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### Transcript of Unreliable Probabilities

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P E T E R G A RD E N F O RS A N D N I L S - E RI C S A H L I N *

U N R E L I A B L E P R O B A B I L I T I E S ,R I SK TA K I N G , A N DD E C I S I O N M A K I N G

1. THE LIMI TATION S OF STRICT BAYESI ANISMA central part of Bayesianism is the doctrine that the decisionmaker's knowledge in a given situation can be represented by asubjective probability measure defined over the possible states of theworld. This measure can be used to determine the expected utility forthe agent of the various alternatives open to him. The basic decisionrule is then that the alternative which has the maximal expectedutility should be chosen.

A fundamental assumption for this strict form of Bayesianism isthat the decision maker's knowledge can be represented by a uniqueprobability measure. The adherents o f this assumption have produceda variety of arguments in favor of it, the most famous being theso-called Dutch book arguments. A consequence of the assumption,in connection with the rule of maximizing expected utility, is that intwo decision situations which are identical with respect to the prob-abilities assigned to the relevant states and the utilities of the variousoutcomes the decisions should be the same.

It seems to us, however, that it is possible to find decision situa-tions which are identical in all the respects relevant to the strictBayesian, but which nevertheless motivate different decisions. As anexample to illustrate this point, consider Miss Julie who is invited tobet on the outcome of three different tennis matches. 1 As regardsmatch A, she is very well-informed about the two players - she knowseverything about the results of their earlier matches, she has watchedthem play several times, she is familiar with their present physicalcondition and the setting of the match, etc. Given all this information,Miss lulie predicts that it will be a very even match and that a merechance will determine the winner. In match B, she knows nothingwhatsoever about the relative strength of the contes tants (she has noteven heard their names before) and she has no other information thatSynthese 53 (1982) 361-386. 0039--7857/82/0533-0361 \$02.60Copyright 1982 by D. Reidel Publishing Co., Dordrecht, Holland, and, Boston, U.S.A.

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362 P E T E R G . A R D E N F O R S A N D N I L S - E R I C S A H L I N

i s r e l ev an t f o r p r ed i c t i n g t h e w i n n e r o f t h e ma t ch . M a t ch C i s s i m i la rt o m a t c h B e x c e p t th a t M i s s J u l ie h a s h a p p e n e d t o h e a r t h a t o n e o ft h e c o n t e s t a n t s i s a n e x c e l l e n t t e n n i s p l a y e r , a l t h o u g h s h e d o e s n o tk n o w a n y t h i n g a b o u t w h i c h p l a y e r i t i s , a n d t h a t t h e s e c o n d p l a y e r i si n d e e d a n a m a t e u r s o t ha t e v e r y b o d y c o n s i d e rs t h e o u t c o m e o f th em a t c h a f o r e g o n e c o n c l u s i o n .

I f p r e s s e d t o e v a l u a t e t h e p r o b a b i l i t i e s o f t h e v a r i o u s p o s s i b l eo u t c o m e s o f t h e m a t c h e s , M i s s J u l i e w o u l d s a y t h a t i n a l l t h r e em a t c h e s , g i v e n t h e i n f o r m a t i o n s h e h a s , e a c h o f t h e p l a y e r s h a s a 5 0%ch an ce o f w i n n i n g . I n th i s s i t u a ti o n a s t r ic t B ay es i a n w o u l d s ay th a tM i s s J u li e s h o u l d b e w i ll in g t o b e t a t e q u a l o d d s o n o n e o f t h e p l a y e r sw i n n i n g i n o n e o f t h e ma t ch es i f an d o n l y i f s h e i s w i l l i n g t o p l ace as i m i l a r b e t i n t h e t w o o t h e r m a t c h e s . I t s e e m s , h o w e v e r , p e r f e c t l yr a t i o n a l i f M i s s J u l i e d e c i d e s t o b e t o n m a t c h A , b u t n o t o n B o r C ,f o r t h e r e a s o n t h a t a b e t o n m a t c h A i s m o r e r e l i a b l e t h a n a b e t o n t h eo t h er s . F u r t h e r m o r e s h e w o u l d b e v e r y s u s p i c i o u s o f a n y o n e o f f e ri n gh e r a b e t a t e q u a l o d d s o n m a t c h C , e v e n i f s h e c o u l d d e c i d e f o rh e r s e l f w h i c h p l a y e r t o b a c k .T h e m a i n p o i n t o f t hi s e x a m p l e i s t o s h o w t h a t t h e a m o u n t a n dq u a l i ty o f i n f o r m a t i o n w h i c h t h e d e c i s i o n m a k e r h a s c o n c e r n i n g t h ep o s s i b l e s t a t e s a n d o u t c o m e s o f t h e d e c i s i o n s i t u a t i o n i n m a n y c a s e si s a n i m p o r t a n t f a c t o r w h e n m a k i n g t h e d e c i s i o n . I n o r d e r t o d e s c r i b et h i s a s p ec t o f t h e d ec i s i o n s i t u a t i o n , w e w i l l s ay t h a t t h e i n f o r ma t i o na v a i la b l e c o n c e r n i n g t h e p o s s i b l e s ta t e s a n d o u t c o m e s o f a d e c i s i o ns i t u a t i o n h as d i f f e r en t d eg r ee s o f e p i s t e m i c r e l i a b i l i t y . T h i s c o n c e p tw i l l b e f u r t h e r e x p l i c a t e d l a t e r . W e b e l i e v e t h a t t h e e p i s t e m i c r e l i -ab i l i t y o f a d ec i s i o n s i t u a t i o n i s o n e i m p o r t a n t f a c t o r w h e n a s s e s s i n gth e r i s k o f t h e d e c i s i o n . I n o u r o p i n i o n , t h e m a j o r d r a w b a c k o f s t r i c tB a y e s i a n i s m is th a t i t d o e s n o t a c c o u n t f o r t h e v a r ia t i o n s o f t h eep i s t emi c r e l i ab i l i t y i n d i f f e r en t d ec i s i o n s i t u a t i o n s .T h e c o n c e p t o f e p i s t e m i c r e l i a b i l i t y i s u s e f u l a l s o i n o t h e r c o n t e x t st h a n d i r e c t d e c i s i o n m a k i n g . I n t h e n e x t s e c t i o n , a f t e r p r e s e n t i n g t h emo d e l s o f d ec i s i o n s i t u a t i o n s , w e w i l l ap p l y t h i s co n cep t i n a d i s -c u s s i o n o f P o p p e r ' s ' p a r a d o x o f id e a l e v i d e n c e ' .

I n o r d e r t o d e t e r m i n e w h e t h e r e m p i r i c a l s u p p o r t c o u l d b e o b t a i n e df o r t h e t h e s i s t h a t t h e ep i s t emi c r e l i ab i l i t y o f t h e d ec i s i o n s i t u a t i o naf fec t s t he dec i s ion , Go ldsmi th and Sah l in [15 ] , [16 ] and [17] per -f o r m e d a s e r i e s o f e x p e r i m e n t s . I n o n e o f th e s e , t e s t s u b j e c t s w e r e

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U N R E L I A B L E P R O B A B I L I T I E S A N D D E C I S I O N M A K I N G 363

first presented with descriptions of a number of events and wereasked to estimate for each event the probability of its occurrence.Some events were of the well-known parlor game type, e.g. that thenext card drawn from an ordinary deck of cards will be a spade; whileother events were ones about which the subjects presumably hadvery limited information, e.g. that there will be a bus strike in Verona,Italy next week. Directly after estimating the probability of an event,subjects were asked to show, on a scale from 0 to 1, the perceivedreliability of their probability estimate. The experiment was con-structed so that for each subject several sets of events were formed,such that all the events in a set had received the same probabilityestimate but the assessed reliability of the various estimates differed.For each set, the subject was then asked to choose between lotterytickets involving the same events, where a ticket was to be con-ceived as yielding a win of 100 SwKr if the event occurred but nomonetary loss if it did not occur. One hypothesis that obtainedsupport in this experiment was that for probabilities other than fairlylow ones, lottery tickets involving more reliable probability estimatestend to be preferred. This, together with the results of similarexperiments, suggested the reliability of probability estimates to be animportant factor in decision making.

The aim of the present paper is to outline a decision theory which isessentially Bayesian in its approach but which takes epistemic reli-ability of decision situations into consideration. We first presentmodels of the knowledge relevant in a decision situation. One devia-tion from strict Bayesianism is that we use a class of probabilitymeasures instead of only one to represent the knowledge of an agentin a given decision situation. Another deviation is that we add a newmeasure which ascribes to each of these probability measures adegree of epistemic reliability. The first step in a decision, accordingto the decision theory to be presented here, is to select a class ofprobability measures with acceptable degrees of reliability on which adecision is to be based. Relative to this class, one can then, for eachdecision alternative, compute the minimal expected utility of thealternative. In the second step the alternative with the largest minimalexpected utility is chosen. 2 This decision theory is then compared tosome other generalized Bayesian decision theories, in particularLevi's theory as presented in [26].

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U N R E L I A B L E PR O B A B I L I T I E S A N D D E C I SI O N M A K I N G 365

o u t t o b e t h e a c t u a l o n e , t h e n t h e a g e n t ' s d e g r e e s o f b e li e f s s a t i s f yK o l m o g o r o f f ' s a x i o m s , i .e ., th e r e i s a u n i q u e p r o b a b i l i t y m e a s u r e t h a td e s c r i b e s t h e s e d e g r e e s o f b e l ie f .

