i e s d e f i n e d on a s i n g l e - a t t r i b u t e consequence s e t , e.g., monetary l o t t e r i e s , v i a an a x i o m a t i z a t i o n o f t h e d e c i s i o n maker's assignment of- c e r t a i n t y e q u i v a l e n t s . We showed, u s i n g t h e mean v a l u e r e p r e s e n t a t i o n theorems o f C h a p t e r 1 , t h a t t h e c e r t a i n t y e q u i v a l e n t M(F) c o r r e s p o n d i n g t o a l o t t e r y F i s g i v e n by: M(F) = v _ 1 ( / R a v d F / / R a d F ) (4.4) f o r some s t r i c t l y p o s i t i v e f u n c t i o n a and 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 v. When a l o t t e r y F i s p r e f e r r e d t o a l o t t e r y G, t h e c e r t a i n t y e q u i - v a l e n t M(F) i s g r e a t e r t h a n t h a t o f G, M(G). I t f o l l o w s ( s i n c e v 1 i s o r d e r - p r e s e r v i n g ) t h a t L o v d F / L a d F > / avdG / / adG . (4.5) K K K K Observe t h a t t h e p r e f e r e n c e r e p r e s e n t e d by t h e above f u n c t i o n a l i s more g e n e r a l t h a n t h a t o f e x p e c t e d u t i l i t y , /„vdF, wh i c h i s a s p e c i a l c a s e o f (4.5) when a i s c o n s t a n t . The p r e s e n t s e c t i o n a p p l i e s t h e r e p r e s e n t a t i o n theorems o f C h a p t e r 2 t o p r o v i d e an a l t e r n a t i v e a x i o m a t i z a t i o n o f a l p h a u t i l i t y t h e o r y U n l i k e t h e c e r t a i n t y e q u i v a l e n t a p p r o a c h , we do not l i m i t o u r s e l v e s t o c h o i c e s i t u a t i o n s where t h e range o f consequences a r e monetary v a l u e s o r some q u a n t i t y o f c e r t a i n commodity. Fo r example, t h e r e l e v a n t consequence s e t f o r a c h i l d c o n t e m p l a t i n g whether t o s t e a l a cake from a b a k e r y may be s t a t u s quo, h a v i n g t h e c ake, g e t t i n g caught i n t h e p r o c e s s . We denote by 3 T h i s was d i s c u s s e d i n Chew and MacCrimmon (197 9 a ) . 87 X (={x,y,z,•••}) t h e consequence s e t c o r r e s p o n d i n g t o a c h o i c e s i t u a t i o n and l _ x (= (P> Q> R,***}) t h e space o f s i m p l e p r o b a b i l i t y measures d e f i n e d on X ( c f . C h a p t e r 1 ) . A s i m p l e p r o b a b i l i t y measure i s c o m p l e t e l y s p e c i f i e d by knowledge o f t h e p r o b a b i l i t i e s o f o c c u r r e n c e o f a f i n i t e number o f consequences. The c h i l d i n t h e 'above example may f e e l t h a t he has an even chance o f g e t t i n g t h e cake w i t h o u t b e i n g c a u g h t . S i m p l e p r o b a b i l i t y measures a r e c o n v e n i e n t r e p r e s e n t a t i o n s o f a c t u a l l o t t e r i e s o r r i s k y d e c i s i o n s when t h e p r o b a b i l i t i e s o f o c c u r r e n c e o f t h e u n d e r l y i n g consequences can be s u b j e c t i v e l y e s t i m a t e d o r d e t e r m i n e d based on symmetry c o n s i d e r a t i o n s , e.g., a game o f c r a p s o r r o u l e t t e . We denote t h e a l t e r n a t i v e o f o b t a i n i n g some consequence x f o r s u r e by 6 . A f i n i t e l o t t e r y P i s t h e n r e p r e s e n t e d as a p r o b a b i l i t y w e i g h t e d combi- n a t i o n o f sure consequences: Our r e p r e s e n t a t i o n f u n c t i o n a l f o r s i m p l e p r o b a b i l i t y measures c o r r e s p o n d - i n g t o t h a t o b t a i n e d from t h e c e r t a i n t y e q u i v a l e n t approach (see e x p r e s - s i o n (4.5)) i s : n (4.6) where p^ i s t h e p r o b a b i l i t y o f o c c u r r e n c e o f t h e consequence x^. E(av,P) / E ( a , P ) , (4.7) where a and v a r e f u n c t i o n s on X, and E(*,P) d e n o t e s t a k i n g e x p e c t a t i o n o f a f u n c t i o n w i t h r e s p e c t t o t h e s i m p l e p r o b a b i l i t y measure P. The theorems o f C h a p t e r 2 show t h a t t h e axioms on t h e s t r i c t p r e f e r - 88 ence b i n a r y r e l a t i o n '-< ' o f a d e c i s i o n maker s t a t e d below 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 r e p r e s e n t i n g h i s p r e f e r e n c e v i a e x p r e s s i o n ( 4 . 7 ) . Axiom U I : O r d e r i n g -< i s a weak o r d e r . The s t r i c t p r e f e r e n c e r e l a t i o n •< i s a weak o r d e r i f i t i s asymme- t r i c ( i . e . , i f a l o t t e r y P i s s t r i c t l y p r e f e r r e d t o a l o t t e r y Q, t h e n t h e c o n v e r s e does n o t h o l d ) , and n e g a t i v e l y t r a n s i t i v e ( i . e . , i f a l o t t e r y P i s n ot s t r i c t l y p r e f e r r e d t o a l o t t e r y Q w h i c h i s i n t u r n n o t s t r i c t l y p r e f e r r e d t o a n o t h e r l o t t e r y R, t h e n P i s n o t s t r i c t l y p r e f e r r e d t o t h e l o t t e r y R ) . Both asymmetry and n e g a t i v e t r a n s i t i v i t y seem l i k e b a s i c c o n s i s t e n c y r e q u i r e m e n t s f o r a s t r i c t p r e f e r e n c e r e l a t i o n t h a t few would want t o v i o l a t e . Axiom U2: S o l v a b i l i t y V P, Q, R £ L X , P -< Q and Q-c R 3 B £ (0,1) 3 BP+(1-B)R ~ Q. Axiom U2 says t h a t whenever a l o t t e r y Q i s between two l o t t e r i e s P and R i n p r e f e r e n c e , t h e n t h e r e i s a m i x t u r e between P and R wh i c h i s i n d i f f e r e n t t o Q. The r e a s o n a b l e n e s s o f t h e above axiom stems from t h e i n t u i t i o n t h a t a m i x t u r e between two l o t t e r i e s i s a l w a y s i n t e r m e d i a t e i n p r e f e r e n c e between them (see t h e Betweenness axiom i n s e c t i o n 4.1 and a l s o Axioms 3:B:a and 3:B:b i n von Neumann and M o r g e n s t e r n ( 1 9 4 7 ) ) . I t f o l l o w s t h a t a m i x t u r e w i t h a g r e a t e r p r o b a b i l i t y w e i g h t on t h e b e t t e r l o t t e r y s h o u l d be p r e f e r r e d t o a n o t h e r m i x t u r e w i t h a s m a l l e r p r o b a b i l i t y w e i g h t on t h e b e t t e r l o t t e r y . T h i s i s t h e s u b s t a n c e o f t h e f o l l o w i n g axiom. 89 Axiom U3: M o n o t o n i c i t y V P, Q e l _ x , P -< Q =* BP+(1-8 ) Q •< y P + ( l - Y ) Q f o r 0 < Y < B < 1. The axioms d i s c u s s e d t h u s f a r a r e s t a n d a r d n o r m a t i v e p r o p e r t i e s common t o e x p e c t e d u t i l i t y t h e o r y . The n e x t axiom w h i c h weakens t h e s u b s t i - t u t i o n p r i n c i p l e o r i t s c l o s e c o u n t e r p a r t , t h e s t r o n g independence p r i n c i p l e , i s t h e o n l y d e p a r t u r e . Axiom U4: Weak Independence v P, Q e I_x, P ~ Q ^ V B £ ( 0 , 1 ) , 3 Y e (0,1) 3 V R e L x , B P + ( l - e ) R ~ Y Q + ( 1 - Y )R - t Weak Independence i s a r e s t a t e m e n t o f t h e Weak S u b s t i t u t i o n axiom (M3) f o r c e r t a i n t y e q u i v a l e n t s . G i v e n two l o t t e r i e s t h a t a r e i n d i f f e r e n t t o each o t h e r , Weak Independence a l l o w s f o r d i f f e r e n t p r o b a b i l i t i e s i n composing each o f t h e s e l o t t e r i e s w i t h a t h i r d l o t t e r y t o p r e s e r v e i n d i f f e r e n c e . However, t h e s e m i x t u r e - p r o b a b i l i t i e s once d e t e r m i n e d must be ind e p e n d e n t o f t h e t h i r d l o t t e r y . We a r e now r e a d y t o i n t e r p r e t t h e r e p r e s e n t a t i o n theorems o f C h a p t e r 2. A d e c i s i o n maker, whose p r e f e r e n c e among f i n i t e l o t t e r i e s d e f i n e d on a consequence s e t X s a t i s f i e s Axioms U1-U4, chooses as i f m a x i m i z i n g t h e f u n c t i o n a l E(ctv,«) / E(ct,«) f o r some s t r i c t l y p o s i t i v e - v a l u e d f u n c t i o n a and r e a l v a l u e d f u n c t i o n v b o t h d e f i n e d on t h e consequence s e t X. 90 When t h e a f u n c t i o n i s c o n s t a n t , t h e r e p r e s e n t a t i o n E ( a v , • ) / E ( a , • ) becomes t h e e x p e c t e d u t i l i t y r e p r e s e n t a t i o n E(v,«) w i t h v assuming t h e r o l e o f a von Neumann-Morgenstern u t i l i t y f u n c t i o n . A l t h o u g h t h e a p p r o a c h t a k e n i n t h i s s e c t i o n does n o t r e q u i r e t h e consequence s e t t o be s i n g l e - d i m e n s i o n a l , t h e c e r t a i n t y e q u i v a l e n t a p p roach has t h e advantage o f b e i n g a b l e t o t r e a t g e n e r a l p r o b a b i l i t y d i s t r i b u t i o n s , e . g . , c o n t i n u o u s random v a r i a b l e o r even unbounded ones l i k e t h e normal d i s t r i b u t i o n . I n t h e n e x t two s e c t i o n s where we d i s c u s s t h e n o r m a t i v e and d e s c r i p t i v e i m p l i c a t i o n s o f a l p h a u t i l i t y t h e o r y , we s h a l l r e p r e s e n t l o t t e r i e s i n terms o f e i t h 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 s o r s i m p l e p r o b a b i l i t y measures. R e p r e s e n t i n g l o t t e r i e s w i t h p r o b a b i l i t y d i s t r i b u t i o n s i s more a p p r o p r i a t e when we d e a l w i t h n u m e r i c a l outcomes, e s p e c i a l l y i f t h e outcome can t a k e on c o n t i n u o u s v a l u e s . O t h e r w i s e , s i m p l e p r o b a b i l i t y measures would be used. 4.3 NORMATIVE IMPLICATIONS D e s p i t e g r o w i n g e v i d e n c e d e m o n s t r a t i n g i t s d e s c r i p t i v e i n a d e q u a c y , e x p e c t e d u t i l i t y r e m a i n s dominant m a i n l y because o f t h e n o r m a t i v e a p p e a l o f i t s u n d e r l y i n g p o s t u l a t e s and t h e e l e g a n c e w i t h w h i c h e x p e c t e d u t i l i t y c h a r a c t e r i z e s s t o c h a s t i c dominance ( i n c r e a s i n g u t i l i t y f u n c t i o n ) , l o c a l ( A r r o w - P r a t t i n d e x ) and g l o b a l (concave u t i l i t y f u n c t i o n ) r i s k a v e r s i o n . We show i n t h i s s e c t i o n t h a t a l p h a u t i l i t y i s a l s o an a t t r a c t i v e 91 n o r m a t i v e t h e o r y b e f o r e c o n s i d e r i n g i t s d e s c r i p t i v e r e l e v a n c e i n t h e f o l l o w i n g s e c t i o n . 4.3.1 R a t i o C o n s i s t e n c y As we n o t e d e a r l i e r , e x c e p t f o r t h e Weak S u b s t i t u t i o n axiom (M3) o r e q u i v a l e n t l y t h e Weak Independence axiom (U4), t h e o t h e r axioms o f a l p h a u t i l i t y a r e s t a n d a r d p r o p e r t i e s common t o e x p e c t e d u t i l i t y . We showed i n C h a p t e r 1 (Lemma 1.2) and C h a p t e r 2 (Lemma 2.2) t h a t Weak Independence t o g e t h e r w i t h M o n o t o n i c i t y i m p l i e s t h e R a t i o C o n s i s t e n c y P r o p e r t y ( C h a p t e r 1, P r o p e r t y 5; Ch a p t e r 2, D e f i n i t i o n 2.6) which i s c r u c i a l t o o u r p r o o f s and t h e assessment o f t h e ct f u n c t i o n . The R a t i o C o n s i s t e n c y p r o p e r t y has been g i v e n a r a t h e r s t r i k i n g g e o m e t r i c a l 4 i n t e r p r e t a t i o n by Weber (1980). C o n s i d e r a s i m p l e x formed by t h e t h r e e l o t t e r i e s P, Q and R i n F i g u r e 4.1. Each l o t t e r y X i n t h e s i m p l e x i s s p e c i f i e d by i t s b a r y c e n t r i c ( a r e a l ) c o o r d i n a t e s (^, B2> 3^) g i v e n by t h e a r e a s o f t h e t r i a n g l e s XQR, RPX and XPQ. Assume t h a t t h e a r e a o f PQR i s 1, X t h e n r e p r e s e n t s a p r o b a b i l i t y m i x t u r e B^P+f^Q+^R among t h e t h r e e v e r t i c e s P, Q and R. The l o t t e r i e s Y(Z) i s a m i x t u r e between P(Q) and R w i t h B(y) we i g h t on P(Q) and l - B ( l - y ) w e i g h t on R. R a t i o C o n s i s t e n c y was o r i g i n a l l y Axiom U5 (see Chew and MacCrimmon, 1979a). I t was a l s o i m p l i c i t i n a s t r o n g e r s t a t e m e n t o f Axiom M3 (see Chew, 1979) . Weber (1980) d e m o n s t r a t e d v i a "an approach t h r o u g h t h e a n a l y s i s o f i s o - p r e f e r e n c e s e t s " t h e redundancy o f R a t i o C o n s i s t e n c y w i t h r e s p e c t t o t h e o t h e r axioms by showing t h a t Theorem 2.1 h o l d s w i t h o u t assuming R a t i o C o n s i s t e n c y . T h i s m o t i v a t e d t h e s t a t e m e n t s o f Lemma 1.2 and Lemma 2.2. 4.1: R a t i o C o n s i s t e n c y i l l u s t r a t e d u s i n g b a r y c e n t r i c c o o r d i n a t e s S 3 93 M i x i n g Y and Z w i t h w e i g h t s v a r y i n g from 0 t o 1 g e n e r a t e s t h e l i n e s e g - ment YZ. Suppose t h e l o t t e r i e s P and Q a r e i n d i f f e r e n t t o each o t h e r , t h e n M o n o t o n i c i t y i m p l i e s t h a t t h e l i n e segment PQ i s an i s o p r e f e r e n c e s e t . F o r each Y on PR, Weak Independence i m p l i e s t h e r e i s an Z on QR such t h a t Y and Z a r e i n d i f f e r e n t . Hence YZ i s an i s o p r e f e r e n c e . P r o j e c t YZ t o meet PQ extended a t t h e p o i n t 0 w i t h c o o r d i n a t e s ( s , l - s , 0 ) (Note t h a t e i t h e r s > 1 (0 i s on t h e l e f t o f P) o r s < 0 (0 i s on t h e r i g h t o f Q ) ) . The R a t i o C o n s i s t e n c y p r o p e r t y s i m p l y r e q u i r e s a l l o t h e r i s o p r e f e r e n c e s e t s such as Y'Z' j o i n i n g p o i n t s on PR t o p o i n t s on QR t o o r i g i n a t e a t t h e same p o i n t 0. That t h i s i s t h e case f o l l o w s from t h e o b s e r v a t i o n t h a t AOZR = AOYR + AYZR, (4.8) w h i c h i s g i v e n by ys = B ( s - l ) + YB- (4.9) S o l v i n g f o r y g i v e s Y = B ( s - l ) / ( s - B ) (4.10) Let x = s - l / s . E x p r e s s i o n (4.10) becomes Y / l - Y B / l - B = T . (4.11) A s i m i l a r o b s e r v a t i o n f o r t h e l i n e segment Y'Z' g i v e s Y ' / l - Y 1 Y / l - Y (4.12) B'/l-B' B/l-B 94 Hence, l i n e segments o r i g i n a t i n g from t h e same p o i n t 0 s a t i s f y t h e R a t i o C o n s i s t e n c y p r o p e r t y ( 4 . 1 2 ) . F o r e x p e c t e d u t i l i t y , t h e m i x t u r e p r o b a b i l i t i e s 8 and Y such t h a t Y i s i n d i f f e r e n t t o Z a r e alw a y s e q u a l ; so t h a t t h e c o n s t a n t x e q u a l s u n i t y . G e o m e t r i c a l l y , t h e l i n e segments YZ and Y'Z' a r e p a r a l l e l . The s i m p l i c a l r e p r e s e n t a t i o n can now be used as i n Weber (1980) t o p r o v i d e an a l t e r n a t i v e p r o o f o f Lemma 1.2 o r e q u i v a l e n t l y Lemma 2.2 w i t h o u t h a v i n g t o s o l v e t h e f u n c t i o n a l e q u a t i o n ( 1 . 