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UBC Theses and Dissertations

Two representation theorems and their application to decision theory Chew, Soo Hong 1980

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TWO AND  REPRESENTATION  THEOREMS  THEIR A P P L I C A T I O N TO D E C I S I O N THEORY  by i^CHEW; SOO HONG M.A.,  Claremont  A T H E S I S SUBMITTED  Graduate  S c h o o l , 1977  I N P A R T I A L FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF  PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES (Interdisciplinary (Mathematics,  Economics,  Management  We a c c e p t t h i s t h e s i s to  the required  Programme)  as  Science)  conforming  standard  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA November- 1980  0  ChewJ, Soo Hong, 1980  In  presenting this  thesis  an advanced degree at the L i b r a r y I  further  for  it  freely  this  thesis for  It  financial  The U n i v e r s i t y  Date  the  requirements  B r i t i s h Columbia,rI for  reference copying o f  o f B r i t i s h Columbia  agree  not  fo  that  and study. this  thesis  by the Head of my Department  gain s h a l l  Interdisciplinary  2075 Wesbrook P l a c e Vancouver, Canada V 6 T 1W5  of  i s understood that copying or  written permission.  of  of  that permission for extensive  representatives.  Department  fulfilment  available  s c h o l a r l y purposes may be granted  by h i s of  the U n i v e r s i t y  s h a l l make  agree  in p a r t i a l  or  publication  be allowed without my  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 . ments and p r o o f s  Part I contains the state-  o f two r e p r e s e n t a t i o n t h e o r e m s .  The f i r s t  theorem, proved  in  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 H a r d y , L i t t l e w o o d a n d 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 .  the  Given  two d i s t r i b u t i o n s  same m e a n s , 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  w i t h another  distribution  i n t h e same p r o p o r t i o n s  o f these  share  to  We w e a k e n t h e q u a s  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 give r i s e  be d i f f e r e n t ,  This  distributions  t h e same mean,  r e g a r d l e s s o f the d i s t r i b u t i o n t h a t they are mixed.with. linearity  with  t o t h e same means  l e a d s t o a more g e n e r a l mean, d e n o t e d b y M  ,  which has t h e form:  %  (  F  )  =  *~V <^dF// c.dF), R  w h e r e <J> i s c o n t i n u o u s positive  (negative)  R  and s t r i c t l y monotone, a i s c o n t i n u o u s  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 .  and s t r i c t l y  The q u a s i l i n e a r  mean, d e n o t e d b y 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 . in  addition, that the M  order.  showed,  mean h a s 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 n d  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 ones) p a r t i a l  We  (including higher  degree  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  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  among  mean o f a d i s t r i b u t i o n  F c a n b e w r i t t e n a s 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 , w h e r e F is  d e r i v e d f r o m F v i a a as t h e R a d o n - N i k o d y m d e r i v a t i v e o f F  with  r e s p e c t t o F. We n o t e d  that the  mean i n d u c e s  an o r d e r i n g among  probability  d i s t r i b u t i o n s v i a t h e m a x i m a n d , / a<}>dF//. adF, t h a t c o n t a i n s t h e R  utility)  R  m a x i m a n d , /^<J)dF, o f t h e q u a s i l i n e a r mean  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  as a s p e c i a l  c h a r a c t e r i z a t i o n o f t h e above  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 m e a s u r e s on a more g e n e r a l  (expected case. represen-  outcome s e t  w h e r e mean v a l u e s may n o t b e d e f i n e d . directly  In t h i s  case, axioms a r e s t a t e d  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 .  several standard solvability  p r o p e r t i e s o f expected  utility,  We r e t a i n e d  n a m e l y weak o r d e r ,  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  Pratt, 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  linearity  i n t h e c o n t e x t o f an o r d e r i n g .  Part  I I o f t h e d i s s e r t a t i o n concerns  decision theory.  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 1 as t h e  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  c e r t a i n t y e q u i v a l e n t o f a m o n e t a r y l o t t e r y F, t h e c o r r e s p o n d i n g binary relation  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  between l o t t e r i e s .  F o r non-monetary  r e p r e s e n t a t i o n theorem o f Chapter agent's preference  (finite)  2.  among l o t t e r i e s  as a l p h a u t i l i t y  This i s l o g i c a l l y  induced  in  equivalent t o saying  preference binary  of the alpha u t i l i t y expected  utility  utility  representation.  relation.  derivations theory.  theory case  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 i tfaced i n the description of  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  Chapter  utility  representation i s a special  comes f r o m d i f f i c u l t i e s  summarized i n Chapter  that  ( c e r t a i n t y e q u i v a l e n t ) axioms  theory i s a g e n e r a l i z a t i o n o f expected  the sense t h a t the expected  we a p p l y t h e  ordering, i s referred t o  t h e c h o i c e a g e n t o b e y s e i t h e r t h e mean v a l u e  Alpha u t i l i t y  lotteries,  preference'  can be r e p r e s e n t e d b y a p a i r o f  the  o r t h e a x i o m s on h i s s t r i c t  as ' s t r i c t  induced  The h y p o t h e s i s , t h a t a c h o i c e  a a n d cj) f u n c t i o n s t h r o u g h theory.  of quasi-  paradox.  These a r e  3.  4 contains the formal o f normative  statements  and d e s c r i p t i v e  We s t a t e d c o n d i t i o n s , t a k e n  o f assumptions and t h e  i m p l i c a t i o n s of alpha  from Chapter  utility  1, f o rconsistency with  iv  stochastic  d o m i n a n c e and  Arrow-Pratt  index  global risk  of local  risk  aversion  aversion.  We  and  a  derived  generalized  a l s o demonstrated  how  1 alpha  utility  theory  that contradict  the  c a n be  consistent with those choice  i m p l i c a t i o n s of expected u t i l i t y ,  e i t h e r s t o c h a s t i c dominance or with  local  a comparison of alpha u t i l i t y  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 ' Several I are  considered  the use level  (Atkinson,  p o i n t i n g out  risk aversion  and  two  theory  other and  of t h i s  1 9 7 0 ) , and The  as  alpha  s i t u a t i o n s i n the an  open q u e s t i o n  stated  an e x t e n s i o n  The  violating  chapter  theories that  prospect  ended have  theory.  theorems of  dissertation.  These  Part  include  a measure o f asymmetry o f utility  by von  representation  sense o f Harsanyi  regarding  can  (1977).  conditions  of Samuelson's  that Arrow's i m p o s s i b i l i t y theorem would h o l d express t h e i r preferences  without  a model o f the 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  (Canning, 1934).  used to rank s o c i a l  e n d e d by  Conclusion  mean as  o f income  with  aversion.  a p p l i c a t i o n s of the r e p r e s e n t a t i o n  i n the  o f the  distribution be  other  risk  phenomena  (1967)  for  also We  comparative  conjecture  i f i n d i v i d u a l s and  Neumann-Morgenstern u t i l i t y  a  society  functions.  v  CONTENTS  INTRODUCTION  PART I  W REPRESENTATO I N THEOREMS 1  2  PART II  1  G E N E R A L I Z I N G T H E QUASI LINEAR MEAN OF HARDY, AND POLYA 7  1.1  INTRODUCTION  1.2  AXIOMS OF MEAN VALUE  1.3  REPRESENTATION THEOREM  1.4  PROPERTIES OF THE M  LITTLEWOOD  7 8 14  , MEAN a<j>  G E N E R A L I Z I N G T H E EXPECTED U T I L I T Y THEOREM 36  29  REPRESENTATION  36  2.1  INTRODUCTION  2.2  PRELIMINARY D E F I N I T I O N S  2.3  AXIOMS  2.4  REPRESENTATION THEOREMS  36  38 40  APPLICATION TO DECISION THEORY BACKGROUND  3  CRITIQUE  59  OF EXPECTED  UTILITY  THEORY  61  61  3.1  INTRODUCTION  3.2  SYSTEMATIC V I O L A T I O N OF THE STRONG PRINCIPLE 64  INDEPENDENCE  vi  4  3.3  CONCURRENCE OF R I S K SEEKING AND R I S K AVERTING BEHAVIOR 70  3.4  SOME PROBLEMS WITH PROBLEM REPRESENTATION  3.5  SUMMARY  A NEW THEORY  77  80  81  4.1  INTERPRETING MEAN VALUE AS CERTAINTY EQUIVALENT  81  4.2  REPRESENTATION OF A PREFERENCE BINARY RELATION  86  4.3  NORMATIVE  4.4  4.5  90  IMPLICATIONS  91  4.3.1  Ratio Consistency  4.3.2  Assessment  4.3.3  S t o c h a s t i c Dominance  100  4.3.4  Global Risk Aversion  103  4.3.5  Local Risk Aversion:  97  The A r r o w - P r a t t  105  Index  108  DESCRIPTIVE IMPLICATIONS 4.4.1  Systematic V i o l a t i o n o f the Strong Axiom 108  Independence  4.4.2  S t o c h a s t i c Dominance  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 : Concurrence 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 P r o b l e m 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  114  C R I T I Q U E OF A L L A I S * THEORY AND PROSPECT THEORY 4.5.1  Allais  4.5.2  Prospect  4.5.3  Comparison  1  Theory Theory 137  124 131  122 124  v i i  CONCLUSO IN 5  CONCLUSION  141  141  5.1  SUMMARY  5.2  EXTENSIONS  143  viii  T A B L E S  R e s u l t s R e l e v a n t t o t h e H I L O L o t t e r y S t r u c t u r e 68  3.1  Summary o f E m p i r i c a l  4.1  A l l o w a b l e Choice P a t t e r n s under Alpha U t i l i t y  4.2  C o m p a r i s o n among T h e o r i e s  139  Theory  110  ix  FIGURES  3.1  A "typical"  3.2  Four d e c i s i o n problems  3.3  The  composition  3.4  The  HILO s t r u c t u r e o f t h r e e consequence  3.5  Standard  lottery  3.6  Examples  of u  3.7  Examples  o f u^,  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  3.9  A sequential representation of lottery  4.1  Ratio Consistency  4.2  Geometric proof o f the Ratio Consistency  4.3  P r o b a b i l i t y E q u i v a l e n t method  4.4  Test  4.5  Preference  4.6  Consistency  4.7  Conditions  4.8  An  4.9  A pair of u  4.10  A p a i r of u  5.1  von Neumann-Morgenstern u t i l i t y  c  64  o f the A (B T  and u ^ and u ^  on d a t a  (based  66  (1972))  75  77  B  92  95  property  98  98 109 115  117  118 d e c i s i o n maker  d e r i v e d f r o m an a l p h a u t i l i t y 3/i»  74  77  and 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 and u  C)  Appendix  f r o m MacCrimmon e t . a l .  f o r S t o c h a s t i c Dominance  function  (1977),  using barycentric coordinates  f o r Local Risk Aversion  g  lotteries  lotteries  P a t t e r n f o r ct(I) < 1  £  65  n  from A l l a i s  on d a t a  illustrated  admissible alpha  (B )  from A  70  (based  o f S u b s t i t u t i o n axiom  An a l p h a  ) lottery  comparison  Conditions  62  function  d e c i s i o n maker  r  f u n c t i o n that d i s c r i m i n a t e s against the r i c h  145  120 120  PREFACE  The p r e f a c e i s perhaps a s u i t a b l e p l a c e f o r an i n f o r m a l d i s c u s s i o n o f 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 o f  u t i l i t y , von Neumann and M o r g e n s t e r n l i k e n e d the " u t i l i t y " ,  l o o s e l y speaking,  o f a l o t t e r y t o the c e n t e r o f g r a v i t y o f a mass d i s t r i b u t i o n . words, the u t i l i t y , u ( x ^ , . . . > 5 P ^ » • • • » P ) » °f X  N  P )» n  the u t i l i t i e s utility  In o t h e r  l o t t e r y , ( x ^ , . . . , x ;p^,...,  a  n  which pays x^ w i t h p r o b a b i l i t y  expected  p^, s h o u l d be the mean o r average o f  a s s o c i a t e d w i t h each p r o b a b l e outcome.  T h i s l e d t o the  expected  expression: n u ( x , , . . . , x ;p,,...,p ) = .S,p.u(x.). ^ 1 n l n i=l i 1 r  r  r  Together w i t h t h e i r e l e g a n t 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 , t h e r e seemed t o be a s t r o n g case f o r the a d o p t i o n o f expected  utility.  v e r s y however c l o u d e d the p i c t u r e somewhat. r e p l a c i n g the p r o b a b i l i t i e s , 1  c  Some, i n c l u d i n g A l l a i s ,  suggested  { p ^ ^ - i * w i t h a more g e n e r a l s e t o f w e i g h t s ,  ^ i ^ i - 1 ' * ^ ' depend on the l o t t e r y , and sum w  The A l l a i s - S a v a g e c o n t r o -  1  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 l n i=l i 1 n l n I r  r  r  v  r  r  A t t h i s l e v e l o f g e n e r a l i t y , the o n l y t e s t a b l e i m p l i c a t i o n i s t h a t the u t i l i t y o f a l o t t e r y i s i n t e r m e d i a t e i n v a l u e between the maximum and minimum a t t a i n a b l e u t i l i t i e s .  T h i s p r o p e r t y , which may  be c a l l e d  the  intermediate  v a l u e p r o p e r t y , i s c o m p a t i b l e w i t h our i n t u i t i o n about the u t i l i t y o f a l o t t e r y as a mean v a l u e . To impose more s t r u c t u r e , one may w  c o n s i d e r r e s t r i c t i n g the <JK weights (Xj,...,x ;p ...,p ) n  1}  n  0) (x ,...,x ;p ,...,p ) = i  1  n  1  ,  n  j ? l j C i > • • • > ' P i ' • • • 'Pnw  x  x  1  n  where { w ^ } ^ ^ i s a s e t o f p o s i t i v e l y v a l u e d weight f u n c t i o n s t h a t depend on  to:  xi the l o t t e r y , ( x ^ , . . . > »P-_> • • • >P ) •  A f u r t h e r r e s t r i c t i o n i s o b t a i n e d by  x  n  n  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 o f  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.  T h i s i m p l i e s t h a t t h e w^ f u n c t i o n s a r e o f t h e form:  w.(x.,...,x ;p,,...,p ) = p.w.(x.;x .,p . ) , 1 1 n l n i1 l - i '- i / ' r  r  r  where x ^ and p ^ denote t h e 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 w i t h t h e i t h component d e l e t e d .  With a b o l d s l e i g h t o f hand, as y e t u n s u b s t a n t i a t e d  by any a p r i o r i r e a s o n , t h e s i n g l e f u n c t i o n a evaluated  values  a r e assumed t o be o b t a i n e d  a t t h e i t h outcome, x^.  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  p a(x ) i  i  u ( x , . . . x ; p , . . . , p ) = J^I f  n  1  } u(x.).  n  j?lPj Of c o u r s e , a n i c e e x p r e s s i o n  T h i s leads t o a f a i r l y  utility: n  1  from a  a ( x  j  )  i s j u s t the f i r s t step.  The next t h i n g  i s t o work back and f o r t h i n o r d e r t o i d e n t i f y a m i n i m a l s e t o f 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 t h e a and u f u n c t i o n s can be c o n s t r u c t e d .  These  p r o p e r t i e s , once found, would t h e n n e c e s s a r i l y be weaker than t h e c o r r e s ponding ones f o r e x p e c t e d u t i l i t y . The  f i r s t p r o o f o f such a r e p r e s e n t a t i o n  theorem i s i n t h e c o n t e x t o f  g e n e r a l i z i n g 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 g i v e n i n C h a p t e r 1.  The c o r r e s p o n d i n g r e s u l t i n terms o f a p r e f e r e n c e  g i v e n i n C h a p t e r 2.  ordering i s  A l t h o u g h t h e 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 o f those o f C h a p t e r 1, a s e l f - c o n t a i n e d t r e a t m e n t o f Chapter 2 i s p r e s e n t e d s o t h a t r e a d e r s who a r e used t o t h e p r e f e r e n c e can s k i p C h a p t e r 1.  Bob Weber p r o v i d e d  an e l e g a n t  o r d e r i n g approach  geometrical i n t e r p r e t a t i o n  o f one o f t h e axioms ( R a t i o C o n s i s t e n c y ) i n an e a r l i e r v e r s i o n o f t h e p r e f e r ence o r d e r i n g theorem and demonstrated t h a t i t was redundant.  This  led to a  xii  much weaker axiom i n the c u r r e n t p r o o f o f the mean value  representation  theorem. As  i s u s u a l , the o r g a n i z a t i o n o f the d i s s e r t a t i o n assumes the normal  trappings  o f academic w r i t i n g .  ously i n Part  Formal r e s u l t s , which are developed r i g o r -  I, precede i n t e r p r e t a t i o n s i n the  context  of decision  i n P a r t I I , d e s i g n e d t o b u i l d a p r i m a f a c i e case f o r the a d o p t i o n more g e n e r a l grounds. has  the  i n Part  expected u t i l i t y h y p o t h e s i s  T h i s d i v i s i o n may  of  on both t h e o r e t i c a l and  the  empirical  cause the appearance o f r e p e t i t i o u s n e s s , but  advantage o f making P a r t II s e l f - c o n t a i n e d . II are the r e s u l t  theory  o f j o i n t work w i t h  Ken  P a r t s o f the  MacCrimmon, my  material  research  supervisor. I would a l s o l i k e t o acknowledge my guidance committee. and  The  Cindy Greenwood f o r h e l p had  been i n s t r u m e n t a l  i n e n a b l i n g me  o f the r e p r e s e n t a t i o n  a l s o i n the u n d e r s t a n d i n g o f some b a s i c mathematics.  from D a n i e l  judgement and  d e c i s i o n - m a k i n g under u n c e r t a i n t y and  i n h i s home.  I have a l s o b e n e f i t e d from d i s c u s s i o n s w i t h  I was  Blackorby  exposed t o r e s e a r c h  Kanemoto's seminars. f o r suggesting choice  to  theorems,  I have b e n e f i t e d  Kahneman's r e s e a r c h with Amos Tversky on the p s y c h o l o g y  h i s work w i t h C h a r l e s  course.  my  numerous i n s t a n c e s when I went t o Shelby Brumelle  c a r r y through the a n a l y s i s i n the p r o o f s and  debt t o the o t h e r members on  a l s o from the  seminars  Dave Donaldson  on the measurement o f i n e q u a l i t y and  i n the economics o f u n c e r t a i n t y d u r i n g  Thanks are a l s o due  of  on  poverty.  Yoshitsugo  t o a non-member, John B u t t e r w o r t h ,  the problem d u r i n g the f i n a l examination o f h i s  information  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  the starting  point f o rthis d i s s e r t a t i o n .  The f i r s t ,  Polya  (1934),  characterization of a rather  class  o f mean v a l u e s  i s an a x i o m a t i c  ^(F)  =  due t o Hardy, L i t t l e w o o d and  c a l l e d t h e q u a s i l i n e a r mean:  CJTVR  cj>dF).  Mjj(F) d e n o t e s 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 probability distribution  F a n d c h a r a c t e r i z e d b y a s t r i c t l y m o n o t o n e f u n c t i o n <J>. Polya proved t h e i r c h a r a c t e r i z a t i o n f o r simple defined  general  on a c o m p a c t i n t e r v a l , .  Hardy, L i t t l e w o o d and  probability distributions  E x a m p l e s 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 h e g e o m e t r i c mean (<j> i s logarithmic),  t h e h a r m o n i c mean (cj> i s o f t h e f o r m ^) a n d t h e r t h  moment mean, a l s o known a s t h e g e n e r a l form  r  second r e p r e s e n t a t i o n  Petersburg's  possesses t o take part  expectation prospects.  he  theorem has i t s g e n e s i s  p a r a d o x -- a n i n d i v i d u a l  chance; thus,  a  r (cj> i s o f t h e  x ).  The  he  mean o f o r d e r  i s not w i l l i n g  i n a l o t t e r y that pays 2  demonstrating t h e l i m i t a t i o n o f using  o f payoffs This  as a general  t o stake  a l l that  dollars with  ^  mathematical  r u l e f o rtheordering  of risky  l e d B e r n o u l l i t o p r o p o s e , i n 1738, t h e e x p e c t a t i o n o f  ' m o r a l 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 .  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  simal  increase  i n t h eworth o f wealth  simal  increase  i n wealth  itself.  1  i nthe S t .  In particular.,  by assuming t h a t an i n f i n i t e -  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 -  but i n v e r s e l y proportional t o wealth  The m o n e t a r y 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  ing  to a lottery  F i s then given by,  u(M(F)) = / u d F ; R  or  alternatively,  M(F) = u Note t h a t t h i s  _ : L  (r udF). R  expression  i s t h e same a s 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  first  axiomatic  function o f payoffs by  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  (1947) p r o v i d e d  map o n a m i x t u r e a minimal  as.the  The  independently  and i n i t i a t e d  s e t (e.g.  t h e space o f p r o b a b i l i t y  function.  expected u t i l i t y  usefulness  the use  distributions)  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  Expected u t i l i t y  Their result  subject  map i s t h e  i s now commonly r e f e r r e d  theorem.  o f q u a s i l i n e a r means n e e d s no e l a b o r a t i o n . ^  theory,  theorem t o decision-making preference  i s given  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  expectation o f a u t i l i t y to  of a  v o n Neumann a n 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  t h e i r " T h e o r y o f Games a n d E c o n o m i c B e h a v i o r "  of t h e term " u t i l i t y " .  to  leading to the expectation  Ramsey ( 1 9 2 6 ) 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 " ,  Morgenstern in  treatment  i . e . , the a p p l i c a t i o n o f the expected by i n t e r p r e t i n g the binary r e l a t i o n  r e l a t i o n and t h e m i x t u r e  utility as a  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 u s e o f t h e r t h moment mean i n s t a t i s t i c s , s e e N o r r i s ( 1 9 7 6 ) , B l a c k o r b y § D o n a l d s o n (1978a) c o n t a i n s e x a m p l e s o f t h e u s e o f q u a s i l i n e a r mean i n t h e m e a s u r e m e n t o f i n c o m e i n e q u a l i t y . W e e r a h a n d i § Z i d e k ( 1 9 7 9 ) 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 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 t h e 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 a r 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 o n l i n e a r s p a c e s 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  o r 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 a s a certainty  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  inception. Samuelson and  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 (1952),  Aumann  H e r s t e i n and M i l n o r  (1963),  Pratt, Raiffa  and S c h l a i f e r  B l a c k w e l l and G i r s h i c k (1961) and DeGroot  utility  theory  to statistical f o r Arrow  Arrow  (1954),  (1964),  (1954),  their  (1970),  by Marschak  Savage  (1970),  foundation  Fishburn  (1953),  DeGroot  the  attention since i t s  decisions.  Anscombe  Jensen  (1971) and o t h e r s .  (1950),  (1967),  Savage  (1970) a p p l i e d  In a d d i t i o n , i tserved  expected as  (1971) and M a r s c h a k and Radner (1972) i n  i n v e s t i g a t i o n o f t h e e c o n o m i c s o f u n c e r t a i n t y , a n d f o r Howard  (1964) and Keeney and R a i f f a  ( 1 9 7 6 ) i n t h e i r w o r k on d e c i s i o n  analysis. Expected u t i l i t y describing  theory,  though, has been l e s s s u c c e s s f u l i n  and e x p l a i n i n g a c t u a l c h o i c e s  F i s c h h o f f and L i c h t e n s t e i n , 1 9 7 7 ) . realized  at the outset  among m u t u a l l y and  E v e n v o n Neumann a n d M o r g e n s t e r n  that expected u t i l i t y  t h a t seem r e l a t i v e l y  M o r g e n s t e r n , 1 9 4 7 , A p p e n d i x A, S e c . 3 ) . challenges, beginning  utility  into question theory,  linked  (Allais,  various 1953), have  o f a key property  independence p r i n c i p l e  lottery using  when e a c h l o t t e r y  t h e same p r o b a b i l i t y .  t o t h e axiom o f q u a s i l i n e a r i t y  o f expected  (Marschak, 1950;  independence p r i n c i p l e r e q u i r e s  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  w i t h an i d e n t i c a l  Subsequently,  the empirical v a l i d i t y  The s t r o n g  f o r gambling per se,  common ( v o n Neumann a n d  with t h e A l l a i s paradox  the strong  Samuelson, 1952). ranking  r u l e s out complementarity  e x c l u s i v e consequences, a u t i l i t y  other behaviors  called  (Edwards, 1961; S l o v i c ,  This  that  i s composed i s closely  o f H a r d y , L i t t l e w o o d and P o l y a  (1934)  }  4  the  "substitution of lotteries"  and  the "sure-thing" p r i n c i p l e Part  I of this  o f P r a t t , R a i f f a and S c h l a i f e r (1964) o f Savage  (1954).  d i s s e r t a t i o n contains  o f two r e p r e s e n t a t i o n t h e o r e m s .  t h e s t a t e m e n t s and p r o o f s  The f i r s t  theorem, proved i n Chapter 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 H a r d y , L i t t l e w o o d a n d P o l y a b y weakening t h e i r axiom o f q u a s i l i n e a r i t y . the with  same m e a n s , 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 another d i s t r i b u t i o n  regardless  i n t h e same p r o p o r t i o n s  axiom by a l l o w i n g t h e p r o p o r t i o n s  same means t o b e d i f f e r e n t . that  This  gives  of these d i s t r i b u t i o n s  s h a r e t h e same mean,  o f t h e d i s t r i b u t i o n t h a t they mixed w i t h .  quasilinearity  M^,  G i v e n two d i s t r i b u t i o n s w i t h  We w e a k e n t h e  that  give r i s e  r i s e t o a more g e n e r a l  i s s p e c i f i e d b y 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  f u n c t i o n , a , and a c o n t i n o u s  mean h a s t h e I n t e r m e d i a t e  n e c e s s a r y and s u f f i c i e n t dominance  (negative)  degree ones) p a r t i a l  The  I n a d d i t i o n , we  Value Property,  conditions f o r consistency  (including higher  mean,  a n d s t r i c t l y m o n o t o n e f u n c t i o n , <J>.  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 . show t h a t t h e  to the  and p r o v i d e  with the stochastic  order.  We a l s o  generalize  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 m e a n s , b y t h e o b s e r v a t i o n the  mean o f a d i s t r i b u t i o n  distribution F  where F  that  F c a n b e w r i t t e n a s t h e q u a s i l i n e a r mean o f a  i s d e r i v e d from F v i a a a s t h e Radon-Nikodym  ct derivative As  of F  with respect  was n o t e d e a r l i e r ,  t o F.  the  d i s t r i b u t i o n s v i a the (expected  mean i n d u c e s an o r d e r i n g  among  u t i l i t y ) maximand,  / *dF. R  Correspondingly,  t h e M ^ mean i n d u c e s a more g e n e r a l  ordering v i a the  5  maximand, / a<f>dF// adF. R  We p r o v e ,  R  i n Chapter  2, a n 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 m e a s u r e s on a more g e n e r a l set  w h e r e mean v a l u e s may n o t be d e f i n e d .  stated directly We r e t a i n  s e v e r a l standard p r o p e r t i e s o f expected  The  of quasilinearity  i n the context  paradoxes i n t h e f i e l d  of decision theory.  a new t h e o r y o f c h o i c e  (called  utility  utility in  Chapter  3 gathers  f i n d i n g s which c o n t r a d i c t  the s u b s t i t u t i o n  i s essentially  i n P a r t I comes  theory)  two s e c t i o n s o f C h a p t e r  3 and 4) o f  t o g e t h e r t h e c r i t i c i s m s and the implications of  4.  expected  s t o c h a s t i c dominance, g l o b a l and l o c a l  of expected  (in particular, utility  theory)  relating  risk  implications (e.g. a v e r s i o n ) and d e s c r i p t i v e  to the descriptive  of alpha u t i l i t y  theory.  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 alternative  utility  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 , the d e r i v a t i o n o f the normative  implications  from  that generalizes  t h e o r y t o p a v e t h e way f o r t h e d e v e l o p m e n t o f a l p h a  the f i r s t  a  Hence, t h e f o r m u l a t i o n o f  theory c o n s t i t u t e s Part I I (Chapters  this dissertation. empirical  alpha u t i l i t y  namely  o f an o r d e r i n g .  