H o w e v e r , a p r e s u p p o s it i o n o f t h is t h e o r e m is t h a t t h e a g e n t b ew i l l in g t o t a k e e i t h e r s i d e o f a b e t , i .e . , i f t h e a g e n t i s n o t w i l l i n g t o b e to n t h e s t a t e s j a t o d d s o f a:b, t h e n h e s h o u l d b e w i ll in g to b e t o nn o t -s j a t o d d s o f b : a . B u t t h i s a s s u m p t i o n m a k e s t o o h e a v y d e m a n d so n p e o p l e ' s w i l li n g n e s s to m a k e b e t s . O n e i s o f t e n n o t w i ll in g t oa c c e p t e i t h e r o f t h e t w o b e t s . 5 I n o u r o p i n i o n , t h i s is e x p l a i n e d b y t h ef a c t t h a t t h e e s t i m a t e d p r o b a b i l i t y o f t h e d i f f e r e n t s t a te s o f n a t u r e a r eu n r e l i a b l e a n d o n e i s n o t w i l li n g t o t a k e t h e r i s k c o n n e c t e d w i t h t h isu n c e r t a i n t y . 6 T h i s c r i ti c is m i s d i r e c t e d a g a i n s t t h e a s s u m p t i o n s b e h i n dt h e D u t c h b o o k t h e o r e m , b u t s i m i l a r c r i t i c i s m c a n b e c o n s t r u c t e da g a in s t o t h e r a r g u m e n t s i n f a v o r o f t h e a s s u m p t i o n o f r e p r e s e n ti n gb e l i e f s b y a u n i q u e p r o b a b i l i t y m e a s u r e .

I n t h i s p a p e r w e w i ll r e l a x t h i s a s s u m p t i o n a n d , a s a f i rs t s t e p i n th ed e s c r i p t i o n o f t h e b e l i e fs w h i c h a r e r e l e v a n t i n a d e c i s i o n s i t u a ti o n ,w e i n s t e a d a s s u m e t h a t t h e b e l i e fs a b o u t t h e s t a te s o f n a t u r e c a n b er e p r e s e n t e d b y a s e t P o f p r o b a b i l i t y m e a s u r e s . T h e i n t e n d e d in t e r -p r e t a t i o n o f t h e s e t ~ i s t h a t i t c o n s i s t s o f a ll e p i s t e m i c a l l y p o s s i b l ep r o b a b i l i ty m e a s u r e s o v e r t h e s t a t e s o f n a t u r e , w h e r e w e c o n c e i v e o f ap r o b a b i l i t y m e a s u r e a s e p i s te m i c a l l y p o s s i b l e i f i t d o e s n o t c o n t r a d i c tt h e d e c i s i o n m a k e r ' s k n o w l e d g e i n t h e g i v e n d e c i s i o n s i t u a t i o n ] I n t h isw a y , w e a s s o c i a t e w i t h e a c h s t a t e s~ a s e t o f p r o b a b i l i ty v a l u e s P ( s j ) ,w h e r e P E ~ . T h e v a l u e s m a y b e c a l le d t h e e p i s t e m i c a l l y p o s s i b l ep r o b a b i l i t i e s o f t h e s t a t e sj. F o r s i m p l i c i t y , w e w i l l a s s u m e t h a t th ep r o b a b i li t ie s o f t h e o u t c o m e s oij a r e i n d e p e n d e n t o f w h i c h a l t e r n a t i v e isc h o s e n , s o t h a t P ( o i i) = P ( s j ) , f o r a ll P ~ ~ a n d a l l a l t e r n a t i v e s a i. S i n c et h is a s s u m p t i o n c a n b e r e l a x e d , t h e d e c i s i o n t h e o r y t o b e p r e s e n t e d c a nb e e x t e n d e d t o t h e m o r e g e n e r a l ca s e . 8T h e i d e a o f r e . p re s e n t in g a s t a t e o f b e l i e f b y a c l as s o f p r o b a b i l i t ym e a s u r e s is n o t n e w b u t h a s b e e n s u g g e s t e d b y v a r i o u s a u t h o r s . 9 I th a s b e e n m o s t e x t e n s i v e l y d i s c u s s e d b y L e v i i n [ 2 6 ] a n d [ 2 7 ] , b u t , a sw i ll b e s e e n i n t h e s e q u e l , h e d o e s n o t u s e t h e c l a s s o f p r o b a b i l i t ym e a s u r e s i n t h e s a m e w a y a s w e d o .

L e v i a l s o a s s u m e s t h a t t h e s e t o f p r o b a b i l i t y m e a s u r e s i s convex,i .e ., t h a t f f P a n d P ' a r e tw o m e a s u r e s i n t h e s e t , t h e n t h e m e a s u r ea P + (1 - a ) . P ' i s a l s o in t h e s e t , f o r a n y a b e t w e e n 0 a n d 1 .1 T h em o t i v a t i o n f o r th i s a s s u m p t i o n i s t h a t if P a n d P ' b o t h a r e p o s s i b l e

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366 P E T E R G ~ . R D E N F O R S A N D N I L S - E R I C S A H L I Nprobability distributions over the states, then any mixture of thesedistributions is also possible. We will discuss the requirement ofconvexity in section 5.If ~ is assumed to be convex, then the set of epistemically possibleprobabilities associated with a state sj by the elements of ~ will forman interval from the lowest probability assigned to s to the highest.Some authors have taken such intervals as basic when describingbeliefs about the states of nature - to each state is assigned a prob-ability interval and this assignment is governed by some consistencyrestrictions. 11 The representat ion by a convex set of probabilitymeasures is, however, more general, since from such a set one canalways compute a unique set of associated intervals, but starting froman assignment of consistent probability intervals, there will in generalbe a large number of convex sets of probability measures that willgenerate the intervals, lzWe believe that not all of an agent's beliefs about the states ofnature relevant to a decision situation can be captured by a set ~ ofprobability measures. As a second element in describing the beliefsrelevant to a decision situation, we introduce a (real-valued) measurep of the epistemic reliabili ty of the probability measures in ~. Even ifseveral probability distributions are epistemically possible, some dis-tributions are more reliable - they are backed up by more informationthan other distributions.

The measure p is intended to represent these different degrees ofreliability. In the introductory examples, Miss Julie ascribes a muchgreater epistemic reliability to the probability distribution where eachplayer has an equal chance of winning in match A where she knows alot about the players than in match /3 where she knows nothingrelevant about the players. In match C, where she knows that oneplayer is superior to the other, but not which, the epistemically mostreliable distributions are the two distributions where one player iscertain to win. Since there are only two relevant states of nature inthese examples, viz. the first player wins (st) and the second playerwins ( sO , a probability distribution can be described simply by theprobability of one of the states. We can then illustrate the epistemicreliability of the various distributions in the three matches bydiagrams as in Figure 1.

Even if examples such as these illustrate the use of the measure p,its properties should be specified in greater detail. Technically, the

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0 0 . 5 1 P ( s 1 ) 0 0 . 5 1 P ( s 1 ) 0 0 . 5 1 P ( s 1 )M o t c h A M a t c h B M a t c h C

F i g . 1 .

o n l y p r o p e r t y o f p t h a t w i l l b e n e e d e d i n t h i s p a p e r i s t h a t t h ep r o b ab i l i t y d i s t r i b u t i o n s i n ~ can b e ordered w i t h r e s p e c t t o t h e i rd e g r e e s o f e p i s t e m i c r e l i a b i l i t y . H o w e v e r , i t s e e m s n a t u r a l t o p o s -t u l a te t h a t O h a s a n u p p e r b o u n d , r e p r e s e n t i n g t h e c a s e w h e n t h ea g e n t h a s c o m p l e t e i n f o r m a t i o n a b o u t a p r o b a b i l i t y d i s t r i b u t i o n , a n d al o w e r b o u n d , r e p r e s e n ti n g t h e c a s e w h e n t h e a g e n t h as n o i n f o r m a t i o na t a l l a b o u t t h e s e d i s t r i b u t i o n s . H o w e v e r , w e w i l l n o t a t t e m p t a f u l ld e s c r i p t i o n o f t h e p r o p e r t i e s o f th e m e a s u r e 0 , s i n c e w e b e l i e v e t h a tt h i s c a n b e d o n e o n l y i n a m o r e c o m p r e h e n s i v e d e c i s i o n t h e o r y i nw h i c h t h e r e l a t i o n s b e t w e e n d i f f e r e n t d e c i s i o n s i t u a t i o n s a r ee x p l o i t e d ] 3A f u n d a m e n t a l f e a t u r e o f th e e p i s t e m i c r e l ia b i li ty o f t h e p r o b a b i l i t yd i s t r i b u t i o n s p o s s i b l e i n a d e c i s i o n s i t u a t i o n , a s w e c o n c e i v e o f t h em e a s u r e , i s t h a t t h e l e s s r e l e v a n t i n f o r m a t i o n t h e a g e n t h a s a b o u t t h es t a t e s o f n a t u r e , t h e l e s s ep i s t emi c r e l i ab i l i t y w i l l b e a s c r i b ed t o t h ed i s t r i b u t i o n s i n ~ . Wh er e l i t t l e i n f o r ma t i o n i s av a i l ab l e , t h e r e f o r e , a l ld i s t r i b u t i o n s w i l l , c o n s e q u e n t l y , h a v e a b o u t t h e s a m e d e g r e e o f e p i s -t e m i c r e li a b il it y . C o n v e r s e l y , i n a d e c i s i o n s i t u a ti o n w h e r e t h e a g e n t isw e l l - i n f o r m e d a b o u t t h e p o s s i b l e s t a t e s o f n a t u r e , s o m e d i s t r i b u t i o n sw i l l t en d t o h av e a co n s i d e r ab l y h i g h e r d eg r ee o f ep i s t emi c r e l i ab i l i t yt h an o t h e r s .