6 ) . The b a s i c i d e a i s t o show, once an i s o p r e f e r e n c e say YZ i s f o u n d , t h a t a l l l i n e segments such as Y'Z' o r i g i n a t i n g from t h e p o i n t 0 o f t h e i n t e r s e c t i o n between t h e e x t e n s i o n s YZ and PQ are a l s o i s o p r e f e r e n c e s . M o n o t o n i c i t y w h i c h i m p l i e s t h a t t h e s e i s o p r e f e r e n c e s a r e n o n - i n t e r s e c t i n g t h e n e n s u r e s t h a t t h e r e a r e no o t h e r i s o - p r e f e r e n c e s . Lemma 2.2 i s p r o v e d below u s i n g t h e s i m p l i c a l a p p r o a c h . Lemma 2.2: O r d e r i n g ( U I ) , M o n o t o n i c i t y (U3) and Weak Independence (U4) i m p l i e s R a t i o C o n s i s t e n c y ( D e f i n i t i o n '2.6). P r o o f : Suppose P ~ Q but n o t (P ~ R) i n F i g u r e 4.2. Then from M o n o t o n i c i t y , PQ i s an i s o p r e f e r e n c e . C o r r e s p o n d i n g t o a m i x t u r e Y = 6P+(1-3)R, Weak Independence i m p l i e s t h a t t h e r e i s a m i x t u r e Z = Y Q + ( 1 - Y ) r such t h a t Y ~ Z> s o t h a t YZ i s a l s o i s o p r e f e r e n c e . L e t 0 be t h e p o i n t o f i n t e r s e c t i o n o f t h e e x t e n s i o n s o f YZ and PQ. (Suppose w i t h o u t l o s s o f g e n e r a l i t y t h a t 0 i s on t h e l e f t o f P ) . R(1,0,1) O(s,l-s,0) P(1,0,0) A Q(0,1,0) Figure 4.2: Geometric proof of the Ratio Consistency property 96 Consider another l i n e segment Y'Z' below YZ o r i g i n a t i n g from the same point 0. We s h a l l show that Y'Z1 i s also an isopre- ference and by the preceding discussion s a t i s f i e s the Ratio Consistency property (4.12). Draw the median from R to bis e c t PQ at A. Draw a l i n e from Y. p a r a l l e l to RA and i n t e r s e c t i n g Y'Z' at S. Produce PS to meet RA at V. Complete the t r i a n g l e PVQ by connecting V and Q. Denote by T the i n t e r s e c t i o n between VQ and Y'Z'. Draw a l i n e from Z p a r a l l e l to RA and int e r s e c t i n g VQ at T'. To see that OST' i s c o l l i n e a r and hence T* must be the same as T, view the figure i n three dimensions with V the top vertex of a tetrahedron with base PQR. The plane through the p a r a l l e l l i n e s YS and ZT' contains 0, S and T'. The plane through PV and Q also contains 0, S and T'. The conclusion that Y'Z' i s also an isopreference follows from applying Weak Independence to S = BP+(1-B)V and T = Y Q + ( 1 -Y )V . A s i m i l a r argument establishes the r e s u l t for the case Y'Z' above YZ. Q.E.D. 4.3.2 Assessment The d e r i v a t i o n o f a p r o c e d u r e f o r e l i c i t i n g a l p h a u t i l i t y f u n c t i o n s d i r e c t l y from t h e p r o o f s o f t h e r e p r e s e n t a t i o n theorems i s a f e a t u r e t h a t i s s h a r e d w i t h e x p e c t e d u t i l i t y and s e t s a l p h a u t i l i t y a p a r t from o t h e r a l t e r n a t i v e t h e o r i e s i n t h e l i t e r a t u r e . T h i s i s i l l u s t r a t e d below v i a a s i m p l e consequence s e t X w i t h d i s - t i n c t outcomes L , I , and H a r r a n g e d i n a s c e n d i n g o r d e r o f p r e f e r e n c e . C h a p t e r 3 s e c t i o n 1 c o n t a i n s a d i s c u s s i o n o f t h e s y s t e m a t i c v i o l a - t i o n s o f t h e S t r o n g Independence p r i n c i p l e by s t a t i n g t h e e m p i r i c a l s t u d i e s on t h e A l l a i s paradox i n terms o f l o t t e r i e s d e f i n e d on such a 3-outcome consequence s e t . Suppose an a l p h a u t i l i t y d e c i s i o n maker chooses among l o t t e r i e s d e f i n e d on such a consequence s e t , t h e n we can measure h i s a and v f u n c t i o n s i n t h e f o l l o w i n g way: Set t h e v - v a l u e s f o r t h e w o r s t and t h e b e s t consequences t o 0 and 1 r e s p e c t i v e l y , i . e . , v ( L ) = 0 and v(H) = 1. The v - v a l u e o f t h e i n t e r m e d i a t e outcome I i s g i v e n by t h e p r o - b a b i l i t y q such t h a t t h e d e c i s i o n maker i s i n d i f f e r e n t between t h e s u r e consequence I and a q chance o f o b t a i n i n g H and 1-q chance o f o b t a i n i n g L, as i l l u s t r a t e d i n F i g . 4.3. 98 Figure 4.3: Probability Equivalent Method v(L) - 0; v(H) = 1; v(I) = q such that P ~ Q. Having constructed the v-function, we form in Fig. 4.4 the follow- ing l o t t e r i e s to determine the a function. Figure 4.4: Test of Substitution Axiom If the decision maker subscribes to the Substitution p r i n c i p l e , P' and Q' would be indifferent whenever 8 equals y; since Q i s constructed to be indifferent to P. If,however,P' and Q' are indifferent with 8 and y 99 unequal, then Ratio Consistency tells us that Y/1-Y — - = T, a constant. (4.13) B/i-e The decision maker's a-function is then given by ct(I)=x and ct(L)=a(H) = l , i.e., assign 1 to the a-values of the best (H) and the worst (L) outcomes of the consequence set X.(See the sufficiency proof of Theorem 2.1 in Chapter 2 for more details). For consequence sets with more than three outcomes, the above proce- dure can be repeated for the other intermediate outcomes. In the case of an' interval of a real line, the v and a functions can be obtained by interpola- ting among a finite number of measurement points. Just as there are many different ways to measure von Neumann-Morgenstern uti l i t y functions, this would also be the case for alpha utility. Since we only skim the issue of assessment here, more work seems to be needed. In the next sub-sections, we obtain conditions for consistency of alpha utility with stochastic domi- nance, local and global risk aversion. 100 4.3.3 S t o c h a s t i c Dominance D e f i n i t i o n 4.1: A d i s t r i b u t i o n G i s s a i d t o s t o c h a s t i c a l l y dominate a n o t h e r d i s t r i b u t i o n F i n t h e f i r s t d e g r e e , d e n o t e d by G \ F, i f G(x) < F ( x ) , V x e R. The above d e f i n i t i o n o f what i s u s u a l l y c a l l e d Stochastic Dominance has i t s o r i g i n i n t h e works o f Hadar and R u s s e l l ( 1969), Lehmann (1955) and Hanoch and Levy (1969). I t i s w e l l known t h a t , G > F =»• / R u d G i / R u d F f o r e v e r y i n c r e a s i n g f u n c t i o n u on R. T h e r e f o r e , e v e r y e x p e c t e d u t i l i t y d e c i s i o n maker would p r e f e r G t o F when G dominates F i n t h e f i r s t d e gree. T h i s i s however not n e c e s s a r i l y t r u e o f e v e r y a l p h a u t i l i t y d e c i s i o n maker. To t h e e x t e n t t h a t c o n s i s t e n c y w i t h f i r s t degree s t o c h a s t i c dominance i s n o r m a t i v e l y d e s i r a b l e (and, i n t h e c o n t e x t o f monetary l o t t e r i e s , p r o b a b l y d e s c r i p t i v e l y v a l i d ) , i t can be imposed as an a d d i t i o n a l r e q u i r e m e n t . The f o l l o w i n g c o r o l l a r y t a k e n from C h a p t e r 1 p r o v i d e s c o n d i t i o n s under w h i c h an a l p h a u t i l i t y d e c i s i o n maker would be c o n s i s t e n t w i t h f i r s t d e gree s t o c h a s t i c dominance. C o r o l l a r y 1.4: Suppose a, v a r e bounded. Then V F, G e D, F > G Q. (F) > fi (G) i f f a ( x ) (v (x) - v ( s ) ) (4.14) i s an i n c r e a s i n g f u n c t i o n V s e R. The f u n c t i o n a l ^ ( F ) r e f e r s t o t h e a l p h a u t i l i t y r e p r e s e n t a t i o n a v d F / / R a d F . i t i s easy t o see t h a t r e l a t i o n (4.14) can be r e s t a t e d , 101 a ( x ) - ( v ( x ) - 0 ( F ) ) (4.15) i s an i n c r e a s i n g f u n c t i o n V F e D- We o b t a i n a " f a m i l i a r " i n t e r p r e t a t i o n o f (4.15) t h r o u g h t h e f o l l o w i n g e x p o s i t o r y d i s c u s s i o n o f t h e n e c e s s i t y p r o o f o f C o r o l l a r y 1.5. Observe t h a t G I F => Fg, i Fg i f 6' > 9 where FQ = (1-0)F+6G V Suppose i s c o n s i s t e n t w i t h >, t h e n fi(F ) i s i n c r e a s i n g i n 9. D i f f e r - e n t i a t i n g w i t h r e s p e c t t o 0 y i e l d s |e nCF 0 ) = / R u ( x ; F e ) d [ G ( x ) - F ( x ) ] > 0 V 6 e (0,1] (4.16) where u ( x ; F Q ) = g [ ^ F -,{v(x) - Q(FQ)} . (4.17) By c o n t i n u i t y o f u ( x ; F ) i n 0, i t f o l l o w s t h a t "Je fi^F^ fi=n+ = / p u ( x ; F ) d [ G ( x ) - F ( x ) ] > 0. (4.18) 9=0 + JR S i n c e G i s any d i s t r i b u t i o n t h a t dominates F, u ( x ; F ) has t o be i n c r e a s i n g f o r e v e r y x e R. The f u n c t i o n u ( x ; F ) has t h e f u n c t i o n a l - a n a l y t i c i n t e r p r e t a t i o n as a Gateaux d e r i v a t i v e ( L u e n b e r g e r , 1969) o f t h e f u n c t i o n a l fi(F) a t F and d T n ( V U + = / R u ( x ; F ) d ( G - F ) i s i n t e r p r e t e d as t h e Gateaux d i f f e r e n t i a l o f fl a t F i n t h e d i r e c t i o n o f G-F. F o r e x p e c t e d u t i l i t y , t h e c o r r e s p o n d i n g Gateaux d e r i v a t i v e and Gateaux d i f f e r e n t i a l a r e u ( x ) and / R u ( x ) d [ G ( x ) - F ( x ) ] r e s p e c t i v e l y . Note t h a t u ( x ) does not depend on F s i n c e t h e e x p e c t e d u t i l i t y r e p r e - 102 s e n t a t i o n J l udF i s l i n e a r i n F. T h i s s u g g e s t s t h e term Lottery Specific Utility Function (abbv. LOSUF) f o r u ( x ; F ) ^ C o r o l l a r y 1.5 has t h e f a m i l i a r i n t e r p r e t a t i o n : A l p h a u t i l i t y i s c o n s i s t e n t w i t h 1 s t degree s t o c h a s t i c dominance i f i t s LOSUF i s i n c r e a s i n g f o r each F. T h i s c o n d i t i o n assumes t h e f o l l o w i n g more manageable form i f we impose 1 s t o r d e r d i f f e r e n t i a - b i l i t y on a and v. av' > max [a' [ v - v ( x ) ] , - a ' [ v ( x ) - v ] ] V x e R, (4.19) where v = Lim v ( x ) and v = Lim v ( x ) . X - w o - X - > - - ° ° A l t e r n a t i v e l y , t h e above can be s t a t e d a s : V x e R, (log cx(x))' < p i f a ' W - ° v ' ( x ) v ( x ) - V (4.20) i f a *(x) < 0 The r a t e o f change o f l o g a i s bounded from above and below by v ' ( x ) / v - v ( x ) and -v' ( x ) / v ( x ) - v r e s p e c t i v e l y . E x p r e s s i o n (4.20) can be used d i r e c t l y t o f i n d a l l t h o s e a g i v e n a c e r t a i n v t h a t w i l l be c o n s i s t e n t w i t h 1 s t degree s t o c h a s t i c dominance. Machina (1980) i n v e s t i g a t e d t h e p r o p e r t i e s o f a t w i c e F r e c h e t - d i f f e r e n t i - a b l e f u n c t i o n a l V ( F ) on D[0,M] w i t h r e s p e c t t o t h e s t o c h a s t i c dominance and g l o b a l r i s k a v e r s i o n p a r t i a l o r d e r s . F r e c h e t d i f f e r e n t i a b i l i t y i s a s t r o n g e r n o t i o n t h a n Gateaux d i f f e r e n t i a b i l i t y and does not admit a n a t u r a l e x t e n s i o n o f t h e a n a l y s i s t o t h e r e a l l i n e . M a c hina c a l l e d t h e f i r s t F r g c h e t d e r i v a t i v e s o f V(F) l o c a l u t i l i t y f u n c t i o n s . The a l p h a u t i l i t y f u n c t i o n a l Q(F) when r e s t r i c t e d t o a compact i n t e r v a l such as [0,M] i s F r e c h e t d i f f e r e n t i a b l e so t h a t we can i d e n t i f y a l o t t e r y s p e c i - f i c u t i l i t y f u n c t i o n w i t h a l o c a l u t i l i t y f u n c t i o n . 103 4.3.4 Global Risk Aversion D e f i n i t i o n 4.2: A d i s t r i b u t i o n G i s s a i d to s t o c h a s t i c a l l y dominate 6 2 another d i s t r i b u t i o n F i n the second degree; denoted by G > F, i f / ^ ( G ( x ) - F ( x ) ) d x < 0 V y £ R (4.21) and / " ( G ( x ) - F ( x ) ) d x = 0. (4.22) When the means ass o c i a t e d with the d i s t r i b u t i o n s F and G e x i s t , c o n d i t i o n (4.22) i m p l i e s that they are equal. Condition (4.21) r e q u i r e s that f o r each x, the mean of G truncated at (-°°,x] i s not l e s s than that 1 2 of F truncted at (-°°,x] . Note a l s o that f o r a l l concave u i n R, F > G i f and only i f / udF > / udG. K K Consequently, an expected u t i l i t y d e c i s i o n maker with a concave u t i l i t y f u n c t i o n always p r e f e r s a d i s t r i b u t i o n F that dominates another d i s t r i b u t i o n G i n the second degree. The normative content of 2nd-degree S t o c h a s t i c Dominance (or Global Risk Aversion) i s derived from the idea that a prudent person should be r i s k averse. D e f i n i t i o n 4.2 extends the d e f i n i t i o n of i n c r e a s i n g r i s k by R o t h s c h i l d and S t i g l i t z (1970) f o r a compact i n t e r v a l . S i m i l a r or r e l a t e d r e s u l t s have been obtained i n Bl a c k w e l l (1951), Hanoch and Levy (1969) and Whitmore (1970) defined third-degree s t o c h a s t i c dominance and r e l a t e d i t to the Arrow-Pratt index. The general kth-degree s t o c h a s t i c dominance was defined i n Chapter 1. However, both d e f i n i t i o n s w i l l not be t r e a t e d here. \ 104 S t r a s s e n (1965). The f o l l o w i n g c o r o l l a r y adapted from C h a p t e r 1, wh i c h i s a r e s t r i c - t i o n o f C o r o l l a r y 1.6 t o t h e case o f second-degree s t o c h a s t i c dominance, p r o v i d e s c o n d i t i o n s under w h i c h an a l p h a u t i l i t y d e c i s i o n maker would be c o n s i s t e n t w i t h second-degree s t o c h a s t i c dominance. C o r o l l a r y 1.6*: Suppose a, v, a', v' a r e bounded and c o n t i n u o u s , z the n V F , G E D , F > G =* n ( F ) > R(G) (4.23) i f f {a (x) [v ( x ) - v (s)] }' i s a d e c r e a s i n g f u n c t i o n V s e R. Note t h a t c o n d i t i o n (4.23) r e q u i r e s t h e LOSUF u ( x ; F ) t o be a concave f u n c t i o n f o r each F. T h i s f u r t h e r c o n f i r m s our i n t u i t i o n t h a t t h e LOSUF p l a y s a von Neumann-Morgenstern u t i l i t y - l i k e r o l e . R e q u i r i n g a, v t o be second d i f f e r e n t i a b l e , t h e c o n d i t i o n (4.23) becomes V x e R, av"+2cx ,v' < min [a" (v-v ( x ) ) , - a " (v ( x ) - v ) ] . (4.24) T h i s c o n d i t i o n i s r e l a t e d t o t h e A r r o w - P r a t t measure f o r a l p h a u t i l i t y d e v e l o p e d i n t h e n e x t s u b s e c t i o n . 