motivation f o r the research contained  expected  utility,  and m o n o t o n i c i t y b u t r e l a x  o f P r a t t , R a i f f a and S c h l a i f e r , which  restatement  case, axioms a r e  i n terms o f p r o p e r t i e s o f the underlying o r d e r i n g .  weak o r d e r , s o l v a b i l i t y principle  In t h i s  outcome  t h e o r i e s t h a t have a t t r a c t e d  significant  The  inadequacy chapter  w i t h two interest.  PART I REPRESENTATION THEOREMS  7  1 GENERALIZING THE  1.1  QUASILINEAR MEAN OF  HARDY/  LITTLEWOOD AND  INTRODUCTION What i s mean v a l u e ?  Conventional  s e n t s , t y p i f i e s o r i n some way tribution. statistics  We  are rescued  t e x t s , by  wisdom t e l l s  from the a m b i g u i t y ,  examples.  Some f a m i l i a r  as  in  dis-  elementary  n o t i o n s o f mean v a l u e mean, h a r m o n i c mean  r o o t - m e a n - s q u a r e 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 of order r ) .  random v a r i a b l e  most a p p r o p r i a t e  'typical'  p e r c a p i t a income and o f whose w e a l t h In  (known a l s o as t h e g e n e r a l mean  O f 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  There are however s i t u a t i o n s  the  value:  n o t a b l y , the discrepancy  'typical'  i s i n t h e hands o f a  sufficient  tailored  to our  s e t o f axioms.  chapter,  satisfy  H a r d y , L i t t l e w o o d and the  needs s u b j e c t t o a n e c e s s a r y  axioms.  We  Polya  quasilinear  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  their  bulk  few.  t h e e x a m p l e s o f mean v a l u e s m e n t i o n e d a b o v e e x c e p t mode w h i c h do n o t  n o t be  between  income f o r a s o c i e t y , the  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  mean, c a n be  used.  f o r w h i c h t h e a r i t h m e t i c mean may  t h e i r p i o n e e r i n g work o f 1934,  showed how  and  cases a l l  f o r median  generalize, i n  and  this  t h e q u a s i l i n e a r mean v i a a w e a k e r s e t o f a x i o m s s t a t e d i n  s e c t i o n 2. for  us t h a t i t r e p r e -  measures the c e n t r a l tendency o f a  i n c l u d e m e d i a n , mode, a r i t h m e t i c mean, g e o m e t r i c and  POLVA  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  sufficient  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 t h e  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  the  8  value  i n s e c t i o n 4.  conclusion o f this  1.2  Applications will dissertation.  AXIOMS OF MEAN V A L U E L e t Dj  2  denote t h e space o f p r o b a b i l i t y  t h e i r mass c o n c e n t r a t e d not  be d i s c u s s e d i n P a r t I I a n d t h e  be b o u n d e d ) .  i n some i n t e r v a l  distributions  J o f the real  with a l l  l i n e R ( J need  We c o n s i d e r a f u n c t i o n a l M whose d o m a i n 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 b e a mean v a l u e ? candidate, motivated  by t h e mean-value theorems o f elementary  A natural calculus,  is: Property  1:  The s u p p o r t  Intermediate  of a distribution  such t h a t every the  Value  JM(F) e conv S u p p ( F ) , V F  Property  F, S u p p ( F ) ,  c o n t a i n i n g Supp(F).  pj .  c o n s i s t s o f each p o i n t x  open s e t c o n t a i n i n g x h a s p o s i t i v e mass,  smallest interval  e  Conv S u p p ( F ) i s  The i n t e r m e d i a t e v a l u e  property  r e q u i r e s t h a t t h e mean o f a d i s t r i b u t i o n b e 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 .  1. b e l o w i s a c o n s e q u e n c e o f t h e i n t e r m e d i a t e v a l u e  A* " 1 0  1  1  ' Consistency  with Certainty  M  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 of probability,  (  5  ) -  X,VJC  Axiom  property.  e J,  f u n c t i o n a t x w h i c h , i n terms  indicates obtaining x with probability  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  1.  i s g i v e n b y A x i o m 2.  _  I n t h i s s e c t i o n , t h e t e r m s a x i o m and p r o p e r t y a r e u s e d 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 fthey 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 t h e o r e m 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  A x i o m 2:  Betweenness  V F, Ge D j ,  i fM(F)< M(G)  t h e n V B e (0,1), M ( 3 F + ( l - B ) G ) e  It  (M(F),M(G)).  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:  Mixture-monotonicity  V F, G e D j ,  i f M ( F ) < M ( G ) , then  M(BF+(1-B)G)  Lemma 1.1: Proof:  Y  i f 1>B>Y*0.  A x i o m 2 <=» P r o p e r t y 2.  Omitted.  A distribution  G i s said  distribution  F i nthe f i r s t  g r e a t e r than  F pointwise.  at  < M( F+(1-Y)G)  some p o i n t , t h e n  t o s t o c h a s t i c a l l y dominate  another  degree, denoted b y G > F, i f G i s always n o t I f i n a d d i t i o n , G i ss t r i c t l y  l e s s than F  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 nthe first  l  d e g r e e , d e n o t e d b y G > F. an  appealing  order  partial  S t o c h a s t i c dominance o f t h e f i r s t  order.  Consistency  degree i s  o f mean v a l u e w i t h t h i s  partial  i s s t a t e d a s P r o p e r t y 3:  Property  3:  Monotonicity  The n e x t  axiom deals w i t h t h e e f f e c t  changes i n t h e c o m p o s i t i o n  A x i o m 3:  V F , G e D j , G > F => M ( G ) > M ( F ) .  o f the underlying  Weak S u b s t i t u t i o n  o n mean v a l u e  of certain  distribution.  V F , G £ D j , i fM(F) = M(G) thenVB  e (0,l)3y  M(BF+(1-6)H)  e (0,1) 9  = M( G+(1-Y)H) . Y  V H E DJ,  10  H a r d y , L i t t l e w o o d and  Polya  called i t quasilinearity.)  (1934) u s e d a s p e c i a l s t a t e d as  Property  case o f Axiom 3  4 below f o r t h e i r  (They  quasi-  l i n e a r mean.  Property  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,  then  He  fjj,  V 6 e  i f M(F)=M(G),  (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 ,  or the 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 mixtures with another  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  r e g a r d l e s s o f the  distribution  axiom a l l o w s m i x t u r e t o be  different.  The  Ratio Consistency The  w e a k e r v e r s i o n i s due  Consistency P r o p e r t y 5:  due  share  implicit  t o Weber, see  geometrical  M(F)  weak  substitution  (Chew, 1 9 7 9 ) ,  o f Weber, M y e r s o n , and interpretation of  section  S u p p o s e F, 3  The  G,  H e Dj  = M(G)  i  and  t M(H),  i  B^,  Proof:  A x i o m s 2 and  S u p p o s e t h a t M(H)  3 imply < M(F)  V "*! 1  Property = M(G)  i  B /l-B 2  =  Milgrom. Ratio  B  2 >  V  1 _ Y  Y^>  Y2  for  i=l,2.  2  2  •  5.  without  3.  e  and  i  Bj/l-Bj  Lemma 1.2:  the  3.1):  M(B F+(l-B )H)=M(Y G+(l-Y )H)  ^  is a  i n a s t r o n g e r statement o f Axiom  C h a p t e r 4,  Ratio Consistency  t h e same mean  called Ratio Consistency  to suggestions  interesting  distributions  t o t h e same mean v a l u e  3 ( I n an e a r l i e r p a p e r  p r o p e r t y was  F o r a d i s c u s s i o n o f an  that give r i s e  following property  c o n s e q u e n c e o f A x i o m s 2 and  o f these  t h a t they mixed w i t h .  proportions  quasilinearity  loss of generality.  (0»1)  11  Axiom 3 = > 3 f :  (0,1) -»• (0,1) g V  B e  (0,1),  M ( B F + ( 1 - B ) H ) = M(£(B)G+(1-£(B))H) ; Lemma 1.1  =*  f is a strictly  T h i s t o g e t h e r w i t h Axioms Therefore, Define  x  they  below t h a t T i s a constant  for  < B + <5 <  F u s i n g Axiom  1  exists.  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  by T(3) =  +  Note t h a t T i s continuous  0<B  3 and 4 i m p l i e s t h a t f "  are continuous  : (0,1) - R  Consider  increasing function.  f  ^g/ilg  ( 3 )  »  a.e,  (1.1)  and d i f f e r e n t i a b l e  a.e..  We  show  t o complete the proof. 1.  I t follows after  substituting  G  3:  M ( ( . « ) F * ( I - ( ^ ) ) H ) . M ( [ . t s^; i ,:i;:; ) . ( B t , i G. ( l l t , ) i ;^_;;i. ( B T T ) H) B  (1.2) But  (B+6)F+(1-(B+5))H =  Therefore  M  BF +  (1-g)  6 1-  L.H.S. o f ( 1 . 2 ) = M ( B F + ( 1 - B )  BMB)  Bx(B)+l-B  after substituting  G +  BT(B)+1-B  G f o r F u s i n g Axiom  A p p l y i n g t h e same a r g u m e n t expression  4 F  ( 1 . 4 ) , we  r  F +  &  l - B - 6 .. , „ H 1-B  _  - T B  +  F  " W  -  H  )  1 # H l-B  3.  f o r the remaining  obtain:  l-B-6 +  F-component i n  (1.3)  (1.4)  12  R.H.S.  6-r(6/e-r(e)+i-e)+eT(e)  =M  6x(S/3x(B)+l-3)+3x(B)+l-(B+6)  1-(3+6)  +  Comparing  6x(6/Bx(B)+l-3)+Bx(B)+l-(3+6) J  expressions  (B+6)T(B+6)  \ H  and e x p r e s s i o n  (1.2)  ( 1 . 5 ) ,  '  C"1  i t follows  = Bx(B) + 6x(6/Bx(B) + 1-6) .  Suppose w i t h o u t l o s s  of generality  that  5  1  that  (1.6)  x i s differentiable  a t 6.  Then Lim  (3+6)T(B^-BT(B)  o+U  o  =  t  (  8  ) + B  t  ,  (  b  )  Lim T 5 / C  =  ).  o+U (1.7)  T h e r e f o r e , t h e r i g h t hand Applying entiable,  limit  o f x a t 0 d e n o t e d by x ( 0 ) e x i s t s , +  t h e same a r g u m e n t 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 we  obtain  f g [ e x ( B ) ] = x ( 0 ) ' a.e. . +  Therefore  x(3)  = x(0 )  and h e n c e  x(B) = T ( 0 )  (1.8)  a.e.,  +  by  +  continuity. Q.E.D.  Finally, in  we r e q u i r e  o u r 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  t h e sense o f Axioms 4 and 5 .  Axiom  4:  Continuity —  I f {F } ,C D n n=l J  converges i n d i s t r i b u t i o n t o  T  6  F e D j and F has compact s u p p o r t , t h e n M(F)  = L i m 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  n  / fdF T  J  of  to F e D i f and o n l y i f J T  converges t o / , f d F V  n  f e C ( J ) , where C ( J ) i s t h e s p a c e  u  a l l bounded c o n t i n u o u s  functions  on J .  N o t e t h a t when J i s u n b o u n d e d , t h e a r i t h m e t i c mean o f F d o e s n o t 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 to C(J).  We i m p o s e t h e c o n d i t i o n  the function  x does n o t b e l o n g  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 f r o m o u r c l a s s o f mean The r e q u i r e m e n t o f C o n t i n u i t y mean o f a d i s t r i b u t i o n distribution  that  is equivalent  i s u s e f u l because i t t e l l s  us t h a t t h e  may b e a p p r o x i m a t e d b y t h e mean o f a  i s close  values.  different  t o i t . When J i s a c o m p a c t i n t e r v a l , A x i o m 4  t o c o n t i n u i t y o f t h e mean v a l u e ,  When J i s u n b o u n d e d , t h e f o l l o w i n g c o n d i t i o n t h e mean v a l u e o f a d i s t r i b u t i o n  M, w i t h  tells  respect  t o t h e I?'-norm.  u s how t o e s t i m a t e  F w i t h o u t compact s u p p o r t  ( i f i t exists)  b y i t s r e s t r i c t i o n t o a c o m p a c t i n t e r v a l , K, d e n o t e d b y F„. K  A x i o m 5:  Extension  L e t {^n^™! such t h a t  D  Lim  The mean v a l u e f o r a d i s t r i b u t i o n the  limit  "• K ^ = 1' F  f  a  increasing  n  f a m i l y o f compact  .= J , t h e n M ( F ) = L i m M ( F  o  r  a  n  y  increasing  K  F w i t h o u t compact s u p p o r t  o f t h e mean v a l u e s o f t h e s e q u e n c e o f t r u n c a t e d family,  n the  e  {K }°°  whose l i m i t  intervals  ),VFeDj. i s given  distributions,  i s J.'  Since  ~ {M(F ) } does n o t always c o n v e r g e , t h e i i n=0 n  s e q u e n c e o f mean v a l u e s ,  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 n e e d 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.  by  14  1.3  PvEPRESENTATION THEOREM We  begin  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  D°[A,B] denotes the  theorem. tions with  Theorem  with  a finite  1.1:  number o f  : D° [ A , B ]  Then M s a t i s f i e s and  of D[A,B] to simple  restriction  distribu-  discontinuities.  ( H a r d y , L i t t l e w o o d S, P o l y a )  Suppose 3M  if  representation  only  R.  A x i o m 1,  Property  3 and  Property  i f 3 cj> : [A,B]->-R, c o n t i n u o u s ,  such t h a t  M(F)  (J  = <J>  V FE  $ dF) ,  4,  strictly  monotone  [A,B].  D  (1.9)  A M o r e o v e r , i f 3 <f>* : [ A , B ]  such t h a t  M(F)  = <{>*  then V x e [ A , B ] ,  •+ R  ( f <j)* dF) , A  1  <J>*(X) = a<j>(x)  V  b>  +  F e D  f°  r  [A,B]  some a,b  with  a i  0. (1.10)  Proof:  (Omitted  In other and it  affine  most g e n e r a l  mean.  Since  a continuous,  transformation,  strictly  (1.10),  Polya  f o r the  is  theorem of Hardy,  extends t h e i r a n a l y s i s from D ° [ A , B ] T  (1.9).  We  shall  arbitrary J follows  to  an M^(F).  Littlewood  to D[A,B].  later.  call  completely  i t i s c o n v e n i e n t t o w r i t e i t as  and  to fJ  by  monotone  i n c r e a s i n g f u n c t i o n <(>, up  f o l l o w i n g theorem g e n e r a l i z e s the  extension  1.2).  consistent,  t h e q u a s i l i n e a r mean M(F)  The  and  certainty  functional of F i s that defined  quasilinear by  i t i s a s p e c i a l case o f Theorem  words, the  quasilinear  specified  since  A  further  15  Theorem  1.2:  S u p p o s e 3 M : D[A,B] -> R.  Then M s a t i s f i e s A x i o m 1, A x i o m 2, A x i o m 3 a n d A x i o m 4 if  a n d o n l y i f 3cp : [ A , B ] ->• R, c o n t i n u o u s , and  a : [ A , B ] -»• R ,  strictly  continuous,  +  monotone,  strictly  positive,  B B s u c h t h a t M ( F ) = <j> (f ctcp d F / J a dF) , V F e D [ A , B ] . A A _1  (1.11)  M o r e o v e r , i f 3<j>* : [ A , B ] -> R, a n d a * : [ A , B ] -»- R*, i  s u c h t h a t M ( F ) = <J>*  then  Vxe[A,B]  ,  B  B  (/ a*cf>*dF/J a * d F ) , V F e D[A,B] A A l ^ ^ ^ r l HU ^ k(<|>(x)-<f>(A)) (<J>(B)-<J)(x))  =  d-")  +  and  a*(x)  for  Proof:  =  some a , b , c , k  c a (x) {k (<j> ( x ) -<|> ( A ) )  + (<J> (B) -<|> ( x ) ) } ,  w i t h a,c i 0, k > 0 .  (Necessity)  Axiom 1 f o l l o w s  immediately.  Axiom 2 f o l l o w s from t h e o b s e r v a t i o n  that,  3 ( / , a dF)<J>fM(F)) + ( 1 - 6 ) (//a dG)4>(M(G)) n \ •e(/ A « dF) + ( 1 - B ) ( / a dG) B  <|>(M(BF+(1-B)-G))=  A  increases Consider Then,  strictly F,G,H  *(M(BF  +  i n 6 when  e D[A,B],  M(F)>M(G).  Suppose M(F) = M(G).  (1-B)H))= ^ l *  dF)KM(F)) + (1-g)(jfr B ( / a dF) + ( l - B ) ( J a B  B  A  <KM( G Y  W i t h  =  ^  d ¥ ) n  +  U  (l-Y)H))  a  dG)  '  V  3  £  ''  [A B]  dH)<j)(M(H)) dH)  (1.13)  16  H e n c e , A x i o m 3. Let  IF >  ^ c o n v e r g e t o F.  interval  U  B d  a  F  n^A  B l ^dF /J a A  n  B  a  d  F  Axiom  a  B dF  A  -1 =»  a r e c o n t i n u o u s on a compact  [A,B], B  -  S i n c e a,<f>,c)>  f  n  n  d  f  B  n  d  -1  B  "  r  ,B  B  (J^acj) d F / / a  d F ) •+ <j,  A  F  dF .  A  n  n  F  B B /a<|> dF /J^  -  (/ a* d F / J a  M ( F ) = cj>  B d  n  A  dF) = M(F) .  4 follows.  (Sufficiency) Define  ^ : [ 0 , l ] + [ A , B ] as f o l l o w s . Kp)  = M(S ), V P p  Axiom  1 =* ^CO) = A  Axiom  2 =* IJJ i s s t r i c t l y  Let  (Pn^-i  Axiom It  c  o  n  v  e  r  and  g  e  o  P-  strictly  increasing.  p n  ) = Lim * ( p ) . n  increasing  I f x = »Kp) > t h e n p = <j>(x) x  Lemma 1.2 i m p l i e s  the existence  depends on x s u c h t h a t , V H  M(36  x  +  and t h e r e -  a) : [ A . B ] -»- [ 0 , l ] w h i c h i s c o n t i n u o u s a n d  M ( 6 ) = x = K p ) = M(S ) = M ( S ~  and  (1.14)  Then Sp^ 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 S .  i> i s c o n t i n u o u s a n d s t r i c t l y  f o r e h a s an i n v e r s e  A  increasing.  p  that  = pSg + ( l - p ) 6 .  p  4^(1) = B.  4 =» K p ) = M ( S ) = L i m M ( S  follows  that  t  e [ 0 , 1 ] , where S  ( x )  ).  (1.15)  o f a s t r i c t l y p o s i t i v e constant e D[A,B],  ( l - B ) H ) = M C g ^ . g j S~  (  x  )  V 6 e ['0,l],  +  H) .  (1.16)  17  C o n s t r u c t a : (A,B) -> (0,°°) b y 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 a r g u m e n t 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) a n d then extends Consider  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 .  g ( x ) = H(h& x  +h&J A  J  = M( "C>0 ^a(x)+l  S  ,+ ? , i <|>(x) a ( x ) + l  2r  M(S { a ( x ) K x ) / 5 ( x ) l }  )  t x } "^ c o n v e r g e t o x e ( A , B ) .  b u t i o n t o %&  + %6 .  = m&  x  + %6 ) = Lim M ( % 6 A  Therefore, g i s continuous It Let  c  o  n  v  e  r  g  = MCS^) = L i m H(h6  +  Xn  = Lim g(x )  distri-  n  X n  +  h&p)  = Lim g (x ) . n  h& ) A  = Lim * ( $ ( x ) / ( 1 + 1/5(x ))) .  n  n  n  L i m a ( x ) = 1, s i n c e i ( B ) = 1.  S i m i l a r l y , we c a n show t h a t L i m ( * ) = 1 as x ^ c o n v e r g e s a  n  We e x t e n d  end-points.  i -  s  then,  n  above.  e  i n (A,B).  ( x } "^ c o n v e r g e t o B f r o m b e l o w ,  =>  ^A  i n (A,B).  f o l l o w s that a i s continuous  Uh)  + n  Axiom 4 i m p l i e s t h a t ,  A  g(x)  +  Then ^ ^ x  n  J  Ka(x)Kx)/a(x) l)  =  +  Let  )  A  A  to  A  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 t h e  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 a n d <f> s a t i s f y c o n d i t i o n ( 1 . 1 1 ) . Let  n { i ^ _ i x  from  be t h e s u p p o r t  of a distribution  ° F i n D [ A , B ] , and r  n represent F i n the form,  F = £ i<5 . » i=l e  x  1  w  n  e  r  e  6. = F ( x ^ ) - F ( x . ) .  M(F)  n = M( I 6 . 6 ) i=l i x  6  =  M (  o a(x )+ze. i  after  l°(xi) *(  s  1  substituting S ^  Repeating  "  x  X l  )  + i  9  £ e 5Cx )+Ee 2  1  1  5 j  x  ) i  •  ^ f o r 6 ^ u s i n g e x p r e s s i o n (.1.16).  (n-1) t i m e s on t h e r e m a i n i n g  = ^ ( E e . a ( x )<j>(x ) / E 6 . a ( x . ) ) :  F i n a l l y , we e x t e n d  j  3  our construction  S u p p o s e F E D [ A , B ] - D°[A,B].  6 ^, i=2,...,n,  yields,  X  ~ - l ,B . ,-B = <j, ( / acJ>dF// a d F ) . A A A  t oF e D[ ,B]. A  Construct the following  sequence  {F } °° i n D°[A,B], n=l n  V  FCAJ«  A  .  fiFCA*  By c o n s t r u c t i o n , F ( x )  I i ^ l „  F(x), V x e { A + ^ ~ I  n  which  F(A.  i s dense i n [A,B].  Therefore  ~1 B B = L i m <j> (/ 54 dF // 5 dF ) n-x=° n n V J  Y  A  —  = $  since  c> ]< ,> f  and  J  y  A  1 B B (/ S$ d F / / a d F ) , A A  a are continuous  A  +  on [A,B],  i  (  B  :i,ne I  {F^} °° c o n v e r g e s n=l  A x i o m 4 =» M ( F ) = L i m M(F )  Y  5  .  +  A  )  /  2  n  , i < 2 }  t o F.  n  19  (Uniqueness) S u p p o s e 3 a : [A.B-] -»• R a n d  cf> : [ A , B ] -»- R, t h a t  +  Then  x = M(6 ) = M(Sx  ( x )  satisfy condition  (1.11)  )  - a(B)c^(B)|(x) + a(A)^>(A)(l-$(x)) " * a(B)$(x) + a(A)(l-$(x)) ' 1  =  r  1  and  V B  U  '  1  /  J  (0,1),  e  *  J  1 L  B a ( x ) ( K x ) + (l-B)a(A)<|>(A) , Ba(x) + (l-B)a(A) J  = M(B6  =  M  x  +  (1-B)6 )  r B a ( x ) (l-B)  U l  +  *  1  1  A  <Kx)  +  Cl-B) , B a ( x ) + (1-B) ° -» A  B5(x)»(x)a(B)(l>(B) + (Ba(x)(l-»(X))-f(l-g))a(A)(|,(A) B 5 ( x ) $ ( x ) a ( B ) + (B5(x)(l-<|)(x)) + ( l - B ) ) a ( A ) ' J  (1.18) after applying Let  a and $ t o t h e e q u a l i t i e s  (1.15) and  (1.16).  a = <j>(B) - <j>(A), b = <KA), c = a (A) ,  and  k = ct(B)/ct(A).  It  i s straightforward  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) =  ca(x){k$(x)+(l-$(x))}.  S u p p o s e a * a n d 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 tion Then  that  satisfy  (1.11). (J>*(x) = - a * { k * * ( . x ) / ( k * $ ( x ) + Q-* ( x ) ) ) } + b* ,  condi-  20  and  a*(x)  = c*o(xHk*$(x) + (l-$(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 c h e c k t h a t , V x e [ A , B ] , 4>*(x) = a ' { k - ( H x ) - K A ) ) / ( k - ( < K x ) - K A ) M K ) - K x ) ) ) } + b '  ,  B  a*(x)  for  = c'a(x){k'(<Kx)-<KA))  a ' = (cf>*(B)-<}>*(A))/a b'  (<£ (B) -<J> ( x ) )  },  = a*/a,  = <j>*(A) = b*.  c' = a * ( A ) / { a ( A ) ( < H B ) - < j > ( A ) ) } = and  +  k' = { a * ( B ) a ( A ) / a * ( A ) a ( B ) }  c*/ca,  = k*/k.  Q.E.D.  Our g e n e r a l i z a t i o n completely this  o f t h e q u a s i l i n e a r mean, d e f i n e d b y (1.1-1), i s  s p e c i f i e d by a p a i r  o f f u n c t i o n s (ct,£>) .  i s t h e m o s t g e n e r a l mean f o r d i s t r i b u t i o n s  that s a t i s f i e s Continuity. by M ^ . class,  Consistency  with  Certainty,  In keeping w i t h precedent, The p a i r  t o T h e o r e m 1.2,  d e f i n e d on a c o m p a c t  Betweenness,  interval  Weak Substitution  we d e n o t e o u r g e n e r a l i z e d mean  o f f u n c t i o n s (a,<j)) d e n o t e s a p a r t i c u l a r member o f t h e  {a,<)>},, o f f u n c t i o n s t h a t y i e l d  a compact i n t e r v a l ,  such  t h e same mean o n D . T  a s [ A , B ] i n T h e o r e m 1.2, we c a n f o r m  i n g s u b c l a s s o f {a,<j)} , c a l l e d k-ratio T  [A,B]  According  = { (a,<j>)  e { a , tj)}  [A,B]  subclass,  f o r k > 0.  : a(B)/a(A) = k) .  When J i s the follow-  and  21  We  k ,k  d e n o t e by  element  ( a ,<f> ) a g e n e r i c  o f {a,<J>}  a (B)=k.  I t c a n be  K  that  [A,B]  element  k rk, (ci , cj> )  and  satisfies:  \  (A)=0,  shown, u s i n g e x p r e s s i o n s  of a k - r a t i o subclass  $(B)=1,  and  (1.12)  oanoniaal  the  [A,B]  a  (A)=l  for  k t h e (j> c o m p o n e n t and a s c a l a r t r a n s f o r m a t i o n  The  c l a s s {ct.cbjj-^  c a n be o b t a i n e d  that the  ( 1 . 1 3 ) ,  a r e r e l a t e d t o e a c h o t h e r v i a an a f f i n e f o r the a  k  and  transformation  component.  from i t s k - r a t i o subclasses  by  taking  t h e i r u n i o n over a l l p o s i t i v e k's. The  o f Theorem 1 . 2 a r e needed t o e x t e n d  following corollaries  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  our  o f {a,<f>} . r  L i, A  to  the i n t e r v a l  [A ,B ] n  1.1:  Corollary  Then  i s d e n o t e d by  n  U  A  < B  Q  Proof:  U, [ A J . B J ]  [A ,B ]  a|(B )  =  K  E  C  0  ! ? 0 1 '  which  k 0  o  K .o*"oBj  0 1 >  '  (1.19)  q^Bp)  (1 -  $*(B ))  ^ ( A Q )  (1  $ K A  (o^ ,^ ) 1  1  n  -  0  (1.20)  ) )  an e l e m e n t  ki ki (ai ,^! ) 1  of {ct^}.-^  1 n  _"l|(Bo)  "^P(A  -, v>\J  1  [A ,B ] 0  where  Q '  {a, aS}. H  LAi,  that  A  ^(Bp)  n  •01  Observe  .  D  L  a^lAoT FfAoT  01  D e n o t e by  r  B  < Bj.  Q  0  and  „ -.  r  L 1 > l-I  Let A j < A  ( a , aS}  where  {a,4)} .  U  0  )  i s a continuous,  &i  E  { a  0  ^ [A°,B ] }  0  (M1  1  1  _ (Bp) (Bp ) * ( 1 - j i . ( B ) ) ) " a l(A )(k $ l(A ) + (1- 1(A )))' 1  0  strictly  1  1  ,  1  n  0  $  1  0  elements  ( U  -'  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  i J  D  -i  22  with  domain  (0,°°) a n d r a n g e  (h  ,h  "01  )  01 Q.E.D.  Corollary  LetA  1.2:  D e n o t e b y ( c L1 ,I^ 1) 1  for  [A..B.1, i  bo2< 01 h  where  and  h i  subclass-  i=l,2.  and h  0  2  >h  0  (1.22)  ,  1  ^ ( A Q )  '  0  0  Construct  2  element o f t h e u n i t a r y  = g J ^ B o H l - ^iHBn)) d.UA )(l - $.UA )) '  0  Proof:  < Bi < B  0  i  a.^Ao)  0 1  < B  0  the canonical  1  the interval  Then  < Ai < A  2  the functions  for  i =  1,2,  £. . : (0,°°) -> ( h . . , h . . ) , f o r i < j , 3> -ij i j 1  via  k. = K. • ( k . ) i ] . i 3  a.HB.JCk.^.lCB.) + a^CA.JCk.^.UA.)  (1-ijL 1  (B.)))  (l-^.^A.)))  +  (1.23)  Note t h a t , by c o n s t r u c t i o n ,  k. { a  '* [A.,B.] 3 3 }  Note a l s o t h a t from  [A.,B.]  i s continuous,  strictly  i n c r e a s i n g , and onto  (0,°°) t o ( h . . ,h. .) . -1J i r  Suppose h  0  2  > h i . 0  P i c k kj_ e [ h , ° ° ) . 12  =* 3  [A.,B.]  k  2  e  Then C i o ^ i )  (0,°°) s u c h t h a t  t;  2 0  (k ) 2  < h  0  < h  1  = £, {k ) l0  1  0  2  .  .  23  B  u  t  { a  '* [A ,B ] }  2  [A ,B ]  2  0  0  [Ai.Bi]  [Ai.Bi]  [A ,  B ]  0  0  [A ,B ] 0  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.  A c c o r d i n g t o C o r o l l a r y 1.1, we have t o r e s t r i c t t h e k - r a t i o correspondQ  ing  t o the  (co '*!) ) 0  0  interval  [AO,BQ] t o  a  g  r  e  [AX,BI].  e  i n t e r v a l t o w i t h i n a range o f v a l u e s i f we want with ( c x i , ^ ) r e s t r i c t e d to 1  1  [AQ.BQ]  for a larger  C o r o l l a r y 1.2 t e l l s us t h a t t h e range o f p e r m i s s i b l e  k g - r a t i o ' s g e t s squeezed as we go from [ A i , B i ] t o a l a r g e r [A ,B ]. 2  2  Theorem  interval  NOW we extend Theorem 1.2 t o t h e case o f a r b i t r a r y i n t e r v a l J .  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 o n l y i f 3 <J> : J -»• R, c o n t i n u o u s , s t r i c t l y and  a : J -*• R , c o n t i n u o u s , s t r i c t l y +  such t h a t M ( F ) = f 3  1  monotone, positive,  (jja<|>dF/JpdF) , V F e D j .  Moreover, i f (a*,<j>*) 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  3  (1.24)  satisfies  The r a t i o fj a^dF/fj adF f o r F w i t h o u t compact s u p p o r t i s d e f i n e d by e x p r e s s i o n (1.25) .  24  condition  ( 1 . 2 4 ) , t h e n V i n t e r v a l [ A,B] C J , 3 , b , c , k a  a, c * 0 and k > 0 a V x e  with  Avrfx! = * 1  j  - 4>(A)) + (cf,(B) - <j>(x))  o f Axiom  1, A x i o m 2 , A x i o m  same a s i n p r o o f o f T h e o r e m 1.2.  ally  b  '  (Necessity)  Verification the  k(d)(x) k ( K x ) - cj) ( A ) )  = ca(x){k(<})(x) - cf>(A)) + (<j> (B) - <|>(x))}.  a*(x)  Proof:  a  d  [A,B] ,  from t h e d e f i n i t i o n  compact  (expression  3 , and A x i o m 4 i s  Axiom 5 f o l l o w s  trivi-  (1.25)) o f M(F) f o r F w i t h o u t  support.  (Sufficiency) I f J i s c o m p a c t , t h e n we a r e d o n e . j-  OO  {K } = { [ A ,B J} n=0 n=0 n  n  such  that  CO  {A } _Q n  be a s e q u e n c e o f i n t e r v a l s  n  00  Otherwise, l e t  CO  _  '-' n^ -o-' "" lB  n  :  n  s a  s  t  r  i-  c  t  ly  decreasing  (increasing)  sequence,  and L i m { K } = J . n  Corollary  { a  where  1.1 s a y s  i^i tA }  i 5  that,  B.]|[Ao,B ] 0  a.^Bo) ( h . ,h .) = ( • ' a.^Ao) 0 1  Corollary  0  1  1.2  {h ^}  00  Q  i=l decreasing.  says  =  k/lh  o i  ,h .) o  (l-^^CBo))  { a  O^O>[;  a^CBo)  0 ) B o  ]  $^(60)  , (l-^.'fAo))  f  ) a.^Ao)  $  i  l  (A ) 0  that,  i s strictly  increasing  a n d { h . } °° i s s t r i c t l y i=l 0  25  Let  C.  1  a n d D.  Then D  = [ ( h .+h . J/2 -01 -0 i + l = (h  i  , ( h .+h . J/2 ], Oi 0 l + l  1  .  0  ^ C. ^  + 1  , h  (h  o i  >  n o i  f o r i = 1,2,3,... .  )  ( h ^ j h ^ J , C^,D^ are s t r i c t l y  Observe t h a t  sets by i n c l u s i o n . therefore,  .  0  C \ i s compact  Since  Lim C . = C  L i m D. c C C L i m  L i m D. = L i m ( h „ . , h „ J .  Hence  L i m ( h . ,h\.) = C .  pick k Define  0  e C  Oi  0 0  on J t h a t s a t i s f i e s  (a,<j>) d e f i n e d M  (1.24),  condition  .  ( a ( x ) , < K x ) ) = (aj°(x) , <J»J°(x)) = (a^fx)  Ca^Cx)  such that  that,  (h..,h.J.  But  construct  f o r each i ,  i tfollows  C  To  sequences o f  t <t> .  Since Di . C . 5 ( h . ,h J V i 1 i l ^ - o i 01  -Oi  decreasing  ^iO^i-  1  =  k  o  forx e  0  forx e [A^BjJ  -  [A ,B„],  , ^ ( x ) )  f o r x e [A. ,B.]  -  [A.^.B.  = 5  0°( o)>  (see expression  B  = l = a5°(A ),  ^i(A )  = 0 =  0  0  , ^ ( x ) )  4i(A ) 0  [A ,B ],  0  $5°(A ), 0  0  (1.23)) ,  26  <f>*i(B ) = 1 = $ ^ 0 0  Observe t h a t agree a t A  0  (a ^ ! l+l 1  1  1  v  , d^it ) ' i + l -*  k • '•"i  1  [A.,B.]  v  E  k• ' ^i ^  1  1  s  i  n  c  e  t  h  e  y  and a t B . 0  Given any d i s t r i b u t i o n t h a t Supp(F) C Then  F w i t h compact s u p p o r t , p i c k  [A^,B^] s u c h  [A , B ] . i  M ( F ) = «j>i k  1  (J® ^ ^ i 1  any d i s t r i b u t i o n  F e Dj,  o b t a i n M(F) , i f i t e x i s t s ,  1  1  dF/J^a^dF) , i  d F / / t dF) .  = f\jja<i>  For  ( B o )  j C  i f Supp(F) i s n o t compact, then from Axiom 5 as  we  follows.  CO  Let  {K ) _Q n  limit  n  be a n i n c r e a s i n g  i s equal t o J .  T h e n A x i o m 5 =*  sequence o f compact  whose  D e n o t e b y F j ^ t h e r e s t r i c t i o n o f F t o K^,  M(F) = L i m M ( F  K  )  = L i m <J. (/ a * d F j ^ / J  a dF^).  _1  When t h e l i m i t  intervals  (1.25) e x i s t s  a n d d o e s n o t d e p e n d on t h e c h o i c e o f  the  sequence, IK i , i t i s d e n o t e d b y tj> n n=0  the  above  l i m i t does n o t always e x i s t .  a r i t h m e t i c mean f o r a c a u c h y  (1.25)  (/ a<J) d F / / a d F ) . <J J T  T  However,  An e x a m p l e i s g i v e n b y t h e  distribution.  (Uniqueness) This follows d i r e c t l y [A,BJ  f r o m a p p l y i n g T h e o r e m 1.2 t o a r b i t r a r y  intervals  in J.  Q.E.D.  27  We  h a v e 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 Betweenness,  functions  Weak  Substitution,  (a,<J>) .  Continuity  For distributions  corresponding necessary  t h e p r o p e r t i e s o f Consistency  line having  means do n o t n e c e s s a r i l y  and s u f f i c i e n t  Extension  and  without  with  Certainty with  a pair  compact s u p p o r t s ,  exist  (see (1.25)).  c o n d i t i o n t h a t ensures  3  of  their A  existence i s given  below.  Corollary  ^ j,C ) F  1.5:  a(  rjj  exists V F e  i f a n d o n l y i f e i t h e r <J> i s b o u n d e d  o r a»<|> i s b o u n d e d .  Proof:  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 To p r o v e  bounded. from  i s bounded Case i ) :  n e c e s s i t y , suppose f o r the p a i r  We may  above.  i s straightforward.  assume, w i t h o u t l o s i n g  T h e r e a r e two c a s e s . from  Then  generality,  As x t e n d s  above, o r i i ) a ( x ) t e n d s  Consider  M  (a,tj>), n e i t h e r <j> n o r a-cj) i s t h a t <j> i s n o t b o u n d e d  t o +«, e i t h e r  i ) a(x)  t o +«>.  9 a(x^)<j>(xp=2 •  a s e q u e n c e {x.  (2 T —r-5 ) = Lim — — acfr 1=1 i Xj/ m-**> 2 : a ( x . ) / 2 i=l ^ \  does n o t  converge.  2  1  J  oo Case i i ) :  Consider  L>^}^_J  a sequence  3  v  Then  1 M . ( 2 . , —r^5 0 otcj) 1=1 „i x± 2 J  v  y  X  m i=l '- i'' a  = Lim  1 = 2 .  <K ^)  m m-x» 2, .  X  ,„i a(x.)/2  r- d o e s n o t  converge.  28  A s i m i l a r argument e s t a b l i s h e s unbounded from  the result  f o r t h e c a s e when 4 i s  below.  Q.E.D.  The  above c o r o l l a r y  i s useful  mean v a l u e a s t h e c e r t a i n t y certainty  i n C h a p t e r 4 when we  interpret  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  e q u i v a l e n t s h o u l d always be  finite.  that  a  29  1.4  PROPERTIES OF THE M  MEAN  ad) ±  O f 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 ( P r o p e r t y 1) e n j o y s property.  Value  property  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  A f t e r a l l , e v e n m e a s u r e s s u c h a s m e d i a n a n d mode, w h i c h  are 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 .  conclusion that M the observation M  rn  a<r  has t h e i n t e r m e d i a t e value p r o p e r t y  The  f o l l o w s from  that, =  c  °  rj  *•  Mx)(<K*) - 4>(c))dF(x)  _  Ax(x)dF(x)  "  U  Hence, Corollary  1.4:  M  satisfies  Property  1 (Intermediate  Value  property)  l Consistency Monotonicity) The  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:  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 .  c o r o l l a r y below gives t h e c o n d i t i o n under which M .  , i s consistent o<J»  .  4  w i t h s t o c h a s t i c dominance (nonstrict) > . C o r o l l a r y 1.5: S u p p o s e a and <f> a r e b o t h b o u n d e d o n J . 1  1  1  Then V F,G e D j , if  and o n l y i f V s e J ,  i s a nondecreasing Proof:  F > G  We s h a l l  M  j,( ) F  a <  I  M  j> ^ ( G  a (  a ( x ) (<|>(x)-<f>(s))  (1-26)  function (nonincreasing function).  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  increasing.  The p a r t i a l o r d e r ' > 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 a b o v e 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 . 1  30  1 Suppose G > F.  (Sufficiency) where  F £(x;F)  Define where  ^(F  = (l-e)F+9G,  Q  V9  e (0,1).  = ( a ( x ) //jCidF}{d>(x)-Q(F) } ,  Q  )  ( f i a s i n p . 22) ( 1 . 2 8 )  = /^(x^^GCx)-F(x)) , = / (G(x)-F(x)dC(x;F ) Q  J  Since  (1.27)  = fj acf>dF//ja dF.  fi(F)  Then,  1 T h e n F Q ' > Fe w h e n e v e r 9' > 6 ,  (1.29)  > 0.  t h e i n t e g r a n d i s n o n n e g a t i v e a n d 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 { / ( G ( x ) - F ( x ) ) d ? ( x ; F ) } d e e  J  M (Necessity)  .(G) > M . ( F ) . S u p p o s e a ( x ) (<j)(x),-<f>(s*)) i s s t r i c t l y  some x * f o r some s * e i n t J . function,  i tis strictly  (x*-£,x*+£).  > o  decreasing at  S i n c e a ( x ) (<}>(x)-c|>(s)) i s a c o n t i n u o u s  d e c r e a s i n g f o r some o p e n  neighbourhood  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 a n y y* > s* a n d compute p * s u c h s* = M  (p*6 *+(l-p*)6 ") y  = p*<5y*  Compute,  /j(G*(x)-F*(x))d<;(x;F*)  But  M ^((l-e)F*+9G*) = M a  E  a<))  ( F * ) , w h e r e x " = x*-hE,.  ( 1 - p * ) 6 ^ , f o r some x ' e  G*  ^-^(F* )  = M  x  Consider  +  that,  [x*,x*+£).  = ( 1 - p * ) (5 (x» ; F * ) - c ( x " ; F * ) ) ' < 0.  ( F * ) i s n o n d e c r e a s i n g i n 9. Q  = / (F*(x)-G*(x))d (x;F* ) > J  ?  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 above i s n o n n e g a t i v e , w h i c h  e  as 0 approaches  i s a contradiction.  possible end-points o f J follows  0.  0  from  The e x t e n s i o n t o  from t h e c o n t i n u i t y and bounded-  n e s s o f a a n d <j>. Q.E.D.  31  . The f u n c t i o n linear  ? ( x ; F ) c a n be u s e d t o g e n e r a t e t h e f o l l o w i n g  functional, 5 * 0 ) = /jC(-;F)d(.) . F  Observe t h a t expression F  (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  i n t h e d i r e c t i o n G-F, w h i c h may b e w r i t t e n a s :  Q  dVfi(F) =  C - )G  e  p  e The as  f u n c t i o n a l C*p(*) and t h e f u n c t i o n t h e G a t e a u x d e r i v a t i v e o f 5. a t F.  r,(«;F) a r e b o t h r e f e r r e d t o We now i n t e r p r e t  condition  (1.26) as f o l l o w s . The  G a t e a u x 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  every F i n D . T  This  generalizes  the corresponding condition  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 G a t e a u x 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> which i s s t r i c t l y  increasing  Another u s e f u l p a r t i a l ' 1 ', d e f i n e d  i r r e s p e c t i v e o f F. order i s second degree s t o c h a s t i c  dominance  by,  2  G > F i f / x(G(y)-F(y))dy where J The  above  X  = { y e J  says  that  < 0, V x e J  and / ( G ( y ) - F ( y ) ) d y T  : y < x }.  = 0 (1.30)  . G dominates F i n t h e second degree i f they  h a v e 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 ) a n d 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 every x i n J . spread  (Rothschild  t h e principle welfare  This  of  i s equivalent'to § Stiglitz,  transfer  i s not diminished  the notion  o f mean  1970) i n u n c e r t a i n t y  (Dalton,  preserving  e c o n o m i c s , and  1920) w h i c h s t a t e s  that  a  society's  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  the poor.  Q u a s i l i n e a r mean M,  i s known t o be  consistent with  d e g r e e s t o c h a s t i c d o m i n a n c e when cj> i s i n c r e a s i n g and d e c r e a s i n g and ?(*;F)  convex.  Having  noted  the  concave  i s t h a t c(-;F) i s concave  (decreasing)  f o r every  F in Dj.  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 developed We  degree s t o c h a s t i c  e n t e r t a i n the conjecture that the corresponding  degree c o n d i t i o n f o r M increasing  or  s i m i l a r i t y b e t w e e n <j> and  i n deriving consistency conditions for f i r s t  d o m i n a n c e , we  i n the next  second  case  The  second  (convex) i f $ i s  v e r i f i c a t i o n of  o f a more g e n e r a l  this  result  paragraph.  begin w i t h the  o f kth  following definition  degree  stochastic  dominance,  G > F i f  / {/ J  d  z  J Z n  _ (-.-{/  n - i  1  )  =  0  j Z 3  {/  j Z 2  (G(z )-F(z ))dz }dz }dz3}---}dz . } 1  2  n  2  and  / J Z ] { / J Z ] _ { « • •{ as a b o v e c  1  When t h e n t h moment a b o u t t h e o r i g i n f o r n = l , . . . , k , then  1  *  f o r n = 2,. . . ,k, <  1  }v •} d z _ ) d z _ k  exists  2  k  l  } < 0,  for distributions  G dominates F i n the kth degree i f t h e i r  V z  k  F and  t h e k t h moment a b o u t t h e  origin  o f G t r u n c a t e d by J  Z  (greater) than  truncated  at J  (even) f o r e v e r y  Z  K  i f k i s odd  g i v e s c o n d i t i o n s on stochastic  Corollary  i s not  a and  less  z^ i n J .  that of F  The  d> f o r c o n s i s t e n c y o f M  following , with kth  G  nth  moments a g r e e f o r n = l , . . . k , and K  c J.  corollary degree  dominance.  1.6:  Suppose a ,  are continuous  and  a',  a " , - " ,  b o u n d e d on  J .  a ^ "  1  ^ ,  and  <J>,  dp',  <(>",•••,  <>f ^  1  '  33  T h e n V F,G e D ,  G > F =*  T  i f V F e D _,  if  and o n l y  is  a nondecreasing  .(G) > M .(F)  M r, ^  k - 1  ^ (x;F) function  i f <> f i s increasing  (nondecreasing) function  i f <f> i s i n c r e a s i n g  (nonincreasing)  ( d e c r e a s i n g ) when k i s o d d , o r is  a nonincreasing  ( d e c r e a s i n g ) when k i s e v e n .  Proof:  Assume w i t h o u t  is  k Suppose G > F .  where h  <£ i s i n c r e a s i n g  and k  even.  (Sufficiency)  T  loss o f generality that  e  k Then F Q , > F  F E (1-9)F+9G, He" CF )  n  f i  Q  = /  J ?  for 0 e  w h e n e v e r 6' > 6,  (0,1).  (x;F )d(G(x)-F(x)) 6  = (-l) /j{/ k  =  Q  J X  a s i n p. 3 2 } d z _ } d c ( x ; F )  {  k  1  E  (-l) / I (x)dc(x;F ) k  J  k  e  > 0, 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  where interval  J  (see expression  1.30), (k-1)  since creasing)  a n d c,  i s nonpositive  i s nondecreasing  (nonin-  f o r k odd (even) V F e D . T  u  It  follows  that,  =»•  M  (Necessity) the  This  fi(G)-fl(F) .(G)  > M  follows  = / J { / T ^ ( x ) d c ( x ; F ) }d0 > 0 J  e  .(F) .  from an argument t h a t  same a s t h e o n e u s e d i n t h e n e c e s s i t y  i sessentially  p r o o f o f C o r o l l a r y 1.5. Q.E. D.  leads  We e n d t h i s  s e c t i o n b y o f f e r i n g a l i n k between M  to a useful  condition  . a n d M.  that  u n d e r 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, c a n b e e x t e n d e d  toM  ,.  We d e r i v e f r o m a d i s t r i b u t i o n  through t h e f u n c t i o n a, another d i s t r i b u t i o n F (x)  = /  a  if  t h e denominator  j X  adF//, adF, u  exists.  In t h i s  t h e f o l l o w i n g manner:  M  F,  a F ,  f o r every x e J , case, M  (1.31)  , a n d M, a r e r e l a t e d i n a<j> cj)  ,(F) = M f F ) . acp cp a  This leads immediately t o : S u p p o s e M ( F ) > M, (F) V F e V C D .  Lemma 1.3: ct  Then i f F  e V whenever F does,  then One  T  X  M  art)  (F) > M (F) V F e V • = ctip  u s e 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 everv * =  F e D, ,, (o, )  s  00  w h e r e M ( F ) = M _ ( F ) = {/°x dF} ^ . r x 0 r  J  to M  a,r  (F) =  r  ,  c»  J  -p  ^1/  00  / a(x)x dF(x)// a(x)dF(x)} 0  / r  0  .  The f u n c t i o n a h a s t h e  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 a s a Radon-Nikodym tzve F F  ct  a  of F  w i t h r e s p e c t t o F.  as an ' i n t e g r a l '  We may, on t h e o t h e r h a n d , c o n s i d e r  o f F t h r o u g h t h e f u n c t i o n OL.  e v e n when / cxdF d o e s n o t e x i s t ? J T  does n o t have compact s u p p o r t  deriva-  C a n we d e f i n e  Our d e f i n i t i o n  of M ( F ) when F ad) v  ( e x p r e s s i o n (1.25)) suggests t h e  following. _  Let  limit f  00  (K } be an i n c r e a s i n g n=l n  i sJ .  Then -fjfdF  = Lim ^K^^  e C°(J) w h e r e C°(J) d e n o t e s  on J .  f a m i l y o f compact i n t e r v a l s d F / J j ^ a dF,  the space  whose  f o r every  o f continuous  functions  35  We h a v e d e f i n e d F  Cfc  so t h a t t h e e q u a l i t y ,  h o l d s e v e n when F d o e s n o t h a v e c o m p a c t s u p p o r t . f o r w a r d t o c h e c k t h a t Lemma 1.3  It i s straight-  holds f o r the extended d e f i n i t i o n  2  36  GENERALIZING T H E E X P E C T E D U T I L I T Y REPRESENTATION THEOREM  2.1  INTRODUCTION The p r e c e d i n g c h a p t e r  axiom o f mean,  quasilinearity.  M^,  represents  maximand, general  / <t>dF.  generalized A s we n o t e d  another  way t o  the in  ordering via  the  maximand,  the  for  the  chapter  case o f  than  the  more  general  getting directly of  real  line.  / a<f>dF/.L D  outcome s p a c e  results  terms  parallel  nonetheless  of  adaptations  1, most  relatively  of  the  definitions  the  the  induces  a more  representation  on a more g e n e r a l  e.g.,  the  may n o t  outcome  fired),  underlying  are  for  space c o n s i s t i n g  ordering.  The  the  a  of  axioms  developments  proofs  here  Chapter  1.  independently  of  given  space  be d e f i n e d  state  2 may b e r e a d  used here  outcome  we n e e d t o  Consequently,  1.  above  c o r r e s p o n d i n g ones i n  Chapter  via  adF.  o f mean v a l u e  the  Chapter of  i n c l u d e d so that  Unlike Chapter  2.2  those  notion  quo and b e i n g  of properties  quasilinear  expected u t i l i t y ,  extending  measures  (consider,  status  the  K  problem of  S i n c e the  a promotion, in  the  simple p r o b a b i l i t y  straightforward  are  treats  the  g e n e r a l i z e d mean, M ^ ,  K  This  mean b y w e a k e n i n g  introduction,  axiomatize  Correspondingly, our  R  quasilinear  explicitly  are  They  are  Chapter  because  they  unfamiliar.  PRELIMINARY DEFINITIONS Definition  2.1:  valued  A simple p r o b a b i l i t y  function  defined  1) . P(A) > 0, V A  C  on the  set  of  all  P(X)  3)  P(AuB) = P(A)+P(B) when A,  4)  P(A)  l;  = 1 for  some f i n i t e  B C X a n d A n B = <j>;  A c  X.  X is  a  s u b s e t s o f X such  X;  2)  =  m e a s u r e P on a s e t  1.  realthat:  37  A simple  p r o b a b i l i t y m e a s u r e P on a s e t X h a s 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 A c X,  and f o r a l l  I P ( x ) w h e r e P(.{x}) i s w r i t t e n a s P ( x ) . xSA  P(A) =  D e f i n i t i o n 2.2:  A p o i n t m a s s , 6 , a t x i s t h e spm w i t h  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  spm Q, BP(A) It  P ( x ) = 1.  X  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 + (l-g)Q(A)  f o r every A C  i s clear that  another  assigns  X.  gP + ( l - g ) Q i s a spm when P, Q a r e spm's.  In  n general,  £  6  i=l  - p  1  i  s  a  s  P  m  i f P-  1  i s a spm  for i =  n I 6 = 1 w i t h & > 0 f o r i = 1, 2, .. . , n . i=l I  let  1,  2,..., n  and  1  {x.} c X be t h e s e t o f p o i n t s i=l n  F o r a spm P on a s e t X,  f o r which P(x.) > 0 f o r  1  i = 1, 2,  n.  n £  I t i s easy t o check t h a t P =  1=1  P  i  = P ( x ) f o r i = 1, 2, i  D e f i n i t i o n 2.4: defined  P H  t o a spm on X i s d e f i n e d  n I p fi , E ( f , P ) i=l I  D e f i n i t i o n 2.5:  , where  E(f,P), of a real-valued function  E(f,P)  For  1  ...,n.  The e x p e c t a t i o n ,  on X r e l a t i v e  p.6  =  =  I xsX  by  f(x)P(x)  n I p.f(x) . i=l  A binary relation  -< on a s e t Y i s a weak o r d e r  if-< i s  asymmetric  ( i . e . V x , y e Y, x -< y =* n o t ( y ^ : x ) ) a n d n e g a t i v e l y  transitive  ( i . e . V x , y , z e Y, n o t ( x •< y ) n n o t ( y ^ z ) =*• n o t  (x-<z)).  f  38  We s u m m a r i z e  some p r o p e r t i e s o f a weak o r d e r , - < , - v i a t h e  S u p p o s e -< i s a weak o r d e r on Y .  Lemma 2.1:  Define binary relations  =< on Y b y x ~ y ** n o t ( x •< y ) n n o t ( y -< x ) , V x , y (x ~ y) , V x , y e Y .  x =< y *» ( x -c y ) u i)  ~ i s an e q u i v a l e n c e  ii)  =S i s c o n n e c t e d  iv)  Y  and  Then,  relation  ( i . e . Vx, y  Y,  €  (x  y)  ( x -< y )  n  ( y ~ z ) =* x -< z , a n d  (x ~ y )  n  ( y -< z ) =• x -< z , V x , y , z  e  u  (y  x)) .  Y.  (Omitted).  In  a preference  x-< y i s r e a d preference' called  2.3  e  ~ ,  =$ i s t r a n s i t i v e ,  iii)  Proof:  following.  c o n t e x t , -< i s c a l l e d  as 'y i s s t r i c t l y  and x  y i s read  'indifference'  'strict  p r e f e r e n c e ' and  p r e f e r r e d t o x'; =5 i s c a l l e d  'weak  a s 'x i s n o t p r e f e r r e d t o y'; ~ i s  and x ~ y i s r e a d  a s 'x i s i n d i f f e r e n t  t o y'.  AXIOMS The  following  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  o f spm's d e f i n e d on a s e t X .  Axiom  1:  Ordering  Axiom  2:  Solvability  -c i s a weak VP, Q, R e |_ =* 33 e 33P  A x i o m 3:  Monotonicity  order. P-< Q a n d Q -< R  (0,1)  + (l-g)R  ~  Q.  VP, Q £ L ^ , P -< Q =• BP + ( l - B ) Q -< y P + for  0  < Y <  g <  1.  (l-Y)Q  on |_^, t h e s e t  39  Axiom  4:  Weak I n d e p e n d e n c e  VP, Q 6  e (o,i) a  =» VB  L»  3 V R £  Axioms  P ~ Q  BP  x  1, 2, a n d 3 a r e s t a n d a r d  ( l - B ) R ~ YQ + ( 1 - Y ) R .  +  properties of a binary relation  be r e p r e s e n t e d b y t h e e x p e c t a t i o n o f a u t i l i t y our  only departure.  Axiom 4 reduces of expected The in  property i s a restatement  2.6:  (Ratio Consistency)  2  Yi,  1  Y  e 2  ^  €  (0,1) 3  for  i = 1, 2, t h e n  Proof:  The p r o o f  Yi  / 1-Yl  Y2 / 1-Y2  Bi  / 1-Bi  B  1, 3 a n d 4 => R a t i o  2  / 1-B  interpretation  2  Consistency.  o f Lemma 2.2 i s e s s e n t i a l l y  o f Lemma 1.2 i n C h a p t e r  the context  I f HP, Q, R  Q a n d B.P + ( l - B . ) R ~ Y.Q + ( l - Y . ) R i i i i  Axioms  identical to that  1.  o f o u r a x i o m s and t h e R a t i o C o n s i s t e n c y  o f choice w i l l  be d e f e r r e d u n t i l  t h e r e p r e s e n t a t i o n theorems o f t h i s  Chapter  property  o f Property 5 o f Chapter  P~  Lemma 2.2:  apply  then  o f a weak o r d e r , -< , on  and 8 i , B ,  in  Axiom 4 i s  a n d Y be i d e n t i c a l ,  that B  that can  utility.  the context  The  function.  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 , which i s another  following  Definition  I f we i n s i s t  (o,i)  G  Y  1 to d e c i s i o n theory.  Chapter  chapter  property  4 w h e r e we  and t h a t o f  40  2.4  REPRESENTATION THEOREMS To f a c i l i t a t e  Definition  2.7:  probability  t h e s t a t e m e n t o f o u r r e p r e s e n t a t i o n t h e o r e m , we h a v e  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 m e a s u r e s on a s e t X .  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 i sdefined by, Vx. y e X , x < y < * ii < ' ' x J  I f -< i s a weak o r d e r ,  J  5 . y  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 :< a n d ^ f r o m -< a s i n Lemma 2 . 1 .  Definition  2.8:  maximal  o n a s e t Y , a n e l e m e n t ye. Y i s a  ( m i n i m a l ) e l e m e n t i f Vx 6 Y , x =§ y ( y ^ x ) .  Theorem 2.1:  Let  a s e t X.  on  L e t -< be a weak o r d e r  be t h e s e t o f s i m p l e  : X ->- R  its  +  exist  defined  t h e induced functions  and a t t a i n s  supremum a n d i n f i m u m o v e r X a n d V P , Q e |_^,  <  0  ~ 4  E(av,P) E(ct.P)  <  E(gv,Q) E(a,Q)  •  ^'  1 J  and o n l y i f -< s a t i s f i e s A x i o m s  and  Then t h e r e  with  a n d v : X ->• R s u c h t h a t v i s n o n c o n s t a n t  r  if  measures  S u p p o s e -< i s a b i n a r y r e l a t i o n on  b i n a r y r e l a t i o n on X d e n o t e d b y -< . a  probability  X contains  1-4  a maximal element x and a minimal  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 then 3 a, b, c, k w i t h  a , c , k > 0 s u c h t h a t Vx £ X ,  theorem,  41  a*(x)  = ca(x){k[v(x)  - v ( x ) ] + [v(x) - v ( x ) ] }  and k[v(x) V  W  =  k[v(x)  a  - v(x)]  - v(x)] [v(X) - v ( x ) ]  +  +  b  -  Proof: Necessity: Let x, x e X  3  Vx e X o b s e r v e  v(x)  = i n f v and v(x) X€X  = sup v X X €  that v(x) < v(x) < v(x) 6  =$6  X  =S 6-  X  X  x =< x =5 x =*• x , x a r e m i n i m a l  and maximal e l e m e n t s 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 -< 6x x ** x Axiom 1 f o l l o w s  -< X .  immediately.  Axiom 2 f o l l o w s f r o m 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) E(a> * (l-B)Q) VP,  Q e L , x  =* ( » ) E(a,P) E  a v  p  5  . c  o  n  t  i  n  u  o  u  s  i  n  D  B-  P -< Q  ^E(av,QJ E(a,Q)  _ E(av,gP + (l-B)Q) E(a,3P + (l-B)Q) =* A x i o m 3.  . 1  , d  e  c  r  e  a  s  e  s  . • .. • « s t r i c t l y m B.  , (2.3) 0  42  Suppose  It  P, Q € L , a n d P ~ Q ~  f f ^ i E(a,P)  = ££™z?l E(a,Q)  f o l l o w s t h a t VR e 1 ^ a n d V B G ( 0 , 1 )  E(gy,BP + ( l - B ) R ) E(a,BP + ( l - B ) R )  w  h  e  r  e  =  E(av,yQ + ( l - y ) R ) E(a,yQ + ( l - y ) R )  c«»p)  v'u-y) =  E  6/(1-6)  E(a,Q) •  H e n c e , A x i o m 4.  Sufficiency: Let  x, x be m i n i m a l and maximal e l e m e n t s o f X , r e s p e c t i v e l y .  D e f i n e VP  [0,1],  e  S p By h y p o t h e s i s  = p 6 - + (1 - p ) 6 . * x ^ x  6^ -< 6-.  I t follows  from Axiom 3 t h a t  S -< S * » 0 < p < q < l . p q V x S X 3 6 -< 6 X  X  and 6  -< 6-, i t f o l l o w s X  X  3q e (0,1) 3 6 -  It  S  (2.4)  from Axiom 2 t h a t  4  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  function v : X  a real-valued  [0,1] i n t h e f o l l o w i n g  manner. v(x) Vx  e  = 0 , v ( x ) = 1,  X - {x,x} , v ( x ) = q  '  n  3 6 ~ S . x q  43  From c o n s t r u c t i o n , V x e X - { x , x } ,  6  Lemma 2.2 => 3 x  x  x ~ v(x)  > 0 3 VR G | ^ a n d B  B 6  x  ^  +  (2.5)  S  (0,1)  E  ~ BxTiTa S x)  R  frrfif  +  (  A  Construct a p o s i t i v e real-valued  ^  R  X  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 Vx e X - {x,x} , a ( x )  G i v e n a spm P E J p.6 . i x . i=l l 1  Applying  ( 2 . 6 ) t o x^,  x ,  x  2  sequentially, i t follows  n  that  f n I  1=1  P,a(x ) p  n  ~  6  S  p a(x,)  i  1  + z  p.  j=2  J  1  v ( x ) v l x  i  +  p  I  '  p.a(x.) +  i=2  j  1  1  -n P^Cxj+p-aCxj + z  p,  j=3  s.  ^  (  i  x  }  ^  P a(x ) 2  2  P a ( x ) + p a ( x ) +_.i 1  1  2  2  n  +  I i=3  L 3P j  P. 3=  _  p a(x ) + p a ( x ) + 1  1  2  7  V  z  9  z  i=3  6  p.  J  x  x  i  6  z J=  1  P a(x ) 1  S 2  P  . J  x  '  i  44  p.acx.)  n I L  — ?  i =i  —  p a (  x)  H e n c e , VP € L v ,  P ~  [ .I i j P W ^ ' l  *C*i> "  S  f o l l o w s f r o m Lemma 2.1  P -< Q ~  s  p.acx.)]  E(av,P)  E(a,P) It  n  S  1  •  ( i v ) t h a t VP,  E(av,P)  S  L^,  Q 6  E(av,Q)  E(a,P)  *  E(S,Q)  E(av,P) E(a,P)  <  E(5v,Q) E(a,Q)  Uniqueness: Suppose H a : x ^ R and  +  and v  : X  a t t a i n s i t s i n f i m u m and  and V P ,  Q e  R  such t h a t v i s nonconstant  supremum o v e r X  t v , y, r e s p e c t i v e l y ,  L , x  i Q ^ ' ^  r  E(av,P) E(a,P)  <  E(gy,Q) E(a,Q)  (y E x ) U -  v  (6 y  =*• v ( x ) = v ( y )  ~  6 ) and x  v /  (y = x) U ^  -  U  "  /  J  elements o f X -  C l e a r l y , y , y a r e m i n i m a l and m a x i m a l  By  a  v  (6- ~ 6-). y x  and v ( x ) = v ( y ) .  construction, V x € X - { x , x l , 6  x  ~S~. v(x)  (see r e l a t i o n  ,. _ v ( x ) a ( x ) v ( x ) + (1 - v ( x ) ) a ( x ) v ( x ) v(.xj = =_ v ( x ) a ( x ) + (1 - v ( x ) ) a ( x ) '  (2.5))  (2.8)  rYl  C2 a-) ' l /  y j  45  a f t e r applying (2.7) to (2.8). Also V f 6 ( 0 , 1 ) ,  3 6  x  Cl-B)«  +  BS(x) S + (1-3)6 g~(g + (i- ) -  x  B  Bct(x)v(x) + (1-B)a(x)v(x) Ba(x) + (l-B)a(x)  C2.10)  (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) a f t e r 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  v  and  (  r x  )  kv(x) — + b kv(x) + (1 - v(x))  -i =  i ~  a  r s  a(x) = cS(x)[kv(x) + (1 - v ( x ) ) ] .  (2.12)  Suppose a*, v* are another p a i r of functions that s a t i s f y the hypotheses of the theorem.  V  and  *W  Then  r!M C  = * ik*v(x) *~r + ( l -~ v( ( x^ ) ) a  +  b  *  a*(x) = c*fi(x)[k*v(x) + (1 - v ( x ) ) ] ,  (2.13)  46  a* = v * ( x ) - v * ( x ) , b* =  where  c* = «*(x)  Finally,  k* = ^  a*(x)  [  .  that,  k'[v(x)  = a'  w  for  and  i t i s s t r a i g h t f o r w a r d t o check  v*fx1  and  ,  v*(x),  - y(x)] = k'[v(x) - v(x)] + v(x) -  :  +  D  v(x)  - v ( x ) ] + v(x) - v(x)>  = c'a(x){k [v(x) 1  (2.14)  a' = ( v * ( x ) - v * ( x ) ) / a = a * / a b' = v * ( x ) = b*  c'  ^SE  =  a  a =  M  *(x) a(x)  ( 1 v(x) - v ( x ) a(x) a*(x)  =  J C  £* c  . 1 a  k^ k Q.E.D.  We  s h o w e d 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  monotonicity,  solvability  a p a i r of functions  the ordering,  a n d weak i n d e p e n d e n c e a x i o m s , i s c h a r a c t e r i z e d b y  (a,v) d e f i n e d  on X.  When a i s c o n s t a n t ,  a t i o n , E(ocv,P)/E(a,P), reduces t o the expected u t i l i t y  our  represent-  representation  Ct  E(v,P). the  I f we d e f i n e  a simple  p r o b a b i l i t y measure P  derived  from P i n  f o l l o w i n g manner, P (A) = E(cd ,P)/E(a,P) a  A  , VA  C  X,  w h e r e 1^ d e n o t e s 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 c a n s t a t e o u r r e p r e s e n t a t i o n  i n the a l t e r n a t i v e fashion  below;  ECv.P"). Our r e p r e s e n t a t i o n  i s then simply  the expectation  of the v-function  47  with respect  t o t h e measure P  ct  d e r i v e d from P v i a t h e a - f u n c t i o n which  i s t h e Radon-Nikodym d e r i v a t i v e o f P  with respect  a  r o l e o f a transparent by c o n s i d e r i n g a uniform  t o P.  We r e n d e r t h e  measure  N  P =  £ i=l  N  '  ^  e  c  o  r  r  e  s  P  o  n  d i g n  representation o f P i s given by,  i N £  E(ctv,P)/E(ct,P) = 1  =  1  ot(x.) ~  v(x.) .  N  .Z.a(x.) J =l 3 M  N  This  i s a weighted average o f ( v ( x ^ ) } ^ _ ^  with weights,  (ctCx^)}/^ .  The s t a t e m e n t o f T h e o r e m 2.1 r e q u i r e s t h e s e t X t o b e b o u n d e d b y a maximal and a m i n i m a l with  element.  The r e m a i n d e r o f t h i s  section deals  t h e e x t e n s i o n o'f T h e o r e m 2.1 t o t h e c a s e w h e r e X h a s n e i t h e r a  maximal n o r a minimal  element.  This p a r a l l e l s  t h e development  towards  t h e p r o o f o f T h e o r e m 1.3 i n C h a p t e r 1.  Definition  2.9:  interval  e  X, s u c h t h a t s  Definition  r  s , t £ X, a n  a maximal element x and a minimal  r e l a t i v e t o a weak o r d e r -<, t h e n 2  P°  [ s , t ] C X i s d e f i n e d by [ s , t ] = {x £ X : s < x , x =st}.  When X c o n t a i n s b o t h  t  on a s e t X-  L e t -< b e a weak o r d e r  2.10:  2  X = [x,x].  •< S j -< t j -< t  2  , then  element x  When s ^ , s , t ^ ,  [Sj.tj] ^  2  [s ,t ]. 2  2  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 represent  weak o r d e r - < o n L  Y  i f (<*,v) s a t i s f i e s  condition (2.1).  a  48  Definition l_x>  2.11:  class representing  {ct>v}x> c o n s i s t s o f a l l p a i r s  Definition  2.12:  k-ratio -< o n L|-  ^-j,  s  Definition  o f the uniqueness  2.13:  w  e  denote  element  by { a , v )  class {a,v)  c o n s i s t i n g o f those p a i r s  A generic  r  a weak o r d e r -< on  ( a , v ) t h a t r e p r e s e n t -< on  L e t s , t £ X 3 s -< t ,  subclass  aft") \ = k.  be  The u n i q u e n e s s  o f {a,v}  , , the [s, t j , representing  r  satisfy  i s denoted  r  r  ,tJ  (a,v) that  k  L^-  k k b y ( a ,v ) .  L e t s , t e X 9 s-< t , t h e p a i r ( c t * , v * ) i s s a i d t o  an ( a , b , c , k )  transformation  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)  and  = ca(x){k(v(x)  v*(x)=a—  k  k[v(x) We d e n o t e  such  - v(s)) + v(t) -v(x)}  K*) '  T a  j  b  j  C  )  , +b  - v(s)] + v(t) - v(x)  a transformation  (a*,v*) =  (  k  by  (<x,v)  on  [ s , t ] C X-  II k , = ^ + {a,v} ^, . L e t ( a , v ) b e an [s,t] kSR [ ,t] element o f { a , v } ,, we c a n t h e n g e n e r a t e a l l o t h e r e l e m e n t s [s, t j It  i s clear that  {ct,v}  r  r  s  r  v i a t h e (a,b,c,k)  transformation.  affine transformation  Note t h a t T  . . . i s an a, D , l , i on v and T, . . i s a positive scalar 1,1,c,l  m u l t i p l e o f a ; a n d t h a t we c a n u s e a u n i t a r y r a t i o p a i r (a*,v*) s { a , v ) r Ls,tj  t o generate  a l l elements  o f {a,v> , [s,tj r  49  since  T a  canonical  j  b  )  C  )  k  (  a  l  .  v  l  )  e  '  { a  k  e  d  e  n  o  t  y  e  I t i s clear that  ( S . v ) , the  b  k  k  k k v ( s ) = 0, v ( t ) = l ,  f  k  each k.  W  k , which s a t i s f i e s [s ,tJ  member o f { a , v }  = 1, a n d & ( t ) = k .  a (s)  ^s t ] '  v }  (a ,v ) k  I n g e n e r a l , a member a , v o f { c t , v } r  i s unique f o r  k  , i s uniquely  LSjtJ  specified by the values v(s),  o f a , v a t . s and t .  