A p r o b l e m w h i c h s t r o n g l y s u p p o r t s o u r t h e s i s t h a t t h e m e a s u r e o fep i s t emi c r e l i ab i l i t y i s a n eces s a r y i n g r ed i en t i n t h e d e s c r i p t i o n o f ad e c i s io n s i t ua t io n i s P o p p e r ' s p a r a d o x o f id e a l e v id e n c e . P o p p e r a s k su s t o co n s i d e r t h e f o l l o w i n g ex amp l e ( [ 3 1 ] , p p . 4 0 7 - 4 0 8 ) :L e t z b e a c e r ta i n p e n n y , a n d l e t a b e t h e s t a t e m e n t ' t h e n t h ( a s y e t u n o b s e r v e d ) t o s s o fz w i ll y ie l d h e a d s ' . W i t h i n t h e s u b j e c t i v e t h e o r y , i t m a y b e a s s u m e d t h a t t h e a b s o l u t e( o r p ri o r ) p r o b a b i l i t y o f t h e s t a t e m e n t a i s e q u a l t o 1 /2 , t h a t i s t o s a y ,

( 1) P ( a ) = 1 /2

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3 6 8 P E T E R G A R D E N F O R S A N D N I L S - E R I C S A H L I N

N o w l e t e b e s o m e s ta t i s t i ca l evidence ; that i s to s a y , a s ta t i s t i ca l repor t , b a s e d u p o nt h e o b s e r v a t i o n o f t h o u s a n d s o r p e r h a p s m i l l i o n s o f t o s s e s o f z ; a n d l e t t h i s e v i d e n c e eb e i d ea l l y f a v ou r a b l e t o t h e h y p o t h e s i s t h a t z i s s t r i c t l y s y m m e t r i c a l . . . . W e th e n h a v en o o t h e r o p t i o n c o n c e r n i n g P (a , e) t h a n t o a s s u m e t h a t

(2 ) P( a , e ) = 1/2T h i s m e a n s t h a t t h e p r o b a b i l i t y o f t o s s i n g h e a d s r e m a i n s u n c h a n g e d i n t h e l i g h t o f th ee v i d e n c e e ; f o r w e n o w h a v e

(3 ) P ( a ) = P ( a , e ).B u t , accord ing to the sub jec t i ve theory , ( 3 ) m e a n s t h a t e i s , o n t h e w h o l e ( a b s o l u t e l y )i r re levan t i n f o rm a t i o n w i t h r e s p e c t t o a .

N o w t h i s is a l it t le s t a r t li n g ; f o r i t m e a n s , m o r e e x p l i c i t l y , t h a t o u r s o - c a l l e d ' degree o fra t ion a l be l ie[ ' in the hypothesis , a , o u g h t to be comple te ly unaf fec ted by the a c -c u m u l a t e d ev iden t ia l know ledge, e ; t h a t t h e a b s e n c e o f a n y s t a t i s t i c a l e v i d e n c ec o n c e r n i n g z j u s t i f i e s p r e c i s e l y t h e s a m e ' d e g r e e o f r a t i o n a l b e l i e f ' a s t h e w e i g h t ye v i d e n c e o f m i l l i o n s o f o b s e r v a t i o n s w h i c h , p r ima f a c i e , s u p p o r t o r c o n f i r m o rs t r e n g t h e n o u r b e l i e f.T h e ' s u b j e c t i v e t h e o r y ' w h i c h P o p p e r i s r e fe r r in g to i n th i s e x a m p l e i sw h a t w e h a v e h e r e c a l l e d s t r i c t B a y e s i a n i s m .

N o w , w i t h t h e a i d o f t h e m o d e l s o f d e c i s i o n s i t u a t i o n s p r e s e n t e da b o v e , w e c a n d e s c r i b e P o p p e r ' s e x a m p l e i n th e f o l l o w i n g w a y . T h e r ei s a s e t ~ o f p o s s i b l e p r o b a b i li t y m e a s u r e s c o n c e r n i n g t h e s t at e s o fn a t u r e d e s c r i b e d b y a a n d n o t - a . I f o n e i s f o r c e d t o s t a t e t h ep r o b a b i l i t y o f a , b e f o r e t h e e v i d e n c e e i s o b t a i n e d , t h e m o s t r e l i a b l ea n s w e r s e e m s t o b e 1 1 2 . T h e d e g r e e o f e p i s t e m i c r e l i ab i l it y o f t h ise s t i m a t e i s , h o w e v e r , l o w , a n d t h e r e a r e m a n y o t h e r a n s w e r s w h i c hs e e m a l m o s t a s r e l i a b l e . A f t e r t h e e v i d e n c e e i s o b t a i n e d , t h e m o s treasonable probabi l i ty as ses sment concerning a i s s t i l l 1 ]2 , but nowt h e d i s t r i b u t i o n a s s o c i a t e d w i t h t h i s a n s w e r h a s a m u c h h i g h e r d e g r e eof epi s temic re l iabi l i ty than be fore , and the o ther d i s tr ibut ions inh a v e c o r r e s p o n d i n g l y l o w e r d e g r e e s o f r e l i a b i l i t y . I t s h o u l d b e n o t e dt h a t t h i s d i s t i n c t i o n b e t w e e n t h e t w o c a s e s c a n n o t b e f o r m u l a t e d w i t hthe a id o f the s e t ~ only , but the measur e p o f epi s te m ic rel iabi li ty i sa l so nec es sar y . 14

W e w i l l c o n c l u d e t h i s s e c t i o n b y b r i e f l y m e n t i o n i n g s o m e r e l a t e da t t em p t s t o e x t e n d m o d e l s o f b e l i e f b y s o m e m e a s u r e o f ' re li a b il it y '. tsA n i n t e r e s t i n g c o n c e p t w a s i n t r o d u c e d b y K e y n e s i n [ 2 5 ] , p . 7 1 :A s t h e r e l e v a n t e v i d e n c e a t o u r d i s p o s a l i n c r e a s e s , t h e m a g n i t u d e o f t h e p r o b a b i l i t y o ft h e a r g u m e n t m a y e i t h e r d e c r e a s e o r i n c r e a s e , a c c o r d i n g a s t h e n e w k n o w l e d g es t r e n g t h e n s t h e u n f a v o u r a b l e o r f a v o u r a b l e e v i d e n c e ; b u t someth ing s e e m s t o h a v e

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U N R E L I A B L E P R O B A B I L I T I E S A N D D E C I S I O N M A K I N G 369i n c r e a s e d i n e i th e r c a s e - w e h a v e a m o r e s u b s t a n t i a l b a s i s u p o n w h i c h t o r e s t o u rc o n c l u s i o n . I e x p r e s s t h i s b y s a y i n g t h a t a n a c c e s s i o n o f n e w e v i d e n c e i n c r e a s e s t h ew e i g h t o f a n a r g u m e n t . N e w e v i d e n c e w il l s o m e t i m e s d e c r e a s e t h e p r o b a b i l it y o f a na r g u m e n t , b u t i t w i ll a l w a y s i n c r e a s e i t s " w e i g h t " .Here Keynes writes about the probability of an argument, while weare concerned with probability distributions over states of nature.Even if the intuitions behind Keynes's 'weight of evidence' and our'epistemic reliability' are related, it is difficult to say how far thisparallel can be drawn.

In [4], pp. 554-555, Carnap discusses 'the problem of the reliabilityof a value of degree of confirmation' which obviously is the same asKeynes's problem. Carnap remarks that Keynes's concept of 'theweight of evidence' was forestalled by Peirce who mentioned it in[30], p. 421, in the following way:, . . t o e x p r e s s t h e p r o p e r s t a te o f b e li e f, n o t o n e n u m b e r b u t tw o a r e r e q u i s i t e , t h e f i r s td e p e n d i n g o n t h e i n fe r r e d p r o b a b i li t y , t h e s e c o n d o n t h e a m o u n t o f k n o w l e d g e o nw h i c h t h a t p r o b a b i l i ty i s b a s e d .The models of the agent's beliefs about the states of nature in adecision situation which have been presented here contain the twocomponents ~ and 0, i.e., the set of epistemically possible probabilitydistributions, and the measure of epistemic reliability. These twocomponents can be seen as an explication of the two numbersrequired by Peirce. ~6

3 . A D E C I S I O N T H E O R YThe models of decision situations which were outlined in the previoussection will be used now as a basis for a theory of decision. Thistheory can be seen as a generalization of the Bayesian rule ofmaximizing expected utility.

A decision, i.e. a choice of one of the alternatives in a decisionsituation, will be arrived at in two steps. The first step consists inrestricting the set ~ to a set of probability measures with a 'satis-factory' degree of epistemic reliability. The intuition here is that in agiven decision situation certain probability distributions over thestates of nature, albeit epistemically possible, are not considered asserious possibilities. For example, people do not usually checkwhether there is too little brake fluid in the.car or whether the wheels

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a r e l o o s e b e f o r e s t a r t i n g t o d r i v e , a l t h o u g h , f o r a ll t h e y k n o w , s u c he v e n t s a r e n o t i m p o s s i b l e , a n d t h e y r e a l i z e t h a t i f a n y s u c h e v e n to c c u r r e d t h e y w o u l d b e i n d a n g e r .

N o w e x a m p l e s o f th i s k i n d s e e m s t o s h o w t h a t w h a t t h e a g e n t d o e si s t o d i s r e g a r d c e r t a i n s t a t e s o f n a t u r e r a t h e r t h a n p r o b a b i l i t y d i s -t r i b u t i o n s o v e r s u c h s t a t e s . B u t i f a c e r t a i n s t a t e o f n a t u r e i s n o tc o n s i d e r e d a s a s e r io u s p o s s i b i l i t y , t h e n t h i s m e a n s t h a t a l l p r o b a b i l i t yd i s t r i b u t i o n s w h i c h a s s i g n t h i s s t a t e a p o s i t i v e p r o b a b i l i t y a r e l e f t o u to f c o n s i d e r a t i o n . A n d t h e r e m a y b e c a s e s w h e n s o m e p r o b a b i l i t yd i s t ri b u t i o n s a r e l e f t o u t o f a c c o u n t e v e n i f a ll r e l e v a n t s t a te s o fn a t u r e a r e c o n s i d e r e d t o b e s e r i o u s p o s s i b i l i t i e s . S o , r e s t r i c t i n g t h e s e t

i s a m o r e g e n e r a l w a y o f m o d e l l i n g th e p r o c e s s t h a n r e s t r ic t i n g t h es e t o f s t a t e s .