105 4.3.5 Local Risk Aversion: The Arrow-Pratt Index Consider an alpha u t i l i t y d e c i s i o n maker with assets x. The Cash Equivalent C ( x ; Z ) corresponding to a r i s k Z i s given by M ( F )-x which X̂" i-t i s the d i f f e r e n c e between the c e r t a i n asset p o s i t i o n M(F ) such that X̂~ Li the d e c i s i o n maker i s i n d i f f e r e n t to t a k i n g the r i s k Z and h i s current asset p o s i t i o n x. The r i s k premium TT(X;Z) i s then defined by TT(X;Z) = E ( Z ) - C ( x ; Z ) (4.25) which i s the d i f f e r e n c e between the a c t u a r i a l value E ( Z ) of the r i s k Z and i t s cash equiyalent C ( x ; Z ) . Since x+ Z and (x+u )+(Z-u) have the same d i s t r i b u t i o n on f i n a l a s s e t s , T and C from expression (4.25) have the p r o p e r t i e s : C(x+y ;Z-p) = C ( x ; Z ) - y , and (4.26) TT(X+U;Z-U) = TT(X;Z) . (4.27) Following P r a t t (1964), we consider the behaviour o f i r ( x ; Z ) f o r an a c t u a r i a l l y f a i r r i s k Z as -*• 0, assuming the t h i r d absolute c e n t r a l moment of Z i s of order o ( a | ) . Thus, v C x - i r ( x ; Z ) ) = fi(Fx+z) = - ^ X . Expanding both sides a f t e r cross m u l t i p l i c a t i o n , we get 106 [v(x) -TTV (X)+0(TT (4.29) T h i s r e d u c e s t o TT(X;Z) hp\ r ( x ) + o ( a | ) , (4.30) where r ( x ) = - | - [ l o g a 2 ( x ) V ( x ) ] ' = - ( ^ f f i + 2 a ' W ), ( 4 i 3 1 ) dx & ' ^ J V V (x) ct(x) As i n e x p e c t e d u t i l i t y , t h e d e c i s i o n maker's r i s k premium f o r a s m a l l , a c t u a r i a l l y n e u t r a l r i s k Z i s a p p r o x i m a t e l y h a l f the v a r i a n c e t i m e s r ( x ) , w h i c h , i n k e e p i n g w i t h p r e c e d e n t , we c a l l t h e A r r o w - P r a t t i n d e x . When Z i s n o t a c t u a r i a l l y f a i r , we o b t a i n from (4.27) I t i s s t r a i g h t f o r w a r d t o check t h a t r ( x ) has t h e f o l l o w i n g a l t e r n a t i v e i n t e r p r e t a t i o n i n terms o f t h e p r o b a b i l i t y premium ( P r a t t , 1964; Arrow, 1971), where % ( l + p ( x ; h ) ) and % ( l - p ( x ; h ) ) a r e t h e p r o b a b i l i t i e s o f o b t a i n i n g x+h and x-h r e s p e c t i v e l y , s u c h t h a t t h e d e c i s i o n maker i s i n d i f f e r e n t between t h e s t a t u s quo x and t a k i n g t h e r i s k . I t i s a l s o s t r a i g h t f o r w a r d t o check t h a t r ( x ) i s i n v a r i a n t under t h e u n i q u e n e s s t r a n s f o r m a t i o n ( e x p r e s s i o n (1.12) and (1.13)) f o r t h e f u n c t i o n s a and v. I n p a r t i c u l a r , i t i s i n v a r i a n t u n d e r an a f f i n e t r a n s f o r m a t i o n f o r v and a s c a l a r TT(X;Z) = ho* r ( x + E ( Z ) ) + o(a|) . (4.32) p(x;h) = % h r ( x ) + 0 ( h 2 ) , (4.33) 107 m u l t i p l e f o r a. I t i s c o m f o r t i n g t o n o t e t h a t r ( x ) has s i m i l a r l o c a l p r o p e r t i e s as e x p e c t e d u t i l i t y . U n l i k e e x p e c t e d u t i l i t y however, we cannot r e c o v e r the f u n c t i o n s a and v o n l y from the knowledge o f r ( x ) p o i n t w i s e . We would no t t h e r e f o r e e x p e c t t o be a b l e t o c h a r a c t e r i z e i n g e n e r a l , g l o b a l r i s k p r o p e n s i t i e s o f an a l p h a u t i l i t y d e c i s i o n maker i n terms o f h i s l o c a l r i s k a v e r s i o n f u n c t i o n r ( x ) . I n p a r t i c u l a r , we n o t e from s u b s e c t i o n 4.3.4 t h a t g l o b a l r i s k a v e r s i o n i n t h e sense o f c o n s i s t e n c y w i t h second degree s t o c h a s t dominance i m p l i e s t h a t r ( x ) > 0 p o i n t w i s e , but t h e c o n v e r s e does n o t i n g e n e r a l h o l d . T h i s r e s u l t seems i n t u i t i v e l y a p p e a l i n g . C o n s i s t e n c y w i t h s econd degree s t o c h a s t i c dominance i m p l i e s p o s i t i v e r i s k premium ir(x;Z) a t any a s s e t p o s i t i o n x, f o r any a c t u a r i a l l y f a i r r i s k Z , w h i c h i n t u r n i m p l i e s t h a t r ( x ) i s p o s i t i v e f o r a c t u a r i a l l y f a i r i n f i n i t e s i m a l r i s k s about x. Even i f r ( x ) i s p o s i t i v e p o i n t w i s e , i n d i c a t i n g a v e r s i o n t o i n f i n i t e s i m a l a c t u a r i a l l y f a i r r i s k s , i t i s s t i l l p o s s i b l e f o r an a l p h a u t i l i t y d e c i s i o n maker t o be r i s k s e e k i n g o v e r some i n t e r v a l . T h i s c o r r e s - ponds t o t h e o b s e r v a t i o n t h a t p e o p l e p u r c h a s e i n s u r a n c e and gamble a t t h e same t i m e (Friedman $ Savage, 1948; M a r k o w i t z , 1952). The i m p l i c a t i o n s o f t h i s and o t h e r e m p i r i c a l l y o b s e r v e d c h o i c e b e h a v i o r f o r a l p h a u t i l i t y w i l l be c o n s i d e r e d i n t h e n e x t s e c t i o n ; where we use t h e c o n s i s t e n c y c o n d i t i o n s d e v e l o p e d i n t h e l a s t s u b s e c t i o n s t o i d e n t i f y r e g i o n s o f l o c a l and g l o b a l r i s k a v e r s i o n . 108 4.4 DESCRIPTIVE IMPLICATIONS We showed i n the p r e c e d i n g s e c t i o n t h a t a l p h a u t i l i t y , l i k e e x p e c t e d u t i l i t y , i s c o m p a t i b l e w i t h s u c h n o r m a t i v e n o t i o n s as s t o c h a s t i c dominance, l o c a l and g l o b a l r i s k a v e r s i o n . We a l s o i l l u s t r a t e d how t h e c o n s t r u c t i v e p r o o f o f our r e p r e s e n t a t i o n theorem f u r n i s h e s a p r o c e d u r e f o r the assessment of the a l p h a u t i l i t y f u n c t i o n s . I n t h i s s e c t i o n , we w i l l show t h a t a l p h a u t i l i t y i s not so g e n e r a l t h a t i t has no t e s t a b l e i m p l i c a t i o n s , n o r i s i t such a m i n u t e d e p a r t u r e from e x p e c t e d u t i l i t y t h a t i t i s • s u s c e p t i b l e t o t h e same s e t o f v i o l a t i o n s . 7 4.4.1 S y s t e m a t i c V i o l a t i o n s of t h e S t r o n g Independence P r i n c i p l e ' The e m p i r i c a l f i n d i n g s v i o l a t i n g t h e i m p l i c a t i o n s o f t h e S t r o n g Independence p r i n c i p l e t h a t stem f r o m t h e A l l a i s paradox were summarized i n t a b l e 3.1 i n terms of t h e HILO s t r u c t u r e o f l o t t e r i e s d e v e l o p e d i n s e c t i o n 3.1. We c o n s i d e r h e r e t h e i m p l i c a t i o n s o f a l p h a u t i l i t y t h e o r y f o r c h o i c e p a t t e r n s f o r the HILO s t r u c t u r e , and d e r i v e some t e s t a b l e p r e d i c t i o n s . A p p l y i n g a l p h a u t i l i t y t h e o r y t o the d e c i s i o n c h o i c e s i n F i g u r e 3.4 we o b t a i n , q 3q gq + ( l - 3 ) a ( I ) v ( I ) 3 + ( l - 3 ) a ( I ) Bq + (1-3) fi(AQ) = V ( I ) fi(BQ) = = B a(I ) v(I) ft(B3).= 6a(I) + (1-3) fl(A^) = v ( l ) Ji(B^) = fi(A6 = 3 a ( I ) v ( D + (1-3) ft(BJb = 3 a(I) + (1-3) H The m a t e r i a l i n t h i s s u b - s e c t i o n appeared i n Chew and MacCrimmon (1979b). 109 F i g u r e 4.5: P r e f e r e n c e p a t t e r n f o r a ( I ) < 1 110 Assuming without loss of generality that ct(L)=ct(H) = l , ( c f . Theorem 2.1), we obtain the following i n e q u a l i t i e s corresponding to preferences i n each decision box. A Q >- B Q o v(I) > q o A I ^ B I ( 4- 3 4) >• B[ ~ v(I) > q ( 6 + - ^ | 1 ) (4.35) AJj >- B* o v(I) > l - ( l . q ) ( p + (4.36) The i n e q u a l i t i e s (4.34), (4.35) and (4.36) are p l o t t e d i n Figure 4.5 fo r the case i n which cx(I) < 1. Note that the four regions, denoted as regions I-IV, correspond to four d i s t i n c t choice patterns. These, along with the choice patterns f o r the case of a(I) > 1, are summarized i n Table 4.1. Table 4.1 Allowable Choice Patterns Under Alpha U t i l i t y Theory Re g io n ^ . ^ ^ < 1 Expected U t i l i t y = 1 > 1 I A ^ A I , A ^ A Q II V V V Ao Regions do not e x i s t V V V Ao III V V V Bo V B r V Bo IV V B r V Bo Apart from the patterns corresponding to regions I and IV (the only ones consistent with expected u t i l i t y ) alpha u t i l i t y theory allows for 4 out of 14 a d d i t i o n a l patterns. These are given by the entries I l l under the case a(I) < 1 and a(I) > 1 for regions I I and I I I . The region I I and region I I I patterns under a(I) > 1 have not been reported in the l i t e r a t u r e (cf. Table 3 . 1 ) . On the other hand, a l l the empirical findings to date of violations of expected u t i l i t y correspond to either of the region I I and I I I patterns with a(I) < 1. In pa r t i c u l a r , both 6 R the standard A l l a i s paradox ( i . e . A B ) and the A l l a i s r a t i o paradox X Li Q ( i . e . B , A ) occur in region I I . The existence of region I I I also has 3 3 1 3 2 some empirical support; note (Table 3 . 1 ) that both the A^, BQ and A^ , B^ vio l a t i o n s have been reported. Before we continue with further implications based on the allowable patterns of choice, we can gain some i n t u i t i o n about alpha u t i l i t y theory i n r e l a t i o n to the HILO structure by examining, i n Figure 4 . 5 , the effects of changes in the parameters 3 and v(I) on the resulting pattern of choices, Looking f i r s t at changes i n 3, we see that i f consequence I i s s u f f i c i e n t l y attractive ( i . e . , v(I) > q/a(I)) then the choice w i l l be A alternative and w i l l be unaffected by changes i n 3. This indicates a basis for choice that may be called the security effect. Correspondingly, i f I i s unattractive ( i . e . , v(I) < 1 - ( l - q ) / a ( I ) ) , then no change in 3 w i l l induce a s h i f t away from the B alternative. This may be called a nothing-to-lose effect. Note that both above regions are the only regions consistent with the substitution p r i n c i p l e . If I i s somewhat at t r a c t i v e ( i . e . , v(I) between q and q/a(I)) then 3 3 decreasing 3 from 1 w i l l cause a switch from A^ to B^. The smaller value of 3 acts as a d i l u t i o n probability to narrow the perceived gap 3 3 between A^ and B^,due to the attractiveness of I } u n t i l f i n a l l y the g attractiveness i s s u f f i c i e n t l y diluted to cause a switch to B . This 1 1 2 8 i s t h e dilution effect. The s p e c i a l case f o r t h e d i l u t i o n from A£l t o 3-7 l , (where 3-̂  = 1.0 and 3 2 << 1-0) has been c a l l e d a c e r t a i n t y e f f e c t (Kahneman and T v e r s k y , 1979). When I i s somewhat u n a t t r a c t i v e ( i . e . , \)(I) i s between 1 - ( l - q ) / a ( I ) and q) a d e c r e a s e i n 3 w i l l 3 3 cause a s w i t c h f r o m t o A^. T h i s i s a r e v e r s e d i l u t i o n e f f e c t . I n a d d i t i o n t o s t u d y i n g t h e e f f e c t of changes i n 3 f o r g i v e n v a l u e s of V(I), f r e s h i n s i g h t can be g a i n e d by e x a m i n i n g t h e e f f e c t o f changes i n v ( I ) f o r g i v e n l e v e l s of 3. F o r 3 = 1, we no l o n g e r have a compound s t r u c t u r e and so r e v e r t t o t h e s i m p l e A^, B ^ c h o i c e w h i c h depends on whether v ( I ) > q. A t low and i n t e r m e d i a t e l e v e l s o f 3, t h e r e g i o n s I I and I I I , g i v i n g r i s e t o t h e p a r a d o x i c a l c h o i c e s can be q u i t e l a r g e . Note t h a t as V ( I ) d e c r e a s e s f r o m b e i n g v e r y a t t r a c t i v e , 3 3 a t some p o i n t t h e A^ c h o i c e w i l l s w i t c h t o B ^ . Then as V ( I ) drops g below q, t h e c h o i c e s i n t h e I and 0 cases change t o B^. and BQ r e s p e c t i v e l y . g F i n a l l y , as v ( I ) d r o p s l o w e r ( i . e . , below q(3 + ( l - 3 ) / a ( I ) ) , A changes 3 H t o B * . Two main i m p l i c a t i o n s s h o u l d be n o t e d f r o m t h e s e o b s e r v a t i o n s . F i r s t , a l p h a t h e o r y can d e s c r i b e a r i c h e r s e t o f p r e f e r e n c e s t h a n can e x p e c t e d u t i l i t y t h e o r y . S p e c i f i c a l l y , i t c o v e r s t h e A l l a i s - t y p e p r e f e r e n c e s ( I - L , L-0, and L-L) and o t h e r o b s e r v e d v i o l a t i o n s (H-0 and H-H). I t a l l o w s f o r a dependence on t h e v a l u e s o f t h e p a r a meter and hence c a p t u r e s t h e o b s e r v e d dilution effect. On t h e o t h e r hand, i t i s n o t so g e n e r a l t h a t i t can d e s c r i b e any p r e f e r e n c e s . Only 6 of t h e p o s s i b l e 16 p r e f e r e n c e p a t t e r n s o f t h e HILO s t r u c t u r e a r e c o n s i s t e n t w i t h a l p h a u t i l i t y t h e o r y (and o n l y 4 p a t t e r n s f o r t h e e m p i r i c a l l y s u p p o r t e d a ( I ) < 1 c a s e ) . As F i g u r e 4.5 1 1 3 s u g g e s t s , our t h e o r y makes v e r y s p e c i f i c p r e d i c t i o n s about t h e p r e f e r e n c e p a t t e r n s and t h e way th e y change as t h e pa r a m e t e r s 8 and V ( I ) changes. In p a r t i c u l a r , m o n o t o n i c i t y r e q u i r e s t h a t p r e f e r e n c e s f o r t h e 0 and I ca s e s agree c o m p l e t e l y , i . e . , >- BQ<=>A^>- B . More i n t e r e s t i n g l y , n o t e t h a t t h e s t a n d a r d A l l a i s paradox (A , B ) o c c u r s i f , and o n l y i f , t h e A l l a i s r a t i o p a radox ( B^, A Q ) o c c u r s . B o t h new r e g i o n s o f p e r m i s s i b l e B 8 c h o i c e ( i . e . , r e g i o n s I I and I I I ) p r o v i d e f o r and B^; hence t h i s p r e v i o u s l y u n r e p o r t e d c a s e w o u l d seem t o be a prime c a n d i d a t e f o r a new "paradox". F u r t h e r , n o t e t h a t as v ( I ) d e c r e a s e s , t h e s w i t c h from 3 3 A t o B o c c u r s f i r s t f o r t h e L c a s e , t h e n f o r t h e I and 0 c a s e s , and f i n a l l y f o r t h e H case. A l l t h e s e i m p l i c a t i o n s from a l p h a u t i l i t y t h e o r y a re e m p i r i c a l l y t e s t a b l e . 114 4.4.2 S t o c h a s t i c Dominance C o n s i s t e n c y w i t h s t o c h a s t i c dominance i s an i n t r i n s i c p r o p e r t y of e x p e c t e d u t i l i t y . T h i s i s not t h e case w i t h a l p h a u t i l i t y . To t h e e x t e n t t h a t i t i s p r e s c r i p t i v e l y d e s i r a b l e and d e s c r i p t i v e l y v a l i d (see C h a p t e r 5 f o r an example of a p o t e n t i a l v i o l a t i o n of s t o c h a s t i c dominance i n t h e c o n t e x t of income d i s t r i b u t i o n s ) , we can r e s t r i c t t h e a l p h a u t i l i t y f u n c t i o n s c o n s i d e r e d t o t h o s e t h a t do n o t v i o l a t e o s t o c h a s t i c dominance v i a t h e f o l l o w i n g c o n s i s t e n c y c o n d i t i o n s t a k e n f r o m C o r o l l a r y 1.4. VxeR, [ l o g a ( x ) ] ' < - [ l o g ( v - v ( x ) ] ' i f a'(x)>0 > - [ l o g ( v ( x ) - v ) ] ' i f a'(x)<0. where V = Lim v ( x ) and v = Lim v( x) (4.37) These c o n d i t i o n s a r e d e p i c t e d g r a p h i c a l l y i n F i g u r e 4.6 f o r a bounded V n o r m a l i z e d t o V=0 and V = l . The s l o p e of l o g a i s bounded by t h e s l o p e of - l o g (1-V) from above and - l o g V from below. The a f u n c t i o n t h a t i s c o m p a t i b l e w i t h (4.37) i s t h e n r e c o v e r e d f r o m the g r a p h of l o g a. The a f u n c t i o n c o n s i d e r e d has a dent i n t h e m i d d l e and i n c r e a s e s i n b o t h t h e p o s i t i v e and t h e n e g a t i v e d i r e c t i o n s i n o r d e r n o t t o e x c l u d e t h e p o s s i b i l i t y of r e g i o n I I and r e g i o n I I I p r e f e r e n c e s i n T a b l e 4.1. ( R e c a l l t h a t t h e c o n d i t i o n f o r t h e e x i s t e n c e of r e g i o n I I and I I I p r e f e r e n c e i s t h a t a ( I ) < q a ( H ) + ( l - q ) a ( L ) ) . The l i m i t i n g b e h a v i o r of a i s bounded above by , and b e l o w by -r. 3 1-v 3 V I t i s i n t e r e s t i n g t o n o t e t h a t a f t e r s a t i s f y i n g s t o c h a s t i c dominance and a l s o t h e " A l l a i s " t y p e p r e f e r e n c e f o r t h e HILO s t r u c t u r e , we a r e s t i l l l e f t w i t h a f a i r l y l a r g e c l a s s o f a f u n c t i o n s . We i n v e s t i g a t e i n t h e n e x t s u b - s e c t i o n , t h e r i s k p r o p e n s i t i e s o f t h e a - l n [ v ( x ) - v ] j 115 3 J - l n [v -v ( F i g . 4 . 6 : C o n s i s t e n c y C o n d i t i o n s f o r S t o c h a s t i c Dominance 116 and u f u n c t i o n s of F i g u r e 4.6. 