b = v(s), c = a(s), k = a(t)/a(s),  then Vx €  ca Cx) [k0 (x) + 1  a(x)  =  v(x)  = a  [s,t].  (1 - v ^ x ) ) ]  1  kv (x) 1  and  k\) (x) + 1 - v Cx) 1  C o r o l l a r y ' 2.1:  Let s ,  t  Q  { a , v}  ,  Q  - U  Q  j  oi  k  f  c  t *  0  /  1  ? 0  ( s  }  defined  0  k  0  r  Q  -k t  -k t , t h e n  , , where  0  (2.15)  }  1 - v  ^ ) (2.16)  1 - v(s ) Q  ] - [ ,t ] L  1 > t l  1  , i  .1 a (s ) Q  0  A; s  j  r ^ ^ tot,v}  v ( s  Q  [ s , t ] J[ s  S  a, , a,, o o  a  a\t ) -01  1  J  + b  J  e X 3  ^  t 1 ~ kefjz,  \s  LS ,t  Proof:  Let a = v(t) -  SQ  Q  /H s ^ t ^ ' Therefore,  o n [ s , t ] r e p r e s e n t s -<on L . L^^jt^J  t h e n a,v  i f  a,v  50  represents ^ o n L  -  Observe t h a t {a,v}  { a . v K  t i =  s  '  Q  U  0  1  > £  0  1  r  t v  s  rs  t 1  ^  =  I! U  (  { a  U  '  v }  _  0 -01* 01 l  C o r o l l a r y 2.1 t e l l s o f a weak o r d e r o n L  (oi.vK  r  l  ,  function  [s ,t ] 0  Q  ,•  {a»v}r° J  u  t o a weak o r d e r o n L r  ^  r  [  0  o r d e r i n g h a s n o t changed on L r L  V  (a ,v ) e {a,v} 1 so t h a t U U [S^tpJ r  7  us i t i s p o s s i b l e t o e x t e n d o u r r e p r e s e n t a t i o n  Q  n  (2-1 )  ^ t1  s  [s ,t ]  n  1 v^Sg))  Hence  ) -  =  w h e r e  i n c r e a s i n g and o n t o  k  { a , v}  on  ,  F s  ^ ' V  £ i s a continuous, s t r i c t l y to  v }  i k ^ ' C S g ) + (1 -  l  (0,oo)  { a  O'V  o — I a (s )  from  , c {a,v}  f  ko f  [Sj.tj]  L  Note t h a t  ,  -i • 0  ^ i f the S l  ,  I t also gives  t l  ]  conditions  .  J  (a ,v ) U U n  n  can be extended  to a  51  member  i.e. (a in  0 J  (ct^vj 6 » [ {a  v}  s  the extended p a i r v )  on [ s , t ] .  0  0  t  ]  SUCh t h a t  1  (a^,vj  defined  These c o n d i t i o n s  Q  terms o f a ( t ) , a ( s ) , v ( t ) , 1  1  1  Q  via  V V = (a ,v )  C  0  0  1  on [ s ^ , t ^ ] a g r e e s  with  ((2.15) and (2,16)) a r e g i v e n  v ( s ) ; and can be  obtained  1  0  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  proof o f  Theorem 2.1.  Corollary  2.2:  -< t j - < t on  Let s, Q  ,t j , t  i  f o r i = 1,2.  02 < 0l l  G  X  3 s  -k s  2  a  n  4; s  1  Q  ^ t  u n i t a r y a,v  Then,  d  *02 *01 >  (2.18)  £;  'i('o)  0 1  a?(t ) 0  i -  «|(t ) 0  and  Proof:  2  , a n d l e t ( *>^*) d e n o t e t h e c a n o n i c a l  l  where  Q  a  2  [s ,t ], i  s ^ s ^ t  for  i = 1,2.  From C o r o l l a r y 2.1,  {a,v} [ s , t ] rs t i 2  {a,v}  2  r  =  U  -  { a  ' r  t l'  v }  s  (a.v} ° U W r V ^ ° r  [ S  °'  t o ]  J  [  S  ,  t  o  1  '  52  k  and  {a,v}  [s ,t ] 2  where  '12  2  .1  and  2  r  l• s  t l  1  a^tp  l - v^ctp  a (s )  1 - v (s )  1  2  Construct the function  k. =  '  v }  v (s )  1  2  { a  ,  a ,  a (s J 2  -  U  1  £. . : (0,°°) -»- f £ . . , £ . . ) , f o r i < i , v i a J ,i - i j i j  C. . ( k . )  a^Ct.)  k.v^t.)  a*(s.)  k.v*(s.) + 1 - v*(s.)  J  3 3 i  i  +  I  -  v^(t.) 3  3 3 i  3 i  i  (2.19)  3 i  Note t h a t , by c o n s t r u c t i o n , k. {a,v}  [s. ,t.] 3 Note a l s o t h a t  {a,v}  3  3  [s.,t.] l i  to  £. . i s c o n t i n u o u s , s t r i c t l y 3 i  ( £ ^ , 1 ^ 3 .  Suppose l  Pick then  k  3 k  1  2  Q1  e  1  ?  >l .  Q2  0  (  e  k  [l ,co) 12  l  ) < l  ( 0 . - )  Q  1  <  3  ?  2  0  l . Q2  (k ) 2  =  3  [s.,t.]  J  J  (0,oo)  ^  e ( ) 10  kl  i n c r e a s i n g , and onto  from  53  ?  {ct,v}  i.e.  {a,v} [s ,t ] 2  2  io V (  [s ,t ] 0  Q  {ct,v}  But  [s ,t ] 2  {a, v}  [s ,t ] 2  {a,v}  2  [Spt^  2  5 (k ) 2 1  2  5 Ck ) = k 21  2  x  But 5 Ck- ) e (A 21  A similar  2  ,A  1 2  )  argument e s t a b l i s h e s £  Thus f a r , we h a v e c o n s i d e r e d  Q 1  < £Q  extending  2  f r o m an i n t e r v a l  Presumably x i s n o t bounded, o t h e r w i s e ,  larger interval.  of X to a  we w o u l d  have c o n s t r u c t e d  (ct,v) on X w i t h T h e o r e m 2.1 o n c e a n d f o r a l l . The  next i n t e r e s t i n g  c a s e t h e n i s when X c o n t a i n s n e i t h e r a m a x i m a l n o r a  minimal element, f o r example, t h e r e a l  line.  With a s t r u c t u r a l  t i o n o n X , we show i n T h e o r e m 2.2 t h a t e v e n i n t h i s sentation exists  Definition 00  a a-v r e p r e -  on L,.  2.14:  ly.}  case,  condi-  L e t •< b e a weak o r d e r  C Y i s cofinal  (coinitial)  i=l some p o s i t i v e  integer i .  on a s e t Y-  A sequence  i f V x € Y , x =$ y . ( y . =< x ) f o r i i w  54  T h e o r e m 2.2:  Let  b e t h e s e t o f spm d e f i n e d  binary relation  on l_x w i t h t h e i n d u c e d  d e n o t e d b y -< .  There e x i s t  such  that  (i)  v(x)  contains  (ii)  VP, Q e  L  W  if  + a  n  d  increasing cofinal  decreasing  an  <3 -< i s a on x  binary relation  f u n c t i o n s a : X~*"R  a strictly  and a s t r i c t l y  o n a s e t X>  coinitial  v  :  X  R  sequence  sequence, and  x  E(a,P)  E(a,Q)  '  and o n l y i f , -< s a t i s f i e s A x i o m s 1-5 X ordered  and  b y -< c o n t a i n s  a strictly  increasing  sequence and a s t r i c t l y  decreasing  coinitial  Moreover, i f a, v and a*, v* b o t h s a t i s f y Vs,  t e X 3 s  a*(x)  and and  v*fx) = a v (x) a  k  [  v  (  x  )  _  v  (  sequence.  ( i ) and ( i i ) , then  t , 3 a , b , c, k w i t h a, c, k > 0 3  = ca(x){k[v(x)  cofinal  Vx e  [s,t],  - v(s)] + [v(t) - v(x)]},  k[y(x) - y(s)] _  s  )  J  +  [  v  (  t  )  v  (  x  )  ]  b.  Proof: Necessity: Let  {d.}°° a n d {e.}°° c v(X) i=0 i=0 1  D  e  a  strictly  decreasing  coinitial  1  sequence and a s t r i c t l y  increasing cofinal  Pick s., t . e X 3 v ( s . ) i i i  = d . a n d v ( t . ) = e. i ^  sequence, r e s p e c t i v e l y . f o r i = 0,1,2, ... i i  55  It  follows that  {s.} i=l  1  coinitial  sequence  decreasing  1  (strictly  V e r i f i c a t i o n o f Axioms P r o o f o f Theorem  ({t.} ) i s a strictly i=l increasing cofinal  1-4 i s s t r a i g h t f o r w a r d  s e q u e n c e ) o f X-  (see  Necessity  2.1).  Sufficiency: oo  Let  {s.} 1  00  ( { t . } ) be a s t r i c t l y i=0 i=0  decreasing  coinitial  sequence  1  (strictly  increasing cofinal  loss o f generality that  s e q u e n c e ) o f X-  s ^ -< t ^ .  Suppose  without  I t i s easy t o check  that  oo  U  X=  [Sj.t ].  i=0 Let  -< on | r  (ex., v.) r e p r e s e n t  f o r i = 0,1,2, ... . i ' i  1  a l w a y s b e done b e c a u s e o f Theorem 2 . 1 .  Corollary  2.1 =*  =  [s.,t.]  a\{t where  )  _  1 - v*(t )  (* ,£ ) = ( oi  U  .  oi  a  (s )  1 - v (s ) 0  <vV o k  a it )  v ^ t )  1  , - f - ^ -• - i _ 2 - ) a (s ) v (s ) 0  C o r o l l a r y 2.2 => "1  0  is strictly  n  strictly  Let  i n c r e a s i n g and {£ .} i s i=i n  i=l  U 1  0  decreasing.  A. - f  0  00  -  0  {A - }  i +  ;°>  i + 1  ,  and B. = ( £ _ . . , £ „ . J l -0,i+l 0,1+1  £  Q  i  +  +  ,  1  I t can  56  Then  B. <~ A. c  Observe t h a t  tt ,i ) Qi  f o r i = 1,2,3, . . . .  Qi  tZ ^,l ), A , B Q  by i n c l u s i o n .  Q i  Since  ares t r i c t l y  i  A^ i s c o m p a c t  decreasing  sequences  f o r each i ,  n e s t e d i n t e r v a l t h e o r e m =* l i m A. = A^ i <j). i - > o o  Since  B  C A C (o j . ) V i 1 ^ l jt - O i , O i  => l i m B. C A C l i m (£.. , J L . ) . l oo . -Oi Oi But  l i m B. = l i m (£..,JL.) l . -Oi Oi  Hence  l i m (£..,£„.) = A -Oi Oi °  To c o n s t r u c t k  n  0  on x t h a t r e p r e s e n t s  (ct,v) d e f i n e d  -< on L ^ ,  pick  e A . 00  k  Define  (a(xj, (xj) v  o  k  o  = ( a (x) , v Q  (x))f o rx € [ s , t ].  Q  Q  l l = ( a (x), v (x))f o rx e t ^ ' ^ ] k  k  1  1  k. (a. ( ), 1  x  such  that  ?  i0 V (  k. v. (x))  forx G  1  =o k  k. and  = 1 = k.  k.  i^V  a  o <o s  3  ~o k  = 0 =  v  [  _  1  v  o < o>  v  o  = 1 =  s  <V  [s.,t.]  -  s 0  »t ], Q  [s _ ,t. i  1  57  Observe  k. . ( a .i +..l , l+l '  that  k. , v. i +, U ) i l+l  k. _  a g r e e a t s ^ and a t t  VP, Q e  B  ?  >  Q  €  [s.,t.]  pick  x  E(a  1  [s.,t.]  k. v  , P)  1  P •< Q ** Efa.  1  k. k. E f a . v. , Q) < L _ J : 1  , P)  E(av,P) < E(a,P) by  I  [s.,t.]' i i k.  then  „, they  .  Q  L ,  L  k. i . . , v. J s i n c e  I  = (a.  1  l  ,  ECa.  1  1  , Q)  EQv,Q) E(a,Q)  construction.  To c o m p l e t e t h e s u f f i c i e n c y  p r o o f , observe that  (v(s.)} 1  ({v(t.)} 1  (strictly  ) is a strictly 1  =  decreasing c o i n i t i a l  i=0  sequence  0  i n c r e a s i n g c o f i n a l sequence) o f v ( x ) -  Uniqueness: This  f o l l o w s d i r e c t l y from a p p l y i n g  intervals  T h e o r e m 2.1 t o a r b i t r a r y  [ s , t ] i n XQ.E.D.  58  PART 11 APPLICATION TO DECISION THEORY  BACKGROUND  A choice situation is  e x i s t s when more t h a n  a v a i l a b l e t o a d e c i s i o n maker.  have a v a l i d  theory  t h e t h e o r y c a n be  compatible  appeal  so  c h o i c e to conform to the theory's  has  been c o n s i d e r e d  researchers  an  example of such  People  3)  to suggest  tend  a key p r o p e r t y of expected  t h e i r choices after being  the  of  actual choices.  $ Tversky,  told  c a l l e d the  change utility  Raiffa  (1968)),  enough e m p i r i c a l  a v e r y good d e s c r i p the i m p l i c a t i o n s of  strong  independence  Many o f them w o u l d n o t  of t h e i r v i o l a t i o n s  of expected  e c o n o m i c s o f u n c e r t a i n t y and  decision theory,  i t has  mistakes  correction.  utility  change  (MacCrimmon,  i n the modeling  i t s application to  1968;  o f phenomena  statistical  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 A departure  the appearance of s e v e r a l recent papers K a r m a r k a r , 1978;  are  1975).  to the success  needing  Yet, there i s  to s y s t e m a t i c a l l y v i o l a t e utility  to  Expected  ( 1 9 6 5 ) and  that i t i s not  p r i n c i p l e or the s u b s t i t u t i o n axiom.  in  chosen.  a t h e o r y b e c a u s e f o r many  l a t t e r requirements.  ( c f . Chapter  t i v e theory.  Due  w i t h the  specifications.  ( e . g . Savage ( 1 9 5 4 ) , MacCrimmon  s a t i s f i e s the  Slovic  be  t h a t a d e c i s i o n maker i s w i l l i n g  his  evidence  that w i l l  for  i s normatively compelling i f the u n d e r l y i n g p o s t u l a t e s  of s u f f i c i e n t  it  of a c t i o n  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  choice situations, The  course  A theory of choice s p e c i f i e s ,  e a c h 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 We  one  Kahneman § T v e r s k y ,  a l t e r n a t i v e t h e o r i e s o f choice to account  from t h i s t r e n d i s evident  (Meginniss,  1979;  as  1977;  M a c h i n a , 1980) for Allais'  Handa,  1977;  proposing  'paradox'  in  and  60 other empirical findings that contradict the implications of  expected  utility. We d e v e l o p  i n Chapter  4 a new t h e o r y o f c h o i c e c a l l e d  theory which g e n e r a l i z e s expected set  o f a x i o m s t h a t weaken  Specifically, called  utility  via. a necessary  the corresponding  the substitution  Weak Independence.  alpha and  utility  sufficient  ones f o r e x p e c t e d  utility.  axiom i s r e p l a c e d by a weaker axiom  Given  two l o t t e r i e s  that are indifferent  e a c h o t h e r , Weak I n d e p e n d e n c e 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 p o s i n g each o f these  lotteries  with a third  lottery  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 the t h i r d  lottery.  (probability) property. unity.  The a x i o m s i m p l y t h a t t h e r a t i o  odds i s c o n s t a n t .  Expected  utility  We c a l l  this  r e s u l t s when t h i s  Our t h e o r y h a s d e s c r i p t i v e r e l e v a n c e  the usual responses  t o preserve  i n comindifference.  m u s t be i n d e p e n d e n t o f o f the mixture  t h e Ratio ratio  to  Consistency  i s identically  i n t h a t i t can represent  g i v e n t o t h e A l l a i s p a r a d o x and i s c o m p a t i b l e  with other reported empirical findings c o n t r a d i c t i n g the i m p l i c a t i o n s of expected  utility.  appealing p a r t i a l aversion.  Y e t , i t c a n be c o n s i s t e n t w i t h such  orders  as s t o c h a s t i c d o m i n a n c e and g l o b a l r i s k  As w i t h e x p e c t e d  utility,  the c o n s t r u c t i v e proof 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 procedure the  alpha u t i l i t y  can  be d e r i v e d .  normatively  f o r t h e assessment o f  f u n c t i o n s , from which 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  C R I T I Q U E OF EXPECTED  3.1  U T I L I T Y THEORY  INTRODUCTION Expected u t i l i t y  since  theory  has a t t r a c t e d c o n s i d e r a b l e  i t s r e v i v a l b y v o n Neumann a n d M o r g e n s t e r n i n t h e i r " T h e o r y o f  Games a n d E c o n o m i c B e h a v i o r " . economics o f u n c e r t a i n t y Diamond and R o t h s c h i l d , 1954;  Blackwell  I t serves as t h e foundation  1976) .  1978), s t a t i s t i c a l  for  Expected u t i l i t y  Kahneman  has been l e s s s u c c e s s f u l  providing  a strong  impetus  f u r t h e r t h e o r e t i c a l development.  f o r expected u t i l i t y .  b a s e d on a r e c e n t  paper  such a v i o l a t i o n extensive  studies  by t h e A l l a i s  losses  a summary,  and g a m b l i n g  The f i r s t  systematic  example o f  paradox, which i n s p i r e d  the concurrence of r i s k - a v e r t i n g  risk-seeking behavior evident  Markowitz  we p r o v i d e  that  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  insurance  First,  independence p r i n c i p l e .  i s provided  follow-up  the empirical findings  (Chew and M a c C r i m m o n , 1 9 7 9 b ) , o f t h e  v i o l a t i o n s of the strong  ing  though i n e x p l a i n -  (Edwards, 1954, 1961; MacCrimmon,  and T v e r s k y , 1 9 7 9 ) ; t h u s ,  pose d i f f i c u l t y  of  (Savage,  ( H o w a r d , 1964; K e e n e y and R a i f f a ,  I n 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  and  decision theory  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;  and d e s c r i b i n g a c t u a l c h o i c e s  1965;  f o rthe  ( A r r o w , 1 9 7 1 ; M a r s c h a k and R a d n e r , 1 9 7 2 ;  D e G r o o t , 1970) and d e c i s i o n a n a l y s i s  ing  attention  i n the prevalence o f the purchase  ( F r i e d m a n and Savage, 1 9 4 8 ) .  (1952) n o t e d p r e v a l e n t among h i s s u b j e c t s .  risk-proneness This  For instance,  for lotteries  observation  involv-  i s a l s o noted by  /  62  Kahneman and T v e r s k y example. utility  F i g u r e 3.1 d i s p l a y s  a typical  their probabilistic  von  correspond  3.1:  d i s c u s s e d above.  to risk-proneness  averting/  The " c o n v e x " ( " c o n c a v e " ) r e g i o n s  (risk-aversion).  A " t y p i c a l " von Neumann-Morgenstern  u(x)  insurance  Neumann-Morgenstern  f u n c t i o n which i s used to account f o r the j o i n t r i s k  seeking behavior  Fig.  (1979), p a r t i c u l a r l y  utility  function  63  S i n c e t h e von Neumann-Morgenstern u t i l i t y and  c o n c a v e a t t h e same t i m e , e x p e c t e d  o f r i s k p r o n e n e s s and r i s k levels.  Whether t h i s  investigated,  equivalent,  f o rthe e l i c i t a t i o n  Different  o f v o n Neumann-  e.g., t h e c e r t a i n t y e q u i v a l e n t , t h e g a i n different  curves  risk-propensities.  A d i f f i c u l t y w i t h expected unexplored  indirect  1977) t o t h e c o n t r a r y .  and t h e c h a i n i n g method, t e n d t o y i e l d  w i t h opposing  concurrence  i s an e m p i r i c a l q u e s t i o n  There i s however  1972; A l l a i s ,  equivalent procedures  Morgenstern u t i l i t y ,  r u l e s out  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  (MacCrimmon, e t . a l .  theoretically  utility  i s a c t u a l l y t h e case  t h a t h a s y e t t o be f u l l y evidence  f u n c t i o n c a n n o t be convex  utility,  one t h a t t o u c h e s  a r e a o f p r o 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 l a r g e l y on t h e  d e c i s i o n m a k e r ' 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 d o m a i n on which a u t i l i t y the  wealth  levels,  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 F r i e d m a n and Savage  (1948),  Pratt  function i s defined.  (1964) and p r a c t i c a l l y  uncertainty, wealth  a l l the l i t e r a t u r e  level? Markowitz  Another d i f f i c u l t y  (1952) and o t h e r s have o b s e r v e d  independent o f the current wealth  i s related  that, p r e f e r e n c e s  to the finding  "customary"  that preferlevels.  (Kahneman a n d  Tversky,  among t w o - s t a g e l o t t e r i e s may d e p e n d on w h e t h e r  t h 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 lent  on t h e e c o n o m i c s o f  o r c h a n g e s 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  ences a r e r e l a t i v e l y  1979)  S h o u l d i t be f i n a l  lotteries  i n their  simple  equiva-  forms. T h i s c h a p t e r e x p a n d s on t h e i s s u e s i n t r o d u c e d a b o v e  d u p l i c a t i n g unduly  the contents  o f other c r i t i q u e s  without  already  cited.  64  We s h a l l  explore the descriptive  of expected u t i l i t y considered  3.2  implications o f our generalization  theory i n the next chapter  i n light  o f t h e examples  here.  SYSTEMATIC VIOLATION OF THE STRONG INDEPENDENCE P R I N C I P L E  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 , the  consider  f o u r d e c i s i o n p r o b l e m s g i v e n i n F i g u r e 3.2.  A : Q  BQ:  $1,000,000  V  f o r sure  10/11 chance o f  $5,000,000  B  1/11 c h a n c e o f $ 0 ,  A  L  :  B :  chance  89/100  chance o f  $0  chance  $5,000,000  10/100  L  90/100  of  chance o f  :  chance  of  $5,000,000  11/100  chance  of  $1,000,000  99/100  chance  of  $5,000,000  1/100  11/100  of  H  89/100  $1,000,000  $0  V  $i,000,000 f o r sure  V  10/100  chance  of  $5,000,000  89/100  chance  of  $1,000,000  1/100  F i g u r e 3.2:  $0  chance o f  $0  chance o f  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  hypothesis, the only permissible  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 , A, , A T  n  0  or B , 'H U  J  B , T  B , T  B . 'oN  If  h a v e c h o s e n A , A , B. H I L and A , w h i c h i s n o t 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  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  u  n  utility.  The c h o i c e o f A j a n d 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. by  the  A  l e s s e r known p a r a d o x , t h e A l l a i s  c h o i c e o f B.  and  L  has  i n Figure 3.2,  lotteries, are  The  violating  choice of A  U  not been s t u d i e d .  ture  A,..  r a t i o paradox, i s given  The  n  i n s i g h t one  r a t h e r than  gains  from the  i s t h a t the v i o l a t i n g p a i r s ( A j , B ),  a l l d e r i v a t i v e s of the b a s i c v i o l a t i o n , A , U  A  r V  B,,  1  L  T  It  f e a t u r e s o f the  i s b a s e d on  d e n o t e d by  L,  s t r u c t u r e i n F i g u r e 3.2  t h r e e consequences $0,' $ 1 , 0 0 0 , 0 0 0 I , and  H respectively.  The  (B  A X  stands  f o r one  AQ(BQ)  a l t e r n a t i v e by  89/100.  This  of the  c o n s e q u e n c e s , L, composition  is illustrated  I , H,  A„  $5,000,000,  a l t e r n a t i v e , where  i s obtained  3.3:  The  composition  lottery  from  o f the A  AQ(BQ)  T  Li  from  (B  L  )  the  probability  case x =  -• I  Figure  versus'  X  f o r the  B^)  are worth n o t i n g .  w i t h consequence x at  i n F i g u r e 3.3  (A^,  U  and  )  binary  A ) and  A ,  V  Several  struc-  (B^,  L  T  ,L  simply considering separate  n  V  B  and  u  L.  x  66  Since A  (B ) i n F i g u r e 3 . 3 h a s t h e same f i n a l  as A ^ ( B j J  i n Figure 3 . 2 , these  Although  it  seems r e a s o n a b l e  sequence v a l u e s  and $5,000,000  t o expect  o f expected  levels.  s p e c t o f t h e i n t e r m e d i a t e consequence I.  A^(B^)  alternative  B^ o f f e r s  The  HILO  .89,  f o r other  l e a d s t o a more  in Fig. 3.4.  i s obtained from the A^(BQ)  3.4:  This  utility  con-  general  A^ i s a s u r e  pro-  a q c h a n c e a t t h e most  l - q chance a t t h e l e a s t p r e f e r r e d outcome  w i t h the x-consequence at p r o b a b i l i t y  Figure  the conse-  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  violations  and o t h e r p r o b a b i l i t y  outcome H and a  probabilities  we h a v e o n l y c o n s i d e r e d  structure of d e c i s i o n problems, i l l u s t r a t e d  preferred  and  l o t t e r i e s are equivalent.  f o r i l l u s t r a t i v e purposes,  quences $0, $1,000,000  outcomes  a l t e r n a t i v e l y by  L.  The  composing  g.  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 " ) given  i n t h e b o x e s o f F i g u r e 3.4, e x h a u s t t h e p o s s i b l e  from t h e b a s i c problem will  be c a l l e d Expected  choices  (denoted  a s "0") .  utility  i n this  theory  lottery  each a l t e r n a t i v e  F o r ease o f r e f e r e n c e  i m p o s e s some s e v e r e  structure.  of A  g  x  this  i s how t h e  i timplies that the choice  one  o f the cases another  o f x a n d B.  case  principle  a t t h e same  g  and  are gener-  o f a l t e r n a t i v e A^ e n t a i l s  the choice g  B_ e n t a i l s t h e c h o i c e O x  of B . '  H e n c e , 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  o f t h e HILO s t r u c t u r e a n d a c h o i c e o f a B a l t e r n a t i v e (such  as t h e s t a n d a r d  Allais  choice of A , B ) v i o X  l a t e s expected  when  alternatives  while the choice o f alternative  a l l values  in  this  on t h e  b e t w e e n two a l t e r n a t i v e s b e p r e s e r v e d  i s c o m p o s e d w i t h a common a l t e r n a t i v e  Since  for  restrictions  The s t r o n g i n d e p e n d e n c e  g  probability.  compositions  t h e "HILO" l o t t e r y s t r u c t u r e .  requires that preference  ated,  and L ( f o r " l o w " ) ,  utility.  In a d d i t i o n t o these  violations  LJ  across  Bl 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 e a c h p r o b l e m s i n c e A^ B  may b e  2  c h o s e n f o r some p a r t i c u l a r some d i f f e r e n t Although  level  level  appeared  2  empirical results  i n the l i t e r a t u r e ) ,  p a r t i c u l a r cases. empirical this  may b e c h o s e n f o r  B -  c a s e and w i t h i n - c a s e c o m b i n a t i o n s not  8^ w h i l e B^  Table  are not a v a i l a b l e f o r a l l the across(since this  lottery  there are results  f o r some o f t h e  3.1 s u m m a r i z e s t h e c h o i c e s  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 has  from t h e main  structure.  From  t a b l e i t s h o u l d b e a p p a r e n t t h a t most o f t h e e f f o r t h a s b e e n  devoted t o studying various versions o f the standard  Allais  paradox,  H-H  Case:  06 S , H  (8  .8' A  0-H  H  Case:  L-L  B„. A„  Case:  (6 >  > 8')  p+ o r>  A^,  0-L  Case:  A  Q  ,  I-L  3  33  i  2  2 1 7T  Aj,  Standard A l l a i s  8')  1— at  O 3  Case:  o»  o> o -S o  I  .2  Paradox  *  Q»  */» -»  —<  "1 -*.  (/» -*•  § 1  o - 3 u i a>  -n m  1  (T>  o o  OU1  O  r o —•  Ul  ui ui  . .  U l UJ  co  U N« M—• ro —  <n 3  Ul 3 11 CO II —' S< —'  89  O  O Ul Ul o  *»• -C* r o  O  tJl  —' ro  ^  0 Tl  T  3 -J a* ro  — —*  T  —•  Ul  CO U l  O  T  ptcy  Ul  TTTT  ni 1  ptcy  o  CO 03 o» a» 3 3  urn  o> o.  —«  O  T  ifl  U3  1  SS  1  I  69  the  I-L c a s e t  Note t h a t w h i l e t h e frequency  6  of choosing the v i o l a t i n g  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 over q u i t e d i f f e r e n t ing  increasing  levels  seems  robust  o f c o n s e q u e n c e s and p r o b a b i l i t i e s .  a t t e n t i o n r e c e n t l y h a s b e e n t h e L-L p a t t e r n , A^  (including the special  case  B^  2  Li  have c o n s i d e r e d s e v e r a l cases  eously are those  o f MacCrimmon and L a r s s o n  Tversky  The f o r m e r  (1979) .  ,  0-L, o f c h o i c e s A , B ) . U  The o n l y s t u d i e s w h i c h  Receiv-  study  simultan-  ( 1 9 7 5 ) a n d Kahneman and  i n t r o d u c e d t h e 0-H a n d t h e H-H  i n t h e c o n t e x t o f n e g a t i v e o u t c o m e s and i s t h e o n l y a t t e m p t  cases  t o map  p a t t e r n s o f p r e f e r e n c e s b e t w e e n A^ a n d B^ f o r v a r i o u s l e v e l s o f 3 (including  the s p e c i a l  "0" case  o f 6 = 1.0) a n d v a r i o u s l e v e l s  of the  i n t e r m e d i a t e outcome, I . It choices  seems c l e a r  from Table  f a r conducted  f o r g a i n s , and t h e H-0  have covered  a n d H-H  "I-L",  "H-I",  t h e I - L , L - 0 , and L-L c a s e s ,  cases, f o r losses.  structure, there are p o t e n t i a l l y  patterns, and  "1-0", "L-0",  The c a s e s  "H-I",  To o b t a i n a more c o m p l e t e  With  t h e HILO  6 distinct binary  violation  "H-0" a n d "H-L", a c r o s s  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 "  problems.  of actual  f o r t h e d e c i s i o n p r o b l e m s i n F i g u r e 3.4 i s i n c o m p l e t e .  Studies thus  lottery  3.1 t h a t o u r u n d e r s t a n d i n g  "1-0", " I - I "  problems  and " L - L " w i t h i n  and "H-L" r e m a i n  unexplored.  p i c t u r e , a systematic study o f choices  r e l a t e d t o F i g u r e 3.4 i s n e e d e d .  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 the 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 the L case.  70  3.3  CONCURRENCE OF R I S K 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 corresponding t o t h e purchase  of lottery  function  t i c k e t s , purchase  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 losses. and  T h i s has been g i v e n adequate  Savage, 1948; M a r k o w i t z ,  (see e.g. F i g . 3 . 1 )  coverage  of insur-  involving  elsewhere  1 9 5 2 ; Kahneman a n d T v e r s k y ,  (Friedman 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 proneness any  and r i s k a v e r s i o n w i t h i n  e x p l a n a t i o n based  utility  t h e same r e g i o n ; t h u s n e g a t i n g  on m o d i f i c a t i o n s  o f t h e von Neumann-Morgenstern  function.  S e v e r a l measurement p r o c e d u r e s von' N e u m a n n - M o r g e n s t e r n u t i l i t y son o f t h e s o r t  to elicit  a decision  f u n c t i o n a r e based  maker's  on l o t t e r y  compari-  g i v e n i n F i g u r e 3.5.  F i g u r e 3.5:  Standard  lottery  comparison  L o t t e r y A i s a s u r e c o n s e q u e n c e o f X , a n amount g r e a t e r t h a n X^ b u t less than X . of  getting X .  L o t t e r y B i s a 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  a n d a 1-p c h a n c e B a t X® a n d X^  71  respectively  but allow X  (X ,p) such t h a t c  and p t o v a r y , then t h e p a i r s  £  lottery A i sindifferent  to lottery B define a  w h i c h we d e n o t e b y u ( X ) , i . e . l o t t e r y A i s i n d i f f e r e n t c  w h e n e v e r p = u (X ) .  We d e n o t e  an a f f i n e  o f numbers  to lottery B  t r a n s f o r m a t i o n o f u (X) b y  u ( X ) , i . e . u ( X ) = a u ( X ) + b , f o r some a , b w i t h a > 0. c  the  c  decision  c  maker  i s an e x p e c t e d u t i l i t y  Morgenstern u t i l i t y  function  A indifferent  to B implies  that  = G (X )u(X°) + ( l - u ( X ) ) u ( x J )  u(X )  = [u(x°)-uCxJ)]a CX D  which  i s an a f f i n e  c  c  c  c  c  Therefore, the function  Certainty  Then  c  + u(X°J  procedure i susually  t o B.  (X , p (X ) ) s u c h t h a t A  Applying expected u t i l i t y  a g a i n , we o b t a i n  relation. 6  u(X?)  Since u(X) i sr e l a t e d p^ * ( X ) i s a l s o  known a s t h e  f i x t h e s u r e amount i n A a t X^ a n d t h e l o s s c  u(X°) = p (X ) u ( X ) + ( 1 - p (XJ)u(X°J L  utility  method.  