D e c i d i n g t o c o n s i d e r s o m e d i s t r i b u t i o n s i n ~ a s n o t b e i n g s e r io u sp o s s i b i l i t i e s m e a n s t h a t o n e t a k e s a r i s k . T h e l e s s i n c l i n e d o n e i s t ot a k e r i s k s , th e g r e a t e r t h e n u m b e r o f d i s t r i b u t i o n s in ~ w i l l b e t h a t a r et a k e n i n t o a c c o u n t w h e n m a k i n g t h e d e c is io n .

A f u n d a m e n t a l q u e s t i o n i s h o w t h e a g e n t d e t e r m i n e s w h i c h p r o b -a b i l i t y d i s t r i b u t i o n s i n 8 ~ a r e ' s a t i s f a c t o r i l y r e l i a b l e ' a n d w h i c h a r en o t . I n o u r v i e w , t h e a n s w e r i s t h a t t h e m e a s u r e p o f e p i s t e m i cr e l ia b i li t y s h o u l d b e u s e d w h e n s e l ec t in g t h e a p p r o p r i a t e s u b s e t ~ ] p oo f ~ . T h e a g e n t s e l e c t s a d e s i r e d l e ve l p o o f e p i s t e m i c r e l i a b i li t y a n do n l y t h o s e p r o b a b i l i t y d i s t r i b u t i o n s in ~ w h i c h p a s s t h i s p - l e v e l ar ei n c l u d e d i n ~[ Po , b u t n o t t h e o t h e r s . 17 ~ 9/p o c a n b e r e g a r d e d a s t h e s e to 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 t h a t t h e a g e n t t a k e s i n t o c o n s i d e r a t i o n i nm a k i n g th e d e c i s io n . A n o b v i o u s r e q u i r e m e n t o n t h e c h o s e n l e v el o fr e l i a b i l it y is , o f c o u r s e , t h a t t h e r e b e s o m e d i s t r i b u t i o n i n ~ w h i c hp a s s e s t h e l e v e l .

W h i c h ' d e s i r e d l e v e l o f e p i s t e m i c r e l i a b i l i t y ' t h e a g e n t w i l l c h o o s ed e p e n d s o n h o w l a r g e th e r i s k s a r e h e i s w i l l in g t o ta k e , T h e m o r e r i s ka v e r s i v e t h e a g e n t i s , t h e l o w e r t h e c h o s e n l e v e l o f e p i s t e m i c r e l i -a b i l i t y w i ll b e. I t i s i m p o r t a n t t o n o t e t h a t t w o a g e n t s i n i d e n t i c a le p i s t e m i c s i t u a t i o n s , h e r e i d e n t i f ie d b y a se t ~ a n d a m e a s u r e p , m a yi n d e e d c h o o s e d i f f e r e n t v a l u e s o f p0 d e p e n d i n g o n t h e i r d i f f e r en t ri s kt a k i n g te n d e n c i e s . T h i s is t h e r e a s o n w h y i t i s a s s u m e d t h a t p y i e ld sa n o r d e r in g o f ~ a n d n o t m e r e l y a d i c h o t o m y .

I f th e a g e n t i s w i l l in g t o t a k e a m a x i m a l r is k a s r e g a r d s w h i c hp r o b a b i l i t y d i s t r i b u t i o n t o c o n s i d e r a s ' s a t i s f a c t o r y ' i t m a y h a p p e nt h a t t h e r e w i ll b e o n l y o n e d i s t ri b u t i o n w h i c h p a s s e s t h e d e s i r e d l e v e l

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of reliability. After such a choice his relevant information about thestates of nature will be of the same type as for the strict Bayesian, i.e.a unique probability measure, but this situation will arise for quitedifferent reasons. TM

The second step in the decision procedure starts with the restrictedset ~Jpo of probability distributions. For each alternative a~ and eachprobability distribution P in PlOp the expected utility elk is computedin the ordinary way. The minimal expected util i ty of an alternative ai,relative to a set ~lpo, is then determined, this being defined as thelowest of these expected utilities elk. Finally the decision is madeaccording to the following rul e: 19The maximin cri terion for expected ut i l i t ies (MMEU): The alternativewith the largest minimal expected utility ought to be chosen.In order to illustrate how this decision procedure works, we willreturn to the introductory examples. For simplicity, let us, for allthree tennis matches, denote by s~ the event that the first player toserve wins the match, and by s2 the event that the other player wins.Now assume that Miss Julie is offered the following bet for each ofthe three matches: She wins 305 if s~ occurs and loses 205 if s2occurs. For each match, she must choose between the alternative a~of accepting the bet and the alternative a2 of declining it. Let usfurthermore assume that Miss Julie's utilities are mirrored by themonetary values of the outcomes.

In match A , where Miss Julie is very well informed about theplayers, she considers that the only probability distribution that sheneeds to take into consideration is the distribution P~, where Pt(s t )=P~(s2) = 0.5. She is will ing to take the (small) risk of letting N/P0consist of this distribution only. The only, and hence minimal, expec-ted utility to compute is then 0.5 30 + 0.5 (-20) for at and 0.5 0 +0.5 0 for a2. Hence, according to MMEU, she should accept the betin match A.

In match /3, where she has no relevant information at all, theepistemic reliability of the epistemically possible distributions is moreevenly spread out. Consequently, Miss Julie is not so willing to leavedistributions out of account when forming the subset ~[P0 as in theprevious case. For simplicity, let us assume that ~[Po = {Pt, P2, P3},where Pt is as before, P2 is defined by P2(s~) = 0.25 and Pz(sz) = 0.75,and P3 is defined by Pa(s~)= 0.75 and P3(s2)= 0.25. The expected

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372 P E T E R G A R D E N F O R S A N D N I L S - E R I C S A H L I Nu t il it ie s f o r t h e a l t e r n a t i v e a~ o f accep t i n g t h e b e t , d e t e r m i n ed b yt h es e t h r e e d i s t r i b u t i o n s , a r e 5 , - 7 .5 an d 1 7.5 r e s p ec t i v e l y ; w h i l e t h eex p e c t e d u t i li ti e s f o r t h e a l t e r n a t i v e a2 o f n o t a ccep t i n g t h e b e t a r e 0i n a l l t h r ee ca s e s . S i n ce t h e mi n i ma l ex p ec t ed u t i l i t y o f a l i s l e s s t h ant h a t o f az, t h e M M E U c r i t e r i o n d e m a n d s t h a t az b e t h e c h o s e na l t e r n a t i v e , i . e . M i s s J u l i e s h o u l d d ec l i n e t h e b e t o f f e r ed .

I n m a t c h C , i t i s r e a s o n a b l e t o a s s u m e t h a t ~/po c o n t a i n s s o m ep r o b ab i l i t y d i s t r i b u t io n w h i ch a s s i g n s s~ a v e r y h i g h p r o b ab i l i t y an ds o m e d i s t r i b u t i o n w h i ch a s s i g n s i t a v e r y l o w p r o b ab i l i t y . A s i mi la ra n a l y s i s a s a b o v e t h e n s h o w s t h a t M i s s J u l i e s h o u l d n o t a c c e p t t h e b e ti n t h i s c a s e e i t h e r .

Th i s ex amp l e can b e h eu r i s t i c a l l y i l l u s t r a t ed a s i n F i g u r e 2 . I n t h i sf i g u r e t h e b r o k en h o r i zo n t a l l i n e i n d i ca t e s t h e d e s i r ed l ev e l o f ep i s -t emic r e l i ab i l i t y .

To g i v e a f u r t h e r i l l u s t r a t i o n o f t h e d ec i s i o n t h eo r y , i t s h o u l d b en o t e d t h a t t h e h y p o t h e s i s f r o m t h e G o l d s m i t h - S a h l i n e x p e r i m e n t s ,m e n t i o n e d i n t h e i n t r o d u c t i o n , i s w e l l e x p l a i n e d b y t h e M M E Uc r i t e r i o n . W h e n a n a g e n t i s a s k e d t o c h o o s e b e t w e e n t i c k e t s i n t w ol o t t e r i e s w h i c h a r e e s t i m a t e d t o h a v e t h e s a m e p r i m a r y p r o b a b i l i t y o fw i n n i n g , h e s h o u l d c h o o s e t h e t i c k e t f r o m t h e l o t t e r y w i t h t h eep i s t emi ca l l y mo s t r e l i ab l e p r o b ab i l i t y e s t i ma t e , s i n ce t h i s a l t e r n a t i v ewi l l have the h igh es t min im al ex pe c t e d u t i li t y . 2 S t il l o the r ap -p l ic a t io n s o f th e d e c i s i o n t h e o r y w i ll b e p r e s e n t e d i n th e n e x t t w os ec t i o n s .

A l im i t in g ca s e o f i n f o r m a t i o n ab o u t t h e s t a t e s o f n a t u r e i n ad ec i s i o n s i t u a t i o n i s t o h av e no i n f o r ma t i o n a t a l l . I n t h e d ec i s i o nm o d e l s p r e s e n t e d h e r e , t h i s w o u l d m e a n t h a t a l l p r o b a b i l i t y d i s -t r i b u t i o n s o v e r t h e s t a t e s a r e e p i s t e m i c a l l y p o s s i b l e a n d t h a t t h e yh av e eq u a l ep i s t emi c r e l i ab i l i t y . I n s u ch a ca s e , t h e mi n i ma l ex p ec t ed

0 0 . 5 1 P ( s 1 ) 0 0 . 5 1 P ( s 1 ) 0 0 . 5 1 P ( s 1 )M a t c h A M o t c h B M a t c h C

Fig . 2 .