4.4.2 L o c a l and G l o b a l R i s k P r o p e r t i e s : C o n c u r r e n c e o f R i s k A v e r t i n g and R i s k S e e k i n g B e h a v i o r L o c a l r i s k a v e r s i o n i n t h e sense t h a t t h e A r r o w - P r a t t i n d e x (see s u b - s e c t i o n 4.3.5) i s n o n p o s i t i v e c o r r e s p o n d s t o t h e o b s e r v a t i o n t h a t p e o p l e t e n d t o a v o i d t a k i n g a s m a l l gamble i n f a v o r of i t s e x p e c t e d r e t u r n . F o r e x p e c t e d u t i l i t y , l o c a l r i s k a v e r s i o n p o i n t w i s e i s e q u i v a l e n t t o h a v i n g a concave von Neumann-Morgenstern u t i l i t y f u n c t i o n w h i c h i s i n t u r n e q u i v a l e n t t o g l o b a l r i s k a v e r s i o n . Thus, t h e b e h a v i o r a l l y p l a u s i b l e l o c a l r i s k a v e r s i o n h y p o t h e s i s i s n o t c o m p a t i b l e w i t h any c o n c u r r e n t r i s k s e e k i n g b e h a v i o r , e.g., t h e p u r c h a s e o f a l o t t e r y t i c k e t . A l p h a u t i l i t y does n o t s h a r e t h e above d i f f i c u l t y s i n c e l o c a l r i s k a v e r s i o n i s n e c e s s a r y b u t n o t s u f f i c i e n t f o r g l o b a l r i s k a v e r s i o n . The c o n d i t i o n f o r l o c a l r i s k a v e r s i o n , namely t h a t the A r r o w - P r a t t i n d e x (= - V ^ - 2 a ( x ) } > i s n o n n e g a t i v e , i s : ( l o g a ( x ) ) ' < -Js(log v ' ( x ) ) \ (4.38) T h i s i s d e p i c t e d g r a p h i c a l l y i n F i g u r e 4.7 f o r t h e V f u n c t i o n o f F i g u r e 4.6. Note t h a t a c o n s t a n t a (which c o r r e s p o n d s t o e x p e c t e d u t i l i t y ) i s n o t a d m i s s i b l e . The f u n c t i o n l o g a has t o d e c r e a s e s u f f i c i e n t l y r a p i d l y n e a r t h e r u i n p o i n t - x r i n o r d e r t o c o r r e c t f o r t h e c o n v e x i t y of v a t t h e same r e g i o n . At t h i s p o i n t , we have a p a i r o f f u n c t i o n s ( a , v) t h a t s a t i s f y s t o c h a s t i c dominance, e x h i b i t l o c a l r i s k a v e r s i o n p o i n t w i s e , and a r e c o m p a t i b l e w i t h the A l l a i s t y p e p r e f e r e n c e f o r t h e HILO s t r u c t u r e . I t r e m a i n s t o c h e c k whether t h e f u n c t i o n s have t h e c o r r e c t g l o b a l r i s k p r o p e r t i e s t h a t c o r r e s p o n d t o a c t u a l c h o i c e b e h a v i o r . F i r s t , we e s t a b l i s h 117 F i g . 4.7: C o n d i t i o n s f o r L o c a l R i s k A v e r s i o n 118 F i g . 4.8: An a d m i s s i b l e a l p h a f u n c t i o n 119 t h a t our a and v f u n c t i o n s d e s c r i b e r i s k s e e k i n g b e h a v i o r by showing t h a t t h e y do n o t s a t i s f y t h e c o n d i t i o n s f o r g l o b a l r i s k a v e r s i o n : VkeR, ( a ( x ) . v ( x ) ) " < 0 i f a"(.x) > 0 < a " ( x ) i f a " ( x ) < 0 (4.39) S i n c e t h e a f u n c t i o n c o n s i d e r e d i s convex, i t s u f f i c e s t o o b s e r v e i n F i g u r e 4.8 t h a t t h e p r o d u c t o:-v a d m i t s a convex r e g i o n n e a r t h e r u i n p o i n t . T h i s means t h a t t h e r e i s a c o n c u r r e n c e of r i s k s e e k i n g and r i s k a v e r t i n g b e h a v i o r f o r t h a t r e g i o n . T h i s c o r r e s p o n d s t o t h e p r e v a l e n c e of r i s k s e e k i n g b e h a v i o r f o r l o s s e s o b s e r v e d by M a r k o w i t z (1952) and r e c e n t l y by Kahneman and T v e r s k y ( 1 9 7 9 ) . However, t h e l o c a l r i s k a v e r s i o n h y p o t h e s i s (see F i g u r e 4.6) r u l e s out t h e p o s s i b i l i t y o f g l o b a l r i s k p r o n e n e s s so t h a t t h e a l p h a u t i l i t y d e c i s i o n maker w i l l a t t h e same t i m e have t h e o p p o s i t e r i s k a v e r s e p r o p e n s i t y f o r some o t h e r gambles. F i n a l l y , we c o n s i d e r t h e m u t u a l i n c o m p a t a b i l i t y o f d i f f e r e n t p r o c e d u r e s f o r t h e assessment of a von Neumann-Morgenstern u t i l i t y f u n c t i o n . Suppose we a p p l y t h e c e r t a i n t y e q u i v a l e n t method t o t h e a l p h a u t i l i t y d e c i s i o n maker c o n s i d e r e d , w i t h s t a t u s quo x=0 and some s u b s t a n t i a l g a i n amount x as t h e e n d p o i n t s of l o t t e r y B i n F i g u r e 3.5. The p a i r s of measurements (x , u (x ) ) t h a t c o r r e s p o n d s t o i n d i f f e r e n c e c c c between g e t t i n g x f o r s u r e i n l o t t e r y A, and g e t t i n g x w i t h u (x ) chance c g c c and g e t t i n g 0 w i t h l - u ^ x ^ ) chance a r e r e l a t e d by t h e f o l l o w i n g e x p r e s s i o n : a ( x ) u (x ) v ( x ) + a ( 0 ) [ l - u (x )]v(0). v ( x c ) = ( 4 > 4 0 ) a ( x ) u (x ) + a ( 0 ) [ l - u (x ) ] £p c c C -̂ A f t e r r e a r r a n g i n g , we g e t : a ( 0 ) [ v ( x ) - v ( 0 ) ] u (x) = (4.41) c a ( 0 ) [ v ( x ) - v ( 0 ) ] + a ( x ) [ v ( x ) - v ( x ) ] 120 F i g . 4.9: A p a i r o f u a n d u d e r i v e d f r o m an a l p h a u t i l i t y d e c i s i o n maker c 1/2 _ ^ r 0 2 F i g . 4.10: A p a i r o f u and u.,. d e r i v e d f r o m g 3/4 an a l p h a u t i l i t y d e c i s i o n maker 121 A l t e r n a t i v e l y , i f we f i x a l o s s amount a t h a l f t h e r u i n p o s i t i o n - x r and use x=0 as t h e i n t e r m e d i a t e amount, we can a p p l y t h e g a i n e q u i v a l e n t method t o o b t a i n Ug below: a ( x g ) p g ( x g ) v ( x g ) + a(-^T)[i-pg(xg)]v(-^) v ( 0 ) = g " g • g' * ^- l *g* g Hence, - 1 - ° ( x ) r v ( x ) - v ( 0 ) , . § T •"2~ F i n a l l y , we c o n s i d e r t h e c h a i n i n g method. W i t h x°=0 and x g=x^ as t h e e n d p o i n t , and p°=h as t h e p r o b a b i l i t y p a r a m e t e r , we o b t a i n t h e sequence ( x ^ , -x.^,...) based on t h e f o l l o w i n g r e l a t i o n . v(x, 4.1) a(x.)v(x.) + a ( 0 ) v ( 0 ) . (4.44) i+1 a ( x . ) + a ( 0 ) 1 We have t h u s d e t e r m i n e d Uj on t h e sequence o f p o i n t s (x , x ,...) g i v e n -2 -L 2 by e x p r e s s i o n ( 4 . 4 4 ) . Changing t h e p r o b a b i l i t y p arameter t o p°=3/4 o ^ r and the l o w e r e n d p o i n t t o x^= ~~2~> we o b t a i n : 3 a ( x . ) v ( x . ) + a ( - f ^ ) v ( - ^ ) . (4.45) v ( x 1 + 1 ) = i 1 1 L _ 3 a ( x ) + a ( 0 ) As b e f o r e , we have d e t e r m i n e d u^/^ on t h e sequence of p o i n t s ( x ^ , x ,...) d e t e r m i n e d by e x p r e s s i o n ( 4 . 4 5 ) . F i g u r e 4.9 shows t h e graphs of u c and ui^ w i t h t h e same axes f o r ease o f c o m p a r i s o n w i t h A l l a i s ' r e s u l t s d i s p l a y e d i n F i g u r e 3.6. S i m i l a r l y , t h e c u r v e s of U3/4 and u g a r e p l o t t e d i n F i g u r e 4.10 f o r c o m p a r i s o n w i t h t h e r e s u l t s of MacCrimmon e t . a l . (1972) i n F i g u r e 3.7. The r a t h e r s t r i k i n g f i t between t h e t h e o r e t i c a l p r e d i c t i o n and t h e e m p i r i c a l measurement s u p p o r t s t h e v a l i d i t y o f a l p h a u t i l i t y t h e o r y ; t h u s p r o v i d i n g a " r a t i o n a l " e x p l a n a t i o n f o r an o t h e r w i s e p u z z l i n g phenomena. 122 4.4.4 Some Problems w i t h Problem R e p r e s e n t a t i o n I n t h i s f i n a l s u b s e c t i o n , we comment b r i e f l y on how a l p h a u t i l i t y h a n d l e s t h e two d i f f i c u l t i e s w i t h p r o b l e m r e p r e s e n t a t i o n d i s c u s s e d i n s e c t i o n 3 o f C h a p t e r 3. Our p o s i t i o n on t h e q u e s t i o n o f whether a l p h a u t i l i t y f u n c t i o n s s h o u l d be d e f i n e d on g a i n s and l o s s e s r e l a t i v e t o a c u stomary w e a l t h l e v e l o r i n terms o f f i n a l a s s e t p o s i t i o n s i s t h e same as t h a t o f e x p e c t e d u t i l i t y : i t depends on t h e p a r t i c u l a r a p p l i c a t i o n . The l a t t e r s h o u l d be adopted i f we a r e i n t e r e s t e d i n m o d e l i n g t h e c h o i c e b e h a v i o r o f t h e "economic" man o r i f a d e c i s i o n a n a l y s t i s h e l p i n g some- one who p r o f e s s e s h i s b e l i e f i n i n t e g r a t i n g p o s s i b l e outcomes i n t o h i s w e a l t h p o s i t i o n p r i o r t o e v a l u a t i o n (Kahneman and T v e r s k y c a l l e d t h i s t h e a s s e t i n t e g r a t i o n p o s i t i o n ) . On t h e o t h e r hand, t h e f o r m e r p o s i t i o n comes i n handy i f we want t o d e s c r i b e t h e a c t u a l c h o i c e b e h a v i o r o f p e o p l e who do n o t conform t o t h e a s s e t i n t e g r a t i o n p o s i t i o n . On t h e o t h e r q u e s t i o n o f whether a t w o - s t a g e l o t t e r y i s e q u i v a l e n t t o i t s s i n g l e - s t a g e d e c o m p o s i t i o n . Kahneman and T v e r s k y showed t h a t f o r a p a r t i c u l a r two-stage l o t t e r y which i s c l o s e l y r e l a t e d t o t h e i r v e r s i o n o f t h e A l l a i s p a radox ( c f . t h e 0-L case i n T a b l e 3.1), a p r o b l e m d e s c r i p t i o n t h a t f o c u s e s t h e s u b j e c t ' s a t t e n t i o n on a c o n d i t i o n a l c o m p arison be- tween one o f t h e b r a n c h e s can e l i c i t a m a j o r i t y p r e f e r e n c e t h a t i s t h e r e v e r s e o f what would be t h e c a s e i f t h e t w o - s t a g e l o t t e r y i s s t a t e d i n terms o f i t s s i n g l e - s t a g e e q u i v a l e n t . E x p e c t e d u t i l i t y cannot be c o n s i s t e n t w i t h t h i s phenomena because t h e s t r o n g independence p r i n c i p l e o r t h e s u r e - t h i n g p r i n c i p l e d i c t a t e s t h a t t h e d i r e c t i o n o f t h e c o n d i t i o n a l p r e f e r e n c e must be t h e same as t h a t between t h e o v e r a l l p r e f e r e n c e . A l p h a u t i l i t y does n o t s h a r e t h i s d i f f i c u l t y , s i n c e t h e c o n d i t i o n a l p r e f e r e n c e f o r A n a g a i n s t B n and t h e 123 p r e f e r e n c e f o r B a g a i n s t A i s a s p e c i a l case o f t h e HILO c h o i c e p a t t e r n J-l Li ( T a b l e 4.1) c o n s i d e r e d i n s u b - s e c t i o n 4.4.1. The phenomena c o n s i d e r e d have i m p l i c a t i o n s f o r s t u d i e s i n d e c i s i o n making i n g e n e r a l . A p a r t from s i t u a t i o n s where a c l e a r n o r m a t i v e p o s i t i o n d i c t a t e s w h i c h i s t h e ' c o r r e c t ' p r o b l e m r e p r e s e n t a t i o n , t h e l e s s o n seems t o be t h a t one s h o u l d be s e n s i t i v e t o t h e c o n t e x t a s s o c i a t e d w i t h a p a r - t i c u l a r c h o i c e s i t u a t i o n . P e o p l e appear t o use d i f f e r e n t schemes t o r e p r e s e n t t h e same c h o i c e s , r e s u l t i n g i n a p p a r e n t i n c o n s i s t e n c y i f t h e r a n k i n g s among a l t e r n a t i v e s a r e d i f f e r e n t f o r d i f f e r e n t p r o b l e m r e p r e s e n t a - t i o n s . 124 4 . 5 CRITIQUE OF ALLAIS' THEORY AND PROSPECT THEORY Hav i n g d e v e l o p e d a l p h a u t i l i t y t h e o r y i n t h e p r e c e d i n g s e c t i o n s , we c o n s i d e r i n t h i s s e c t i o n two o t h e r a l t e r n a t i v e s t o e x p e c t e d u t i l i t y t h a t have a t t r a c t e d s i g n i f i c a n t i n t e r e s t . - 4 . 5 . 1 A l l a i s ' T heory A l l a i s (1953) assumed t h e e x i s t e n c e o f a f u n c t i o n a l V(F) t h a t r e p r e s e n t s p r e f e r e n c e among p r o b a b i l i t y d i s t r i b u t i o n s . Such an approach has t h e immediate i m p l i c a t i o n s t h a t p r e f e r e n c e thus r e p r e s e n t e d s a t i s f i e s s e v e r a l s t a n d a r d 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 : c o m p l e t e n e s s , t r a n s i t i v i t y , c o m b i n a t i o n and c o m p o s i t i o n . A l l a i s f u r t h e r assumed t h a t p r e f e r e n c e s h a r e s a n o t h e r p r o p e r t y o f e x p e c t e d u t i l i t y , c a l l e d c o n s i s t e n c y w i t h s t o c h a s t i c dominance. T h i s a s s u m p t i o n r e s t r i c t s t h e p r e f e r e n c e f u n c t i o n a l V ( F) t o t h o s e t h a t i n c r e a s e when t h e u n d e r l y i n g d i s t r i b u t i o n i n c r e a s e s i n the s t o c h a s t i c dominance n sense. F o r f i n i t e l o t t e r i e s o f the form F = £ p.S^., t h i s c o n d i t i o n has i = l 1 X l t h e f o l l o w i n g s i m p l e form, n V(E p.6 x.) i n c r e a s e s i n each x.. (4.46) i = l U n l i k e e x p e c t e d u t i l i t y , t h e t h e o r y o u t l i n e d above i s n o t an a x i o m a t i c t h e o r y i n the sense t h a t t h e e x i s t e n c e o f t h e r e p r e s e n t a t i o n V(F) i s a s s e r t e d r a t h e r t h a n b e i n g a consequence o f the assumed p r o p e r t i e s o f t h e u n d e r l y i n g p r e f e r e n c e . Note a l s o t h a t t h e a d o p t i o n o f a more g e n e r a l r e p r e s e n t a t i o n i s t r a d e d a g a i n s t t h e c o n v e n i e n c e o f h a v i n g a s i m p l e von Neumann-Morgenstern u t i l i t y f u n c t i o n w h i c h c a p t u r e s o ur i n t u i t i o n about d i m i n i s h i n g m a r g i n a l u t i l i t y and o f f e r s a s i m p l e c h a r - a c t e r i z a t i o n o f r i s k p r o n e n e s s ( a v e r s i o n ) v i a t h e c o n v e x i t y ( c o n c a v i t y ) 125 o f t h e u t i l i t y f u n c t i o n . In o r d e r t o o b t a i n a u t i l i t y - l i k e f u n c t i o n w i t h o u t r e s o r t i n g t o e x p e c t e d u t i l i t y , A l l a i s r e v i v e d t h e F r i s c h (1926) n o t i o n o f q u a r t e n a r y p r e f e r e n c e among i n t e r v a l s o f w e a l t h and a s s e r t e d t h e e x i s t e n c e o f a c a r d i n a l u t i l i t y o f w e a l t h he termed p s y c h o l o g i c a l v a l u e (denoted by s ) . I n o t h e r words, g e t t i n g $100 a t s t a t u s quo i s " b e t t e r " t h a n g e t t i n g $100 a f t e r you have j u s t r e c e i v e d $1000 i f t h e d i f f e r e n c e i n p s y c h o l o g i c a l v a l u e from s t a t u s quo t o g e t t i n g $100, s ( $ 1 0 0 ) - s ( $ 0 ) , i s g r e a t e r t h a n t h e c o r r e s p o n d i n g d i f f e r e n c e g o i n g from $1000 t o $1100, s ( $ 1 1 0 0 ) - s ( $ 1 0 0 0 ) . A c c o r d i n g t o A l l a i s , a c h o i c e a g e n t ' s p r e f e r e n c e depends on t h e mathe- m a t i c a l e x p e c t a t i o n ( f i r s t moment), t h e d i s p e r s i o n (second moment) and i n g e n e r a l t h e shape 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 o f p s y c h o l o g i c a l v a l u e s . T h i s l e d him t o a s s e r t 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 a l V ( F ) e v a l u a t e d a t a d i s t r i b u t i o n F can be s t a t e d i n terms o f some f u n c t i o n a l h ( F - ) o f t h e d i s t r i b u t i o n o f p s y c h o l o g i c a l v a l u e s , F_, as f o l l o w s : s V ( F ) = h C F i ) . (4.47) n F o r a f i n i t e l o t t e r y , F (= £ p . 