indifferent  following  (3.1)  c  amount i n B a t X^ a n d o b t a i n t h e p a i r s  the  ,  u ( X ) i s a von Neumann-Morgenstern  We c a n a l t e r n a t i v e l y  remains  , or alternatively  t r a n s f o r m a t i o n o f u (X ) . c c  The a b o v e m e a s u r e m e n t Equivalent  c  Suppose  m a x i m i z e r w i t h v o n Neumann-  u(X ) c  function.  u(X).  function  6  6  6  6  , or alternatively  *>  - u(X°)  t o p * ( x ) t h r o u g h an a f f i n e  a v o n Neumann-Morgenstern u t i l i t y  transformation, function;  and t h i s  72  e l i c i t a t i o n procedure S i m i l a r l y , we using  t h e Loss  pairs  (X  ,p  otherwise utility  i s called  can  determine  Equivalent  Fig.  is  a  v o n  -  3.5  by h o l d i n g c o n s t a n t  the p a i r s  I f i n s t e a d we  candidates  Chaining  Methods  (P^,Xj),  ( P ^ , X ^ ) and  We  getting X  apply  expected  e q u i v a l e n t methods a r e o b t a i n e d  h o l d p^  (X , X ) , X. g  (X  , X ) and £  constant, then  C  (p^,X-^) r e s p e c t i v e l y . g  first  i.e. fixing  2  e r e n t t o p^  c h a n c e o f g e t t i n g X^  from  (X  C  , X ) g  C  there are  three  f o r a d d i t i o n a l m e t h o d s known c o l l e c t i v e l y  amount X an in practice.  the  as  the  pairs  Of t h e t h r e e c a s e s ,  we  ( p ^ , X ^ ) , w h i c h i s more o f t e n u s e d  such t h a t a sure consequence o f X i - A w o u l d be i n d i f f 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^, determine n  2  Determine a t h i r d  the  f o r sure.  w h i c h 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  d e s c r i b e only the  process.  versus  the  Neumann-Morgenstern u t i l i t y f u n c t i o n .  loss  x.  to obtain  XJ  c e r t a i n t y , g a i n and  remaining  X^  u(X°)  The  respectively.  and  utility  obtain:.  -=—^jvT  V  method.  Neumann-Morgenstern  fixing  t o g e t t i n g X^  u(X°)  1_  von  Equivalent  t h a t a p (X ) c h a n c e o f X  i s indifferent  Therefore,  the  method by  (X ) ) s u c h  a g a i n and  Gain  the  T h u s , we  following u(X.  v a l u e X^ b y  and  o b t a i n i n g X^  r e p l a c i n g X^  obtain a decreasing  by  X  sequence  2  otherwise and  in  repeating  (X^,X ,X.j, 2  B. the  . ..)  with  property, + 1  (3.4)  ) = p°u(X.) + (l-p°)u(X°).  Assigning a r b i t r a r y values to  u  ( ^ )  a n  d  a greater value to  u(X^),  73  we  observe that expression  utility  on  the  decreasing  denote the u t i l i t y fixed probability  (3.4)  determines the  set of points  function obtained p°  and  fixed  von  (X^,X2,X2>  u s i n g the  . . . ) . We  we  have noted e a r l i e r ,  expected u t i l i t y  l o s s amount X^ b y  u  0  p  functions u  'true' , u  Up  one  o f them s u f f i c e s .  Q  w o u l d be  affine transformations  c o n d u c t e d i n 1952  Allais  o f each o t h e r  (1979, A p p e n d i x C ) ,  found t h a t the u  and  u,  C  same s u b j e c t s a r e is  given  thesis, bal  i n Figure the  curves  J6  so t h a t  any  Note t h a t under the  c o n v e x r e g i o n n e a r X=0  i n an  experiment  obtained  from  o f u^  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  A fairly  expected u t i l i t y  i s not  risk  typical  compatible  seeking  --  people tend  intuition  region of  tance  hypoglo-  u C  about the psychology o f l o t t e r y  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  chance o f a l a r g e g a i n ; w h i l e  plot  w i t h the  ~"2  corresponds to our  the  2  g e n e r a l l y very d i f f e r e n t . 3.6.  , u g  C  and  with  (X).  i f t h e d e c i s i o n maker i s a  maximizer, then the u t i l i t y  shall  c h a i n i n g method  a As  Neumann-Morgenstern  the  concavity of u  o f i n d i v i d u a l s t o engage i n s y m m e t r i c  of a  purchase small  r e a f f i r m s the r e l u c -  x  2  bets.  76  I n an o n g o i n g s t u d y  on t h e r i s k  attitudes of top-level  m a n a g e r s c a r r i e d o u t b y MacCrimmon and o t h e r s  [1972),  chaining  g a i n e q u i v a l e n t s w e r e among t h e m e t h o d s u s e d t o a s s e s s Morgenstern u t i l i t y curves  u  and u  functions.  obtained  g a i n e q u i v a l e n t method of u with  n e a r X=0  Fig.  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 o f Ug v e r s u s Even  though  tion  that the convexity  the  hypothesis,  The same i n c o n s i s t e n c y  d i r e c t i o n , t o t h e convex  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  p r o n e n e s s and r i s k scant  Note again  a t t h e same r e g i o n .  pair of  t h e c h a i n i n g method and  i s i n c o n s i s t e n t , under the expected u t i l i t y  the concavity o f u  and  v o n Neumann-  3.7 d i s p l a y s a t y p i c a l  from a s u b j e c t u s i n g  respectively.  business  convex  region  region.  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 a v e r s i o n w i t h i n t h e same r a n g e o f w e a l t h  levels i s  a n d f r a g m e n t a r y , what we a l r e a d y know a b o u t t h e a c t u a l a p p l i c a o f d i f f e r e n t methods t o e l i c i t  expected u t i l i t y  utility  functions suggests  does n o t account f o r t h e r e s u l t s .  This  that  suggests  that  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  risk  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 functions  i s warranted.  utility  77  3.4  SOME PROBLEMS WITH PROBLEM  REPRESENTATION  P r o b l e m r e p r e s e n t a t i o n a n d i t s i n f l u e n c e on p r e f e r e n c e s relatively  u n t o u c h e d 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 .  Normatively,  a d e c i s i o n m a k e r ' s p r e f e r e n c e s h o u l d n o t d e p e n d o n t h e way 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 ability this  is a  alternatives the d e s i r -  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  may n o t b e t h e c a s e i s d e m o n s t r a t e d b y Kahneman and T v e r s k y  t h r o u g h a c l a s s o f phenomena t e r m e d  Isolation Effects.  (1979)  Consider the  c h o i c e b e t w e e n A and B i n F i g u r e 3.8.  $4000  $0  Fig.  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  lotteries  $4000  $0  Fig.  3.9:  A sequential representation of lottery  B  78  I f you  a r e an e x p e c t e d  does n o t  utility  d e p e n d on how  d e c i s i o n maker, then your  the p r o b a b i l i t i e s  of f i n a l  preference  outcomes are  obtained,  s o t h a t l o t t e r y C i n F i g . 3.9  i s equivalent to l o t t e r y B i n Fig.  i.e.  B s h o u l d be  p r e f e r e n c e b e t w e e n A and  f e r e n c e b e t w e e n A and subjects  ( n = 95  C.  Kahneman and  i n t h e same d i r e c t i o n  Tversky  found  ) , 6 5 % p r e f e r l o t t e r y B t o l o t t e r y A.  m o d a l p r e f e r e n c e p a t t e r n b e t w e e n l o t t e r y A and group o f s u b j e c t s i s found w i t h n=141). for to  The  t o be  l o t t e r y A i s reproduced  a l o t t e r y t h a t p a y s $4000 w i t h and  the  the  first  stage, there i s a p r o b a b i l i t y winning  a n y t h i n g , and  move i n t o t h e s e c o n d s t a g e .  s t a g e you  Y o u r c h o i c e m u s t be before  A;  preference  (4000,.80)  refers  $0 w i t h  proba-  .2  $3000.  of  .75  to  reach  the  end of  .25  second  first  above f o c u s e s  $0 w i t h t h e ' same p r o b a b i l i t y  (3000)  made b e f o r e  ( 4 0 0 0 , . 8 0 ) and  most s u b j e c t s t h e n  another  C to  a probability  I f you  and  t h e outcome o f t h e  problem d e s c r i p t i o n  c h o i c e between  the  have a c h o i c e between (4000,.80)  The  and  of  f o l l o w i n g t w o - s t a g e game.  t h e game w i t h o u t to  .8 p r o b a b i l i t y  pre-  group  t h e modal  below.  (3000) d e n o t e s t h e s u r e consequence o f  Consider In  (78% p r e f e r  that e l i c i t e d  as  However,  lottery C for  the opposite  problem d e s c r i p t i o n  l o t t e r y C versus  bility,  f o r one  3.8,  t h e game s t a r t s , i . e . , stage  i s known.  t h e s u b j e c t s ' a t t e n t i o n on  (3000) r a t h e r t h a n .75.  Kahneman and  the  the  common o u t c o m e  Tversky  conjectured  i g n o r e t h e common o u t c o m e - p r o b a b i l i t y  component  under the  a b o v e p r o b l e m r e p r e s e n t a t i o n s o t h a t t h e i r c h o i c e becomes  identical  t o t h a t between  ( 4 0 0 0 , . 8 0 ) and  (3000).  that  79  Another k i n d of i s o l a t i o n is  related  relatively  to Markowitz  (1952)'s  considered  P r o b l e m 1.  A:  different  1000.  You  a r e now  (1000,.50)  P r o b l e m 2. given C:  levels.  You  and  B:  a r e now  (-1000,.50)  and  own,  D:  you  c h o o s e A o r C. between  People,  ( 1 0 0 0 , . 5 0 ) and  (-1000,.50) and  ( 5 0 0 ) , and  (-500), w i t h the  with  an a l t e r n a t i v e  expected  utility,  o u t c o m e s as g a i n s A l l o w i n g the i n the  and  up  with a u t i l i t y  choice problems  i f you  choose B or  a choice  final  Kahneman and  are  D,  a  current wealth  outcome p o s i t i o n n o r m a l l y Tversky  choice  between  lump sums o f $1000 i n P r o b l e m 1  integrated into their  to the  Note>  w i t h $1000 o r $2000 more i f y o u  P r o b l e m 2 as  and  levels. associated  proposed that people  perceive  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 .  r e f e r e n c e p o i n t t o be  context  B i n P r o b l e m 2.  h o w e v e r , seem t o p e r c e i v e P r o b l e m 1 as  $2000 i n P r o b l e m 2 s a f e l y As  have been  (-500)  $1500 r i c h e r  have even chance o f e n d i n g  have been  asked t o choose between  e q u i v a l e n t i . e . you  to  you  own,  o u t c o m e s , t h e two  o r you  the  (500)  however, t h a t i n terms o f f i n a l either  are  They p r e s e n t e d  m a j o r i t y o f s u b j e c t s c h o s e A i n P r o b l e m 1 and  are  Tversky  asked t o choose between  In a d d i t i o n t o w h a t e v e r you 2000.  Kahneman and  groups o f s u b j e c t s .  In a d d i t i o n t o w h a t e v e r you  given  by  observation that preferences  independent of current wealth  f o l l o w i n g p r o b l e m s t o two  The  effect  of the choice  d e t e r m i n e d by  s i t u a t i o n he  t h e d e c i s i o n maker  f a c e s , expected  utility,  f u n c t i o n d e f i n e d -on c h a n g e s i n a s s e t p o s i t i o n  the reference p o i n t , i s compatible  relative  w i t h t h e modal p r e f e r e n c e s  in  80  P r o b l e m s 1 a n d 2. when t h e c h o i c e  3.5  The r e f e r e n c e p o i n t n e e d n o t b e t h e s t a t u s quo e s p e c i a l l y situation  c o n t r a d i c t s the  principle (section  briefly  some o f t h e l i t e r a t u r e  i m p l i c a t i o n s o f expected u t i l i t y .  under the headings:'  systematic  violations  I t has been  of the strong  aversion  ( s e c t i o n 3.4).  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  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 m e t h o d s o f m e a s u r i n g a  Neumann-Morgenstern u t i l i t y  isolation  classified  independence  3 . 3 ) , a n d some p r o b l e m 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  phenomena c o n s i d e r e d  f u n c t i o n , a n d Kahneman and T v e r s k y ' s  effects.  In t h e next 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  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 We t h e n e x p l o r e respect  on e m p i r i c a l e v i d e n c 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  modifications, von  g a i n o f $1000.  SUMMARY  We h a v e r e v i e w e d  The  involves' a sure  theory  I as a t h e o r y  by  of choice.  the 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 with  t o t h e phenomena c o n s i d e r e d  i n this  critique.  that  4  81  A NEW THEORY  We d e v e l o p utility sections  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  theory which g e n e r a l i z e s expected  u t i l i t y by i n t e r p r e t i n g , i n  1 a n d 2, t h e r e p r e s e n t a t i o n t h e o r e m s o f C h a p t e r s  t e r m s o f c h o i c e among l o t t e r i e s . implications  o f our theory  alpha u t i l i t y of expected  We e x p l o r e i n s e c t i o n 3 some  and g l o b a l r i s k  i s considered  i s compatible  utility  1 and 2 i n normative  including consistency conditions with  s t o c h a s t i c dominance and l o c a l of descriptive v a l i d i t y  alpha  aversion.  i n s e c t i o n 4 w h e r e we show t h a t  w i t h t h e phenomena r e v i e w e d  (Chapter  3).  The q u e s t i o n  i n the critique  We e n d t h e c h a p t e r w i t h a c o m p a r i s o n  o f o u r 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  Chapter  1, we p r o v e d  a r e p r e s e n t a t i o n t h e o r e m o f a mean  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 and  sufficient  s e t o f axioms.  The p r e s e n t  section  value  necessary  explores the impli-  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 t h e o r e m f o r c h o i c e among l o t t e r i e s b y 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 .  In the ensuing  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  single-attribute  c o n s e q u e n c e s e t , e.g., m o n e t a r y g a i n s , c a n b e r e p r e s e n t e d b y p r o b a b i l i t y distributions  d e f i n e d on t h e r e a l  sequence space i s c o n s i d e r e d The  2  (The c a s e  o f more g e n e r a l  con-  i n s e c t i o n 4.2)  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  preference is  line.  over  lotteries.  The f o l l o w i n g  able to assign, corresponding T h i s was d i s c u s s e d i n Chew  axiom a s s e r t s t h a t he  t o a n y l o t t e r y F, a c e r t a i n t y  (1979).  equivalent  82  M(F)  w h i c h i s an amount s u c h t h a t t h e d e c i s i o n m a k e r 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 sure  a n d t a k i n g t h e l o t t e r y F.  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  A x i o m MO:  Existence  V F e D,  A x i o m MO r u l e s o u t i n f i n i t e possibility The  not  an i n t r i n s i c p r o p e r t y  mean d o e s n o t a l w a y s  Axiom M l :  on t h e r e a l  line.  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 type  a x i o m s M1-M5  We d i d n o t s t a t e e x i s t e n c e  defined  M(F) e x i s t s .  o f a St. Petersburg  following five  (lotteries)  D denotes t h e  paradox. are taken  directly  f r o m C h a p t e r 1.  (MO) a s a n a x i o m i n C h a p t e r 1 b e c a u s e i t i s 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  exist.  Certainty Consistency  M(6  ) = x  V x £ R.  X  It the  isdifficult  t o take  certainty equivalent  normatively  A x i o m M2:  appealing  Betweenness  of ^  i s s u e w i t h Axiom Ml which r e q u i r e s i . e . g e t t i n g x f o rsure,  i s x.  that Another  axiom i s t h e f o l l o w i n g :  V F, G e D, i f M ( F ) < M ( G ) ,  then  V B € ( 0 , 1 ) , M ( B F + ( 1 - B ) G ) €E ( M ( F ) , M ( G ) ) .  A x i o m 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 two  l o t t e r i e s be i n t e r m e d i a t e  of the respective l o t t e r i e s . quasilinearity  o f a mixture  of  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 The n e x t a x i o m w e a k e n s t h e a x i o m o f  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  A x i o m M3:  Weak S u b s t i t u t i o n  V  F,  G e D,  then V 3 V  S u p p o s e F and i.e.  indifferent  lotteries 3 F + a ;third the  ties  ( 1 - 3 ) H and  lottery  H  be  the  yG+(l-y)H of  principle  n e x t two  A x i o m M4  certain  i n the  A x i o m M4:  independent of  mixture  e q u i v a l e n t s or  indifference  the  third  lottery  l i m i t i n g o p e r a t i o n s on approximation of  If ^  F n  probability  e.g.  uniform  compact s u p p o r t ,  = Lim  M(F  see  certainty  F  'close'  K,  is  rela-  distributions. discrete  distribution.  to  F  then  ).  axiom s t a t e d below a s s e r t s t h a t the  of  behaved  l o t t e r i e s w i t h numerous  d i s c u s s i o n o f A x i o m M4  interval  introduced  ^ _ ^ converges i n d i s t r i b u t i o n  F has  interval  is well  probabili-  H.  equivalent  the  weakens  probabilities  certainty  a more t e c h n i c a l  F i f the  l o t t e r i e s , with  n o t i o n of  Continuity  equivalents  compound  assumptions which are  M(F)  lottery  Y  Weak s u b s t i t u t i o n  certainty  a continuous d i s t r i b u t i o n  , t r u n c a t e d by  mixtures or  respective  y.  M(yG+(l- )H).  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  and  For  (0,1)  same c e r t a i n t y  sense t h a t the  of  £  M(3F+(1-3)H) =  the  6 and  y  M(G),  technical  p e r m i t s the  o u t c o m e s by  3  =  axioms are  e n s u r e t h a t our to  e D,  C o n s i d e r the  at p r o b a b i l i t i e s  same.  3 e (0,1)  l o t t e r i e s w i t h the  each o t h e r .  o n c e d e t e r m i n e d be The  tive  two  y that preserve equality  need not  to  to  substitution  3 and  G are  H  i f M(F)'  truncation K  C h a p t e r 1.  equivalent of to  that of  the  is sufficiently  Our a  last lottery  original large.  84  A x i o m M5:  Extension  Let  —  {K  }  n  o o  n=  .be  an  increasing family 0  compact i n t e r v a l s such t h a t Then M(F)  = L i m Y[{Jc  ), V  of  J  Lim K = n-*=° n  F e  R.  D.  n  We  are  certainty  now  r e a d y t o r e s t a t e T h e o r e m 1.3  o f Chapter 1 i n terms  equivalents.  Suppose a d e c i s i o n maker a s s i g n s M(F)  corresponding to a l o t t e r y  M(F)  satisfies  A x i o m s M1-M5  s t r i c t l y p o s i t i v e l y valued i n g f u n c t i o n v such t h a t  M(F)  = v  -1  F,  then the  i f and  only  equivalent assignment  i f there  f u n c t i o n a and  V F G  (/ ctvdF//  a certainty  exist a  a strictly  [A,B]  condition  C R,  such t h a t  there  adF)  .  (4.1)  V x S  v*(x)  a  (4.1), then, for every  exist  a*(x)  constants  that  interval  a,b,c,k w i t h  a,c,k  >  0  [A,B], k[v(xj k[v( ) x  and  increas-  D,  M o r e o v e r , 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 satisfies  of  - v(A)] - v(A)] + v(B)  = ca(x)[k(v( ) x  - v(A))]  -  v( )  + v(B)  + b ,  x  - v( )] . x  (4.2)  of  85  Note t h a t t h e e x p r e s s i o n general  than  for certainty  the corresponding M(F)  = v  _ 1  expression  cases,  then  (4.3) The u n i q u e n e s s r e l a t i o n  (4.2) sub-  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 To a v o i d t h e p o s s i b i l i t y  t y p e p a r a d o x , we h a v e a s s e r t e d t h r o u g h  certainty equivalent  utility,  R  s c a l a r t r a n s f o r m a t i o n f o r a. Petersburg  f o r expected  ( 4 . 1 ) i s more  (/ vdF)  w h i c h r e s u l t s when a i s c o n s t a n t . sumes a s s p e c i a l  equivalent  M ( F ) must be f i n i t e  o f any S t .  A x i o m MO t h a t t h e  f o r a n y l o t t e r y F.  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  We c a n  that either v i s  b o u n d e d o r a.v i s b o u n d e d .  T h u s , we h a v e o b t a i n e d  o f expected  alpha u t i l i t y ,  i n the sense t h a t the  over  i s represented  utility,  called  d e c i s i o n maker's p r e f e r e n c e general  expression  additional  b y a more  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 b y an  alpha-function.  We s h a l l  s h a p e s o f a and v f u n c t i o n s t i l l section contains a p a r a l l e l the  lotteries  a generalization  defer discussion of the actual  the sections a f t e r next.  The  development o f the above t h e o r y  r e p r e s e n t a t i o n t h e o r e m s o f C h a p t e r 2.  next  using  86 4.2  REPRESENTATION OF  In the lotteries lotteries, certainty  A PREFERENCE BINARY RELATION  preceding  defined  on  v i a an  s e c t i o n , we  a x i o m a t i z a t i o n of the showed, u s i n g  t h e o r e m s o f C h a p t e r 1, t h a t t h e F i s given  M(F)  = v  _ 1  t h e mean v a l u e  certainty equivalent  (/ avdF  / LadF  K  K  the preference  >  The  K  present  represented  3  This  was  c e r t a i n t y equi(since v  1  is  by  the  (4.5)  a b o v e f u n c t i o n a l i s more  /„vdF, w h i c h i s a s p e c i a l c a s e  contemplating  having  the  discussed  theorems  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  s i t u a t i o n s where the  s t a t u s quo,  function  .  s e c t i o n a p p l i e s the r e p r e s e n t a t i o n  c e r t a i n t y equivalent  for a child  the  It follows  K  a p p r o a c h , we  of  do  not  limit  of  utility  ourselves  theory to  range of consequences are monetary v a l u e s  some q u a n t i t y o f c e r t a i n c o m m o d i t y . set  increasing  constant.  Chapter 2 to provide U n l i k e the  M(G).  / avdG / / adG  than that of expected u t i l i t y ,  when a i s  choice  t h a n t h a t o f G,  strictly  that  LovdF  (4.5)  corresponding  (4.4)  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,  general  M(F)  R  p o s i t i v e f u n c t i o n a and  i s greater  of-  representation  / / adF)  R  v.  Observe t h a t  for  by:  some s t r i c t l y  order-preserving)  theory  d e c i s i o n maker's assignment  for  v a l e n t M(F)  utility  a s i n g l e - a t t r i b u t e c o n s e q u e n c e s e t , e.g., m o n e t a r y  e q u i v a l e n t s . We  to a l o t t e r y  developed alpha  For  example, the r e l e v a n t  whether to s t e a l  i n Chew and  MacCrimmon  consequence  a c a k e f r o m a b a k e r y may  cake, g e t t i n g caught i n the process.  (1979a).  We  or  denote  be by  87  X (={x,y,z,•••}) the 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 s p a c e o f s i m p l e p r o b a b i l i t y m e a s u r e s d e f i n e d on X ( c f . C h a p t e r specified  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  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  number o f c o n s e q u e n c e s .  The c h i l d  i n t h e 'above e x a m p l e  of a  finite  may f e e l  he h a s an e v e n c h a n c e o f g e t t i n g t h e c a k e w i t h o u t b e i n g c a u g h t . probability or  measures a r e convenient  representations of actual  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  c o n s e q u e n c e s c a n be s u b j e c t i v e l y  finite  estimated or determined  lotteries  b a s e d on We  denote  o f o b t a i n i n g some c o n s e q u e n c e x f o r s u r e b y 6 .  l o t t e r y P i s then r e p r e s e n t e d as a p r o b a b i l i t y  n a t i o n o f sure  Simple  of the underlying  s y m m e t r y 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 . the a l t e r n a t i v e  that  weighted  A combi-  consequences: n (4.6)  where p^ i s t h e p r o b a b i l i t y  of  occurrence  of  t h e consequence x^.  Our  representation functional  ing  t o t h a t o b t a i n e d from  sion  f o r simple p r o b a b i l i t y  the certainty  measures  e q u i v a l e n t approach  correspond-  (see expres-  (4.5)) i s :  E(av,P) / E(a,P),  (4.7)  w h e r e a a n d v a r e f u n c t i o n s o n X, a n d E ( * , P ) denotes t a k i n g expectation o f a f u n c t i o n with 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 m e a s u r e P.  The t h e o r e m s o f C h a p t e r  2 show t h a t t h e a x i o m s on t h e s t r i c t  prefer-  88  ence b i n a r y and  sufficient  Axiom U I :  The tric  relation  '-< ' o f a d e c i s i o n m a k e r s t a t e d b e l o w a r e n e c e s s a r y  f o rrepresenting  Ordering  strict  h i s preference  -< i s a weak  preference  (i.e.,if a lottery  relation  •< i s a weak o r d e r  P i s strictly  preferred  preferred  preferred  consistency  i f i t i s asymme-  to a lottery  transitive  Q, t h e n t h e  (i.e.,if a lottery  P  t o a l o t t e r y Q which i s i n t u r n not s t r i c t l y  t o another l o t t e r y  lottery R).  (4.7).  order.  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 i s not s t r i c t l y  v i aexpression  R, t h e n P i s n o t s t r i c t l y  Both asymmetry and n e g a t i v e requirements f o r a s t r i c t  transitivity  preference  preferred seem l i k e  relation  that  to the basic  few would  want t o v i o l a t e .  A x i o m U2:  Solvability  V  P, Q, R £ L , P X  3B 3  -< Q a n d Q-c  R  £ (0,1)  BP+(1-B)R ~  Q.  A x i o m U2 s a y s t h a t w h e n e v e r a l o t t e r y Q i s b e t w e e n two l o t t e r i e s P and  R i n preference,  indifferent the  t o Q.  i n t u i t i o n that  in preference  then there  i s a m i x t u r e between P and R which i s  The r e a s o n a b l e n e s s o f t h e a b o v e a x i o m s t e m s f r o m 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  b e t w e e n them  ( s e e t h e B e t w e e n n e s s a x i o m i n s e c t i o n 4.1 a n d  a l s o A x i o m s 3:B:a a n d 3:B:b i n v o n Neumann a n d M o r g e n s t e r n follows that a mixture with  a greater  lottery  to another mixture with  should  be p r e f e r r e d  w e i g h t on t h e b e t t e r axiom.  intermediate  lottery.  This  (1947)).  It  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 a smaller  probability  i s the substance o f the f o l l o w i n g  89  A x i o m U3:  Monotonicity  V P, Q e l _ , P -< Q x  =* B P + ( 1 - 8 ) Q  •< y P ( l - Y ) Q +  f o r 0 < Y < B < 1.  The a x i o m s 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 to  expected u t i l i t y theory.  common  The n e x t a x i o m w h i c h w e a k e n s t h e s u b s t i -  t u t i o n p r i n c i p l e or i t s close counterpart, the strong principle,  i s the only  A x i o m U4:  Weak I n d e p e n d e n c e  independence  departure.  P, Q e I_ , P ~ Q  v  x  ^VB  £ (0,1),  BP+(l-e)R  ~  3 Y e (0,1) 3 V R e L , x  YQ+(1-Y)Rt  Weak I n d e p e n d e n c e 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 a x i o m (M3) for to  certainty  equivalents.  G i v e n two l o t t e r i e s  are indifferent  e a c h o t h e r , Weak I n d e p e n d e n c e a l l o w s f o r d i f f e r e n t  composing each o f t h e s e l o t t e r i e s indifference.  with a t h i r d  lottery.  probabilities in  l o t t e r y t o preserve  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  must be i n d e p e n d e n t o f t h e t h i r d the  that  once d e t e r m i n e d  We a r e now r e a d y t o i n t e r p r e t  r e p r e s e n t a t i o n t h e o r e m s 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 lotteries  d e f i n e d on a c o n s e q u e n c e s e t X  finite  satisfies  A x i o m s U1-U4, c h o o s e s a s 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,«)  for  some s t r i c t l y  / E(ct,«)  positive-valued  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 o n t h e c o n s e q u e n c e s e t X.  90  a function  When t h e  becomes t h e  i s constant,  expected u t i l i t y  t h e r o l e o f a von Although  the approach taken  the  distributions, ones l i k e we  advantage of being  can  we  4.3  continuous  the  probability  s e c t i o n s where  or simple  i n terms of e i t h e r  p r o b a b i l i t y measures. i s more  Otherwise, simple  Representing  appropriate  outcomes, e s p e c i a l l y i f the  values.  alpha  outcome  probability  measures  used.  NORMATIVE IMPLICATIONS  Despite  growing evidence demonstrating  expected u t i l i t y  remains dominant m a i n l y  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 utility local  lotteries  with probability distributions  on  w o u l d be  I n t h e n e x t two  d e s c r i p t i v e i m p l i c a t i o n s of  shall represent  deal with numerical  take  s e c t i o n does not r e q u i r e  able to t r e a t general  the normal d i s t r i b u t i o n .  probability distributions  when we  function.  