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utility of an alternative is obtained from the distr ibution which assignsthe probability 1 to the worst outcome of the alternative. This is,however, jus t another way of formulating the classical maximin rule,which has been applied in what traditionally has been called 'decisionmaking under uncertainty' (a more appropriate name would be'decision making under ignorance'). Hence, the classical maximinturns out to be a special case of the MMEU criterion.

At the other extreme, having f u l l information about the states ofnature implies that only one probability distribution is epistemicallypossible} ~ In this case the MMEU criterion collapses into theordinary rule within strict Bayesianism, i.e. the rule of maximizingexpected utility, which has been applied to what traditionally, butsomewhat misleadingly, has been called 'decision making under risk'.

The decision theory which has been presented here thus covers thearea between the traditional theories of 'decision making under un-certainty' and 'decision making under risk' and it has these theories aslimiting cases.

4 . R E L A T I O N T O E A R L I E R T H E O R I E S

Several authors have proposed decision theories which are based onmore general ways of representing the decision maker's beliefs aboutthe states than what is allowed by strict Bayesianism. The mostdetailed among these is Levi's theory ([261 and [27]), which will bediscussed in a separate section. In this section we will compare thepresent theory with some earlier statistical decision theories.

In [38], Wald formulates a theory of 'statistical decision functions'where he considers a set [1 of probability measures and a 'risk'function. He says that 'the class f~ is to be regarded as a datum of thedecision problem' (p. 1). He also notes that the class II will generallyvary with the decision problem at hand and that in most cases 'will bea proper subset of the class of all possible distribution functions' (p.1). In the examples, II is often taken to be a parametric family ofknown functional form. A risk function is a function which deter-mines the 'cost' of a wrong decision. Such a function can be seen asan inverted utility function restricted to negative outcomes. On thisinterpretation it is easier to compare Wald's theory to the presentdecision theory.

Wald suggests two alternative decision rules. The first is the tradi-

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374 PETER GARDENFOR S AND NILS-ER IC SAI -1 LIN

tional Bayesian method when an 'a priori' probability measure in l~can be selected, or, as Wald puts it, when 'it exists and is known tothe experimenter' (p. 16). The second case is when the entire set t2 isemployed to determine the decision. Wald suggests that in such casesone should 'minimize the maximum risk'. In our terminology, usingutility functions instead of risk functions, this is the same as maxi-mizing the minimal expected utility with respect to the set IL

If Wald's theory is interpreted as above, the difference between hisand our theory is mainly of epistemological character. Since Walddoes not say anything about how ~ is to be determined it is difficult totell whether it corresponds to our set ~ of epistemically possibleprobability functions or to the set ~]po of 'reliable' functions. Inparticular, he does not introduce any factor corresponding to themeasure p of epistemic reliability, nor does he associate the choice ofiq with any form of risk taking.

Hurwicz [22] apparently interprets Wald's set 1) as correspondingto our set ~9. He notes that sometimes some of the distributions in l)are more 'likely' than others (p. 343). For example, assume that f~consists of all normal distributions with mean zero and standarddeviation tr. In a particular decision situation evidence at hand maysupport the assumption that or is considerably small. It thus seemsreasonable to select a proper subset O0 of I) which is restricted tothose distributions with standard deviation less than or equal to somevalue or0.Hurwicz assumes that such a subset ~0 ~-~t=~~ n his terminology) of f~can be selected in any decision situation. He then suggests a 'general-ized Bayes-minimax porinciple' which amounts to using f~0 as thebase when maximizing the minimal expected utility (minimizing themaximal risk). Obviously, the set Oo corresponds closely to our set~/P0. The main difference between Hurwicz' theory and the presentone is that he does not give any account of how the set 120 is to bedetermined. In particular he, as Wald, does not introduce any factorcorresponding to the measure p.In [21], Hodges and Lehmann suggest an alternative to Wald'sminimax solution which they call a 'restricted Bayes solution'. It is ofinterest here since it is adopted by Ellsberg [8] as a solution to his'paradox'. Let us start by considering Ellsberg's problem. 22

Ellsberg ([8] , pp. 653-654) asks us to consider the followingdecision problem. Imagine an urn known to contain 30 red balls and

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UNRE LIAB LE PROBA BILI TIES AND DECISION MAKING 375

60 black and yellow balls, the latter in unknown proportion. One ballis to be drawn at random from the urn. In the first situation you areasked to choose between two al ternatives a~ and av I f you choose a~you will receive \$100 if a red ball is drawn and nothing if a black oryellow ball is drawn. If you choose a2 you will receive \$100 if a blackball is drawn, otherwise nothing. In the second situation you areasked to choose, under the same circumstances, between the twoalternatives a3 and a4. If you choose a3 you will receive \$100 if a redor a yellow ball is drawn, otherwise nothing and if you choose a4 youwill receive \$100 if a black or yellow ball is drawn, otherwisenothing. This decision problem is shown in the following decisionmatrix.Red Black Yellow

aa[-'~-~ \$100 \$0 \$0a2 \$0 \$100 \$0a3 \$100 \$0 \$100\$0 \$100 \$100

The most frequent pattern of response to these two decisionsituations is that al is preferred to a2 and a4 is preferred to a3. AsEllsberg notes, this preference pattern violates Savage's 'sure thingprinciple' (postulate P2 in [35]), which requires tha t the preferenceordering between al and a2 be the same as the ordering between a3and an.When applying the present decision theory to this problem, themain step is to determine the set ~/po. The set ~ should be the samein the two decision situations, since they do not differ with respect tothe information about the states. Now, unless the decision makerbelieves that he is being cheated about the content of the urn, 9 ismost natural ly taken as the class of distributions (1/3, x, 213- x),where x varies from 0 to 2/3.

If the decision maker chooses a low po, ~/po will contain most ofthe distributions in this class. For simplicity, let us for the momentassume that ~[Po = ~9. With this choice, the minimal expected utilitiesof the alternatives are 113-u(\$100), 1-u(\$0), 1/3. u(\$100) and23. u(\$100), for al, a2, a3 and an, rospectively. Assuming that1/3. u(\$100) is greater than u(\$0), the MMEU criterion requires that a~

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U N R E L I A B L E PR O B A B I L I T I E S A N D D E C I SI O N M A K I N G 377S e c o n d l y , e v e n i f w e i d e n t if y y 0 w i t h ~ /P 0 , E U s b e r g e x p l o i ts t h ed eg r ee p ~ o f ' co n f i d en ce ' , w h i ch i s d e f i n ed f o r o n l y o n e d i s t r i b u t i o ny 0, i n a w a y t h a t d i ff e r s c o n s i d e r a b l y f r o m o u r u s e o f t h e m e a s u r e p ,w h i c h i s a s s u m e d t o b e d e f i n e d f o r a l l d i s t r i b u t i o n s i n ~ . I np a r t ic u l a r w e n e e d n o t a s s u m e t h a t p g i v e s a n u m e r i c a l v a l u e , o n ly t h a ti t o r d e r s t h e d i s t r i b u t i o n s i n @. Th i r d l y , s i n ce t h e d ec i s i o n r u l e s a r ed i f f e r e n t , t h e t h e o r i e s w i l l r e c o m m e n d d i f f e r e n t d e c i s i o n s i n m a n ys i tu a t io n s . T h e m o s t im p o r t a n t d i s a g r e e m e n t h e r e i s t h a t w e r e j e c t t h en e e d o f a n e s t i m a t e d d i s t r i b u ti o n y 0. W e b e l i e v e t h a t o n c e t h e s e t ~ / p 0h a s b e e n s e l e c t e d , t h e d i s t r i b u t i o n w i t h t h e h i g h e s t d e g r e e o f r e l i -a b i li ty ( c o r r e s p o n d i n g t o E U s b e r g ' s y 0) d o e s n o t p l a y a n y o u t s t a n d in gr o l e i n t h e d ec i s i o n mak i n g .

T h i s d i f f e r e n c e b e t w e e n t h e t w o t h e o r i e s i s i n p r i n c i p l e t e s t a b l e ,a s s u m i n g t h a t p~ is n o t a l w a y s c l o s e to z e r o . T h e e x p e r i m e n t p e r -f o r m e d b y B e c k e r a n d B r o w n s o n [ 2 ] i s r e l e v a n t h e r e . T h e y o f f e r e ds u b j e c t s a m b i g u o u s b e t s d i f f e r i n g i n t h e r a n g e o f p r o b a b i l i t i e s p o s -s i b l e b u t o f e q u a l e x p e c t e d v a l u e a n d f o u n d t h a t s u b j e c t s w e r e w i l l i n gt o p a y m o n e y f o r o b t a i n i n g b e t s w i t h n a r r o w e r r a n g e s o f p r o b a b i l i t y .T h i s f i n d i n g s e e m s t o h i g h l i g h t t h e i m p o r t a n c e o f a m e a s u r e o fr e l i a b i l i t y . B u t t h e y d i d n o t , h o w e v e r , o b t a i n s u p p o r t f o r E l l s b e r g ' sh y p o t h es i s t h a t t h e d i s t r i b u t i o n yO i s r e l ev an t f o r t h e d e c i s i o n m akin g~

5. A COMPARI SON WITH LEVI 'S THEORY

I n t h i s s e c t i o n w e c o m p a r e o u r t h e o r y w i t h L e v i ' s t h e o r y w h i c h i sp r e s en t ed i n [ 2 6 ] an d e l ab o r a t ed o n i n [ 2 7 ] . Lev i s t a r t s o u t f r o m ad e s c r i p t i o n o f t h e d e c i s i o n m a k e r X ' s i n f o r m a t i o n a t t h e ti m e t a b o u tt h e s t a t e s o f n a tu r e . T h i s in f o r m a t i o n i s c o n t a i n e d i n a c o n v e x s e t B x , to f p r o b ab i l i t y d i s t r ib u t i o n s . T h e d i s t r i b u t i o n s i n B x, a r e , a cco r d i n g t oL e v i , t h e ' p e r m i s s i b l e ' d i s t r i b u t i o n s . A s t o t h e m e a n i n g o f " p e r -mi s s i b l e " , h e o f f e r s o n l y i n d i r ec t c l a r i f i c a t i o n b y i n d i ca t i n g t h e co n -n e c t i o n s b e t w e e n p e r m i s s i b i l i t y a n d r a t i o n a l c h o i c e . I n o r d e r t o c o m -p a r e t h e t h e o r i e s , w e w i l l h e r e a s s u m e t h a t t h e s e t B x , t c o r r e s p o n d s t ot h e s e t @/P 0 ( o r i ts co n v ex h u l l) a s p r e s en t ed i n s e c t i o n 3 . 24

L e v i a l s o g e n e r a l iz e s t h e t r a d it io n a l w a y o f r e p r e s e n t i n g t h e u t i li ti e so f t h e o u t c o m e s b y i n t r o d u c i n g a c l a s s G o f 'p e r m i s s i b le ' u t i li tym e a s u r e s , s u c h t h a t n o t a l l o f t h e s e m e a s u r e s n e e d b e l i n e a r t r a n s -f o r m a t i o n s o f o n e a n o t h e r .