6 x . ) , t h e c o r r e s p o n d i n g F- i s g i v e n by: i = l F s E . " P i 6 s ( x . ) - ' ( 4 - 4 8 ) 1=1 1J A f t e r r e s c a l i n g , we o b t a i n e d t h e f u n c t i o n a l h below. h ( 6 - ( x ) ) = s ( x ) . (4.49) An i m p o r t a n t p r o p e r t y o f h, wh i c h we w i l l d e r i v e s h o r t l y , was however not n o t e d by A l l a i s . S i n c e s i s an i n t e r v a l s c a l e , t h e f u n c t i o n a l h must r e p r e s e n t t h e same p r e f e r e n c e u n d e r an a f f i n e t r a n s f o r m a t i o n f o r s. C o n s i d e r a l o t t e r y F and i t s c e r t a i n t y e q u i v a l e n t M ( F ) . I t f o l l o w s t h a t h ( F _ ) = h ( 6 _ ( M ( F ) ) ) = s ( M ( F ) ) . (4.50) 126 Under t h e a f f i n e t r a n s f o r m a t i o n as + b, where a>0-, h(F - . ) = h(6 . ) v as+b' a s ( M ( F ) ) + b ' = as(M(F))+b = a h ( F _ ) + b . (4.51) I t t u r n s out t h a t t h e above p r o p e r t y o f h e n a b l e s us t o draw v e r y , s p e - c i f i c c o n c l u s i o n s about t h e a d m i s s i b l e f u n c t i o n a l forms o f h. C o n s i d e r 1 N a u n i f o r m l o t t e r y F_ = _ S 6 A p p l y i n g A c z e l ' s theorem ( A c z e l , s s ( - x i J 1966, p. 236) t o i i e v a l u a t e d a t F_ y i e l d s : - 1 N h ( N . S 1 6 5 ( x . ) ) E y + 0 g N ( § ( x i } - y ' ^ V " ^ 1 = 1 1 — " a — a 1 N 2 1 n 2 where y = ^ I s ( x ) , a = - I ( s ( x j - y ) , i = l 1=1 and g^ i s an a r t i b r a r y symmetric f u n c t i o n , p r o v i d e d a > 0. (4 .52) D e f i n e t h e f u n c t i o n a l g on t h e space o f u n i f o r m d i s t r i b u t i o n s as f o l l o w s : 1 N g ( N 1 6y > = % C y i ' V - ( 4 - 5 3 ) 1=1 i _ 1 N 1 N I t f o l l o w s t h a t 1 N h ( r r E 6_ , J = y + a g ( i j 6 _ , , ) . (4.54) '•N. . s ( x . ) K 6^N. , s ( x . ) - y i = l l i = l v l a Note t h a t t h e above r e s u l t can be a p p l i e d t o f i n i t e l o t t e r i e s w i t h r a t i o n a l p r o b a b i l i t i e s by t a k i n g N t o be t h e l e a s t common d e n o m i n a t o r . Hence Lemma 4.1: h(F_ ) = y + a g ( F _ ) ' (4.55) a n where F_ = £ p.8_, .,, w i t h p. r a t i o n a l , s i = 1 r i s ( x i ) ' F i n - 2 n 2 y = Z p s ( x ) , o = l p . [ s ( x )-y] i = l 1 i = l 1 1 127 and g an a r b i t r a r y f u n c t i o n a l on t h e space o f f i n i t e d i s t r i b u t i o n s w i t h r a t i o n a l p r o b a b i l i t y w e i g h t s . W i t h o u t g e t t i n g i n t o d e t a i l s , we remark t h a t t h e e x t e n s i o n o f Lemma 4.1 t o remove t h e r e s t r i c t i o n o f r a t i o n a l w e i g h t s i s s t r a i g h t f o r w a r d . We need h t o be c o n t i n u o u s w i t h r e s p e c t t o weak convergence ( c f . C h a p t e r 2 Axiom M4), and the n n o t e t h a t any f i n i t e d i s t r i b u t i o n i s t h e weak l i m i t o f a sequence o f f i n i t e d i s t r i b u t i o n s w i t h r a t i o n a l w e i g h t s . T h i s argument can be extended t o i n c l u d e t h e case o f d i s - t r i b u t i o n s w i t h compact s u p p o r t s s i n c e t h e y a r e weak l i m i t s o f sequences o f f i n i t e d i s t r i b u t i o n s . F u r t h e r e x t e n s i o n t o t h e case o f non-compact s u p p o r t can be o b t a i n e d t h r o u g h an a s s u m p t i o n t h a t i s s i m i l a r t o Axiom M5 o f C h a p t e r 2. I n a r e c e n t p a p e r , A l l a i s (L979) c o n s i d e r e d t h e f o l l o w i n g f u n c t i o n a l form f o r h. h( F _ ) = y > w ( F § _ y ) (4.56) He the n e x p r e s s e d w i n terms o f the n o r m a l i z e d c e n t r a l moments o f F_ such t h a t s-y 9 i n* i m? where m = ( s - y ) n d F . n R The r e s u l t a n t e x p r e s s i o n f o r h, h(F s-) = y + f(l4> 1 7 - y y however does n o t s a t i s f y p r o p e r t y ( 4 . 5 1 ) . T h e r e f o r e i t i s n o t a d m i s s i b l e , The i d e a o f r e p r e s e n t i n g a d i s t r i b u t i o n v i a i t s moments can be a p p l i e d t o t h e s t a n d a r d i z e d d i s t r i b u t i o n i n e x p r e s s i o n (4.55) . We d e f i n e 128 t h e f u n c t i o n f i n ' t h e f o l l o w i n g way: f ( m 3 , i 4 , i n , . . . ) = g ( F g _ y ) (4.58) a where m i s t h e nth-moment o f F. , . n s-y a Note t h a t t h e v a l u e o f f i s n o t a f f e c t e d by an a f f i n e t r a n s f o r m a t i o n o f §. A key i d e a i n A l l a i s ' c r i t i c i s m o f e x p e c t e d u t i l i t y i s t h a t i t n e g l e c t s t h e h i g h e r moments o f t h e p s y c h o l o g i c a l v a l u e s. Presumably, t h e more moments we i n c l u d e , t h e c l o s e r does t h e r e s u l t i n g h a p p r o x i m a t e a c t u a l p r e f e r e n c e . As a f i r s t s t e p , one i s tempted t o c o n s i d e r o n l y t h e f i r s t two moments. The g e n e r a l form o f h f o r t h i s c a s e i s o b t a i n e d by s e t t i n g t h e f u n c t i o n f i n (4.58) e q u a l t o some c o n s t a n t X, i . e . , h ( F _ ) = u + X o. (4.59) I t i s however e a s y t o see t h a t h above w i t h a n o n z e r o X v i o l a t e s t h e as s u m p t i o n o f c o n s i s t e n c y w i t h s t o c h a s t i c dominance. C o n s i d e r F s E p 6 s ( x ) + ( 1 _ p ) 6 s ( y ) 5 w i t h § W > § <- y- 1 , D i f f e r e n t i a t i n g ^ ( p 5 s ( x ) + ( 1 _ p ) 6 s ( y ) - ) W i t h r e s P e c t t 0 P y i e l d s : | ^ { p s ( x ) + ( l - p ) s ( y ) + A [ p l 2 ( l - p ) i ' 2 ( s ( x ) - s ( y ) ) ] } = ( s ( x ) - s ( y ) ) [ l + | ( l - 2 p / p l s ( l - p ) 1 2 ) ] . (4.60) T h i s d e r i v a t i v e i s n e g a t i v e f o r any n e g a t i v e ( p o s i t i v e ) v a l u e o f X, when p i s s u f f i c i e n t l y c l o s e t o z e r o ( o n e ) . Hence, h r e s t r i c t e d t o t h e f i r s t two moments o f s cannot be c o n s i s t e n t w i t h s t o c h a s t i c dominance even f o r l o t t e r i e s w i t h o n l y two outcomes. We a r e c o m p e l l e d t o i n c l u d e t h e h i g h e r moments i f we want t o get away from e x p e c t e d u t i l i t y b u t s t a y w i t h i n A l l a i s ' framework. The most g e n e r a l h t h a t depends on t h e f i r s t t h r e e moments i s g i v e n by 129 h ( F . ) = y + o f ( m 3 / a 3 ) . (4.61) 3 A l i n e a r a p p r o x i m a t i o n o f f by X + y m /a p r o d u c e s : h ( F _ ) = y + Xo + y m 3/a 2. (4.62) T h i s i s e s s e n t i a l l y t h e same e x p r e s s i o n adopted by Hagen (1979) . Hagen showed t h a t t h e A l l a i s t y p e c h o i c e b e h a v i o r i s c o m p a t i b l e w i t h a p o s i t i v e y as l o n g as t h e magnitude o f X i s n o t t o o l a r g e . The p o s i t i v e skewness dependence matches our i n t u i t i o n about p e o p l e ' s p r e f e r e n c e e v i d e n t i n t h e p r e v a l e n c e o f l o t t e r y t i c k e t p u r c h a s e . . The q u e s t i o n o f whether th e above form o f h i s c o n s i s t e n t w i t h s t o c h a s t i c dominance r e m a i n s . C o n s i d e r a g a i n t h e two-outcome l o t t e r y F- = p6-, . + ( l - p ) 6 _ . . w i t h s s (x j s (_yj s ( x ) g r e a t e r t h a n s ( y ) . The f u n c t i o n a l h e v a l u a t e d at F_ g i v e s : h ( p 8 _ . , + ( l - p ) 6 _ , O ^ s ( x ) v y j s{y)J = s ( y ) + (p + X [ p ( l - p ) ] ^ + y [ p 2 + ( l - p ) 2 ] } [ s ( x ) - s ( y ) ] . (4.63) I n o r d e r t h a t t h e d e r i v a t i v e s o f h w i t h r e s p e c t t o p i s always p o s i t i v e , we a g a i n r e q u i r e X t o be z e r o . The dependence o f h on a i s t h e n subsumed 2 m t h e d e n o m i n a t o r o f t h e m^/a t e r m . The r e s u l t a n t h w i t h o u t t h e Xo t e r m i s g i v e n by: H ( P < 5 s ( x ) + C 1 - P ) < 5 s ( y ) ) = § i y ) + { p + ^ [ P 2 ^ 1 " ? 2 ) ] } [ 5 ( x ) - s ( y ) ] . (4.64) As p t e n d s t o one, h c o n v e r g e s t o s ( x ) + y [ s ( x ) - s ( y ) ] . Y e t , when p e q u a l s one, h i s e q u a l t o s ( x ) w h i c h i s l e s s t h a n s ( x ) + y [ s ( x ) - s ( y ) ] . T h i s i m p l i e s t h a t t h e l o t t e r y p8 + ( l - p ) 6 i s p r e f e r r e d t o g e t t i n g t h e x y h i g h e r amount x f o r s u r e when p i s s u f f i c i e n t l y c l o s e t o u n i t y . The c o n c l u s i o n t h a t fi v i o l a t e s s t o c h a s t i c dominance s t i l l h o l d s f o r n e g a t i v e y i f we c o n s i d e r t h e b e h a v i o r o f i i as p tends t o z e r o . I n s t e a d o f e x t e n d i n g o u r a n a l y s i s t o i n c l u d e h i g h e r moments, we s h a l l pause t o t a k e s t o c k o f what we l e a r n e d . I n p l a c e o f t h e e x p e c t a t i o n o f a u t i l i t y f u n c t i o n , A l l a i s a s s e r t e d t h a t p r e f e r e n c e i s r e p r e s e n t e d by 130 a f u n c t i o n a l t h a t depends on t h e f i r s t moment, t h e d i s p e r s i o n and i n g e n e r a l , t h e shape o f t h e d i s t r i b u t i o n o f a p s y c h o l o g i c a l v a l u e f u n c t i o n , w h i c h i s o b t a i n e d from comparison among h y p o t h e t i c a l changes i n w e a l t h p o s i t i o n . Based on t h e p r o p e r t y o f the p s y c h o l o g i c a l v a l u e f u n c t i o n as an i n t e r v a l s c a l e , we o b t a i n e d a r e s t r i c t i o n on t h e c l a s s o f a d m i s s i b l e p r e f e r e n c e f u n c t i o n a l s . I t t u r n s o u t t h a t t h e p a r t i c u l a r form o f dependence on t h e moments o f t h e p s y c h o l o g i c a l v a l u e c o n s i d e r e d by A l l a i s i s not a d m i s s i b l e . N e x t , we c o n s i d e r e d t h o s e a d m i s s i b l e f u n c t i o n a l s t h a t depend o n l y on t h e f i r s t two moments. The r e s u l t a n t f u n c t i o n a l f o r m, t h e sum o f t h e f i r s t moment and the s t a n d a r d d e v i a t i o n s c a l e d by some c o n s t a n t , i s shown t o be i n c o n s i s t e n t w i t h s t o c h a s t i c dominance. E x t e n d i n g t h e a n a l y s i s t o i n c l u d e t h e t h i r d moment, we o b t a i n e d t h e f u n c t i o n a l form t h a t Hagen (1979) c o n s i d e r e d i n a r e c e n t p a p e r . T h i s f u n c t i o n a l form i s however a g a i n shown t o v i o l a t e s t o c h a s t i c dominance. I t i s not known whether t h e problem w i t h s t o c h a s t i c dominance can be a v e r t e d by i n c o r p o r a t i n g even h i g h e r moments. The d i f f i c u l t y w i t h t h e f i r s t t h r e e moments s u g g e s t s however t h a t t h e p s y c h o l o g i c a l v a l u e a s s u m p t i o n would n o t l e a d t o a " c l e a n " way t o c h a r a c t e r i z e p r e f e r e n c e s t h a t e x p e c t e d u t i l i t y f a i l s t o c a p t u r e . I n t h e n e x t s u b - s e c t i o n , we i n t r o d u c e p r o s p e c t t h e o r y , w h i c h r e p r e s e n t s a d i f f e r e n t a p p roach t o the, pr o b l e m o f d e s c r i p t i v e v a l i d i t y o f e x p e c t e d u t i l i t y f i r s t i d e n t i f i e d by A l l a i s v i a t h e famed A l l a i s p a r a d o x . 131 4.5.2 P r o s p e c t Theory P r o s p e c t t h e o r y , d e v e l o p e d by Kahneman and T v e r s k y (1979), d i s t i n g u i s h e s two phase s i n t h e c h o i c e p r o c e s s : an e d i t i n g phase i n whi c h p r o b l e m r e p r e s e n t a t i o n r u l e s c a l l e d e d i t i n g o p e r a t i o n s a r e 7 a p p l i e d t o t h e o f f e r e d p r o s p e c t s f o l l o w e d by an e v a l u a t i o n phase. We f i r s t i n t r o d u c e t h e s e e d i t i n g o p e r a t i o n s b e f o r e d i s c u s s i n g t h e i r r e l a t i o n t o the form o f t h e e v a l u a t i o n f u n c t i o n s . C o d i n g . The p e r c e p t i o n o f outcomes i n terms o f g a i n s and l o s s e s r e l a t i v e t o some r e f e r e n c e w e a l t h l e v e l . T h i s i s u s u a l l y t a k e n t o be t h e c u r r e n t a s s e t p o s i t i o n , i n wh i c h c a s e g a i n s and l o s s e s a r e t h e a c t u amounts t o be r e c e i v e d o r p a i d . C a n c e l l a t i o n . T h i s r e f e r s t o t h e p o s s i b l e d i s c a r d i n g o f common components t h a t a r e s h a r e d by t h e o f f e r e d p r o s p e c t s . An example i s t h e c a n c e l l a t i o n o f a common p r o b a b i l i t y - o u t c o m e p a i r , w h i c h i s a r e s t a t e m e n t o f Savage's s u r e - t h i n g p r i n c i p l e . S e g r e g a t i o n . An o f f e r e d p r o s p e c t w i t h a r i s k l e s s component such as a minimum g a i n ( l o s s ) i s decomposed i n t o a r i s k l e s s component and t h e p r o s p e c t w i t h t h e r i s k l e s s component t a k e n from each outcome. C o m b i n a t i o n . The p r o b a b i l i t i e s a s s o c i a t e d w i t h e q u a l outcomes a r e combined t o y i e l d a s i n g l e outcome w i t h p r o b a b i l i t y g i v e n by t h e sum o f t h e r e s p e c t i v e p r o b a b i l i t i e s . D e t e c t i o n o f Dominance. The dominated p r o s p e c t s a r e e l i m i n a t e d f r o m t h e c h o i c e s e t p r i o r t o e v a l u a t i o n . S i m p l i f i c a t i o n . T h i s r e f e r s t o t h e p o s s i b l e s i m p l i f i c a t i o n o f p r o s p e c t s by r o u n d i n g o f f p r o b a b i l i t y o r outcome v a l u e s . 132 The c o d i n g and t h e c a n c e l l a t i o n o p e r a t i o n s a r e prompted by t h e e m p i r i c a l e v i d e n c e d e s c r i b e d i n C h a p t e r 3 s e c t i o n 4. As a r e s u l t o f c o d i n g , t h e outcome v a l u e s o f o f f e r e d p r o s p e c t s a r e always s t a t e d i n terms o f g a i n s and l o s s e s . A l t h o u g h more r e m a i n s t o be known about t h e c o n d i t i o n s under w h i c h c a n c e l l a t i o n a p p l i e s , i n c o r p o r a t i n g such an o p e r a t i o n does y i e l d an a d d i t i o n a l degree o f freedom i n t h e d e s c r i p t i o n o f c h o i c e phenomena. Even l e s s i s known about t h e s i m p l i f i c a t i o n o p e r a t i o n , w h i c h seems t o be a p l a u s i b l e p r o b l e m r e p r e s e n t a t i o n h e u r i s t i c . The o t h e r e d i t i n g o p e r a t i o n s a r e r e l a t e d t o t h e e v a l u a t i o n phase, w h i c h c o n c e r n s t h e way a v a l u e f u n c t i o n v ( x ) o f the outcome x and a d e c i s i o n w e i g h t f u n c t i o n Tr(p) o f t h e p r o b a b i l i t y o f an outcome p combine t o o b t a i n t h e o v e r a l l v a l u e o f a p r o s p e c t . The v a l u e f u n c t i o n i s con- cave f o r g a i n s and convex f o r l o s s e s . The IT f u n c t i o n has t h e f o l l o w i n g p r o p e r t i e s : 1) TT i n c r e a s e s f r o m TT(0)=0 t o TT(1) = 1. 2) Tr(p)>p, f o r s m a l l p. 3) T r ( p ) + T r ( l - p ) < l , f o r p e ( 0 , 1 ) . 