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  theory,  lotteries  in this  assuming  single-dimensional, the c e r t a i n t y equivalent  d i s c u s s t h e n o r m a t i v e and  utility  E(av,•)/E(a,•)  r e p r e s e n t a t i o n E(v,«) w i t h v  Neumann-Morgenstern u t i l i t y  c o n s e q u e n c e s e t t o be a p p r o a c h has  the r e p r e s e n t a t i o n  the  because of the normative  elegance with which  c h a r a c t e r i z e s s t o c h a s t i c dominance  (Arrow-Pratt  aversion.  We  index)  show i n t h i s  and  global  i t s d e s c r i p t i v e inadequacy, appeal  expected  (increasing u t i l i t y function),  (concave u t i l i t y  section that alpha u t i l i t y  function) i s a l s o an  risk attractive  91  normative  theory before  following  section.  4.3.1  Ratio As  considering i t s descriptive relevance  Consistency  we n o t e d  earlier,  except  f o r t h e Weak S u b s t i t u t i o n  o r e q u i v a l e n t l y t h e Weak I n d e p e n d e n c e a x i o m alpha u t i l i t y  1 (Lemma 1.2) a n d C h a p t e r  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  crucial  (Chapter  1, P r o p e r t y 5; C h a p t e r  t o our proofs  Ratio Consistency  and  the  a x i o m (M3)  (U4), t h e o t h e r axioms o f  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  showed i n C h a p t e r  Property  i n the  utility.  We  2 (Lemma 2.2) t h a t Weak  implies the Ratio 2, D e f i n i t i o n  Consistency  2.6)  which i s  a s s e s s m e n t o f t h e ct f u n c t i o n .  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  The  geometrical  4 interpretation by  b y Weber  the three lotteries  simplex  i s specified  g i v e n by t h e areas area  (1980).  Consider  a simplex  P, Q a n d R i n F i g u r e 4.1.  by i t s b a r y c e n t r i c ( a r e a l )  of the triangles  o f PQR i s 1, X t h e n  formed  Each l o t t e r y coordinates  XQR, RPX a n d XPQ.  between P(Q) and R w i t h B ( y ) weight  The l o t t e r i e s  (^,  B> 2  3^)  Assume t h a t t h e  represents a p r o b a b i l i t y mixture  among t h e t h r e e v e r t i c e s P, Q a n d R.  X i n the  B^P+f^Q+^R  Y(Z) i s a  mixture  on P ( Q ) a n d 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 A x i o m U5 ( s e e Chew a n d M a c C r i m m o n , 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 A x i o m M3 ( s e e Chew, 1979) . Weber (1980) d e m o n s t r a t e d v i a " a n a p p r o a c h 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 Consistency w i t h r e s p e c t t o t h e o t h e r a x i o m s b y s h o w i n g t h a t T h e o r e m 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 . This motivated t h e statements o f Lemma 1.2 a n d Lemma 2.2.  4.1:  Ratio Consistency  illustrated  using barycentric coordinates  S3  93  Mixing  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  ment YZ.  Suppose t h e l o t t e r i e s  then M o n o t o n i c i t y set.  P and Q a r e i n d i f f e r e n t t o each  implies that the line  i s a n Z on QR  H e n c e YZ i s a n i s o p r e f e r e n c e .  P r o j e c t YZ t o m e e t PQ e x t e n d e d 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 (Note t h a t e i t h e r right  o f Q)).  s > 1 (0 i s o n t h e l e f t  The R a t i o C o n s i s t e n c y  isopreference  (s,l-s,0)  o f P) o r s < 0 (0 i s on t h e  property  simply requires a l l other  s e t s s u c h a s 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. observation  other,  segment PQ i s a n i s o p r e f e r e n c e  F o r e a c h Y o n PR, Weak I n d e p e n d e n c e i m p l i e s t h e r e  such t h a t Y and Z a r e i n d i f f e r e n t .  seg-  That t h i s  i s t h e case f o l l o w s from t h e  that  AOZR = AOYR + A Y Z R ,  (4.8)  which i s g i v e n by  ys  Solving  = B ( s - l ) + YB-  for y  (4.9)  gives  Y = B(s-l)/(s-B)  Let x = s - l / s .  Expression  (4.10)  ( 4 . 1 0 ) becomes  Y/l-Y =  B/l-B  A similar  observation  Y'/l-Y  1  (4.11)  T .  for the line  segment Y'Z' g i v e s  Y/l-Y (4.12)  B'/l-B'  B/l-B  94  Hence, l i n e Ratio  segments o r i g i n a t i n g  Consistency  probabilities equal;  property  f r o m t h e same p o i n t 0 s a t i s f y t h e  (4.12).  For expected u t i l i t y ,  8 and Y such t h a t Y i s i n d i f f e r e n t  so t h a t t h e c o n s t a n t  x equals  unity.  the mixture  t o Z a r e always  Geometrically,  the line  s e g m e n t s YZ a n d Y'Z' a r e p a r a l l e l . The  simplical  representation  c a n now b e u s e d a s 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  Lemma 2.2 w i t h o u t  having  The  basic  t o solve  o f Lemma 1.2 o r e q u i v a l e n t l y  the functional  equation  i d e a i s t o show, o n c e an i s o p r e f e r e n c e  that a l l line  s e g m e n t s s u c h a s Y'Z' o r i g i n a t i n g  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 isopreferences.  Monotonicity  s a y YZ i s f o u n d , from t h e p o i n t  0  YZ a n d PQ a r e a l s o  which i m p l i e s t h a t these  are n o n - i n t e r s e c t i n g then ensures t h a t there preferences.  (1.6).  Lemma 2.2 i s p r o v e d b e l o w u s i n g  isopreferences  a r e no o t h e r  iso-  the simplical  approach.  Lemma 2.2:  Ordering  (UI), Monotonicity  implies Ratio Consistency Proof:  Suppose P ~ Q  implies that there  of  s  o  Y = 6 P + ( 1 - 3 ) R , Weak I n d e p e n d e n c e  i sa mixture  Z = YQ+(1-Y)  t h a t YZ i s a l s o i s o p r e f e r e n c e .  intersection o f the extensions  without  '2.6).  PQ i s a n i s o p r e f e r e n c e .  Corresponding t o a mixture  Z>  (Definition  b u t 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 ,  Y ~  (U3) a n d Weak I n d e p e n d e n c e (U4)  r  such  that  L e t 0 be t h e p o i n t  o f YZ a n d PQ.  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  (Suppose of P).  R(1,0,1)  O(s,l-s,0)  P(1,0,0)  F i g u r e 4.2:  A  Geometric p r o o f o f the R a t i o C o n s i s t e n c y  Q(0,1,0)  property  96  C o n s i d e r another the same p o i n t 0.  l i n e segment Y'Z' below YZ o r i g i n a t i n g We s h a l l  show t h a t Y'Z  1  from  i s a l s o an i s o p r e -  f e r e n c e and by t h e p r e c e d i n g d i s c u s s i o n s a t i s f i e s the R a t i o Consistency property PQ a t A. Y'Z'  Draw t h e median from R t o b i s e c t  Draw a l i n e from Y. p a r a l l e l t o RA and i n t e r s e c t i n g  a t S.  Produce PS t o meet RA at V.  PVQ by c o n n e c t i n g VQ and Y'Z'. VQ a t T'.  (4.12).  V and Q.  Complete t h e t r i a n g l e  Denote by T the i n t e r s e c t i o n between  Draw a l i n e from Z p a r a l l e l  t o RA and i n t e r s e c t i n g  To see t h a t OST' i s c o l l i n e a r and hence T* must  be the same as T, view t h e f i g u r e i n t h r e e dimensions with V the t o p v e r t e x o f a t e t r a h e d r o n with base PQR. through The  the p a r a l l e l  plane through  The plane  l i n e s YS and ZT' c o n t a i n s 0, S and T'.  PV and Q a l s o c o n t a i n s 0, S and T'.  The  c o n c l u s i o n t h a t Y'Z' i s a l s o an i s o p r e f e r e n c e f o l l o w s from a p p l y i n g Weak Independence t o S = BP+(1-B)V and T = Y Q + ( 1 - Y ) V . A s i m i l a r argument e s t a b l i s h e s the r e s u l t Y'Z'  f o r t h e case  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 functions directly is  from  This tinct  a p a r t from  alpha  utility  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  a f e a t u r e t h a t i s shared  utility  for eliciting  w i t h expected  utility  and s e t s  other a l t e r n a t i v e theories i n the  alpha  literature.  i s i l l u s t r a t e d below v i a a simple consequence s e t X w i t h outcomes  Chapter  L , I , and H a r r a n g e d  3 section  i n ascending  order of preference.  1 contains a discussion of the systematic  t i o n s of the Strong  Independence p r i n c i p l e  c h o o s e s among l o t t e r i e s can measure  d e f i n e d on  an a l p h a u t i l i t y  decision  d e f i n e d on s u c h a c o n s e q u e n c e s e t , t h e n  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  Set the v - v a l u e s  f o r the worst  way:  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 b a b i l i t y q such indifferent  I i s g i v e n by t h e p r o -  t h a t t h e d e c i s i o n maker i s  between t h e sure consequence I  a n d a q c h a n c e o f o b t a i n i n g H a n d 1-q of  viola-  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 p a r a d o x i n t e r m s o f l o t t e r i e s a 3-outcome c o n s e q u e n c e s e t . S u p p o s e  dis-  chance  o b t a i n i n g L, a s i l l u s t r a t e d i n F i g . 4.3.  such maker we  98  Figure 4.3:  v(L) - 0;  P r o b a b i l i t y Equivalent Method  v(H) = 1;  v ( I ) = q such that P ~ Q. Having constructed the v-function, we form i n F i g . 4.4 the f o l l o w ing l o t t e r i e s to determine the a f u n c t i o n .  Figure 4.4:  Test of S u b s t i t u t i o n Axiom  I f the d e c i s i o n maker subscribes to the S u b s t i t u t i o n p r i n c i p l e , P' and Q' would be i n d i f f e r e n t whenever 8 equals y; since Q i s constructed to be i n d i f f e r e n t to P.  If,however,P' and Q' are i n d i f f e r e n t with 8 and y  99  unequal, then Ratio Consistency tells us that Y/1-Y  — - = T, a constant. B/i-e  (4.13)  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 procedure 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 interpolating among a f i n i t e number of measurement points.  Just as there are many  different ways to measure von Neumann-Morgenstern u t i l i t y functions, this would also be the case for alpha u t i l i t y .  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 u t i l i t y with stochastic dominance, local and global risk aversion.  100  4.3.3  Stochastic  D e f i n i t i o n 4.1:  Dominance  A distribution G  i s said to  another d i s t r i b u t i o n F i n the G(x)  < F(x),  V  x e  has  i t s o r i g i n i n the  and  H a n o c h and  Levy  w o r k s o f H a d a r and  =»•  every increasing  i s however not  the  extent that  normatively  R  The  ( 1 9 6 9 ) , Lehmann  that,  Therefore, every expected  G t o F when G d o m i n a t e s F i n t h e  consistency (and,  with  i n the  i t can  be  of every alpha u t i l i t y first  alpha u t i l i t y  degree s t o c h a s t i c  Corollary  1.4:  decision  iff  context of monetary l o t t e r i e s ,  i m p o s e d as  R  are  G e D,  a ( x ) (v ( x ) an  an  additional  probably  requirement.  consistent  with  under first  bounded. F > G  Q. (F) > fi (G)  - v (s))  increasing  functional ^(F)  avdF// adF.  maker.  dominance.  Then V F,  The  degree.  degree s t o c h a s t i c dominance i s  d e c i s i o n m a k e r w o u l d be  Suppose a, v  is  utility  first  f o l l o w i n g c o r o l l a r y taken from Chapter 1 p r o v i d e s c o n d i t i o n s  w h i c h an  (1955)  R  necessarily true  descriptively valid),  Dominance  / udG i / udF  f u n c t i o n u on R.  desirable  Russell  I t i s w e l l known  d e c i s i o n maker would p r e f e r  To  G \ F, i f  R.  (1969).  G > F  This  d e g r e e , d e n o t e d by  a b o v e d e f i n i t i o n o f w h a t i s u s u a l l y c a l l e d Stochastic  The  for  first  s t o c h a s t i c a l l y dominate  (4.14)  function V  r e f e r s to the  i t i s e a s y t o see  that  s e  R.  alpha u t i l i t y  representation  r e l a t i o n (4.14) can  be  restated,  101  a(x)-(v(x) is  We o b t a i n  discussion  G I F  Suppose  an i n c r e a s i n g  a "familiar"  expository that  - 0(F))  respect  nCF ) 0  V F e D-  of thenecessity  proof of Corollary  i f 6' > 9 w h e r e F with  = / u(x;F )d[G(x) - F(x)] > R  e  F  -,{v(x) - Q(F )} Q  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  ^^ F  fi=n+ 9=0  +  e v e r y x e R.  i n 9.  Differ-  V 6 e (0,1]  (4.16)  (1-0)F+6G  V  J p  that  0  .  (4.17)  that  = /Ru ( x ; F ) d [ G ( x ) - F ( x ) ]  Since G i s any d i s t r i b u t i o n for  Observe  to 0 yields  Q  fi  1.5.  >, t h e n fi(F ) i s i n c r e a s i n g  where u ( x ; F ) = g [ ^  "Je  =  Q  i s consistent  |e  function  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  => F g , i Fg  entiating with  (4.15)  > 0.  (4.18)  d o m i n a t e s F, u ( x ; F ) h a s t o b e i n c r e a s i n g  The f u n c t i o n u ( x ; F ) h a s 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 a n d  dT is  n (  VU+  =  / u(x;F)d(G-F) R  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 G-F.  For expected u t i l i t y ,  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  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  a r e u ( x ) and / u ( x ) d [ G ( x ) - F ( x ) ]  Note t h a t u ( x ) does n o t depend  R  on F s i n c e  respectively.  t h e expected u t i l i t y  repre-  102  s e n t a t i o n J l u d F i s l i n e a r i n F. Utility  Function  ( a b b v . LOSUF) f o r u ( x ; F ) ^  interpretation: dominance the  Alpha u t i l i t y  with  1 s t degree  f o r e a c h F.  This  stochastic  condition  f o r m i f we i m p o s e 1 s t o r d e r  assumes  differentia-  on a and v.  av'  > max [ a ' [ v - v ( x ) ] , - a ' [ v ( x ) - v ] ]  where  v = Lim v ( x ) and X-wo  Alternatively,  V  -  (4.19)  X->--°°  (log cx(x))' < p  The r a t e o f c h a n g e o f l o g a i s b o u n d e d  used d i r e c t l y  e R,  v = Lim v ( x ) .  i f  v'(x) v(x) -  v'(x)/v-v(x)  x  t h e above c a n be s t a t e d a s :  V x e R,  with  Specific  C o r o l l a r y 1.5 h a s t h e f a m i l i a r  i sconsistent  i f i t s LOSUF i s i n c r e a s i n g  f o l l o w i n g more m a n a g e a b l e  bility  s u g g e s t s t h e t e r m Lottery  This  and -v' ( x ) / v ( x ) - v to find  if  '  W  -°  (4.20)  a*(x) < 0  from above and below by  respectively.  a l l those a given  1st degree s t o c h a s t i c  V  a  Expression  a certain v that  (4.20) can be will  be  consistent  dominance.  M a c h i n a (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 d o m i n a n c e 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 . Frechet 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 n o t admit a natural extension o f the analysis t o t h e r e a l l i n e . Machina c a l l e d t h e f i r s t Frgchet 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 functions. 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 c o m p a c t i n t e r v a l s u c h a s [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 s o t h a t we c a n i d e n t i f y a l o t t e r y s p e c i fic u t i l i t y function with a local u t i l i t y function.  103  4.3.4  Global Risk  D e f i n i t i o n 4.2:  Aversion  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  6  2  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 second degree; denoted by G > F, if /^(G(x)-F(x))dx < 0 and  Vy £ R  (4.21)  / " ( G ( x ) - F ( x ) ) d x = 0.  (4.22)  When t h e means a s s o c i a t e d w i t h t h e 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 t h a t they a r e e q u a l .  Condition  (4.21) r e q u i r e s  t h a t f o r each x, t h e mean o f G t r u n c a t e d at (-°°,x] i s n o t l e s s than t h a t 2  1  o f F t r u n c t e d a t (-°°,x] .  Note a l s o t h a t f o r a l l concave u i n R, F > G  i f and o n l y i f / udF > / udG. K  K  C o n s e q u e n t l y , an expected u t i l i t y d e c i s i o n maker w i t h a concave utility  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 t h a t dominates a n o t h e r  d i s t r i b u t i o n G i n t h e second degree.  The n o r m a t i v e c o n t e n t  S t o c h a s t i c Dominance ( o r G l o b a l R i s k A v e r s i o n ) t h a t a prudent person s h o u l d be r i s k  o f 2nd-degree  i s d e r i v e d from t h e i d e a  averse.  D e f i n i t i o n 4.2 extends t h e d e f i n i t i o n o f 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 o b t a i n e d i n B l a c k w e l l (1951), Hanoch and Levy (1969) and  Whitmore (1970) d e f i n e d t h i r d - d e g r e e s t o c h a s t i c dominance and r e l a t e d i t t o t h e A r r o w - P r a t t i n d e x . The g e n e r a l kth-degree s t o c h a s t i c dominance was d e f i n e d i n Chapter 1. However, b o t h d e f i n i t i o n s w i l l n o t be t r e a t e d here.  \  104  Strassen  (1965).  The  f o l l o w i n g c o r o l l a r y a d a p t e d f r o m C h a p t e r 1, w h i c h i s a  tion of Corollary  1.6 t o t h e c a s e o f s e c o n d - d e g r e e s t o c h a s t i c  provides conditions consistent  with  Corollary  1.6*:  u n d e r w h i c h an a l p h a u t i l i t y  second-degree s t o c h a s t i c  Suppose  iff  V  Note that function plays  condition  This  > R(G)  i s a decreasing  t h e LOSUF  (4.23)  function  u ( x ; F ) t o be a concave  further confirms our i n t u i t i o n  a von Neumann-Morgenstern Requiring  =* n ( F )  and c o n t i n u o u s ,  s e R.  (4.23) r e q u i r e s  f o r e a c h F.  dominance.  z F > G  {a (x) [v ( x ) - v ( s ) ] }'  dominance,  d e c i s i o n maker w o u l d be  a, v , a ' , v' a r e bounded  V F , GED,  then  restric-  utility-like  that  t h e LOSUF  role.  a, v t o be s e c o n d d i f f e r e n t i a b l e ,  the condition  (4.23)  becomes  V  This  x e R, av"+2cx v'  condition  developed  ,  < min [a" (v-v ( x ) ) ,  - a " (v ( x ) - v ) ] .  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  i n the next  subsection.  f o r alpha  (4.24)  utility  105  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  Consider Equivalent  Index  an a l p h a u t i l i t y d e c i s i o n maker w i t h a s s e t s x.  C(x;Z)  corresponding  t o a r i s k Z i s g i v e n by M ( F  ) - x which  X^" i s t h e d i f f e r e n c e between t h e c e r t a i n a s s e t p o s i t i o n M(F  The Cash  i-t  ) such t h a t  X^~  Li  the d e c i s i o n maker i s i n d i f f e r e n t t o t a k i n g t h e r i s k Z and h i s c u r r e n t a s s e t p o s i t i o n x.  The r i s k premium  TT(X;Z) =  E(Z)  -  TT(X;Z)  i s t h e n d e f i n e d by (4.25)  C(x;Z)  which i s t h e d i f f e r e n c e between t h e a c t u a r i a l v a l u e E ( Z ) o f t h e r i s k Z  and i t s cash e q u i y a l e n t C ( x ; Z ) .  Since  x+Z  and ( x + u ) + ( Z - u ) have t h e  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 , TT and C from e x p r e s s i o n  (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)  F o l l o w i n g P r a t t (1964), we c o n s i d e r t h e b e h a v i o u r 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 t h e t h i r d  c e n t r a l moment o f Z i s o f o r d e r o ( a | ) .  vCx-ir(x;Z)) =  Expanding both  fi(F ) x+z  =  o f ir(x;Z)  -  Thus,  ^  X  .  s i d e s a f t e r c r o s s m u l t i p l i c a t i o n , we g e t  absolute  106  [v(x) -TTV (X)+0(TT (4.29)  This  reduces t o  hp\ r ( x ) + o ( a | ) ,  TT(X;Z)  (4.30)  r(x) = -| - [ l o g a ( x ) V ( x ) ] ' = - ( ^ f f i 2  where  dx  '  &  ^  J  V  2 a +  ' W ),  ( 4 i 3 1  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 m a k e r ' s r i s k p r e m i u m f o r a s m a l l , actuarially neutral risk which, i n keeping with Z i snot actuarially  TT(X;Z) It  p r e c e d e n t , we c a l l  fair,  ho*  r  we o b t a i n  Arrow,  from  the Arrow-Pratt  times r ( x ) ,  index.  When  (4.27)  (4.32)  + o(a|) .  (x+E(Z))  i s straightforward  alternative 1964;  =  Z i s approximately h a l f the variance  t o check t h a t r ( x ) has t h e f o l l o w i n g  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 , 1971),  p(x;h) = %hr(x)  where % ( l + p ( x ; h ) )  and  + 0(h ) ,  (4.33)  2  %(l-p(x;h))  are the p r o b a b i l i t i e s o f obtaining  x+h a n d 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 m a k e r 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 also  t o check t h a t r ( x ) i s i n v a r i a n t under t h e uniqueness (expression it  (1.12)  and (1.13)) f o r t h e f u n c t i o n s  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  straightforward  transformation  a and v.  Inp a r t i c u l a r ,  f o r v and a s c a l a r  107  multiple  for  It  a.  i s comforting  expected u t i l i t y .  t o n o t e t h a t r ( x ) has  Unlike expected u t i l i t y  the  f u n c t i o n s a and  not  t h e r e f o r e e x p e c t t o be  propensities  o f an  v o n l y from the  alpha  aversion function r ( x ) . global risk  aversion  however,  This  we  cannot  We  i n the  would risk  d e c i s i o n maker i n terms o f h i s l o c a l  I n p a r t i c u l a r , we  note from s u b s e c t i o n  sense of c o n s i s t e n c y w i t h  seems i n t u i t i v e l y  the  risk  4.3.4  that  second degree  stochast  c o n v e r s e does n o t  appealing.  as  recover  able to characterize i n general, global  utility  result  local properties  knowledge o f r ( x ) p o i n t w i s e .  dominance i m p l i e s t h a t r ( x ) > 0 p o i n t w i s e , b u t general hold.  similar  in  Consistency  with  second 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  actuarially  fair  Z , which i n turn  risk  implies that r(x) i s positive for actuarially  fair  a b o u t x.  indicating aversion  Even i f r ( x ) i s p o s i t i v e p o i n t w i s e ,  infinitesimal utility  same t i m e  be  and  observation  risks,  risk  other  o v e r some i n t e r v a l .  that people purchase insurance M a r k o w i t z , 1952).  i n t h e n e x t s e c t i o n ; w h e r e we last  subsections  use  risks  p o s s i b l e f o r an  e m p i r i c a l l y observed choice behavior  developed i n the aversion.  i t is still  seeking  ( F r i e d m a n $ S a v a g e , 1948;  considered  risk  fair  d e c i s i o n m a k e r t o be  ponds t o t h e  this  actuarially  infinitesimal  the  and The  alpha  This  corres-  gamble a t  the  implications of  f o r alpha  utility  consistency  to i d e n t i f y regions  to  of l o c a l  will  conditions and  global  108  4.4 D E S C R I P T I V E  IMPLICATIONS  We showed i n t h e p r e c e d i n g expected u t i l i t y ,  section  i s compatible with  that  alpha u t i l i t y ,  such normative notions  s t o c h a s t i c d o m i n a n c e , l o c a l and g l o b a l r i s k a v e r s i o n . illustrated furnishes In t h i s that  how t h e c o n s t r u c t i v e  like  We  as also  proof of our representation  theorem  a procedure f o r the assessment of the a l p h a u t i l i t y  s e c t i o n , we w i l l  show t h a t  alpha u t i l i t y  i s n o t so  general  i t h a s 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 s u c h a m i n u t e  from expected u t i l i t y  that  functions.  departure  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  violations.  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 Strong Independence The e m p i r i c a l  findings violating  Independence p r i n c i p l e in  table  stem  the i m p l i c a t i o n s of the Strong  from the A l l a i s  paradox were summarized  3.1 i n t e r m s o f 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. for  that  7 Principle'  choice  We  consider  patterns  here the i m p l i c a t i o n s  of alpha u t i l i t y  f o r t h e HILO s t r u c t u r e , a n d d e r i v e  some  theory  testable  predictions. Applying we  alpha u t i l i t y  theory t o the d e c i s i o n choices  i n Figure  obtain, fi(A ) Q  = V(I)  fi(B )  = q  Q  = Ba(I)v(I) 6 a ( I ) + (1-3)  fl(A^)  ft(B ).= 3q 3  = v(l)  Ji(B^) = gq + 3 +  fi(A6 =  3 a ( I ) v ( D + (1-3) 3 a ( I ) + (1-3)  The m a t e r i a l (1979b).  in this  ft(BJb  sub-section  H  =  Bq +  (l-3)a(I)v(I) (l-3)a(I) (1-3)  a p p e a r e d i n Chew and  MacCrimmon  3.4  109  Figure  4.5:  Preference  pattern  for a(I) <  1  110 Assuming without we  l o s s o f g e n e r a l i t y t h a t ct(L)=ct(H) = l ,  o b t a i n the f o l l o w i n g i n e q u a l i t i e s  decision  Q  AJj  for  2.1),  c o r r e s p o n d i n g t o p r e f e r e n c e s i n each  box.  A  The  ( c f . Theorem  o  v(I) > q  B[  ~  v(I) > q ( 6  B*  o  v(I) > l - ( l . q ) ( p  >-  B  >•  >-  Q  inequalities  (4.34),  +  I  A  ^  B  I  ( 4  -^|1)  3 4  )  (4.35)  (4.36)  +  (4.35) and  the case i n which cx(I) < 1.  r e g i o n s I-IV,  o  (4.36) are p l o t t e d i n F i g u r e  Note t h a t the f o u r r e g i o n s , denoted  correspond t o f o u r d i s t i n c t  choice p a t t e r n s .  4.5 as  These, along  with the c h o i c e p a t t e r n s f o r the case o f a ( I ) > 1, are summarized i n Table  4.1.  Table  4.1  A l l o w a b l e Choice P a t t e r n s Under Alpha U t i l i t y  Re g io n  ^.^^  Expected U t i l i t y = 1  < 1  A^  I  V V V o V V V o  II  A  I  A^  A  Regions do not  > 1  Q  V V V o V r V o A  B  exist  B  V r V o B  IV  Apart  B  from the p a t t e r n s c o r r e s p o n d i n g t o r e g i o n s I and  o n l y ones c o n s i s t e n t w i t h expected for  A ,  B  III  Theory  utility)  4 out of 14 a d d i t i o n a l p a t t e r n s .  IV ( t h e  alpha u t i l i t y theory  These are g i v e n by the  allows  entries  Ill  under the case a ( I ) < 1 and a(I) > 1 f o r 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 i n the l i t e r a t u r e (cf. Table 3 . 1 ) .  On the other hand, a l l the e m p i r i c a l  f i n d i n g s to date of v i o l a t i o n s of expected u t i l i t y correspond of the region I I and I I I patterns with a(I) < 1 . 6  the standard A l l a i s paradox ( i . e . A  X  Q  to e i t h e r  In p a r t i c u l a r , both  R  B ) and the A l l a i s r a t i o paradox Li  ( i . e . B , A ) occur i n region I I . The existence of region I I I also has 3132  3  some e m p i r i c a l support; note (Table 3 . 1 ) that both the A^, B Q and A^ , B^ v i o l a t i o n s have been reported. Before we continue with f u r t h e r i m p l i c a t i o n s 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 e f f e c t s of changes i n the parameters 3 and v ( I ) on the r e s u l t i n g 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 a t t r a c t i v e ( i . e . , v ( I ) > q/a(I)) then the choice w i l l be A a l t e r n a t i v e and w i l l be unaffected by changes i n 3.  This i n d i c a t e s a b a s i s f o r choice  that may be c a l l e d the security  Correspondingly, i f I i s  effect.  u n a t t r a c t i v e ( i . e . , v ( I ) < 1 - ( l - q ) / a ( I ) ) , then no change i n 3 w i l l induce a s h i f t away from the B a l t e r n a t i v e . nothing-to-lose  effect.  This may be c a l l e d a  Note that both above regions are the only  regions consistent with the s u b s t i t u t i o n p r i n c i p l e . If I i s somewhat a t 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 p r o b a b i l i t y to narrow the perceived gap 3  3  between A^ and B^,due to the a t t r a c t i v e n e s s of I u n t i l f i n a l l y the }  g  a t t r a c t i v e n e s s i s s u f f i c i e n t l y d i l u t e d t o cause a switch to B .  This  112  is  t h e dilution  effect.  The  special  case  <<  has  f o r the d i l u t i o n  8  from  A£l  to  3-7 l  ,  ( w h e r e 3-^ = 1.0  effect  (Kahneman and  ( i . e . , \)(I)  and  3  Tversky,  i s between 1 -  of  1979).  t o A^.  fresh  insight  h a v e a compound s t r u c t u r e and  c a n be  I I and  be  quite large.  q)  gained  by  o f 3.  For  I I I , giving  At rise  low  and  b e l o w q, Finally, to  3  will  effect.  examining  given  the  effect  no  longer  3 = 1, we  B^  choice  intermediate levels  of  t o the p a r a d o x i c a l choices  N o t e t h a t as V ( I ) d e c r e a s e s  some p o i n t t h e A^  in 3  of changes i n 3 f o r  3  at  a decrease  s o r e v e r t t o t h e s i m p l e A^,  w h i c h d e p e n d s on w h e t h e r v ( I ) > q. the regions  certainty  This i s a reverse d i l u t i o n  changes i n v ( I ) f o r g i v e n l e v e l s  3,  a  When I i s somewhat u n a t t r a c t i v e  a d d i t i o n to studying the e f f e c t of V(I),  been c a l l e d  3  cause a s w i t c h from  values  1-0)  ( l - q ) / a ( I ) and  3  In  2  from b e i n g v e r y  can  attractive,  3  switch to B^.  choice w i l l  Then as V ( I ) d r o p s g t h e c h o i c e s i n t h e I and 0 c a s e s c h a n g e t o B^. and B Q r e s p e c t i v e l y . g as v ( I ) d r o p s l o w e r ( i . e . , b e l o w q(3 + ( l - 3 ) / a ( I ) ) , A changes H  B*.  