A n a l t e r n a t i v e a i i s s a i d t o b e E - a d m i s s i b l e i f an d o n l y i f t h e r e i s

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378 P E T E R G A R D E N F O R S A N D N I L S - E R i C S A H L I N

some probability distribution P in Bx, t and some utility function u inG such that the expected utility of al relative to P and u is maximalamong all the available alternatives. A first requirement on the alter-native to be chosen in a given decision situation is then that it shouldbe E-admissible.

The second step in Levi's decision procedure concerns the oppor-tunity to defer decision between two or more E-admissible alter-natives. He argues that a rational agent should "keep his optionsopen" whenever possible. An alternative is said to be P - a d m i s s i b l e i fit is E-admissible and it is 'best' with respect to E-admissible optionpreservation. Levi does not, however, explicate what he means by'best' here, and we will ignore the effects of this requirement here,since we have not imposed any structure on the set of alternatives. 25

Let us say that a P-admissible alternative ai is s ecu r i t y o p t ima lrelat ive to a utility function u if and only if the minimum u-valueassigned to some possible outcome oii of ai is at least as great as theminimal u-value assigned to any other P-admissible alternative. Levithen, finally, calls an alternative S - a d m i s s i b l e if it is P-admissible andsecurity optimal relative to s o m e utility function in G.

Levi states (in [27], p. 412) that he "cannot think of additionalcriteria for admissibility which seem adequate" so he, tentatively,assumes that all S-admissible alternatives are 'admissible' for the finalchoice, which then, supposedly, is determined by some randomdevice.In order to illustrate the differences between the decision theorypresented in the previous section and Levi 's theory, we will considerthe following example which contains two states and three alter-natives:

S I S2 1l a l -1 0 12a~ 11 - 9aa 0 0

In this matrix the numbers denote the utilities of the outcomes.Assume that the set ~/po, which here is identified with Levi's set B x , t,consists of the two probability distributions P, defined by P ( s l ) = 0.4and P(s2)=0.6, and P', defined by P'(sl)=0.6 and P ' ( s 2 ) = 0 . 4 ,together with all convex combinations of P and P' .

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U N R E L I A B L E PR O B A B I L I T I E S A N D D E C I SI O N M A K I N G 379

T h e m i n i m a l e x p e c t e d u t i li ty o f a ~ i s - 1 . 2 a n d t h e m i n i m a l e x p e c t e du t i li t y o f a2 i s - 1 .0 . Th e mi n i ma l ex p e c t e d u t i li t y o f a3 i s o f co u r s e 0 ,s o M M E U r e q u ir e s t h a t a 3 b e c h o s e n .

I n co n t r a s t t o th i s , o n l y a~ an d a2 a r e E - ad m i s s i b l e . P - a d m i s s i b i l i t yh as n o e f f e c t h e r e , b u t a2 i s s ecu r i t y o p t i ma l r e l a t i v e t o t h e u t i l it ym eas u r e g i v en in t h e ma t r i x , s o a2 i s t h e o n l y S - ad m i s s i b l e a l t e r n a t i v e .T h u s , a c c o r d i n g t o L e v i ' s th e o r y , a 2 s h o u l d b e c h o s e n .

W h e n t h e u n c e r t a i n t y a b o u t th e s t a t e s o f n a t u r e i n t h is d e c i s i o ns i tu a t io n , r e p r e s e n t e d b y ~ /P 0 , is c o n s i d e r e d , w e b e l i e v e t h a t a 3 i si n t u i t iv e l y t h e b e s t a l te r n a t i v e . A g a i n s t th i s it m ay b e a r g u ed t h a tu s i n g a m a x i m i n p r i n c i p l e i s u n n e c e s s a r i l y r i s k a v e r s i v e . I t s h o u l d b er e m e m b e r e d , h o w e v e r , th a t w h e n r e s t r ic t in g ~ t o ~ /P 0 t h e a g e n t i sa l r ead y t ak i n g a r i s k an d h i s ch o i ce o f ~ / p 0 i n d i ca t e s t h a t h e i s n o tw i ll in g t o t a k e a n y f u r t h e r e p i s t e m i c r i sk s . O n t h e o t h e r h a n d , L e v i ' sr e q u i r e m e n t o f E - a d m i s s i b i l i t y h a s t h e c o n s e q u e n c e t h a t , i n m a n yc a s e s , t h e c h o i c e s m a d e b y h is t h e o r y s e e m u n r e a l is t ic a l ly o p t im i s t ic .

A s t r a n g e f e a t u r e o f L e v i ' s t h e o r y i s t h a t i f t h e p r e v i o u s d e c i s i o ns i t u a t io n i s r e s t r i c t ed t o a ch o i ce b e t w ee n a2 an d a3, t h en h i s t h eo r yr e c o m m e n d s c h o o s i n g a 3 i n s t e a d o f a 2! I n t h e ir c h a p t e r o n i n d iv i d u a ld e c i s i o n m a k i n g u n d e r u n c e r t a i n t y , L u c e a n d R a i f fa ( [2 9 ], p p . 2 8 8 -2 9 0 ) i n t r o d u ces t h e co n d i t i o n o f independence of irrelevant alter-natives w h i c h i n i t s s i m p l e s t f o r m d e m a n d s t h a t i f a n a l t e r n a t i v e i sn o t o p t i m a l i n a d e c i s i o n s i t u a t i o n i t c a n n o t b e m a d e o p t i m a l b ya d d i n g n e w a l t e r n a t i v e s t o t h e s i t u a t i o n . T h e e x a m p l e p r e s e n t e d h e r es h o w s t h a t L e v i ' s t h e o r y d o e s n o t s a t i s f y t h is c o n d i t io n s i n c e in th ed ec i s i o n s i t u a t i o n w h e r e a2 an d a3 a r e t h e o n l y av a i l ab l e a l t e r n a t i v e sa n d w h e r e a3 i s o p t i m a l a c c o r d i n g t o L e v i ' s t h e o r y , a 2 c a n b e m a d eo p t i m a l b y a d d i n g a l . 26 I t i s e a s y t o s h o w , h o w e v e r , t h a t t h e M M E Uc r i te r i o n w h i c h h a s b e e n p r e s e n t e d h e r e s a t i s fi e s t h e c o n d i t i o n o fi n d e p e n d e n c e o f i r r e le v a n t a l t e rn a t i v e s . 27L e v i ' s t h e o r y a l s o s e e m s t o h a v e p r o b l e m s i n e x p l a i n i n g s o m e o ft h e e x p e r i m e n t a l r e s u l t s c o n s i d e r e d i n t h i s p a p e r . I f L e v i ' s t h e o r y i sa p p l i e d t o E l l s b e r g ' s t w o d e c i s i o n s i t u a t i o n s a s p r e s e n t e d e a r l i e r , i tg i v e s t h e r e s u l t t h a t b o t h a l t e r n a t i v e s i n t h e t w o s i t u a t i o n s a r eS - ad m i s s i b l e , an d h e n ce t h a t a~ i s eq u a l l y g o o d a s a2 an d a3 i s eq u a l l yg o o d a s a4 . Th i s co n t r a s t s w i t h E l l s b e r g ' s f i n d i n g s w h i ch a r e i na c c o r d a n c e w i th th e r e c o m m e n d a t i o n s o f th e p r e s e n t t h e o r y . S i m ila rc o n s i d e r a t i o n s a p p l y t o t h e e x p e r i m e n t s p r e s e n t e d i n B e c k e r a n dB r o w n s o n [2].

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3 8 0 P E T E R G A R D E N F O R S A N D N I L S - E R I C S A H L 1 N

D e c i s i o n t h e o r i e s o f t h e k i n d p r e s e n t e d i n t h i s p a p e r a r e b a s e d o ns e v e r a l i d e a l i z a t i o n s a n d t h e y w i l l u n a v o i d a b l y b e e x p o s e d t o s o m er e f r a c t o r y e m p i r i c a l m a t e r i a l . W e b e l i e v e , h o w e v e r , t h a t t h e c o n -s i d e r a t i o n s o f t h i s s e c t i o n s h o w t h a t t h e d e c i s i o n t h e o r y p r e s e n t e d i ns ec t i o n 3 i s a mo r e r ea l i s t i c t h eo r y t h an Lev i ' s .