4) TT(pq)/TT(p) < TT(pqr)/TT(pr) , f o r p , q , r e ( 0 , l ] . P r o s p e c t t h e o r y i s d e v e l o p e d f o r s i m p l e p r o s p e c t s o f t h e form, P = P 6 x + q 6 y + ( l - p - q ) 6 o , (4.65) w h i c h have a t most two n o n - z e r o outcomes. I f t h e outcomes a r e s t r i c t l y p o s i t i v e ( n e g a t i v e ) and p and q add t o u n i t y , t h e s i m p l e p r o s p e c t i s known as a s t r i c t l y p o s i t i v e ( n e g a t i v e ) p r o s p e c t . A s i m p l e p r o s p e c t i s r e g u l a r i f i t i s n e i t h e r s t r i c t l y p o s i t i v e n o r s t r i c t l y n e g a t i v e . F o r r e g u l a r p r o s p e c t s , t h e o v e r a l l v a l u e V i s o b t a i n e d from t h e s c a l e s v and T i n t h e f o l l o w i n g manner. 133 V ( p 6 x + q 6 y + ( l - p - q ) 6 0 ) = T:(p)v(x)+TT(q)v(y). (4.66) U n l i k e t h e von Neumann-Morgenstern u t i l i t y , t h e v a l u e f u n c t i o n v ( x ) i s a r a t i o s c a l e , i . e . v v a n i s h e s a t t h e r e f e r e n c e p o i n t . T h i s p r o p e r t y was n o t e d i n Edwards ( 1 9 6 1 ) . There a r e however some d i f f i c u l t i e s w i t h t h e use o f the above e x p r e s s i o n , stemming from t h e n o n l i n e a r i t y o f IT. S i n c e TT(p)+TT(s-p) i s n o t e q u a l t o TT(p')+TT(s-p'), V ( p 6 x + ( s - p ) 6 x + 6 0 ) j V ( p ' 6 x + ( s - p ' ) 6 x + 6 0 ) . (4.67) In o t h e r words, two l o t t e r i e s each y i e l d i n g outcome x w i t h p r o b a b i l i t y s a r e n o t e q u i v a l e n t i n p r e f e r e n c e . To g e t around t h i s , t h e c h o i c e agent i s assumed t o a p p l y t h e c o m b i n a t i o n e d i t i n g o p e r a t i o n p r i o r t o e v a l u a t i o n . The o t h e r d i f f i c u l t y n e c e s s i t a t e s t h e d e t e c t i o n - o f - d o m i n a n c e o p e r a t i o n . C o n s i d e r t h e f o l l o w i n g c o m p a r i s o n : T r ( p ) v ( x ) + T T ( s - p ) v ( x + e ) ? TT(S)V(X) (4.68) Suppose t h e L.H.S. i s l e s s (more) th a n t h e R.H.S. f o r e e q u a l t o z e r o , then t h e i n e q u a l i t y would s t i l l h o l d f o r a s m a l l b u t p o s i t i v e ( n e g a t i v e ) e. Thus, dominance i s v i o l a t e d . E l i m i n a t i n g t he dominated a l t e r n a t i v e b e f o r e e v a l u a t i o n c i r c u m v e n t s t h i s p r o b l e m , b u t n o t c o m p l e t e l y . Suppose Tr(p)v(x)+7T(s-p)v(x+e°)<Tr(s)v(x), ' (4.69) - . . o f o r some p o s i t i v e £ . We can f i n d q°<s such t h a t T T ( p ) v ( x ) + T T ( s - p ) v ( x + e 0 ) < T r ( q 0 ) v ( x + e ° ) < i T ( s ) v ( x ) . (4.70) T h i s has t h e i m p l i c a t i o n t h a t PV(S-P)«X+£O+(1-S)«0 - < q 0 6 x + £ o + l l - q ° ) V and p s 6 x + ( l - s ) 6 0 >• q % + e o + d - q ° ) 6 0 , which i s b o t h n o r m a t i v e l y and e m p i r i c a l l y u n t e n a b l e . 134 v F o r t h e s t r i c t l y p o s i t i v e and s t r i c t l y n e g a t i v e p r o s p e c t s , we w i l l r u n i n t o t h e same p r o b l e m w i t h v i o l a t i o n s o f dominance i f we adopt e x p r e s s i o n ( 4 . 6 6 ) . I n s t e a d , Kahneman and T v e r s k y p r o p o s e d t h a t p e o p l e decompose a s t r i c t l y p o s i t i v e ( n e g a t i v e ) p r o s p e c t P = p6 + ( l - p ) 6 w i t h x y y>x>0 (0>x>y) i n t o a r i s k l e s s component 6 and a r i s k y component P' = P5Q+(1-P)<S^_ x and e v a l u a t e t h e s e g r e g a t e d p r o s p e c t v i a t h e f o l l o w i n g e x p r e s s i o n : V ( p 6 x + ( 1 - P ) 6 y ) = v ( x ) + T T ( l - p ) [ v ( y ) - v ( x ) ] . (4.71) The u s e o f a d i f f e r e n t e x p r e s s i o n t o e v a l u a t e s t r i c t l y p o s i t i v e o r s t r i c t l y n e g a t i v e p r o s p e c t s l e a d s t o a new d i f f i c u l t y w h i c h i s n o t c o v e r e d by t h e e d i t i n g o p e r a t i o n s . C o n s i d e r t h e r e g u l a r p r o s p e c t P = p S x + q 5 x + e + ( l - p - q ) 6 0 , w i t h x,£>0. V(P) = T r ( p ) v ( x ) + T r ( q ) v ( x + e ) < TT(p)v(x )+Tr(l-p)v(x+e) . S i n c e V i s concave f o r p o s i t i v e x, V(P) < T T ( p ) v ( x ) + T r ( l - p ) [v(x)+ev' (x) ] . (4.72) Choose £Q such t h a t '0 - v ' ( x ) T h i s i m p l i e s t h a t , 1-TT(P)-TT(1-P) TT(l-p) (4.73) p 6 x + q 6 x + £ o + ( 1 - p " q ) 6 0 -< V V q £[0, 1-p). F o r t h e c a s e q = l - p , we a p p l y e x p r e s s i o n (4.69) and o b t a i n : V(p) = v ( x ) + T T ( l - p ) [ v ( x + e - v ( x ) ] > v ( x ) . (4.74) T h i s i m p l i e s t h a t p6 + ( l - p ) 5 >- 5. r x ^ * J x+e The above i m p l i c a t i o n i s r a t h e r p a t h o l o g i c a l . No m a t t e r how c l o s e q g e t s t o 1-p, t h e L.H.S. i s s t r i c t l y worse t h a n 6 . Y e t , when q i s 135 e q u a l t o 1-p, t h e L.H.S. i s s t r i c t l y p r e f e r r e d t o 6 . We can get some f e e l i n g f o r t h e pr o b l e m by e s t i m a t i n g £ q f o r some r e a s o n a b l e v a l u e s o f x, p and TT(P). Suppose x=$100, p=.50 and TT(0 .50) = .45 . (Note t h a t p r o p e r t y (3) o f t h e IT f u n c t i o n i m p l i e s t h a t T  (0.5)<0 .5) . We have t h a t , [1-2TT(0.5)] < v ( 1 0 ) [1-2TT(0.5)] o - i U x TT(0.5) v'(10) TT(0.5) • l ^ - / i > J s i n c e t h e c o n c a v i t y o f v i m p l i e s t h a t x < v ( x ) . A c o n s e r v a t i v e e s t i m a t e f o r SQ i s t h e n g i v e n by e = 10 x .1 i .45 = 22 o " • C o n s i d e r t h e f o l l o w i n g c h o i c e p r o b l e m s . p l E • 5 6 $ i o o + • 4 5 6 $ 1 2 2 + • 0 5 6 $ o v s . 6 $ 1 0 0 P 2 E • 5 6 $ i o o + • 4 9 6 $ 1 2 2 + • o l 6 $ o v s . 6 $ 1 0 0 P 3 ~= • 5 6 $ i o o + • 4 9 9 6 $ 1 2 2 + . 0 0 1 8 $ 0 v s . 6$100 P r o s p e c t t h e o r y would p r e d i c t t h a t P^, P^, P^ a r e a l l s t r i c t l y worse th a n "SjjjTQQ- M o r e over, t h e d i r e c t i o n o f p r e f e r e n c e remains unchanged as l o n g as t h e p r o b a b i l i t y o f o b t a i n i n g $122 i s l e s s t h a n 0.5! Note t h a t t h e i n e q u a l i t y , x < v ( x ) , used t o o b t a i n e i s h i g h l y c o n s e r v a t i v e V ( x ) 0 s i n c e v ' ( x ) i s a d e c r e a s i n g f u n c t i o n f o r a concave v ( x ) . We can a r r i v e a t t h e above c o n c l u s i o n w i t h a g e n e r a l l y much h i g h e r £ q g i v e n a s p e c i f i c v ( x ) . I f v i s bounded, t h e n v ' ( x ) t e n d s t o z e r o so t h a t £ q can be made a r b i t r a r i l y l a r g e i f we c o n s i d e r a s u f f i c i e n t l y l a r g e x. P r o s p e c t t h e o r y b u i l d s on t h e fo r m o f t h e e v a l u a t i o n f u n c t i o n f i r s t s u g g e s t e d by Edwards (1955) . I t t r e a t s s y s t e m a t i c a l l y s e v e r a l c l a s s e s o f c h o i c e phenomena ( c f . C h a p t e r 3) t h a t v i o l a t e t h e i m p l i c a t i o n s o f e x p e c t e d u t i l i t y . However, t h e n o n l i n e a r i t y o f t h e d e c i s i o n w e i g ht f u n c t i o n TT, which c o n s t i t u t e s i t s main d e v i a t i o n from e x p e c t e d u t i l i t y , 136 g e n e r a t e s some s e r i o u s d i f f i c u l t i e s f o r p r o s p e c t t h e o r y . The immediate ones, namely t h e v i o l a t i o n o f the c o m b i n a t i o n p r i n c i p l e and s t o c h a s t i c dominance, were c i r c u m v e n t e d v i a t h e c o m b i n a t i o n and t h e d e t e c t i o n - o f - d o m i n a n c e e d i t i n g o p e r a t i o n s . Two problems however remain. One o f t h e problems i s t h e i m p l i c a t i o n t h a t t h e r e i s always a p r o s p e c t Q wh i c h i s s t r i c t l y worse t h a n some p r o s p e c t P but s t r i c t l y b e t t e r t h a n a n o t h e r p r o s p e c t P' t h a t dominates P. The o t h e r p r o b l e m has t o do w i t h t h e use o f a d i f f e r e n t e v a l u a t i o n f u n c t i o n f o r t h e s t r i c t l y p o s i t i v e ( n e g a t i v e ) p r o s p e c t s . T h i s g i v e s r i s e t o d i s c o n t i n u i t y i n t h e p r e f e r e n c e r e p r e s e n t e d w h i c h l e a d s t o u n t e n a b l e p r e d i c t i o n s o f a c t u a l c h o i c e b e h a v i o r . o 137 4.5.3 Comparison We c o n c l u d e t h e s e c t i o n w i t h a c o m p a r i s o n o f a l p h a u t i l i t y w i t h A l l a i s ' t h e o r y and p r o s p e c t t h e o r y . The a l p h a u t i l i t y r e p r e s e n t a t i o n fi(F) can be s t a t e d i n terms t h e e x p e c t a t i o n o f a " v a l u e " f u n c t i o n , v , ct w i t h r e s p e c t t o an 'a-weighted' d i s t r i b u t i o n F g i v e n by: pa(x) = /JW / / ^ a d F , where a i s a s t r i c t l y p o s i t i v e f u n c t i o n . The r o l e o f a becomes c l e a r e r n i f we c o n s i d e r a s i m p l e d i s t r i b u t i o n , F = E p.6 . In t h i s c a s e , i = l 1 x i fl(F) = / R v d F a n = E q ( F ) v ( x ) , i = l where q (F) = p a C x ^ / S p . a ( x ) . j = l J L i k e p r o s p e c t t h e o r y , fi(F) i s o b t a i n e d from t h e v ( x ^ ) ' s v i a a s e t o f " d e c i s i o n w e i g h t s " , ^4^^ n^_2» w i t h each q^ b e i n g a n o n l i n e a r f u n c t i o n o f t h e p r o b a b i l i t y p^ o f o b t a i n i n g t h e i t h outcome, x^. We s h o u l d however n o t e t h r e e d i s t i n c t i o n s . The q^ w e i g h t s sum t o u n i t y but n o t t h e TT(P ) w e i g h t s o f p r o s p e c t t h e o r y . In a d d i t i o n t o p^, q^ depends on t h e r e s t o f the P j ' s and a l l t h e x ^ ' s . F i n a l l y , t h e q^-weights has t h e c o m b i n a t i o n p r o p e r t y , s i n c e i f t h e j t h and k t h outcomes are b o t h e q u a l t o x, th e n fi fp.6 +p. 6 + E p. 6 .) • r i x r k x • / • , i x i [ p ' . a ( x ) + p a ( x ) ] v ( x ) + E p a ( x . ) v ( x ) n E p . a ( x . ) 138 .(Pi+Pk)aU)v(X) + 1 Pia(xi)v(xi) = iili n E p . a ( x . ) i = l 1 1 = n ( ( p + P k ) 6 x + E p 6 ). Some o f A l l a i s ' i d e a s can a l s o be e x p r e s s e d i n terms o f t h e a l p h a u t i l i t y r e p r e s e n t a t i o n . We can w r i t e fi(F) as t h e sum o f v , — ct t h e f i r s t moment o f v w i t h r e s p e c t t o F, and the d e v i a t i o n term / D ( v - v ) d F shown below: £}(F) = v + / R ( v - v ) d F a , where v" = .Lvd'F. K Thus, t h e p r e f e r e n c e o f an a l p h a u t i l i t y d e c i s i o n maker depends on t h e f i r s t moment o f v , and a l s o t h e d i s t r i b u t i o n o f t h e d e v i a t i o n , ( v - v ) , o f v from v t h r o u g h t h e a - w e i g h t e d d i s t r i b u t i o n F a . A c c o r d i n g t o t h e p r o o f s o f the r e p r e s e n t a t i o n theorems i n C h a p t e r s 1 and 2, t h e v - f u n c t i o n , much l i k e t h e von Neumann-Morgenstern u t i l i t y , can be c o n s t r u c t e d from p r e f e r e n c e c o m p a r i s o n s u s i n g a s t a n d a r d l o t t e r y . A l l a i s may i n s i s t however t h a t v be a p s y c h o l o g i c a l v a l u e f u n c t i o n t h a t i s d e t e r m i n e d from i n t r o s p e c t i v e c o m p arison among h y p o t h e t i c a l changes i n w e a l t h p o s i t i o n . I n t h i s c a s e , we can a p p l y t h e p r i n c i p l e o f Occam's r a z o r t o weed out t h e " p s y c h o l o g i c a l v a l u e " a s s u m p t i o n s i n c e i t i s r e d u n d a n t . F i n a l l y , we summarize v i a T a b l e 4.2 some o f t h e s a l i e n t f e a t u r e s o f t h e t h e o r i e s t r e a t e d i n t h i s c h a p t e r . A '+' s i g n means t h e c o r r e s p o n d i n g t h e o r y i s c o m p a t i b l e w i t h t h e p r o p e r t y r e f e r r e d t o . O t h e r w i s e , we use a '-' s i g n . 139 P r o p e r t i e s T r a n s i t i v i t y Dominance C o m b i n a t i o n C o n t i n u i t y A l l a i s » P aradox E x p e c t e d U t i l i t y + + + + - A l p h a U t i l i t y + + + + + A l l a i s Theory + - + + + P r o s p e c t Theory - + + - + T a b l e 4.2: Comparison among T h e o r i e s A n o v e l t y o f P r o s p e c t t h e o r y i s t h e e x p l i c i t use o f e d i t i n g o p e r a t i o n s p r i o r t o e v a l u a t i o n . Two o f t h e s e o p e r a t i o n s a re however needed t o ensure c o n s i s t e n c y w i t h dominance and c o m b i n a t i o n . I f A l l a i s ' t h e o r y were t o adopt t h e d e t e c t i o n - o f - d o m i n a n c e e d i t i n g o p e r a t i o n , i t would e x h i b i t s y s t e m a t i c i n t r a n s i t i v i t y , as i s t h e case f o r p r o s p e c t t h e o r y ( c f . e x p r e s s i o n ( 4 . 7 0 ) ) . S i n c e l o t t e r i e s a r e s t a t e d i n terms o f g a i n s and l o s s e s , the c o d i n g o p e r a t i o n seems a r e a s o n a b l e h y p o t h e s i s about how p e o p l e p e r c e i v e monetary outcomes. Kahneman and T v e r s k y (1979) have p r o v i d e d p r e l i m i n a r y e m p i r i c a l e v i d e n c e i n s u p p o r t o f two o t h e r e d i t i n g o p e r a t i o n s , namely c a n c e l l a t i o n and s e g r e g a t i o n . These may be adopted by a l p h a u t i l i t y t h e o r y i f f u t u r e e m p i r i c a l s t u d i e s a s c e r t a i n t h e i r v a l i d i t y as b e h a v i o r a l h y p o t h e s i s . 140 CONCLUSION 5. CONCLUSION 141 5.1 SUMMARY P a r t I o f t h i s d i s s e r t a t i o n c o n t a i n s t h e s t a t e m e n t s and p r o o f s o f two r e p r e s e n t a t i o n theorems. The f i r s t g e n e r a l i z e s the q u a s i l i n e a r mean, M^, o f Hardy, L i t t l e w o o d and P o l y a : V F ) = l * ~ V R 0 d F ) , (5.1) where 0 i s c o n t i n o u s and s t r i c t l y monotone, and F i 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 . We have weakened a c h a r a c t e r i s t i c p r o p e r t y o f t h e q u a s i - l i n e a r mean, M^, c a l l e d q u a s i l i n e a r i t y t o o b t a i n a more g e n e r a l mean v a l u e , M ,, w h i c h i s c h a r a c t e r i z e d by an a d d i t i o n a l f u n c t i o n a. The mean 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 F has t h e f o l l o w i n g form: M a 0 ( F ) = 0~V Ra0dF/.T RadF) f (5.2) where a i s c o n t i n u o u s and s t r i c t l y p o s i t i v e ( n e g a t i v e ) . Through t h e s t r i c t i n e q u a l i t y , <, b i n a r y r e l a t i o n , t h e mean i n d u c e s a b i n a r y r e l a t i o n , •-< , among p r o b a b i l i t y d i s t r i b u t i o n s : F -< G <^> M a 0 ( F ) < M a 0 ( G ) . (5.3) The R.H.S. i s e q u i v a l e n t t o / Ra0dF// RadF < 7 R a 0 d G / / R a d G , (5.4) f o r a s t r i c t l y i n c r e a s i n g 0. Note t h a t t h e o r d e r i n g r e p r e s e n t e d by (5.4) i s more g e n e r a l t h a n t h a t r e p r e s e n t e d by t h e e x p e c t e d u t i l i t y r e p r e s e n t a t i o n , f^tfdF. F o r c o n v e n i e n c e , we use fi(F) t o l a b e l t h e r e p r e s e n t a t i o n f u n c t i o n a l , / RajzSdF// RadF. An a l t e r n a t i v e a p p r o a c h t o o b t a i n t h e r e p r e s e n t a t i o n , ( 5 . 