Two First,  main i m p l i c a t i o n s  alpha theory  expected  utility  preferences H-H).  can  describe a richer  theory.  ( I - L , L-0,  s h o u l d be n o t e d  Specifically,  and  L-L)  and  I t a l l o w s f o r a d e p e n d e n c e on  hence c a p t u r e s On  the  observed  preferences.  i t covers  the values  the  than  can  Allais-type  violations  (H-0  of the parameter  and and  effect. so g e n e r a l t h a t i t can d e s c r i b e  O n l y 6 o f t h e p o s s i b l e 16  f o r the e m p i r i c a l l y  observations.  set of preferences  preference p a t t e r n s of  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 patterns  these  other observed  dilution  the o t h e r hand, i t i s not  from  supported  theory  a(I) < 1 case).  (and As  any  the  only  Figure  4 4.5  113  suggests,  o u r t h e o r y makes v e r y  p a t t e r n s and t h e way t h e y In  change as t h e p a r a m e t e r s 8 and V ( I ) c h a n g e s .  particular, monotonicity  cases note  agree completely, that the standard  the A l l a i s  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  requires that preferences  f o r t h e 0 and I  >- B <=>A^>- B .  interestingly,  i.e., Allais  r a t i o paradox  Q  paradox  (B^, A ) Q  (A , B ) o c c u r s  occurs.  Both B  choice  (i.e.,  regions  previously unreported new  "paradox".  I I and I I I ) p r o v i d e f o r  i f , and o n l y i f ,  new r e g i o n s o f p e r m i s s i b l e 8 and B^; h e n c e  c a s e w o u l d seem t o b e a p r i m e c a n d i d a t e  Further, note  t h a t as v ( I ) d e c r e a s e s ,  this for a  the switch  from  3  3  A  More  to B  finally theory  occurs  first  f o r the L case,  f o r t h e H case. are e m p i r i c a l l y  A l l these testable.  then  f o r t h e I and 0 c a s e s , and  i m p l i c a t i o n s from alpha  utility  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 d o m i n a n c e i s an i n t r i n s i c  of expected  utility.  This  i s not  the case w i t h a l p h a u t i l i t y .  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 (see Chapter  descriptively  5 f o r an e x a m p l e o f a p o t e n t i a l v i o l a t i o n  dominance i n the  context  the alpha u t i l i t y  property  o f i n c o m e d i s t r i b u t i o n s ) , we  f u n c t i o n s considered to those  To  the  valid  of s t o c h a s t i c can  restrict  t h a t do n o t  violate  o  s t o c h a s t i c dominance v i a the 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 from C o r o l l a r y VxeR,  1.4.  [loga(x)]'  < -  [log (v-v(x)]'  > -  [ l o g ( v ( x ) - v ) ] ' i f a'(x)<0.  where V = Lim v(x)  normalized  t o V=0  (4.37)  x  and  s l o p e o f - l o g (1-V) that  i s compatible  log  a.  in  both  The  exclude Table II  and  with  (Recall  investigate  f o r a bounded  The  from the graph  a dent i n the middle  and  i n order not  a l s o the  to note "Allais"  left with a fairly i n the next  and  1-v  that after  to in  region limiting  -r.  b e l o w by 3  The  of increases  region I I I preferences  i s that a(I) < qa(H)+(l-q)a(L)). ,  the  a function  t h a t the c o n d i t i o n f o r the e x i s t e n c e of  is interesting  are s t i l l  recovered  o f r e g i o n I I and  of a i s bounded above by  d o m i n a n c e and we  - l o g V from below.  the negative d i r e c t i o n s  3  It  i n F i g u r e 4.6  s l o p e of l o g a i s bounded by  (4.37) i s then  and  I I I preference  behavior  The  a f u n c t i o n c o n s i d e r e d has  the p o s s i b i l i t y  4.1.  V=l.  f r o m a b o v e and  the p o s i t i v e  i f a'(x)>0  and v = L i m v ( )  These c o n d i t i o n s are d e p i c t e d g r a p h i c a l l y V  taken  V  satisfying stochastic  type preference  f o r t h e HILO  l a r g e c l a s s of a f u n c t i o n s .  s u b - s e c t i o n , the r i s k  structure, We  p r o p e n s i t i e s of the  a  115 -ln[v(x)-v]  j  3 J - l n [v -v (  Fig.  4.6:  Consistency  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 functions  of Figure  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 : Concurrence of Risk 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 Local r i s k (see  aversion  sub-section  4.3.5) i s n o n p o s i t i v e  t h a t people tend return.  i n the sense that  to avoid  t o having  with  ticket. risk The  (log  2  a ( x )  pointwise i s  seeking  aversion.  aversion hypothesis  }  risk  >  i s  i s not  a t t h e same this  compatible  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  f o r global risk  a v e r s i o n , namely t h a t  nonnegative,  i s  since  the  r  i n order  local  aversion.  Arrow-Pratt  : (4.38)  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  a (which corresponds t o expected u t i l i t y )  -x  function  Thus, t h e  The f u n c t i o n l o g a h a s t o d e c r e a s e s u f f i c i e n t l y  near the r u i n point  4.6.  i snot  rapidly  to correct f o rthe convexity  of v  region. p o i n t , we h a v e a p a i r o f f u n c t i o n s  s t o c h a s t i c dominance, e x h i b i t compatible  risk  g r a p h i c a l l y i n Figure  a constant  admissible.  At  of i t s expected  a ( x ) ) ' < -Js(log v ' ( x ) ) \  i s depicted  Note t h a t  aversion  i s necessary but not s u f f i c i e n t  (= - V ^ -  to the observation  does n o t s h a r e t h e above d i f f i c u l t y  condition for local  index  This  risk  Alpha u t i l i t y  aversion  risk  to global risk  plausible local  any c o n c u r r e n t  local  index  a concave v o n Neumann-Morgenstern u t i l i t y  which i s i n turn equivalent behaviorally  corresponds  t a k i n g a s m a l l gamble i n f a v o r  For expected u t i l i t y ,  equivalent  the Arrow-Pratt  with  the A l l a i s  local  risk  type preference  ( a , v) t h a t  satisfy  a v e r s i o n p o i n t w i s e , and a r e f o r t h e HILO s t r u c t u r e .  remains t o check 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 properties that  correspond  to actual choice behavior.  It  risk  F i r s t , we e s t a b l i s h  117  Fig.  4.7:  Conditions f o r Local Risk  Aversion  118  Fig.  4.8:  An  admissible alpha  function  119  t h a t o u r a and v f u n c t i o n s d e s c r i b e r i s k that they  do n o t s a t i s f y  seeking  behavior  the conditions f o rglobal risk  VkeR, ( a ( x ) . v ( x ) ) " < 0  by showing aversion:  i f a"(.x) > 0  < a"(x) i fa"(x) < 0 Since  the a function considered  Figure  4.8 t h a t t h e p r o d u c t  T h i s means t h a t t h e r e behavior seeking by  This  seeking  (1979).  Finally,  we c o n s i d e r  averse  risk  substantial  a t t h e same  f o r some o t h e r  gambles.  amount x  The  p a i r s of measurements  and  getting 0 with l-u^x^)  utility  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  d e c i s i o n maker c o n s i d e r e d ,  gain  risk  the mutual i n c o m p a t a b i l i t y of d i f f e r e n t  S u p p o s e we a p p l y  alpha u t i l i t y  of r i s k  aversion  procedures f o r t h e assessment of a von Neumann-Morgenstern function.  averting  of global  d e c i s i o n maker w i l l propensity  point.  (1952) and r e c e n t l y  However, t h e l o c a l  proneness so t h a t t h e alpha u t i l i t y risk  and r i s k  corresponds to the prevalence  ( s e e F i g u r e 4.6) r u l e s o u t t h e p o s s i b i l i t y  time have t h e o p p o s i t e  to observe i n  admits a convex r e g i o n n e a r t h e r u i n  f o r l o s s e s observed by Markowitz  Kahneman a n d T v e r s k y  hypothesis  i s convex, i t s u f f i c e s  i s a concurrence of r i s k  f o rthat region. behavior  o:-v  (4.39)  with  as t h e e n d p o i n t s  s t a t u s quo x=0 and some  of l o t t e r y  B i n Figure  3.5.  (x , u (x ) ) t h a t corresponds 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, a n d g e t t i n g x w i t h u ( x ) c h a n c e c g c c  a(x v(x ) c  ) u (x ) v ( x ) + a ( 0 ) [ l - u (x  expression:  )]v(0).  =  (  a(x  4  >  4  0  )  ) u (x ) + a ( 0 ) [ l - u (x ) ] £p  After  chance a r e r e l a t e d by t h e f o l l o w i n g  c  c  C  ^-  r e a r r a n g i n g , we g e t : a( 0 ) [ v ( x ) - v ( 0 ) ] u  (x) = c  a( 0)[v(x)-v(0)]+a(x  )[v(x  )-v(x)]  (4.41)  120  Fig.  4.9:  _^r 2  A pair of u  c  andu  1/2  d e r i v e d f r o m an a l p h a u t i l i t y  0  Fig.  4.10:  A pair  of 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 m a k e r  decision  maker  121  Alternatively, -x  and  r  use  x=0  i f we  as t h e  f i x a l o s s amount a t h a l f  intermediate  e q u i v a l e n t method t o o b t a i n Ug a  v(0)  (  g  x  )  p  g  (  x  g  )  v  (  g  x  )  can  apply  position  the  gain  below: a(  +  amount, we  the r u i n  -^T [i-p (x )]v(-^) )  g  g  = g"g•  g'  *  ^-  *g*  l  g  Hence, -  - °( )  1  §  the endpoint,  the sequence  v  r  T  F i n a l l y , we as  v(x)-v(0)  x  .  •"2~  consider the p°=h  and  ,  as  ( x ^ , -x.^,...)  c h a i n i n g method.  W i t h x°=0 and  t h e p r o b a b i l i t y p a r a m e t e r , we  b a s e d on  a(x.) 1  +  d e t e r m i n e d Uj  (4.44)  a(0)  have thus  by  expression  (4.44).  Changing the p r o b a b i l i t y  the lower  endpoint  o t o x^=  -2  ) =  i  on  the sequence of p o i n t s  ^r ~~2~>  we  +  have d e t e r m i n e d u^/^  d e t e r m i n e d by  expression  o f U3/4  comparison w i t h the r e s u l t s rather striking  on  p a r a m e t e r t o p°=3/4  (4.45)  the sequence of p o i n t s  c  results  and  (x^, x  u  g  and  ui^  ,...)  w i t h t h e same a x e s f o r  displayed i n Figure  3.6.  a r e p l o t t e d i n F i g u r e 4.10  for  o f MacCrimmon e t . a l . (1972) i n F i g u r e  f i t between the t h e o r e t i c a l  e m p i r i c a l measurement s u p p o r t s thus  given  (4.45).  ease of comparison w i t h A l l a i s '  The  ,...)  L_  shows t h e g r a p h s o f u  the curves  2  a(0)  As b e f o r e , we  Similarly,  -L  obtain:  1  1 3a(x)  F i g u r e 4.9  (x , x  + a ( - f^)v(- ^ ) .  3a(x.)v(x.) 1 + 1  obtain  a(x.)v(x.) + a ( 0 ) v ( 0 ) .  (x,i + 14.1)  v(x  g  the f o l l o w i n g r e l a t i o n .  We  and  x =x^  the v a l i d i t y  p r e d i c t i o n and  of a l p h a u t i l i t y  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  otherwise  3.7.  the theory;  p u z z l i n g phenomena.  122  4.4.4  Some P r o b l e m s w i t h In t h i s  handles the  final two  Problem  subsection,  difficulties  s e c t i o n 3 o f C h a p t e r 3.  Our  utility  be  functions  customary wealth  should level  or  Representation  we  comment b r i e f l y  p o s i t i o n on defined  on  the  question  gains  and  i n terms o f f i n a l  The  are  behavior one  who  wealth the  of the  be  i f we  his belief  p e o p l e who  do n o t  the  question  On  particular  two-stage l o t t e r y which  the  paradox  that  focuses  t w e e n one reverse  the  subject's  terms o f i t s s i n g l e - s t a g e Expected u t i l i t y strong  0-L  the  between the difficulty,  choice  i s helping  Tversky c a l l e d  o t h e r hand, the  the  actual choice  some-  this  former p o s i t i o n behavior  of  integration position.  Kahneman and  is  equivalent  T v e r s k y showed t h a t f o r a  i s c l o s e l y r e l a t e d to t h e i r version  case i n Table 3.1),  elicit  of  a problem d e s c r i p t i o n  a c o n d i t i o n a l comparison  a majority preference  case i f the  c a n n o t be  that  beis  the  two-stage l o t t e r y i s s t a t e d  consistent with  or the  in  overall preference.  t h i s phenomena b e c a u s e  sure-thing principle dictates that  conditional preference  s i n c e the  the  equivalent.  independence p r i n c i p l e  the d i r e c t i o n of the  the  a t t e n t i o n on  o f t h e b r a n c h e s can  o f w h a t w o u l d be  a  the p a r t i c u l a r a p p l i c a t i o n .  of whether a two-stage l o t t e r y  ( c f . the  to  i n t e r e s t e d i n modeling the  asset  to i t s single-stage decomposition.  Allais  in alpha  losses r e l a t i v e  (Kahneman and  want t o d e s c r i b e  conform t o the  other  discussed  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  integration position).  comes i n h a n d y i f we  utility  of whether  or i f a d e c i s i o n analyst  p o s i t i o n p r i o r to evaluation  asset  On  adopted  " e c o n o m i c " man  professes  alpha  asset p o s i t i o n s i s  i t d e p e n d s on  should  how  with problem representation  same a s t h a t o f e x p e c t e d u t i l i t y : latter  on  must be  Alpha u t i l i t y  conditional preference  the  does not for A  n  same as share  against  that  this B  n  and  the  the  123  preference (Table  for B  J-l  against  4.1) c o n s i d e r e d  A  Li  i n sub-section  The phenomena c o n s i d e r e d making i n g e n e r a l .  i s a s p e c i a l c a s e o f t h e HILO c h o i c e 4.4.1.  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  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 which i s the ' c o r r e c t ' problem r e p r e s e n t a t i o n , the lesson t o b e t h a t one s h o u l d t i c u l a r choice represent rankings tions .  pattern  be s e n s i t i v e t o t h e c o n t e x t  situation.  associated with a par-  People appear t o use d i f f e r e n t  t h e same c h o i c e s , r e s u l t i n g  seems  schemes t o  i n apparent i n c o n s i s t e n c y i f the  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 problem  representa-  124 4.5  CRITIQUE OF A L L A I S ' THEORY AND PROSPECT THEORY Having  developed  alpha u t i l i t y  theory  i n the preceding sections,  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 h a t have a t t r a c t e d  significant  4.5.1  utility  -  A l l a i s ' Theory Allais  (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 )  represents preference has  interest.  t o expected  among p r o b a b i l i t y  distributions.  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  several  standard  transitivity, Allais of expected  p r o p e r t i e s o f expected  combination  utility:  assumption r e s t r i c t s  S u c h an a p p r o a c h  represented  satisfies  completeness,  and c o m p o s i t i o n .  f u r t h e r assumed t h a t p r e f e r e n c e utility, called  thus  that  shares  another  property  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.  the preference  f u n c t i o n a l V(F)  t o those  This  that  increase  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 sense.  For finite  i n c r e a s e s i n t h e s t o c h a s t i c dominance n o f t h e form F = £ p.S^., t h i s c o n d i t i o n h a s i=l  lotteries  1  X  l  the f o l l o w i n g simple form, n V(E p.6 .) i n c r e a s e s i n each x.. i =l  (4.46)  x  U n l i k e expected theory  utility,  the theory outlined  a b o v e i s n o t an a x i o m a t i c  i n t h e 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  asserted r a t h e r than being the underlying preference.  a c o n s e q u e n c e o f t h e assumed p r o p e r t i e s o f N o t e a l s o t h a t t h e a d o p t i o n o f a more  general representation i s traded  against t h e convenience  s i m p l e v o n Neumann-Morgenstern u t i l i t y intuition  o f r i s k proneness  a  f u n c t i o n which captures our  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  acterization  of having  and o f f e r s a s i m p l e  (aversion) v i a the convexity  char-  (concavity)  125 of the u t i l i t y f u n c t i o n . In  order t o obtain a u t i l i t y - l i k e  expected u t i l i t y , preference  Allais  revived the Frisch  among i n t e r v a l s  cardinal u t i l i t y  f u n c t i o n without  of wealth  resorting to  (1926) n o t i o n o f q u a r t e n a r y  and a s s e r t e d t h e e x i s t e n c e o f a  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 w o r d s , g e t t i n g $100 a t s t a t u s q u o 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  value the  f r o m s t a t u s quo t o g e t t i n g $ 1 0 0 , s ( $ 1 0 0 ) - s ( $ 0 ) ,  corresponding  According matical in  d i f f e r e n c e going  to Allais,  evaluated h(F-)  s($1100)-s($1000).  m o m e n t ) , t h e d i s p e r s i o n ( s e c o n d moment) and  l e d him t o a s s e r t t h a t the preference  of psychological functional V(F)  a t a d i s t r i b u t i o n F c a n be s t a t e d i n t e r m s o f some  functional  of the distribution V(F)=hCF ). i  a finite  F  s  lottery,  E  o f p s y c h o l o g i c a l v a l u e s , F_, a s f o l l o w s : s (4.47) n F (= £ p . 6 . ) , t h e c o r r e s p o n d i n g F- i s g i v e n b y : i=l x  . 1=1 " P i s ( x . ) 1J  h(6important  '  6  A f t e r r e s c a l i n g , we o b t a i n e d  An  than  depends on t h e mathe-  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 This  i s greater  f r o m $1000 t o $ 1 1 0 0 ,  a choice agent's preference  expectation (first  values.  For  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  ( x )  -  4 8 )  t h e f u n c t i o n a l h below.  )=s(x).  (4.49)  p r o p e r t y o f h , w h i c h we w i l l  noted by A l l a i s .  ( 4  S i n c e s i s an i n t e r v a l  d e r i v e s h o r t l y , was h o w e v e r n o t scale,  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 . Consider  a l o t t e r y F and i t s c e r t a i n t y h(F_)=h(6_  ( M ( F ) )  )=s(M(F)).  equivalent M(F).  I t follows that (4.50)  126 Under t h e a f f i n e h(F v  transformation  a s + b , w h e r e a>0-,  - . ) = h(6 .) as+b' as(M(F))+b' = as(M(F))+b = ah(F_) b.  (4.51)  +  It turns cific  o u t t h a t t h e above p r o p e r t y  conclusions  about t h e a d m i s s i b l e  1 a uniform  s  1  very,spe-  f u n c t i o n a l forms o f h.  Applying s (  1966, p. 236) t o i i e v a l u a t e d - 1 N.  us t o draw  Consider  N  F_ = _ S 6  lottery  h (  o f h enables  A c z e l ' s theorem  (Aczel,  - iJ x  a t F_  yields:  N  S  6 1  =  5(x.)  )  E  y  +  0  g  N  (  §  (  x  i  -  }  y  ^V"^  '  — a  — " a  1  1  1 2 1 2 where y = ^ I s ( x ) , a = - I (s(x j-y) , i=l 1=1 a n d g ^ i s an a r t i b r a r y s y m m e t r i c f u n c t i o n , N  provided Define  a > 0.  (4 .52)  t h e f u n c t i o n a l g on t h e s p a c e o f u n i f o r m 1 g (  It  n  N  distributions  as f o l l o w s :  N  y >  1 6  1=1 follows that  %  =  i  C y  _ 1 1 N h ( r r E 6_, J '•N. . s ( x . ) i=l l  V -  i'  ( 4  N 1 = y + a g ( i j 6_, , ^N. , s(x.)-y i=l l  N  K  ).  6  -  5 3 )  (4.54)  v  a Note t h a t t h e above r e s u l t rational  probabilities  c a n be a p p l i e d t o f i n i t e  by t a k i n g N t o be t h e l e a s t  lotteries  with  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 w h e r e F_ = £ p . 8 _ , .,, w i t h p . s i s(x )' i r  i  =  1  F  rational,  i  2 2 y = Z p s(x ), o = l p . [ s ( x )-y] i=l i=l n  n  1  1  1  127  and  g an a r b i t r a r y  distributions Without g e t t i n g to  into  functional  with rational  probability  weights.  d e t a i l s , we r e m a r k t h a t t h e e x t e n s i o n o f Lemma 4.1  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  We n e e d h t o be c o n t i n u o u s Chapter  on t h e s p a c e o f f i n i t e  weights  i s straight  forward.  w i t h r e s p e c t t o weak c o n v e r g e n c e ( c f .  2 A x i o m M 4 ) , and t h e 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  weights.  a r g u m e n t c a n be e x t e n d e d t o i n c l u d e t h e c a s e  This  tributions of f i n i t e support  w i t h compact s u p p o r t s distributions.  c a n be o b t a i n e d  A x i o m M5 o f C h a p t e r  distributions  with  of dis-  s i n c e t h e y a r e weak l i m i t s o f s e q u e n c e s  F u r t h e r e x t e n s i o n t o t h e case through  rational  o f non-compact  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  2.  In a r e c e n t paper, 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  functional  form f o r h. h(F_) He t h e n F_ s-y  expressed  such  = y > w(F _ ) §  w i n terms o f the normalized  moments o f  9  i m?  n  where m n resultant  =  (s-y) dF . n  R  expression f o r h,  h(F -) = y s  +  f(l4> y  however does n o t s a t i s f y p r o p e r t y The  central  that  i *  The  (4.56)  y  1 7 y (4.51).  Therefore  i ti s not admissible,  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 c a n be a p p l i e d  to 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 expression  (4.55) .  We  define  128  the  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 ,  i  3  ,  4  i ,...)  = g(F _ )  n  g  (4.58)  y  a where m  i s t h e nth-moment o f F . , . s-y  n  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 b y an a f f i n e  transformation  o f §. A key idea i n A l l a i s '  criticism  o f expected  utility  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  i s that i t  s.  Presumably,  t h e m o r e moments we i n c l u d e , t h e c l o s e r d o e s t h e r e s u l t i n g h a p p r o x i m a t e actual preference. the by  first  t w o moments.  assumption  ^  The g e n e r a l  form o f h f o r t h i s  only  case i s obtained  t o some c o n s t a n t  X, i . e . ,  = u + X o.  (4.59)  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  It  s  s t e p , one i s t e m p t e d t o c o n s i d e r  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 h(F_)  F  As a f i r s t  E  ( p 5  p 6  s(x)  s(x)  +  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. +  (  ( 1 _ p  |^  1 _ p  )  ) 6  6  s(y)  s(y)-  {ps(x)  )  5  W  i  w  i  t  h  t  h  r  §  e  s  W  P  >  e  c  t  t  §<  - -  0  P  y  1,  erentiating  yields:  (l-p)s(y) A[p (l-p) ' (s(x)-s(y))]} l 2  +  D i f f  Consider  i  2  +  = (s(x)-s(y))[l+|(l-2p/p (l-p) )] . l s  This d e r i v a t i v e i s negative  f o r any n e g a t i v e  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 first  (one).  (4.60)  1 2  (positive) value  o f X,  Hence, h r e s t r i c t e d  to the  t w o moments o f s c a n n o t 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 d o m i n a n c e  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 h e h i g h e r moments i f we want t o g e t away f r o m e x p e c t e d within Allais'  framework.  t h r e e moments i s g i v e n b y  to include  utility  but stay  The most g e n e r a l h t h a t depends on t h e f i r s t  129  h(F.)  = y + of(m /a ).  A l i n e a r approximation h(F_)  (4.61)  3  3  o f f by X + y m /a  = y + Xo  3  produces:  + y m /a .  (4.62)  2  3  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 a d o p t e d b y Hagen showed t h a t t h e A l l a i s as  type  of X i s not too  l o n g as t h e m a g n i t u d e  dependence matches our i n the prevalence  choice behavior  intuition  (1979) .  i s compatible  large.  about p e o p l e ' s  of l o t t e r y t i c k e t purchase..  The  with a positive y  positive  preference The  Hagen  skewness  evident  question of  the 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  whether  remains.  Consider  a g a i n t h e t w o - o u t c o m e l o t t e r y F- = p 6 - , . + ( 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 a t F_ g i v e s : h(p8_. , + ( l - p ) 6 _ , O ^ s(x) s{y) v  = s(y)  y j  J  (p + X [ p ( l - p ) ] ^ +  +  y[p  2  ( l - p ) ] } [s(x)-s(y)].  In order t h a t the d e r i v a t i v e s of h w i t h r e s p e c t t o p i s always we  a g a i n r e q u i r e X t o be  zero.  The  (4.63)  2  +  positive,  d e p e n d e n c e 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  Xo  term i s given H ( P < 5  As p t e n d s p equals This  The  r e s u l t a n t h without  s(x)  +  C 1 - P ) < 5  s(y)  )  =  § i y )  +  { p  +  h converges t o s(x) + y  ^[P ^ "? )]} 2  h i s equal t o s(x) which i s l e s s than  implies that the  l o t t e r y p8  c o n c l u s i o n t h a t fi v i o l a t e s i f we  x  +(l-p)6  y  Instead of extending s h a l l pause to take  2  [5(x)-s(y)]. Yet,  when  i s p r e f e r r e d to g e t t i n g the  o f ii as p t e n d s  close to u n i t y .  to  holds  The  for negative  zero.  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,  s t o c k o f what we  function, Allais  (4.64)  s(x) + y [ s ( x ) - s ( y ) ] .  s t o c h a s t i c dominance s t i l l  consider the behavior  of a u t i l i t y  1  [s(x) - s ( 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  y  the  by:  t o one,  one,  term.  learned.  In p l a c e o f the  asserted that preference  we  expectation  i s represented  by  130  a functional  t h a t d e p e n d s on t h e f i r s t  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 which i s obtained position.  of a psychological value function,  f r o m c o m p a r i s o n among h y p o t h e t i c a l c h a n g e s i n w e a l t h  B a s e d on t h e p r o p e r t y o f t h e 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 preference  moment, t h e d i s p e r s i o n and i n  s c a l e , we o b t a i n e d  functionals.  a restriction  on t h e c l a s s  of admissible  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 b y A l l a i s i s not  admissible.  N e x t , we c o n s i d e r e d  d e p e n d o n l y on t h e f i r s t the  sum o f t h e f i r s t  some c o n s t a n t , Extending  those  two moments.  moment  admissible functionals that  The r e s u l t a n t f u n c t i o n a l  and t h e s t a n d a r d  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 form t h a t Hagen (1979) c o n s i d e r e d  functional  form  be  d e v i a t i o n s c a l e d by  i s shown t o b e 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 d o m i n a n c e .  functional  It  i s however a g a i n  i n a recent paper.  shown t o v i o l a t e  This  s t o c h a s t i c dominance.  i s n o t known w h e t h e r t h e p r o b l e m w i t h s t o c h a s t i c d o m i n a n c e c a n  a v e r t e d b y i n c o r p o r a t i n g e v e n h i g h e r moments.  the  form,  first  t h r e e moments s u g g e s t s  The d i f f i c u l t y  with  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 w o u l d 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 that  expected  utility  introduce prospect  fails  t o capture.  In the next  theory, which represents  problem of d e s c r i p t i v e v a l i d i t y A l l a i s v i a t h e famed A l l a i s  o f expected  paradox.  s u b - s e c t i o n , we  a different utility  a p p r o a c h t o the,  first  identified  by  131  4.5.2  Prospect Prospect  Theory  theory, developed  b y Kahneman a n d T v e r s k y  d i s t i n g u i s h e s two phases i n t h e c h o i c e p r o c e s s : which problem 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 to the offered prospects introduce these the  editing  an e d i t i n g phase i n  editing  operations  f o l l o w e d b y an e v a l u a t i o n p h a s e . operations  before  We  are  7  applied  first  discussing their relation to  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 . Coding.  The p e r c e p t i o n o f outcomes i n t e r m s o f g a i n 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 be  (1979),  the current  asset p o s i t i o n ,  level.  This  and l o s s e s  i s u s u a l l y taken t o  i n which case gains  and l o s s e s a r e t h e a c t u  a m o u n t s t o be r e c e i v e d o r p a i d . Cancellation.  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 shared cancellation  Segregation. as a minimum g a i n  An o f f e r e d p r o s p e c t  with a riskless  component  ( l o s s ) i s decomposed i n t o a r i s k l e s s component t a k e n  The p r o b a b i l i t i e s  are combined t o y i e l d  restatement  principle.  with the riskless  Combination.  the  An e x a m p l e i s t h e  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  of Savage's s u r e - t h i n g  prospect  by t h e o f f e r e d prospects.  from each  such  component and t h e  outcome.  a s s o c i a t e d w i t h equal  a s i n g l e outcome w i t h p r o b a b i l i t y  outcomes  g i v e n by  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.  