6 . CONCLUSION

Th e s t a r t i n g - p o i n t o f t h i s p ap e r i s t h a t i n man y d ec i s i o n s i t u a t i o n t h ea s s u m p t i o n , m a d e i n th e s t r i c t f o r m o f B a y e s i a n i s m , t h a t t h e b e li e f so f a n a g e n t c a n b e r e p r e s e n t e d b y a s i n g l e p r o b a b i l i t y d i s t r i b u t i o n i su n r e a li s ti c . W e h a v e h e r e p r e s e n t e d m o d e l s o f b e l ie f w h i c h c o n t a i nf i r s t l y , a c l a s s o f p r o b ab i l i t y d i s t r i b u t i o n s , an d , s eco n d l y , a meas u r e o ft h e ep i s t emi c r e l i ab i l i t y o f t h e s e p r o b ab i l i t y d i s t r i b u t i o n s . S ev e r a la u t h o r s b e f o r e u s h a v e s u g g e s t e d t h a t a c l a s s o f p r o b a b i l i t y d i s -t r i b u t i o n s s h o u l d b e e x p l o i t e d w h e n d e s c r i b i n g t h e b e l i e f s o f t h eag en t . A ma i n t h e s i s o f t h i s p ap e r i s t h a t t h i s i s n o t s u f f i c i en t , b u t ana s s e s s m e n t o f th e i n f o r m a t i o n o n w h i c h t h e c l a s s o f p r o b a b i l i t yd i s tr i b u ti o n s i s b a s e d i s a l so n e c e s s a r y . W e h a v e h e r e t r ie d t o c a p t u r et hi s a s s e s s m e n t b y t h e m e a s u r e p o f e p i s t e m i c r e l ia b i li ty . W i th t h e a i do f th is m e a s u r e w e c a n a c c o u n t f o r o n e f o r m o f r i s k ta k i n g in d e c i s io ns i t u a t i o n s .

O n t h e b a s i s o f t h e m o d e l s o f t h e b e l i e f s w h i c h a r e r e l e v a n t i n ad e c i s i o n s i tu a t io n w e h a v e f o r m u l a t e d a d e c i s i o n t h e o r y . W e h a v ea r g u e d t h a t t h i s t h e o r y h a s m o r e d e s i r a b l e p r o p e r t i e s a n d i s b e t t e rs u p p o r t e d t h a n o t h e r d e c i s i o n t h e o r i e s w h i c h a l s o g e n e r a l i z e t h et r a d i t i o n a l B a y e s i a n t h e o r y .T h e M M E U c r i t e r i o n , w h i c h h a s b e e n s u g g e s t e d h e r e a s t h e m a i nr u l e o f t h e d ec i s i o n t h eo r y , i s g en e r a l l y ap p l i cab l e t o d ec i s i o n s i t u a -t io n s w h e r e t h e p o s s i b l e o u t c o m e s a r e n o n - n e g a t iv e f r o m t h e p o i n t o fv i e w o f t h e d e c i s i o n m a k e r . H o w e v e r , t h e r e a r e s it u a ti o n s w h e r e t h eM M E U c r it e ri o n s e e m s t o b e t o o c a u ti o u s. F o r e x a m p l e , t h e' r e f l e c t i o n e f f e c t ' a n d t h e ' i s o l a t i o n e f f e c t ' s u g g e s t e d b y K a h n e m a na n d T v e r s k y [ 24 ] c a n n o t b e e x p l a i n e d d i r e c t ly w i t h th e d e c i s i o n t h e o r yo f th is p a p e r . W e b e l i e v e t h a t in o r d e r to c o v e r th e s e p h e n o m e n a a m o r eg e n e r a l a n d c o m p r e h e n s i v e d e c i s i o n t h e o r y i s n e e d e d w h i c h i n c l u d e sr e f e r e n c e s t o th e d e c i s i o n m a k e r ' s ' le v e l s o f a s p i r a ti o n '. A s p e c i a l c a s eo f t h e e f f e c t s o f l e v e l s o f a s p i r a t i o n w o u l d b e t h e ' s h i f t s o f r e f e r e n c ep o i n t' d i s c u s s e d b y K a h n e m a n a n d T v e r s k y . I n t ro d u c i n g ' le v e ls o f

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U N R E L I A B L E P R O B A B I L I T I E S A N D D E C I S I O N M A K I N G 3 81

a s p i r a t i o n ' m e a n s t h a t t h e p a r t o f t r a d i t i o n a l B a y e s i a n t h e o r y w h i c hr e f e r s t o u t i l i t i e s h a s t o b e c o n s i d e r a b l y e x t e n d e d a n d m o d i f i e d .

L u n d U n i v e r s i t y , S w e d e n

N O T E S* T h e o r d e r o f t h e a u t h o r s ' n a m e s i s b a s e d o n l y u p o n a g e a n d ( o r ? ) w i s d o m . T h e a u t h o r sw i s h t o t h a n k R o b e r t G o l d s m i t h , S 6 r e n H a l ld 6 n , B e n g t H a n s s o n a n d I s a a c L e v i f o r h e l p f u lc r it i ci s m a n d c o m m e n t s .

T h i s e x a m p l e w a s c h o s e n i n o r d e r t o s i m p l i f y t h e e x p o s i t i o n . W e b e l i e v e , h o w e v e r ,t h a t s i m i l a r e x a m p l e s c a n b e f o u n d w i t h i n m a n y a r e a s o f d e c i s i o n m a k i n g , e . g . m e d i c a ld i a g n o s i s a n d p o r t f o l i o s e l e c t io n .2 T h i s de c i s ion r u l e i s a ge ne r a l i z a t i on o f t he r u l e sugge s t e d by G ~ i r de n f o r s i n [ 13 ] , p .169.3 T h i s m e a s u r e i s a s s u m e d t o b e u n i q u e u p t o a p o s i t i v e l i n e a r t r a n s f o r m a t i o n . I n [ 2 6 ]a n d [ 2 7 ] , L e v i h a s g e n e r a l i z e d a n o t h e r d i m e n s i o n o f t h e t r a d i t i o n a l B a y e s i a n d e c i s i o nt h e o r y b y a l l o w i n g s e t s o f u t i l it y fu n c t i o n s w h i c h a r e n o t l i n e a r t r a n s f o r m a t i o n s o f e a c ho t h e r . W e b e l i e v e t h a t t h i s g e n e r a l i z a t i o n i s b e n e f i c i a r y i n s o m e c o n t e x t s , b u t w e w i lln o t d i s c u s s i t f u r t h e r i n t h e p r e s e n t p a p e r . C f . n o t e 2 0 .4 I t i s i n t e r e s t i n g to n o t e t h a t d e F i n e t t i i n [ 10 ], p . 6 2 , n o t e ( a ), h a s r e c o g n i z e d s o m ep r o b l e m i n s u c h a n o p e r a t i o n a l d e f i n i t i o n a s a w a y o f r e p r e s e n t i n g t h e a g e n t ' s b e l i e f s .s T h e c e n t r a l a x i o m i s t h e s o - c a l l e d c o h e r e n c e c r i t e r i o n w h i c h a s s u m e s t h a t i f t h e a g e n ti s w i ll i n g t o b e t o n s t a t e s i a t t h e l e a s t o d d s o f a : b , t h e n h e s h o u l d b e w i l l in g t o b e t o nn o t - s j a t o d d s o f b : a . T h e f i r s t o f t h e s e b e t t i n g r a t i o s w i ll t h u s b e e q u a l t o o n e m i n u st h e s e c o n d b e t t i n g r a t i o , i. e. a l ( a + b ) = 1 - b / ( a + b ) . S m i t h [ 3 7], a m o n g o t h e r s , p o i n t so u t t h a t th i s a s s u m p t i o n n e e d n o t b e s a t i s fi e d . A n a g e n t m a y v e r y w e l l b e w i l li n g t o b e to n l e a s t o d d s o f a : b f o r s j, b u t a t t h e s a m e t i m e b e t o n l e a s t o d d s o f c : d a g a i n s t s~,w h e r e a / ( a + b ) 1 - c / ( c + d ) , w h i c h c o n t r a d i c t s t h e c o h e r e n c e c r it e ri o n .6 T h i s a s p e c t o f r i s k t a k i n g w i ll b e f u r t h e r d i s c u s s e d i n t h e n e x t s e c t i o n .7 I n t h e p r e s e n t p a p e r w e d o n o t a i m a t a n e l a b o r a t e a n a l y s is o f t h e c o n c e p t o fk n o w l e d g e , b u t w e t a k e t h i s a s a p r i m i t i v e n o t i o n .8 I n t h i s p a p e r w e u s e a d e c i s i o n t h e o r y s i m i l a r t o S a v a g e ' s [ 35 ]. W e t h u s d e l i b e r a t e l ye x c l u d e p r o b l e m s c o n n e c t e d w i t h c o n d i t i o n a l p r o b a b i l i t i e s a n d p r o b a b i l i t i e s o f c o n -d i t i o n a l s. O n e r e a s o n f o r t h i s i s t h a t i t i s r a t h e r s t r a i g h t f o r w a r d t o g e n e r a l i z e a d e c i s i o nt h e o r y b a s e d o n s u c h p r o b a b i l i t i e s i n t h e s a m e w a y a s w e h a v e g e n e r a l i z e d S a v a g e ' st h e o r y . T h e s e c o n d r e a s o n i s t h a t L u c e a n d K r a n t z [ 2 8 ] h a v e s h o w n t h a t i n d e c i s i o ns i t u a t i o n s w i t h o n l y f i n i t e l y m a n y s t a t e s a n d o u t c o m e s i t i s p o s s i b l e t o t r a n s l a t e ad e c i s i o n s i t u a t i o n c o n t a i n i n g c o n d i t i o n a l p r o b a b i l i t i e s in t o a S a v a g e t y p e s i t u a t i o n ( a n dv ic e ve r sa ) . F o r a d i s c us s ion o f t h i s r e su l t , c f . J e f f r e y [ 23 ] . A s i s e a s i l y s e e n , t h i s r e su l ta l s o h o l d s f o r s e t s ~ o f p r o b a b i l i t y m e a s u r e s .9 S e e , e . g . , D e m ps te r [ 5 ] , G ood [ 18 ] , S m i th [ 36 ] a nd [ 37 ] .10 L e v i [ 2 7 ], p . 4 0 2 , r e q u i r e s t h a t t h e s e t o f ' p e r m i s s i b l e ' p r o b a b i l i t y m e a s u r e s b e c o n v e x .T h e i n t e r p r e t a t i o n o f ' p e r m i s s i b l e ' i s d i s c u s s e d i n s e c t i o n 4 . I n t h i s c o n n e c t i o n i t i si n t e r e s t i n g to n o t e t h a t S a v a g e [ 35 ], p . 5 8 , n o t e ( + ) , m e n t i o n s t h a t " o n e t e m p t i n g