4 ) , i s g i v e n i n C h a p t e r 2. I n s t e a d 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 c o n s i d e r s i m p l e p r o b a b i l i t y measures d e f i n e d on some a r b i t r a r y s e t X - (A s i m p l e 142 p r o b a b i l i t y measure i s a convex l i n e a r c o m b i n a t i o n o f a f i n i t e number o f p o i n t masses i n X )• Axioms were s t a t e d d i r e c t l y i n terms o f a b i n a r y r e l a t i o n , -< , t o o b t a i n t h e c o r r e s p o n d i n g fi r e p r e s e n t a t i o n f o r s i m p l e p r o b a b i l i t y measures: fi(P) = E(av,P)/E ( c t,P) , (5 .5 ) where P i s a s i m p l e p r o b a b i l i t y measure on a s e t X , a n a " a and v are r e a l - v a l u e d f u n c t i o n s on X - T h i s i s c o n t r a s t e d w i t h t h e approach based on axioms on mean v a l u e s i n C h a p t e r 1. The r e a s o n why t h e mean v a l u e a p p r o a c h cannot be e x t e n d e d t o s i m p l e p r o b a b i l i t y measures on an a r b i t r a r y s e t X i s t h a t mean v a l u e i t s e l f may n o t be d e f i n e d i n X « As an example, c o n s i d e r t h e outcome s e t X - { s t a t u s quo, b e i n g promoted, b e i n g f i r e d ) . B e i n g a b l e t o d e a l w i t h an a r b i t r a r y outcome s e t such as the above example i s an advantage f o r t h e s i m p l e p r o b a b i l i t y measure approach. There a r e however c e r t a i n drawbacks. W i t h o u t f u r t h e r s t r u c t u r a l a s s u m p t i o n s on X . j w e cannot d i s c u s s t h e n o t i o n s o f c o n t i n u i t y and d i f f e r e n t i a b i l i t y o f t h e a and v f u n c t i o n s n o r can we g e n e r a l i z e t h e fi r e p r e s e n t a t i o n t o i n c l u d e more g e n e r a l p r o b a b i l i t y measures. P a r t I I o f t h e d i s s e r t a t i o n c o n c e r n s one s p e c i f i c a r e a o f a p p l i c a t i o n — d e c i s i o n t h e o r y . I n t e r p r e t i n g t h e M A 0 ( F ) mean o f C h a p t e r 1 as t h e c e r t a i n t y e q u i v a l e n t o f a monetary l o t t e r y F, t h e c o r r e s p o n d i n g i n d u c e d b i n a r y r e l a t i o n , K , has t h e n a t u r a l i n t e r p r e t a t i o n as ' s t r i c t p r e f e r e n c e ' between l o t t e r i e s . , F o r non-monetary ( f i n i t e ) l o t t e r i e s , we a p p l y t h e r e p r e s e n t a t i o n theorem o f C h a p t e r 2. The h y p o t h e s i s t h a t t h e fi r e p r e s e n t a t i o n o f e i t h e r a p proach r e p r e s e n t s a c h o i c e a g e n t ' s p r e f e r e n c e among l o t t e r i e s i s r e f e r r e d t o as a l p h a u t i l i t y t h e o r y . T h i s i s l o g i c a l l y 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 c h o i c e agent obeys e i t h e r t h e mean v a l u e ( c e r t a i n t y e q u i v a l e n t ) axioms o r t h e axioms ;on t h e '-< ' ( s t r i c t p r e f e r e n c e ) b i n a r y r e l a t i o n . 143 A l p h a u t i l i t y t h e o r y i s a g e n e r a l i z a t i o n o f e x p e c t e d u t i l i t y t h e o r y i n t h e sense t h a t t h e e x p e c t e d u t i l i t y r e p r e s e n t a t i o n i s a s p e c i a l case o f t h e a l p h a u t i l i t y r e p r e s e n t a t i o n , and t h a t a l p h a u t i l i t y assumes a weaker f o r m o f a k e y p r o p e r t y o f e x p e c t e d u t i l i t y , c a l l e d s u b s t i t u t a b i l i t y ( P r a t t , R a i f f a and S c h l a i f e r , 1964), w h i c h i s e s s e n t i a l l y t h e same as t h e q u a s i l i n e a r i t y p r o p e r t y o f Hardy, L i t t l e w o o d and P o l y a . (A c l o s e c o u n t e r p a r t o f s u b s t i t u t a b i l i t y i s t h e s t r o n g independence p r i n c i p l e o f Marschak (1950) and Samuelson ( 1 9 5 2 ) ) , The m o t i v a t i o n f o r g e n e r a l i z i n g e x p e c t e d u t i l i t y comes f r o m d i f f i c u l - t i e s i t f a c e d i n t h e d e s c r i p t i o n o f c e r t a i n c h o i c e phenomena, e s p e c i a l l y t h e A l l a i s p a r a d o x . These a r e summarized i n C h a p t e r 3. C h a p t e r 4 c o n t a i n s t h e f o r m a l s t a t e m e n t s o f a s s u m p t i o n s and t h e d e r i v a t i o n s o f n o r m a t i v e and d e s c r i p t i v e i m p l i c a t i o n s o f a l p h a u t i l i t y t h e o r y . We s t a t e d c o n d i t i o n s , t a k e n from C h a p t e r 1, f o r c o n s i s t e n c y w i t h s t o c h a s t i c dominance and g l o b a l r i s k a v e r s i o n and d e r i v e d a g e n e r a l i z e d A r r o w - P r a t t i n d e x o f l o c a l r i s k a v e r s i o n . We a l s o d e m o n s t r a t e d how a p a i r o f a and v f u n c t i o n s t h a t s a t i s f y b o t h s t o c h a s t i c dominance and l o c a l r i s k a v e r s i o n can be c o n s i s t e n t w i t h t h o s e c h o i c e phenomena, summarized i n C h a p t e r 3, t h a t c o n t r a d i c t s t h e i m p l i c a t i o n s o f e x p e c t e d u t i l i t y . The c h a p t e r ended w i t h a c o m p a r i s o n o f a l p h a u t i l i t y w i t h two o t h e r t h e o r i e s t h a t have a t t r a c t e d a t t e n t i o n ; namely, A l l a i s ' t h e o r y and p r o s p e c t t h e o r y . 5.2 EXTENSIONS We c o n c l u d e by p o i n t i n g out some p o t e n t i a l a r e a s o f a p p l i c a t i o n . The q u a s i l i n e a r mean was g i v e n an i n t e r p r e t a t i o n as t h e ' e q u a l l y - d i s - t r i b u t e d - e q u i v a l e n t ' l e v e l o f income c o r r e s p o n d i n g t o an income d i s t r i b u t i o n by A t k i n s o n (1970) i n h i s p a p e r on t h e measurement o f i n e q u a l i t y . We may use t h e M ^ mean as a more g e n e r a l model o f 144 e q u a l l y - d i s t r i b u t e d - e q u i v a l e n t income. I s t h e r e any need f o r a more g e n e r a l measure? C o n s i d e r two s o c i e t i e s w i t h income d i s t r i b u t i o n s F and G g i v e n by: ) F E ° - 5 0 6 $ l , 0 0 0 + ° - 5 0 6 $ 2 , 0 0 0 > M D G E °- 5 0 6$i , o o o + ° - 4 9 6 $ 2 , o o o + ° - o l 6 $ i t o o o , o o o - To many ( i n c l u d i n g p r o b a b l y Mao Tse-Tung), s o c i e t y F f a r e s b e t t e r t h a n s o c i e t y G. But t h e G d i s t r i b u t i o n s t o c h a s t i c a l l y dominates t h e F d i s t r i b u t i o n , so t h a t M^(F) i s l e s s t h a n M^(G). T h e r e f o r e , t h e measure f a i l s to r e f l e c t t h e r e l a t i v e w e l f a r e o f t h e two s o c i e t i e s f o r t h o s e who b e l i e v e t h a t s o c i e t y F i s b e t t e r o f f . T h i s does n o t pose any d i f f i c u l t y f o r t h e measure s i n c e c o n s i s t e n c y w i t h s t o c h a s t i c dominance i s n o t an i n t r i n s i c p r o p e r t y . The d e p a r t u r e o f M , from M, can be made c l e a r e r i f we c o n s i d e r a N s o c i e t y o f N i n d i v i d u a l s w i t h incomes, { x ^ } ^ _ ^ . The c o r r e s p o n d i n g e q u a l l y - d i s t r i b u t e d - e q u i v a l e n t i s g i v e n by: N N M a«S ( F ) = * ( E a ( x i ) v ( x i ) / Z a O r > ) - C 5 - 6 ) i = l i = l A r e m a r k a b l e f e a t u r e o f (5.6) i s t h e p r e s e n c e o f c o m p l e m e n t a r i t y a c r o s s incomes o f d i f f e r e n t i n d i v i d u a l s . That t h i s i s a d e s i r a b l e p r o p e r t y i s r e i n f o r c e d by t h e f a c t t h a t an i n d i v i d u a l p e r c e i v e s c o n c u r r e n t l y t h e incomes o f o t h e r i n d i v i d u a l s i n t h e d i s t r i b u t i o n whereas o n l y one o f a s e t o f m u t u a l l y e x c l u s i v e outcomes w i l l o b t a i n i n a l o t t e r y . We can t h i n k o f t h e r o l e o f a as a s s i g n i n g d i s c r i m i n a t o r y w e i g h t s on i n d i v i d u a l s based on t h e i r a t t a i n e d incomes. Mao Tse-Tung's ( h y p o t h e t i c a l ) p r e f e r e n c e f o r s o c i e t y F may t h e n be e x p l a i n e d i n terms o f a d e c r e a s i n g d i s c r i m i n a t i o n f u n c t i o n a t h a t t r e a t s w e a l t h y i n d i v i d u a l s l e s s ' e q u a l l y ' t h a n t h e p o o r e r f o l k s . 145 In a r e c e n t p a p e r on t h e measurement o f p o v e r t y , B l a c k o r b y and Donald- son (1978b) a p p l i e d A t k i n s o n ' s e q u a l l y - d i s t r i b u t e d - e q u i v a l e n t t o t h e 'ce n s o r e d ' d i s t r i b u t i o n , i . e . , t h e income d i s t r i b u t i o n t r u n c a t e d a t some e x o g e n e o u s l y e s t a b l i s h e d p o v e r t y l i n e . T h i s i s tantamount t o h a v i n g a measure w i t h an a t h a t i s c o n s t a n t up t o t h e p o v e r t y l i n e and z e r o beyond. I t i s n a t u r a l t o suggest t h a t a d e c r e a s i n g a w i t h an i n f l e x i o n p o i n t a t t h e p o v e r t y l i n e (see F i g u r e 5.1) would i n t e g r a t e t h e c o n t r i b u t i o n due t o t h e whole d i s t r i b u t i o n and a t t h e same ti m e be p a r t i c u l a r l y r e p r e - s e n t a t i v e o f t h e p o o r e r f o l k s w i t h incomes below t h e p o v e r t y l i n e . p o v e r t y l i n e Income F i g . 5.1: An a l p h a f u n c t i o n t h a t d i s c r i m i n a t e s a g a i n s t t h e r i c h The mean can a l s o be u s e d t o g e n e r a t e measures o f i n e q u a l i t y as f o l l o w s : R e l a t i v e I n e q u a l i t y Index = 1 - M^/y, and A b s o l u t e I n e q u a l i t y Index = y - M ^ , 146 where y d e n o t e s t h e a r i t h m e t i c mean. The r e l a t i v e i n d e x i s due t o A t k i n s o n (1970) and t h e a b s o l u t e i n d e x i s due t o Kolm (1976a,b). When t h e d i s t r i b u t i o n o f income i s c o m p l e t e l y e q u a l , b o t h i n d i c e s e q u a l z e r o . The c o e f f i c i e n t of v a r i a t i o n , an o c c a s i o n a l measure of income i n e q u a l i t y , i s r e l a t e d t o by: \. c o e f f i c i e n t o f v a r i a t i o n = {M ,(F) - y c } a , 1>1 r (5.7) where M g t ( F ) = { / Q X S + t d F / / ~ x S d F } / ( i . e . a ( x ) = x S and 0(x) = x l ) . As an e q u a l l y - d i s t r i b u t e d - e q u i v a l e n t , M i s u n d e s i r a b l e because i t i s always h i g h e r t h a n t h e a r i t h m e t i c mean e x c e p t a t e q u a l i t y , and so enco u r a g e s i n e q u a l i t y . A l s o , i t s w e i g h t i n g f u n c t i o n a ( x ) =• x a s s i g n s p r o g r e s s i v e l y more w e i g h t t o t h e more w e a l t h y . The M g t mean s u g g e s t e d above may be o f use i n s t a t i s t i c s . We o f f e r some examples. M = s t a n d a r d d e v i a t i o n • c o e f f i c i e n t o f skewness, and M = s t a n d a r d d e v i a t i o n • ( c o e f f i c i e n t o f K u r t o s i s + 3 ) 2 . An e q u a l i t y t h a t g e n e r a l i z e s t h e w e l l known r e s u l t t h a t t h e p r o d u c t o f t h e a r i t h m e t i c mean and harmonic mean i s e q u a l t o t h e square o f t h e g e o m e t r i c mean f o r two p o s i t i v e numbers, may be s t a t e d i n terms o f ^ t 2t a S f°ll° w s- M (F) = M, ( F ) , ( g e o m e t r i c mean) (5.8) - t , 2 t l o g x^ ^ b where F = £ ^ 5 Y , w i t h > 0, when t h e f r e q u e n c y p o l y g o n o f l o g x^ i s s y m m e t r i c a l about t h e a x i s o f o r d i n a t e a t -j^E l o g x^. Canning (1934) p r o p o s e d the use of 147 { M - t , 2 t ^ / M l o g x ^ > ( 5 ' 9 ) as a d e s c r i p t i v e measure o f asymmetry. H a r s a n y i (1977) a p p l i e d e x p e c t e d u t i l i t y t o an i n d i v i d u a l making a m o r a l judgement about a l t e r n a t i v e s o c i a l s i t u a t i o n s . Making a mo r a l judgement, i n t h i s c a s e , means making a h y p o t h e t i c a l b e s t c h o i c e under t h e a s s u m p t i o n t h a t t h e i n d i v i d u a l assumes t h e p o s i t i o n o f any one member o f t h e s o c i e t y w i t h e q u a l chance. A s o c i a l s i t u a t i o n X, w h i c h i s a l i s t i n g o f t h e N p e r s o n s ' s t a t e s , would be p e r c e i v e d as a l o t t e r y t h a t a s s i g n s t h e i n d i v i d u a l t o any p a r t i c u l a r i n d i v i d u a l ' s p o s i t i o n w i t h 1/N chance. An e x p e c t e d u t i l i t y d e c i s i o n maker would t h e n maximize t h e e x p e c t e d u t i l i t y c o r r e s p o n d i n g t o s o c i a l s i t u a t i o n X . T h i s i s s i m p l y t h e N a r i t h m e t i c a v e r a g e , E u . ( X ) / N , o f h i s von Neumann-Morgenstern u t i l i t y , i = l 1 u ^ ( X ) , f o r t a k i n g t h e p o s i t i o n o f t h e i t h _ p e r s o n . An a l p h a u t i l i t y d e c i s i o n maker w i l l however maximize a s l i g h t l y more g e n e r a l e x p r e s s i o n : N N E a (X)v ( X ) / E a ( X ) , (5.10) i = l i = l where o t ^ ( X ) d e n o t e s h i s a v a l u e f o r b e i n g i n i t h p e r s o n ' s shoes w i t h i n t h e s o c i a l s i t u a t i o n X . T h i s i s a w e i g h t e d average o f the v ^ ( X ) ' s w i t h w e i g h t s g i v e n by t h e OL(X)'S. E x p r e s s i o n (5.10) may be i n t e r p r e t e d as t h e average o f t h e v . ( X ) ' s b u t t h e a l p h a u t i l i t y d e c i s i o n maker u s e s 1 N a ' b i a s e d ' e s t i m a t e , a . ( X ) / E a . ( X ) , f o r h i s p r o b a b i l i t y o f b e i n g t h e 1 i = l 1 i t h p e r s o n . A g a i n , t h e a l p h a u t i l i t y e x p r e s s i o n has t h e advantage o f a l l o w i n g f o r c o m p l e m e n t a r i t y , w h i c h i s s u p p o r t e d by o u r i n t u i t i o n about how m o r a l judgements a r e made. S e v e r a l q u e s t i o n s remain u n t o u c h e d . There a r e o t h e r s i t u a t i o n s where t h e r e p r e s e n t a t i o n f u n c t i o n s have t h e a d d i t i v e s t r u c t u r e o f e x p e c t e d u t i l i t y . One example i s t h e a v e r a g i n g o f i n d i v i d u a l u t i l i t y 148 f u n c t i o n s u s i n g a s e t o f f i x e d w e i g h t s t o o b t a i n a group u t i l i t y f u n c t i o n (Keeney and R a i f f a , 1976). A n o t h e r example i s t h e t i m e - h o n o r e d p r a c t i c e o f d i s c o u n t i n g a t i m e s t r e a m o f u t i l i t i e s (Koopmans, 1972). The i n t u i t i o n t h a t c o m p l e m e n t a r i t y s h o u l d n o t be r u l e d out i s s t r o n g i n b o t h c a s e s . F o r t h e s e and o t h e r s i m i l a r examples, the a l p h a u t i l i t y r e p r e s e n t a t i o n p r o v i d e s a u s e f u l f i r s t c a n d i d a t e f o r a d e p a r t u r e from t h e a d d i t i v e s t r u c t u r e i n o r d e r t o i n c o r p o r a t e c o m p l e m e n t a r i t y a c r o s s a t t r i b u t e s . The l a s t decade saw a tremendous growth i n t h e l i t e r a t u r e on t h e m i c r o e c o n o m i c s o f u n c e r t a i n t y . P r a c t i c a l l y a l l o f t h e s e works a r e based on t h e a s s u m p t i o n t h a t c h o i c e agents maximize e x p e c t e d u t i l i t y . We have n o t i n v e s t i g a t e d what new r e s u l t s would f o l l o w o r what o l d r e s u l t s ar e r o b u s t i f we i n t r o d u c e a l p h a u t i l i t y c h o i c e a g e n t s . One example i s c o m p a r a t i v e r i s k a v e r s i o n . The c o n d i t i o n f o r one e x p e c t e d u t i l i t y m a x i m i z e r t o always be more r i s k a v e r s e t h a n a n o t h e r i s g i v e n i n P r a t t (1964) . An e q u i v a l e n t r e s u l t appeared i n Hardy, L i t t l e w o o d and P o l y a (1934) i n t h e form o f an i n e q u a l i t y between two q u a s i l i n e a r means. The c o r r e s p o n d i n g r e s u l t f o r a l p h a u t i l i t y c h o i c e agents o r t h e mean, however, remains an open q u e s t i o n . A n o t h e r example i s Samuelson's (1967) c o n j e c t u r e , w h i c h was c o n f i r m e d by K a l a i and S c h m e i d l e r ( 1 9 7 7 ) , t h a t Arrow's i m p o s s i b i l i t y theorem w i l l h o l d i n a c a r d i n a l s e t u p where i n d i v i d u a l s and s o c i e t y e x p r e s s t h e i r p r e f e r e n c e s by von Neumann-Morgenstern u t i l i t y f u n c t i o n s . I t i s perhaps s a f e t o c o n j e c t u r e t h a t Arrow's i m p o s s i b i l i t y theorem would s t i l l h o l d i f i n d i v i d u a l s and s o c i e t y e x p r e s s t h e i r p r e f e r e n c e s v i a a l p h a u t i l i t y f u n c t i o n s . 