from t h e choice  s e tp r i o r  Simplification. prospects  by rounding  The d o m i n a t e d p r o s p e c t s  are eliminated  to evaluation.  This r e f e r s t o the possible s i m p l i f i c a t i o n o f off probability  o r outcome v a l u e s .  132  The empirical  c o d i n g and evidence  coding, the  the  cancellation  o p e r a t i o n s a r e p r o m p t e d by  d e s c r i b e d i n Chapter  3 s e c t i o n 4.  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  terms o f gains  and  losses.  Although  an  o f c h o i c e phenomena. w h i c h seems t o be The  to  i n c o r p o r a t i n g such  i s known a b o u t t h e  an  description  simplification  operation,  o p e r a t i o n s are r e l a t e d t o the e v a l u a t i o n phase, a value  f u n c t i o n v(x)  o f t h e o u t c o m e x and  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 b t a i n the  the  a p l a u s i b l e problem r e p r e s e n t a t i o n h e u r i s t i c .  t h e way  d e c i s i o n weight  of  known a b o u t  a d d i t i o n a l degree o f freedom i n the  Even l e s s  other e d i t i n g  which concerns  a result  are always s t a t e d i n  more r e m a i n s t o be  c o n d i t i o n s under which c a n c e l l a t i o n a p p l i e s , o p e r a t i o n does y i e l d  As  the  overall  c a v e f o r g a i n s and  value  of a prospect.  convex f o r l o s s e s .  The  The  o f an o u t c o m e p value  a combine  function i s  TI f u n c t i o n h a s  the  con-  following  properties:  TT(0)=0  1) TT i n c r e a s e s f r o m 2)  Tr(p)>p, f o r s m a l l  3)  Tr(p)+Tr(l-p)<l,  4)  TT(pq)/TT(p)  Prospect P  theory  = 6 P  x +  q6  y +  is  f o r pe(0,  , f o r p,q,r  e(0,l].  f o r simple prospects  of the  form, (4.65)  o  n o n - z e r o outcomes. p  and  positive  q add  following  I f the  to unity,  the  (negative) prospect.  For r e g u l a r prospects, the o v e r a l l TT i n t h e  1).  TT(pqr)/TT(pr)  regular i f i t i s neither strictly  v and  = 1.  (l-p-q)6 ,  ( n e g a t i v e ) and  known a s a s t r i c t l y  TT(1)  p.  i s developed  w h i c h h a v e a t most two positive  <  to  manner.  p o s i t i v e nor  outcomes a r e  strictly  simple prospect A  simple  strictly  value V i s obtained  from  is  prospect negative. the  scales  133  V(p6  x +  q6  y +  = T:(p)v(x)+TT(q)v(y).  (l-p-q)6 ) 0  U n l i k e t h e vonNeumann-Morgenstern u t i l i t y , a ratio  scale,  was n o t e d  i . e . v vanishes  i n Edwards  (1961).  thevalue  TT(p)+TT(s-p) V(p6  x +  x +  6 ) 0  property  thenonlinearity  with  o f IT.  TT(p')+TT(s-p'),  to  j V(p'6  In o t h e r words, two l o t t e r i e s  x +  (s-p')6  each y i e l d i n g  s arenot equivalent i npreference. agent  This  T h e r e a r e h o w e v e r some d i f f i c u l t i e s  i s n o t equal  (s-p)6  function v(x) i s  at the reference point.  t h e u s e o f t h e above e x p r e s s i o n , stemming from Since  (4.66)  x +  6 ).  (4.67)  0  outcome x w i t h  To g e t a r o u n d t h i s ,  i s assumed t o a p p l y t h e c o m b i n a t i o n  editing  probability  the choice  operation p r i o r to  evaluation. The  other d i f f i c u l t y  operation.  Consider  necessitates the  the following  Tr(p)v(x)+TT(s-p)v(x+e) S u p p o s e t h e L.H.S. i s l e s s then e.  t h e i n e q u a l i t y would  TT(S)V(X)  (more) t h a n  evaluation circumvents  (4.68)  t h e R.H.S. f o r e e q u a l  Eliminating  t h e dominated  . . some p o s i t i v e  We c a n f i n d  '  q°<s s u c h  This has t h e i m p l i c a t i o n  0  that  PV(S-P)« O (1-S)« - < q X+£  which i s both  x +  (4.69)  that 0  ps6  Suppose  o £ .  TT(p)v(x)+TT(s-p)v(x+e )<Tr(q )v(x+e°)<iT(s)v(x) .  and  (negative)  alternative  t h i s problem, b u t n o t completely.  Tr(p)v(x)+7T(s-p)v(x+e°)<Tr(s)v(x), for  t o zero,  hold f o ra small but positive  Thus, dominance i s v i o l a t e d .  before  comparison: ?  still  detection-of-dominance  (l-s)6  +  0  >• q %  0  0 6 x + £  o+ll-q°)V  o d-q°)6 , +  + e  0  n o r m a t i v e l y and e m p i r i c a l l y  untenable.  (4.70)  134  v  For the s t r i c t l y run  into  positive  and s t r i c t l y  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  expression  (4.66).  Instead,  decompose a s t r i c t l y  y>x>0 (0>x>y) i n t o P' = P5Q+(1-P)<S^_ following  positive  and e v a l u a t e  prospect  v i a the  negative  x  expression t o evaluate  prospects  by the e d i t i n g  P = pS +q5  x + e  operations. 0  =  strictly  l e a d s t o a new d i f f i c u l t y  +(l-p-q)6 , with  V(P)  (4.71)  y  use o f a d i f f e r e n t  covered  people  expression: x  strictly  adopt  component  V(p6 +(1-P)6 ) = v(x)+TT(l-p)[v(y)-v(x)]. The  will  P = p6 + ( l - p ) 6 with x y  and a r i s k y  the segregated  we  proposed that  (negative) prospect  component 6  prospects,  o f d o m i n a n c e i f we  Kahneman a n d T v e r s k y  a riskless x  negative  Consider  positive or which i s not  the regular  prospect  x,£>0.  Tr(p)v(x)+Tr(q)v(x+e)  < TT(p)v(x)+Tr(l-p)v(x+e) . Since V i s concave f o r p o s i t i v e V(P) C h o o s e £Q s u c h  < T T ( p ) v ( x ) + T r ( l - p ) [ v ( x ) + e v ' (x) ] .  1-TT(P)-TT(1-P) TT(l-p)  x  + q 6  x+£  + ( 1 o  - " p  q ) 6  0  F o r t h e c a s e q = l - p , we a p p l y V(p)  -< V  V q £[0, 1 - p ) .  expression  = v(x)+TT(l-p) [v(x+e-v(x)]  (4.69) and o b t a i n : >v(x).  (4.74)  implies that p6 r  The  (4.73)  implies that, p 6  This  (4.72)  that  '0 - v ' ( x ) This  x,  +(l-p)5 >x ^ * x+e J  above i m p l i c a t i o n  q gets  5.  i s rather pathological.  t o 1-p, t h e L.H.S. i s s t r i c t l y  No m a t t e r  worse than  6 .  how  close  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 feeling x,  f o r t h e problem by estimating  p a n d TT(P).  property  i  U  [1-2TT(0.5)] TT(0.5)  x  the concavity  A conservative e  P  2  P  estimate  long  TT(0  .50) = .45 .  (Note  i m p l i e s t h a t TT ( 0 . 5 ) < 0 .5) .  v(10)  that  We h a v e  [1-2TT(0.5)] TT(0.5) •  that,  l^-/i>J  v'(10)  f o rS  Q  i sthen given by  •  4 5 6  $122  +  •  0 5 6  $o  vs.  6  $100  • $ioo  +  •  4 9 6  $122  +  •  o l 6  $o  vs.  6  $100  +  •  4 9 9 6  vs.  6  $100  5 6  E  problems.  +  Prospect theory  as  f o r some r e a s o n a b l e v a l u e s o f  • $ioo 5 6  E  3 ~= •  t h a n "SjjjTQQ-  <  We c a n g e t some  of v implies that x < v(x) .  Consider the f o l l o w i n g choice  l  q  to 6 .  = 10 x .1 i .45 = 22 " •  o  p  £  S u p p o s e x = $ 1 0 0 , p=.50 a n d  (3) o f t h e T I function o -  since  preferred  5 6  $ioo  $122  would p r e d i c t t h a t  .0018  +  P^,  P^,  $ 0  P^ a r e a l l s t r i c t l y  Moreover, t h e d i r e c t i o n o f preference  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  worse  remains unchanged  $122 i s l e s s t h a n 0.5!  Note  that  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) s i n c e v'(x) i s a decreasing f u n c t i o n f o r a concave v ( x ) . We c a n a r r i v e a t 0  the  above c o n c l u s i o n  v(x).  Ifv  a g e n e r a l l y much h i g h e r  i s bounded, then v'(x)  made a r b i t r a r i l y  large  Prospect theory first  with  i f we c o n s i d e r  o f choice  q  given  a sufficiently  q  c a n be  l a r g e x. function  I ttreats systematically  phenomena ( c f . C h a p t e r 3) t h a t v i o l a t e  of expected u t i l i t y .  a specific  so that £  b u i l d s on t h e f o r m o f t h e e v a l u a t i o n  s u g g e s t e d b y Edwards (1955) .  classes  tends t o zero  £  several  the implications  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  f u n c t i o n TT, w h i c h c o n s t i t u t e s i t s m a i n d e v i a t i o n f r o m e x p e c t e d  weight  utility,  136  generates  some s e r i o u s d i f f i c u l t i e s  ones, namely t h e v i o l a t i o n  The i m m e d i a t e  o f t h e 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 editing operations.  f o rprospect theory.  v i athe combination  Two p r o b l e m s h o w e v e r r e m a i 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 One o f t h e p r o b l e m s  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 prospect Q which  worse than  b e t t e r than another prospect  some p r o s p e c t P b u t s t r i c t l y  t h a t d o m i n a t e s P. evaluation  The o t h e r p r o b l e m  function f o rthe s t r i c t l y  gives r i s e to discontinuity untenable predictions  o  i s strictly P'  h a s t o do w i t h t h e u s e o f a d i f f e r e n t positive  (negative) prospects.  i n the preference represented which  o f actual choice behavior.  This  leads t o  137  4.5.3  Comparison We  Allais' fi(F) c a n  conclude theory  the s e c t i o n w i t h a comparison o f alpha u t i l i t y  and p r o s p e c t  theory.  The a l p h a u t i l i t y  with  representation  be s t a t e d i n t e r m s 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 (x)  'a-weighted'  = /JW  pa  F  g i v e n by:  / /^adF,  where a i s a s t r i c t l y if  distribution  positive  function.  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 ,  The r o l e o f a becomes c l e a r e r n F = E p.6 . In t h i s case, i=l i 1 x  fl(F) = / v d F  a  R  n = E q i=l where q  (F)v(x ),  (F) = p a C x ^ / S p . a ( x j=l  ) .  J  Like prospect  t h e o r y , fi(F) i s o b t a i n e d  of "decision weights", of the p r o b a b i l i t y note  rest  combination to  x,  n  The q ^ w e i g h t s  theory.  We s h o u l d  Finally,  d e p e n d s on t h e  t h e q^-weights has t h e  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 a r e b o t h  then fi f p . 6 +p. 6 i x k x r  r  + E p. 6 .) • • /• , i x i  [p'.a(x)+p  a(x)]v(x)  n E p.a(x.)  + E  however  sum t o u n i t y b u t n o t t h e TT(P  I n a d d i t i o n t o p ^ , q^  and a l l t h e x ^ ' s .  s  a nonlinear function  p^ o f o b t a i n i n g t h e i t h outcome, x^.  of prospect  of the P j '  the v(x^)'s v i a a set  ^4^^ ^_2» w i t h e a c h q ^ b e i n g  three d i s t i n c t i o n s .  weights  from  p a(x.)v(x )  equal  )  138  .(P P ) U) (X) +  a  i  v  +  P a(x )v(x )  1  k  i  i  i  iili  =  n E p.a(x.) i=l 1  =  n p ((  + P k  Some o f A l l a i s ' alpha  utility  )6  x  p 6  E  +  ideas  1  can  ).  a l s o be  representation.  We  expressed  can w r i t e  fi(F)  i n terms o f as t h e  sum  the  of  v,  — the  first  moment o f v w i t h  shown b e l o w : £}(F)  = v  +  respect  t o F,  and  the  d e v i a t i o n term  ct  / (v-v)dF D  / (v-v)dF , a  R  w h e r e v" = .Lvd'F. K  Thus, the p r e f e r e n c e the  first  o f an  moment o f v ,  (v-v), of v  alpha  and  a l s o the  o f the  representation  t h e v - f u n c t i o n , much l i k e  may  insist  determined in  the von  from p r e f e r e n c e  distribution  of the  F .  According  a  1 and  Neumann-Morgenstern u t i l i t y , a standard  a psychological value  t o weed o u t  the  can  apply  "psychological  value"  2, can  lottery.  function that  f r o m i n t r o s p e c t i v e c o m p a r i s o n among h y p o t h e t i c a l I n t h i s c a s e , we  on  deviation,  theorems i n Chapters  comparisons using  h o w e v e r t h a t v be  wealth p o s i t i o n .  razor  d e c i s i o n maker depends  from v through the a-weighted d i s t r i b u t i o n  to the proofs  constructed  utility  be Allais  is  changes  t h e p r i n c i p l e o f Occam's  assumption since  i t is  redundant. Finally,  we  s u m m a r i z e v i a T a b l e 4.2  of the  theories treated  theory  i s compatible with  a  '-'  sign.  in this the  chapter.  property  some o f t h e A  '+'  salient  s i g n means t h e  referred to.  features corresponding  O t h e r w i s e , we  use  139  Properties  Expected  Allais » Paradox  Transitivity  Dominance  +  +  +  +  -  Utility  Combination  Continuity  Alpha  Utility  +  +  +  +  +  Allais  Theory  +  -  +  +  +  -  +  +  -  +  Prospect  Theory  Table  4.2:  C o m p a r i s o n among  A novelty o f Prospect  theory  prior  Two o f t h e s e  to evaluation.  i s the e x p l i c i t  ensure c o n s i s t e n c y w i t h dominance  (cf. and  systematic  expression  intransitivity,  (4.70)).  losses, the coding  how p e o p l e  Since  and c o m b i n a t i o n .  o p e r a t i o n s , namely  adopted by a l p h a u t i l i t y their  validity  I f Allais'  as i s t h e case  lotteries  f o r prospect  outcomes.  cancellation  theory  theory  theory  are s t a t e d i n terms o f gains  o p e r a t i o n seems a r e a s o n a b l e  p e r c e i v e monetary  operations  e d i t i n g o p e r a t i o n , i t would  hypothesis  Kahneman and T v e r s k y  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 editing  use o f e d i t i n g  o p e r a t i o n s a r e however needed t o  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 exhibit  Theories  i n support  and s e g r e g a t i o n .  about (1979)  o f two o t h e r T h e s e may be  i ffuture empirical studies ascertain  as b e h a v i o r a l h y p o t h e s i s .  140  CONCLUSION  141  5. CONCLUSION  5.1  SUMMARY  Part  I of this dissertation  o f two r e p r e s e n t a t i o n  theorems.  contains  t h e s t a t e m e n t s and p r o o f s  The f i r s t  generalizes  the quasilinear  mean, M^, o f H a r d y , L i t t l e w o o d a n d P o l y a :  V ) F  =  l*~V 0dF),  where 0 i s c o n t i n o u s distribution.  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  We h a v e w e a k e n e d a c h a r a c t e r i s t i c p r o p e r t y  l i n e a r mean, M^, value, M  (5.1)  R  called quasilinearity to obtain  a 0  R  R  where a i s c o n t i n u o u s and s t r i c t l y inequality,  <, b i n a r y  F -< G <^> M The  a 0  (5.2) (negative).  r e l a t i o n , the  r e l a t i o n , •-< , among p r o b a b i l i t y (F) <M  a 0  The  form:  f  positive  mean  f u n c t i o n a.  d i s t r i b u t i o n F has the f o l l o w i n g  M ( F ) = 0~V a0dF/.T adF)  strict  a more g e n e r a l  ,, w h i c h i s c h a r a c t e r i z e d b y a n a d d i t i o n a l  mean o f a p r o b a b i l i t y  of the quasi-  Through t h e  mean i n d u c e s a  binary  distributions: (G).  (5.3)  R.H.S. i s e q u i v a l e n t t o  / a0dF// adF < 7 a 0 d G / / a d G , R  for a strictly (5.4)  R  R  i n c r e a s i n g 0.  i s more g e n e r a l  representation An i s given simple  by t h e expected  by  utility  / ajzSdF// adF. R  R  approach t o obtain the  i n C h a p t e r 2.  represented  F o r c o n v e n i e n c e , we u s e fi(F) t o l a b e l t h e  functional,  alternative  Note t h a t t h e o r d e r i n g  than that represented  r e p r e s e n t a t i o n , f^tfdF.  (5.4)  R  Instead  representation,  of probability  p r o b a b i l i t y measures d e f i n e d  (5.4),  d i s t r i b u t i o n s , we  o n some a r b i t r a r y s e t X -  consider  (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 of 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  number  i n terms o f a  binary  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  simple  p r o b a b i l i t y measures: fi(P) = E ( a v , P ) / E ( c t , P ) ,  where P i s a s i m p l e real-valued on  (5.5)  p r o b a b i l i t y measure on a s e t ,  f u n c t i o n s on -  This  X  a x i o m s on mean v a l u e s  i s contrasted  i n C h a p t e r 1.  approach cannot be extended t o s i m p l e set X  i s t h a t mean v a l u e  consider Being  t h e outcome s e t  able  t o deal  with  X  - {status  .j  w  e  and v a r e  a  the approach based  The r e a s o n why t h e mean  i t s e l f may n o t be d e f i n e d quo, b e i n g  value  in X «  As an e x a m p l e ,  promoted, being  fired).  an a r b i t r a r y o u t c o m e s e t s u c h a s t h e a b o v e  There a r e however c e r t a i n drawbacks. X  with  "  p r o b a b i l i t y m e a s u r e s on an a r b i t r a r y  e x a m p l e i s an a d v a n t a g e f o r t h e s i m p l e  on  a n a  X  cannot d i s c u s s t h e n o t i o n s  p r o b a b i l i t y measure  approach.  Without f u r t h e r s t r u c t u r a l  assumptions  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 a n d v f u n c t i o n s n o r c a n 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 Part decision  p r o b a b i l i t y measures.  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 theory.  Interpreting the  of a monetary l o t t e r y has ,  A  0(F)  o f C h a p t e r 2. represents  (finite)  lotteries,  The h y p o t h e s i s  a choice  alpha u t i l i t y  mean o f C h a p t e r  F, t h e c o r r e s p o n d i n g  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  F o r non-monetary  as  M  This  induced  preference'  we a p p l y  t h e c h o i c e a g e n t o b e y s e i t h e r t h e mean v a l u e o r t h e a x i o m s ;on t h e '-< ' ( s t r i c t  preference)  relation,K  between  equivalent  theorem  o f e i t h e r approach i s referred to to saying  that  (certainty equivalent) binary  ,  lotteries.  the representation  among l o t t e r i e s  i s logically  of application  —  1 as t h e c e r t a i n t y e q u i v a l e n t  binary  t h a t t h e fi r e p r e s e n t a t i o n  agent's preference  theory.  area  relation.  axioms  143  Alpha u t i l i t y in  the  theory  i s a g e n e r a l i z a t i o n of expected  sense t h a t the expected  of the alpha u t i l i t y  utility  and  r e p r e s e n t a t i o n , and  Schlaifer,  that alpha u t i l i t y utility, called  counterpart of s u b s t i t u t a b i l i t y  The ties  ( 1 9 5 0 ) and  the A l l a i s  of normative  and  stated c o n d i t i o n s , taken  d o m i n a n c e and  global risk  index of l o c a l v  risk  ended w i t h a  5.2  of  1,  aversion.  We  difficul-  especially  3.  o f a s s u m p t i o n s and  the d e r i v a t i o n s  theory.  for consistency with stochastic  a l s o d e m o n s t r a t e d how  s t o c h a s t i c d o m i n a n c e and  Arrow-Pratt  a p a i r of a local  risk  and  aversion  c h o i c e phenomena, summarized i n C h a p t e r  i m p l i c a t i o n s of expected  a t t e n t i o n ; namely, A l l a i s '  utility.  w i t h two  The  chapter  other theories that  t h e o r y and  prospect  theory.  EXTENSIONS We  The  comes f r o m  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  comparison of alpha u t i l i t y  have a t t r a c t e d  (A c l o s e  c h o i c e phenomena,  statements  from Chapter  consistent with those  that contradicts the  utility  i m p l i c a t i o n s of alpha u t i l i t y  f u n c t i o n s t h a t s a t i s f y both  can be  substitutability  Polya.  summarized i n Chapter  formal  descriptive  a  (1952)),  These are  4 c o n t a i n s the  assumes  s t r o n g independence p r i n c i p l e  i n the d e s c r i p t i o n of c e r t a i n  paradox.  Chapter  We  Samuelson  i s the  motivation f o r g e n e r a l i z i n g expected  i t faced  case  1 9 6 4 ) , 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 H a r d y , L i t t l e w o o d and  Marschak  theory  representation i s a special  weaker form o f a key p r o p e r t y o f expected (Pratt, Raiffa  utility  conclude  by p o i n t i n g out  q u a s i l i n e a r mean was  tributed-equivalent'  level  d i s t r i b u t i o n by A t k i n s o n inequality.  We  may  given  use  some p o t e n t i a l  an i n t e r p r e t a t i o n  areas  as t h e  o f income c o r r e s p o n d i n g  (1970) i n h i s p a p e r  of  t o an  application. 'equally-disincome  on t h e m e a s u r e m e n t  t h e M ^ mean as a more g e n e r a l m o d e l  of  of  3,  144  equally-distributed-equivalent general  measure?  income.  Is there  C o n s i d e r two s o c i e t i e s w i t h  G given by:  a n y n e e d f o r a more  income d i s t r i b u t i o n s F and  )  F  E  °-  5 0 6  G  E  °-  5 0 6  $l,000  °-  +  $i,ooo  +  5  °-  0  6  4 9 6  $2,000> $2,ooo  M  +  D  °-  o l 6  $i ooo,ooot  To many ( i n c l u d i n g p r o b a b l y Mao T s e - T u n g ) , s o c i e t y F f a r e s b e t t e r 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  distribution,  so t h a t M^(F) i s l e s s than M^(G).  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 t h o s e who b e l i e v e any  difficulty  that  for the  measure s i n c e  departure o f M  (F)  =  *  A remarkable feature  (  E  consistency  with  stochastic  , f r o m M, c a n be made c l e a r e r i f we c o n s i d e r  N  incomes, {x^}^_^.  equally-distributed-equivalent N a«S  T h i s does n o t pose  property.  society o f N individuals with  M  Therefore, the  o f t h e two s o c i e t i e s f o r  society F i sbetter off.  dominance i s n o t an i n t r i n s i c The  than  ( i=l a  x i  )  o f (5.6)  v  The c o r r e s p o n d i n g  i s given by: N (  x  i  )  /  Z  Or>)i=l  C - )  a  5  i s the presence o f complementarity  incomes o f d i f f e r e n t i n d i v i d u a l s . r e i n f o r c e d by the f a c t t h a t  a  6  across  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  an i n d i v i d u a l p e r c e i v e s  concurrently the  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  set o f mutually e x c l u s i v e  We c a n  think  o f t h e r o l e o f a as a s s i g n i n g  b a s e d on t h e i r a t t a i n e d for  outcomes w i l l o b t a i n  incomes.  folks.  discriminatory  w e i g h t s on i n d i v i d u a l s  Mao T s e - T u n g ' s ( h y p o t h e t i c a l )  s o c i e t y F may t h e n be e x p l a i n e d  function a that  i n a lottery.  i n terms o f a decreasing  t r e a t s wealthy i n d i v i d u a l s less  'equally'  preference  discrimination  than t h e poorer  145  In son  a r e c e n t p a p e r on t h e m e a s u r e m e n t o f p o v e r t y ,  (1978b) a p p l i e d A t k i n s o n ' s  'censored'  distribution,  equally-distributed-equivalent  i . e . , t h e income d i s t r i b u t i o n  exogeneously e s t a b l i s h e d poverty  line.  m e a s u r e w i t h an a t h a t i s c o n s t a n t beyond. point due  I t i s n a t u r a l t o suggest  at the poverty  line  sentative o f the poorer  5.1:  The as  to the  t r u n c a t e d a t some  up t o t h e p o v e r t y  inflexion  i n t e g r a t e thec o n t r i b u t i o n  a n d a t t h e same t i m e b e p a r t i c u l a r l y  f o l k s w i t h incomes below t h e p o v e r t y  repre-  line.  Income  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  mean c a n a l s o b e 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  follows: Relative  I n e q u a l i t y Index = 1  Absolute  I n e q u a l i t y Index = y -  - M ^ / y , and M^,  a  l i n e and z e r o  t h a t a d e c r e a s i n g a w i t h an  poverty line  Fig.  and D o n a l d -  This i s tantamount t o having  ( s e e F i g u r e 5.1) w o u l d  t o t h e whole d i s t r i b u t i o n  Blackorby  146 w h e r e y d e n o t e s t h e a r i t h m e t i c mean. Atkinson  (1970) and t h e a b s o l u t e  The r e l a t i v e  i n d e x i s due t o K o l m  the  distribution  The  c o e f f i c i e n t o f v a r i a t i o n , an o c c a s i o n a l  is  o f income i s c o m p l e t e l y e q u a l ,  related to  g  t  ( F ) = {/QX  o f v a r i a t i o n = {M  S + t  dF//~x dF} S  /  is  always higher  progressively  offer  An  t  \. -y } , r a  c  S  and  0(x) = x  i s undesirable  Also,  i t s weighting  l  ) .  because i t  f u n c t i o n a ( x ) =• x  mean s u g g e s t e d a b o v e may b e o f u s e i n s t a t i s t i c s .  assigns  We  some e x a m p l e s . M  = standard deviation  • c o e f f i c i e n t o f skewness, and  M  = standard deviation  • (coefficient of Kurtosis  generalizes  t h e w e l l known r e s u l t t h a t  a r i t h m e t i c mean and h a r m o n i c mean i s e q u a l  t 2t  a  S  + 3) . 2  the product of  t o t h e square o f t h e  g e o m e t r i c mean f o r t w o p o s i t i v e n u m b e r s , may b e s t a t e d ^  (5.7)  more w e i g h t t o t h e more w e a l t h y .  equality that  the  zero.  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 , a n d s o  encourages i n e q u a l i t y .  g  both indices equal  (i.e. a(x) = x  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  M  When  measure o f income i n e q u a l i t y ,  ,(F) 1>1  As  The  (1976a,b).  by:  coefficient  where M  i n d e x i s due t o  i n terms o f  f°ll° w s  M  ( g e o m e t r i c mean) ^  (5.8)  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  -t,2t  where F = £ ^ 5  ordinate  Y  ( F ) = M, (F), l o g x^ , with  a t -j^E l o g x ^ .  b  > 0,  Canning  (1934) proposed  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 m e a s u r e o f a s y m m e t r y . Harsanyi a moral  (1977) a p p l i e d e x p e c t e d  judgement about a l t e r n a t i v e  utility  social  t o an i n d i v i d u a l  situations.  making  Making a moral  j u d g e m e n t , i n t h i s c a s e , means m a k i n g a h y p o t h e t i c a l b e s t c h o i c e u n d e r the assumption member o f is  that the individual  the s o c i e t y w i t h equal  a listing  o f the N persons'  that assigns the i n d i v i d u a l with  1/N c h a n c e .  the expected  chance.  A social  situation  X,  s t a t e s , would be p e r c e i v e d as a  t o any p a r t i c u l a r  An e x p e c t e d  utility  assumes t h e p o s i t i o n o f any one  utility  corresponding  individual's  lottery  position  d e c i s i o n maker would t h e n  to social  situation X .  which  maximize  This i s simply the  N arithmetic average, E u . ( X ) / N , o f h i s von Neumann-Morgenstern u t i l i t y , i=l 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 1  d e c i s i o n m a k e r w i l l h o w e v e r m a x i m i z e 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 s h o e s w i t h i n the  situation X .  social  weights  This i s a weighted  g i v e n b y t h e OL(X)'S.  Expression  average o f the v ^ ( X ) ' s w i t h ( 5 . 1 0 ) may be i n t e r p r e t e d a s  the average o f t h e v . ( X ) ' s but t h e alpha u t i l i t y 1  a  'biased' estimate,  a.(X)/ 1  ith  person.  Again,  N E a i=l  how m o r a l  f o r h i s p r o b a b i l i t y of being the  . ( X ) , 1  the alpha u t i l i t y  a l l o w i n g f o rcomplementarity,  d e c i s i o n maker u s e s  which  e x p r e s s i o n has t h e advantage o f  i s supported  by our i n t u i t i o n  about  j u d g e m e n t s a r e made.  S e v e r a l q u e s t i o n s remain  untouched.  There a r e o t h e r  situations  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 expected  utility.  One e x a m p l e i s t h e a v e r a g i n g  of individual  utility  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 weights t o o b t a i n a group u t i l i t y (Keeney of  and R a i f f a ,  1976).  Another  example i s t h e time-honored  d i s c o u n t i n g a time stream o f u t i l i t i e s  that  complementarity  s h o u l d n o t be r u l e d  F o r t h e s e and o t h e r s i m i l a r e x a m p l e s , provides a useful in  first  based  have n o t are  comparative to  always  equivalent  risk  result  alpha u t i l i t y  Another  aversion.  The  appeared  in  a cardinal  condition  are  choice agents. f o r one  One  example i s  expected u t i l i t y  (1964) .  The  An  corresponding  mean, h o w e v e r , r e m a i n s was  result  an open q u e s t i o n .  confirmed  (1977), t h a t Arrow's i m p o s s i b i l i t y theorem and  maximizer  P o l y a (1934) i n t h e  (1967) c o n j e c t u r e , w h i c h  s e t u p where i n d i v i d u a l s  We  f o l l o w o r what o l d r e s u l t s  q u a s i l i n e a r means.  or the  by von Neumann-Morgenstern u t i l i t y  will  by hold  society express t h e i r preferences  functions.  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 and  structure  on  a l l o f these works  i n H a r d y , L i t t l e w o o d and  c h o i c e agents  Schmeidler  in the l i t e r a t u r e  averse than another i s given i n P r a t t  example i s Samuelson's  K a l a i and  cases.  attributes.  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