149 R E F E R E N C E S A c z e l , J . ( 1 9 6 6 ) , F u n c t i o n a l E q u a t i o n s and T h e i r A p p l i c a t i o n , Academic P r e s s , New York and London. A l l a i s , M. ( 1 9 5 3 ) , "Le comportement de l'homme r a t i o n n e l devant l e r i s q u e " , E c o n o m e t r i c a 21, No. 4, 503-546. A l l a i s , M. ( 1 9 7 9 ) , "The S o - C a l l e d A l l a i s Paradox and r a t i o n a l d e c i s i o n s Under C e r t a i n t y , " i n M. A l l a i s and 0. Hagen, eds., 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 and t h e A l l a i s P aradox, R e i d e l P u b l i s h i n g Company, 437-682. A l l a i s , M. $ 0. Hagen ( 1 9 7 9 ) , 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 £ t h e A l l a i s P a r a d o x , D. R e i d e l P u b l i s h i n g Co. Anscombe, F . J . $ R . J . Aumann ( 1 9 6 3 ) , "A D e f i n i 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 y , " A n n a l s o f M a t h e m a t i c a l S t a t i s t i c s , V o l . 34, No. 1, 199-205. Arrow, K . J . ( 1 9 7 1 ) , Theory o f R i s k B e a r i n g , Markham P u b l i s h i n g Company, C h i c a g o . A t k i n s o n , A.B. (1970) "On t h e Measurement o f I n e q u a l i t y , " J o u r n a l o f Economic Theory, 2, 244-263. B e c k e r , G.M. & C G . M c C l i n t o c k ( 1 9 6 7 ) , " V a l u e : B e h a v i o r a l D e c i s i o n T h e o r y , " Annual Review o f P s y c h o l o g y , V o l . 18, 239-286. B e r n o u l l i , D. (1954) "Speciman t h e o r i a e novae de mensura s o r t i s , " C o m e n t a r i i academiae s c i e n t i a r u m i m p e r i a l a s P e t r o p o l i t a n a e , V o l . 5, 175-192; t r a n s l a t e d a s , " E x p o s i t i o n o f a New Theory on t h e Measurement o f R i s k , " E c o n o m e t r i c a , V o l . 22, 23-26. B e n - t a l , A. (1977) "On G e n e r a l i z e d Means and G e n e r a l i z e d Convex F u n c t i o n s , " J o u r n a l o f O p t i m i z a t i o n Theory and A p p l i c a t i o n s , V o l . 21, No. 1, J a n u a r y , 1-13. B l a c k o r b y , C. and D. Donaldson ( 1 9 7 8 a ) , "A T h e o r e t i c a l Treatment o f E t h i c a l I n d i c e s o f A b s o l u t e I n e q u a l i t y " f o r t h c o m i n g i n I n t e r n a t i o n a l Economic Review, Department o f Economics D i s c u s s i o n Paper 78-03, 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 . B l a c k o r b y , C. and D. Donaldson (1978b), " E t h i c a l I n d i c e s f o r t h e Measure- ment o f P o v e r t y , Department o f Economics D i s c u s s i o n P a p e r 78-04, 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 . B l a c k w e l l , D. ( 1 9 5 1 ) , "Comparison o f E x p e r i m e n t s , " P r o c e e d i n g s o f Symposium on M a t h e m a t i c a l S t a t i s t i c s and P r o b a b i l i t y , 2nd, B e r k e l e y , J . Neyman and L.M. Lecam, eds., U n i v e r s i t y o f C a l i f o r n i a P r e s s , 93-102. 150 B l a c k w e l l , D. and M.A. G i r s h i c k ( 1 9 5 4 ) , Theory o f Games and S t a t i s t i c a l D e c i s i o n s , W i l e y § Sons, I n c . C a n n i n g , J.B. "A Theorem C o n c e r n i n g A C e r t a i n F a m i l y o f Averages o f a C e r t a i n Type o f F r e q u e n c y D i s t r i b u t i o n , " E c o n o m e t r i c a 2, 442. The a b s t r a c t i s a r e p o r t e r ' s summary o f an u n p u b l i s h e d p a p e r . Chew, S.H. ( 1 9 7 9 ) , "A G e n e r a l i z a t i o n o f the Q u a s i l i n e a r Mean o f Hardy, L i t t l e w o o d and P o l y a , " I n s t i t u t e o f A p p l i e d M a t h e m a t i c s and S t a t i s t i c s T e c h n i c a l R e p o r t 79-39, 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 . Chew, S.H. and K.R. MacCrimmon (19 7 9 b ) , " A l p h a U t i l i t y , L o t t e r y C o m p o s i t i o n and t h e A l l a i s P a r a d o x , " F a c u l t y o f Commerce and B u s i n e s s A d m i n i - s t r a t i o n W orking Paper #686, 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 . Chew, S.H. and K.R. MacCrimmon (1 9 7 9 a ) , " A l p h a U t i l i t y Theory: A G e n e r a l i z a t i o n o f E x p e c t e d U t i l i t y , " F a c u l t y o f Commerce and B u s i n e s s A d m i n i s t r a t i o n Working Paper #669, 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 . D a l t o n , H. ( 1 9 2 0 ) , "The Measurement o f I n e q u a l i t y o f Incomes," Economic J o u r n a l 20, 348-361. DeGroot, M.H. ( 1 9 7 0 ) , O p t i m a l S t a t i s t i c a l D e c i s i o n s , M c G r a w - H i l l Book Co., New Y o r k . Diamond, P. § M. R o t h s c h i l d (1978) U n c e r t a i n t y i n Economics, Academic P r e s s , New Y o r k . Edwards, W. ( 1 9 5 4 ) , "The Theory o f D e c i s i o n M a k i n g , " P s y c h o l o g i c a l B u l l e t i n , V o l . 51, 380-417. Edwards, W. ( 1 9 5 5 ) , "The P r e d i c t i o n o f D e c i s i o n s Among B e t s , " J o u r n a l o f E x p e r i m e n t a l P s y c h o l o g y 60, 265-277. Edwards, W. ( 1 9 6 1 ) , " B e h a v i o r a l D e c i s i o n T h e o r y , " Annual Review o f P s y c h o l o g y 12, 473-498. F i s h b u r n , P.C. ( 1 9 7 0 ) , U t i l i t y T h e o r y f o r D e c i s i o n M a k i n g , John W i l e y , New Y o r k . F r i e d m a n , M. § L . J . Savage ( 1 9 4 8 ) , "The U t i l i t y A n a l y s i s o f C h o i c e I n v o l v i n g R i s k , " J o u r n a l o f P o l i t i c a l Economy, V o l . 56, 279-304. F r i s c h , R. ( 1 9 2 6 ) , "Suv une Probleme d'Economie P u r e , " Norsk M a t h e m a t i s h F o r e n i n g s S k r i f t e r , 1. Hadar, J . and W.R. R u s s e l l (1969), " R u l e s f o r O r d e r i n g U n c e r t a i n P r o s p e c t s , " AER 49, 25-34. 151 Hagen, 0. ( 1 9 7 9 ) , "Towards a P o s i t i v e Theory o f P r e f e r e n c e s Under R i s k , " i n 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 and t h e A l l a i s P a r a d o x , M. A l l a i s and 0. Hagen, eds., R e i d e l P u b l i s h i n g Company, 271-302. Handa, J . ( 1 9 7 7 ) , " R i s k , P r o b a b i l i t i e s and a New Theory o f C a r d i n a l U t i l i t y , " J o u r n a l o f P o l i t i c a l Economy, V o l . 85, No. 1, F e b r u a r y , 97-122. Hanoch, G. and C. Levy ( 1 9 6 9 ) , " E f f i c i e n c y A n a l y s i s o f C h o i c e s I n v o l v i n g R i s k , " Review o f Economic S t u d i e s 36, 335-346. Hardy, G.H., J.E. L i t t l e w o o d § G. P o l y a ( 1 9 3 4 ) , I n e q u a l i t i e s , Cambridge U n i v e r s i t y P r e s s . H a r s a n y i , J.C. (1 9 7 7 ) , R a t i o n a l B e h a v i o r and B a r g a i n i n g E q u i l i b r i u m i n Games and S o c i a l S i t u a t i o n s , Cambridge: Cambridge U n i v e r s i t y P r e s s . H e r s t e i n , I.N. 5 J . M i l n o r ( 1 9 5 3 ) , "An A x i o m a t i c Approach t o M e a s u r a b l e U t i l i t y , " E c o n o m e t r i c a , V o l . 2 1 ( 2 ) , A p r i l . Howard, R.A. ( 1 9 6 4 ) , "The F o u n d a t i o n s o f D e c i s i o n A n a l y s i s , " IEEE T r a n s . SSC-4:211-19. J e n s e n , N.E. ( 1 9 6 7 ) , "An I n t r o d u c t i o n t o B e r n o u l l i a n U t i l i t y . I . U t i l i t y F u n c t i o n s , " Swedish J o u r n a l o f Economics. Kahneman, D. § A. T v e r s k y ( 1 9 7 9 ) , " P r o s p e c t Theory: An A n a l y s i s o f D e c i s i o n Under R i s k , " E c o n o m e t r i c a , March, 263-291. K a l a i , E. and D. S c h m e i d l e r ( 1 9 7 7 ) , " A g g r e g a t i o n P r o c e d u r e f o r C a r d i n a l P r e f e r e n c e s : A F o r m u l a t i o n and P r o o f o f Samuelson's I m p o s s i b i l i t y C o n j e c t u r e , " E c o n o m e t r i c a 45, No. 6, September, 1431-1437. Karmarkar, U.S. ( 1 9 7 8 ) , " S u b j e c t i v e l y Weighted U t i l i t y : A D e s c r i p t i v e E x t e n s i o n o f E x p e c t e d U t i l i t y M o d e l , " OBHP, V o l . 21, 61-72. Keeney, R.L. & H. R a i f f a ( 1 9 7 6 ) , D e c i s i o n s w i t h M u l t i p l e O b j e c t i v e s : R e f e r e n c e s and V a l u e T r a d e o f f s , W i l e y § Sons, I n c . Kolm. S.Ch. ( 1 9 7 6 a ) , "Unequal I n e q u a l i t i e s I , " J o u r n a l o f Economic Theory 12, J u n e , 416-442. Kolm, S.Ch. (1976b), "Unequal I n e q u a l i t i e s I I , " J o u r n a l o f Economic Theory 13, August, 82-111. Koopmans, T.C. (1 9 7 2 ) , " R e p r e s e n t a t i o n o f P r e f e r e n c e O r d e r i n g s Over Time," i n D e c i s i o n and O r g a n i z a t i o n , C.B. McGuire and R. Radner, eds., N o r t h H o l l a n d P u b l i s h i n g Company, Amsterdam. K r a n t z , D.H., D.R. Luce § A. T v e r s k y ( 1 9 7 1 ) , F o u n d a t i o n s o f Measurement V o l . 1, Academic P r e s s , New Yor k § London. 152 Lehman'n, E.L. ( 1 9 5 5 ) , " O r d e r e d F a m i l i e s o f D i s t r i b u t i o n s , " A n n a l s o f M a t h e m a t i c a l S t a t i s t i c s 26, 399-419. L i c h t e n s t e i n , S. § P. S l o v i c ( 1 9 7 1 ) , " R e v e r s a l o f P r e f e r e n c e Between B i d s and C h o i c e s i n Gambling D e c i s i o n s , " J o u r n a l o f E x p e r i m e n t a l P s y c h o l o g y 89, 46-55. L u e n b e r g e r , D.G. ( 1 9 6 9 ) , O p t i m i z a t i o n by V e c t o r Space Methods, New Y o r k : John W i l e y § Sons. MacCrimmon, K.R. ( 1 9 6 5 ) , "An E x p e r i m e n t a l S t u d y o f t h e D e c i s i o n - M a k i n g B e h a v i o r o f B u s i n e s s E x e c u t i v e s , " u n p u b l i s h e d d i s s e r t a t i o n , U n i v e r - s i t y o f C a l i f o r n i a , Los A n g e l e s . MacCrimmon, K.R. ( 1 9 6 8 ) , " D e s c r i p t i v e and N o r m a t i v e I m p l i c a t i o n s o f t h e D e c i s i o n Theory P o s t u l a t e s , " i n K. B o r c h and J . M o s s i n , e d s . , R i s k and U n c e r t a i n t y , S t . M a r t i n ' s P r e s s , New Y o r k , 3-32. MacCrimmon, K.R., John F. B a s s l e r £ W i l l i a m T. S t a n b u r y ( 1 9 7 2 ) , U n p u b l i s h e d R e s u l t s , R i s k S t u d y P r o j e c t , U n i v e r s i t y o f B r i t i s h C o l u mbia. MacCrimmon, K.R. § S. L a r s s o n ( 1 9 7 9 ) , " U t i l i t y T heory: Axioms V e r s u s ' P a r a d o x e s ' , " i n M. A l l a i s and 0. Hagen, eds., E x p e c t e d U t i l i t y and t h e A l l a i s P a r a d o x , H o l l a n d : D. R i e d e l . M a c h i n a , M.J. ( 1 9 8 0 ) , " E x p e c t e d U t i l i t y A n a l y s i s W i t h o u t t h e Independence Axiom," u n p u b l i s h e d w o r k i n g p a p e r , Department o f Economics, U n i v e r s i t y o f C a l i f o r n i a - San Di e g o . M a r k o w i t z , H. ( 1 9 5 2 ) , " T h e . U t i l i t y o f W e a l t h , " J o u r n a l o f P o l i t i c a l Economy, V o l . 60, 151-158. Marschak, J . ( 1 9 5 0 ) , " R a t i o n a l B e h a v i o r , U n c e r t a i n P r o s p e c t s , and Me a s u r e a b l e U t i l i t y , " E c o n o m e t r i c a 18, 111-141. Marschak, J . § R. Radner ( 1 9 7 2 ) , Economic Theory o f Teams, Y a l e U n i v e r s i t y P r e s s . M e g i n n i s s , J.R. ( 1 9 7 7 ) , " A l t e r n a t i v e s t o t h e E x p e c t e d U t i l i t y R u l e , " u n p u b l i s h e d Ph.D. d i s s e r t a t i o n , U n i v e r s i t y o f C h i c a g o . von Neumann, J . £ 0. M o r g e n s t e r n ( 1 9 4 7 ) , Theory o f Games and Economic B e h a v i o r , P r i n c e t o n U n i v e r s i t y P r e s s , 2nd E d i t i o n , P r i n c e t o n . N o r r i s , N i l a n ( 1 9 7 6 ) , " G e n e r a l Means and S t a t i s t i c a l T h e o r y , " The Ame r i c a n S t a t i s t i c i a n , F e b r u a r y , V o l . 30, No. 1. P r a t t , J . ( 1 9 6 4 ) , " R i s k A v e r s i o n i n t h e S m a l l and i n t h e L a r g e , " E c o n o m e t r i c a , V o l . 32, No. 1-2, J a n u a r y - A p r i l , 122-136. 153 P r a t t , ' J.W., H. R a i f f a § R. S c h l a i f e r ( 1 9 6 4 ) , "The F o u n d a t i o n s o f D e c i s i o n s Under U n c e r t a i n t y : An E l e m e n t a r y E x p o s i t i o n , " JASA 59, 353-375. P r e s t o n , M.G. § P. B a r a t t a , "An E x p e r i m e n t a l Study o f t h e A u c t i o n V a l u e o f An U n c e r t a i n Outcome," J o u r n a l o f P s y c h o l o g y 61, 183-193. R a i f f a , H. ( 1 9 6 8 ) , D e c i s i o n A n a l y s i s : I n t r o d u c t o r y L e c t u r e s on C h o i c e Under U n c e r t a i n t y , R e a d i n g , M a s s a c h u s e t t s : A d d i s o n - W e s l e y . R a i f f a , H. $ R. S c h l a i f e r ( 1 9 6 1 ) , A p p l i e d S t a t i s t i c a l D e c i s i o n Theory, D i v i s i o n o f R e s e a r c h , H a r v a r d B u s i n e s s S c h o o l , B o s t o n , 356. Ramsey, F.P. ( 1 9 3 1 ) , " T r u t h and P r o b a b i l i t y , " (1926) i n The F o u n d a t i o n s o f M a t h e m a t i c s , R.B. B r a i t h w a i t e , ed., H u m a n i t i e s P r e s s . R o t h s c h i l d , M. $ J.E. S t i g l i t z ( 1 9 7 0 ) , " I n c r e a s i n g R i s k I . A D e f i n t i o n , " J o u r n a l o f Economic Theory 2, 225-243. Samuelson, P.A. ( 1 9 5 2 ) , " P r o b a b i l i t y , U t i l i t y , and t h e Independence Axiom, E c o n m e t r i c a 20, 670-78. Samuelson, P. ( 1 9 6 7 ) , "Arrow's M a t h e m a t i c a l P o l i t i c s , " i n Human V a l u e s and Economic P o l i c y , S. Hook, ed., New York: New York U n i v e r s i t y P r e s s , 41-51. Savage, L . J . ( 1 9 5 4 ) , The F o u n d a t i o n s o f S t a t i s t i c s , W i l e y , New York. S l o v i c , P., B. F i s c h h o f f § S. L i c h t e n s t e i n ( 1 9 7 7 ) , " B e h a v i o r a l D e c i s i o n T h e o r y , " A n n u a l Review o f P s y c h o l o g y , 39. S l o v i c , P. § A. T v e r s k y ( 1 9 7 4 ) , "Who A c c e p t s Savage's Axiom?" B e h a v i o r a l S c i e n c e 19, 368-373. Weber, R. ( 1 9 8 0 ) , P e r s o n a l Communication, Weerahandi, S. and J . Z i d i k ( 1 9 7 9 ) , "A C h a r a c t e r i z a t i o n o f t h e Genera*! Mean," f o r t h c o m i n g i n t h e C a n a d i a n J o u r n a l o f S t a t i s t i c s , Department o f M a t h e m a t i c s D i s c u s s i o n Ppaer, 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 . Whitmore, G.A. (1970), " T h i r d - D e g r e e S t o c h a s t i c Dominance," American Economic Review 60, 457-459.

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