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382 P E T E R G _ ~ R D E N F O R S A N D N I L S - E R I C S A H L I N

r e p r e s e n t a t i o n o f th e u n s u r e i s t o r e p l a ce t h e p e r s o n ' s s i n g le p r o b a b i l i t y m e a s u r e P b y a s e to f s u c h m e a s u r e s , e s p e c i a l l y a c o n v e x s e t " .11 See , e .g . , De m ps te r [5], Ed m an [6], Ekel iSf [71 , Gi i rde nfo rs [13], Go od [181, Ha l ld6 n[19] , Smith [361, and [37] .t2 W e s a y t h a t a s e t o f p r o b a b i l i t y i n t e r v a l s a s s o c i a t e d w i t h t h e s t a t e s o f a d e c i s i o ns i t u a t i o n i s co n s i s t en t i f a n d o n l y i f , f o r a n y s t a t e s~ a n d f o r a n y n u m b e r x w i t h i n t h ei n t e r v a l a s s o c i a t e d w i t h s i , t h e r e i s a c o m b i n a t i o n o f n u m b e r s , w h i c h l i e w i t h i n t h ei n t e r v a l s a s s o c i a t e d w i t h t h e r e m a i n i n g s t a t e s , s u c h t h a t t h e s u m o f x a n d t h e s en u m b e r s e q u a l 1. L e v i [ 2 61 , p . 4 16 - 4 1 7 , g i v e s a n e x a m p l e w h i c h s h o w s t h a t t h e r e m a yb e t w o d e c i s i o n s i t u a t i o n s w i t h t h e s a m e a l t e r n a t i v e s , s t a t e s a n d o u t c o m e s , b u t w i t hd i f f e r e n t s e t s o f ' p e r m i s s i b l e ' p r o b a b i l i t y m e a s u r e s , w h i c h g i v e d i f fe r e n t d e c i s io n s w h e nh i s d e c i s i o n t h e o r y i s u s e d , a l t h o u g h t h e i n t e r v a l s t h a t c a n b e a s s o c i a t e d w i t h t h e s t a t e s a r eide n t i c a l .t3 A n i n t e r e s t i n g p o s s i b i l it y is t o t a k e p t o b e a s e c o n d o r d e r p r o b a b i l i t y m e a s u r e , i .e , t ol e t 0 b e a p r o b a b i l i t y m e a s u r e d e f i n e d o v e r t h e s e t ~ o f e p i s t e m i c a l l y p o s s i b l ep r o b a b i l i t y m e a s u r e s . I f ~ i s f in i te t h e r e s e e m t o b e n o p r o b l e m s c o n n e c t e d w i t h s u c h am e a s u r e . B u t i f ~ i s t a k e n t o b e a c o n v e x s e t o f p r o b a b i l i t y m e a s u r e s a n d w e a t th es a m e t i m e w a n t a l l m e a s u r e s i n ~ t o h a v e a n o n - z e r o s e c o n d o r d e r p r o b a b i l i t y , w e r u ni n t o p ro b l e m s . H o w e v e r , n o t h i n g t h a t w e h a v e s a i d e x c l u d e s t h e p o s s i b il i ty o f t a k i n g pt o b e a n o n - s t a n d a r d p r o b a b i l it y m e a s u r e . F o r a d i s c u s s i o n o f s u c h m e a s u r e s s e eB e r n s t e i n a n d W a t t e n b e r g [ 3 1 .1 4 W e c a n c o m p a r e t h i s e x a m p l e w i t h t h e d i f f e r e n c e b e t w e e n m a t c h A a n d m a t c h B i nt h e i n t r o d u c t o r y e x a m p l e . T h e r e l ia b i li ty m e a s u r e c o n n e c t e d w i t h m a t c h A w h i c h i sd e p i c t e d i n F i g u r e l , c a n b e s e e n a s c o r r e s p o n d i n g t o t h e r e li a b i l it y m e a s u r e a f t e r t h ee v i d e n c e e h a s b e e n o b t a i n e d , a n d t h e m e a s u r e c o n n e c t e d w i t h m a t c h B a s c o r -r e s p o n d i n g t o t h e r e l i a b i l i t y m e a s u r e b e f o r e e i s o b t a i n e d . I d e a s s i m i l a r t o t h o s ep r e s e n t e d h e r e h a v e b e e n d i s c u s s e d b y B a r - H i l l e l [ 1 ] a n d R o s e n k r a n t z [ 3 2 1 .t~ M o d e l s o f b e l i e f si m i l a r t o o u r s h a v e b e e n p r e s e n t e d i n te r m s o f f u z z y s e t t h e o r y b y [ 39 ]a n d [1 11 . H o w e v e r , w e w i ll n o t c o n s i d e r t h i s t h e o r y i n t h e p r e s e n t p a p e r .16 I n t h e q u o t a t i o n a b o v e i t i s o b v i o u s t h a t P o p p e r u s e s t h e 4 r a d i t io n a l d e f in i t i o n ofr e l e va nc e , i . e . , e i s re levan t t o a i f a n d o n l y i f P ( a ) ~ P ( a , e ) . W e b e l i e v e t h a t t h i sd e f i n it i o n i s t o o n a r r o w . I n s t e a d w e p r o p o s e t h e d e f i n i ti o n t h a t e i s r e l e v a n t t o a i ffP ( a ) ~ P ( a , e ) o r t h e e v i d e n c e e c h a n g e s t h e d e g r e e o f e p i s t e m i c r e l i a b i l i ty o f P .K e y n e s i s a l s o d i s s a ti s f ie d w i t h t h e t r a d i t i o n a l d e f in i t i o n o f ' r e l e v a n c e ' . H e w a n t s t ot r e a t ' w e i g h t o f e v id e n c e ' a n d ' r e l e v a n c e ' a s c o r r e la t i v e t e r m s s o t h a t " t o s a y t h a t a n e wp i e c e o f e v i d e n c e i s ' r e l e v a n t ' is t h e s a m e t h i n g as t o s a y t h a t it i n c r e a s e s t h e ' w e i g h t 'o f t h e a r g u m e n t " ([2 5] , p. 7 2). F o r a p r o o f t h a t K e y n e s ' s d e f i n i ti o n o f 'r e l e v a n c e ' l e a d st o a t r i v ia l i z a t io n re s u l t , a n d f o r a d i s c u s s i o n o f s o m e g e n e r a l r e q u i r e m e n t s o n ad e f i n i t i o n o f ' r e l e v a n c e ' t h e r e a d e r i s r e f e r r e d t o G ~ d e n f o r s [ 1 2 ] ,17 T h e ' d e s i r e d l e v e l o f e p i s t e m i c r e l i a b i l i t y ' c a n b e i n t e r p r e t e d i n t e r m s o f l e v e l s o fa s p i r a t i o n s o t h a t t h e p 0 c h o s e n b y t h e d e c i s i o n m a k e r i s h i s l e v e l o f a s p i r a t i o n a sr e g a r d s e p i s t e m i c r e l ia b i l it y .1~ W e h a v e m e n t i o n e d t h a t a n a g e n t i s t a k i n g a r i s k b y n o t t a k i n g a l l e p i s t e m i c a l l yp o s s i b l e m e a s u r e s u n d e r c o n s i d e r a t i o n . ' R i s k ' is a n o t i o n o f g r e a t c o m p l e x i t y a n d t h el i t e r a t u r e i s f l o o d e d w i t h p a p e r s t r y i n g t o c a p t u r e a l l a s p e c t s o f r i s k i n o n e m e a s u r e .H o w e v e r , w e d o n o t b e l i e v e t h a t t h i s i s p o s s i b l e a n d w e w i l l g i v e a b r i e f e x p l a n a t i o n

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U N R E L I A B L E P R O B A B I L I T I E S A N D D E C I S I O N M A K I N G 3 8 3

w h y . L e t u s , a s a n e x a m p l e , r e t u r n t o M i s s J u l i e . I t s e e m s r e a s o n a b l e t h a t s h e p e r c e i v e sa g r e a t er r i s k in b e t t in g o n m a t c h B ( a n d a n e v e n g r e a t e r o n e i n m a t c h C ) t h a n i n m a t c hA . S h e m a y v e r y w e l t , i f f o r c e d t o d o s o , e s t i m a t e t h a t e a c h p l a y e r h a s a 5 0 % c h a n c e o fw i n n i n g i n b o t h m a t c h A a n d B , b u t s t i ll r e g a r d m a t c h B a s r is k i e r . T h i s i s d u e t o t h ef a c t t h a t h e r s t a t e o f k n o w l e d g e i s v e r y d i f f e r e n t i n t h e t w o d e c i s i o n s i t u a t i o n s a n d i ts h o w s t h a t t h e d e g r e e o f e p i s t e m i c r el i a b i li t y i s a n i m p o r t a n t f a c t o r w h e n d e t e r m i n i n gt h e r i s k i n v o l v e d i n a d e c i s i o n s i t u a t i o n . T h i s a s p e c t o f r i s k t a k i n g h a s n o t b e e nc o n s i d e r e d i n t h e t r a d i t i o n a l th e o r i e s o f r i s k .T h i s d i m e n s i o n o f r i s k t a k i n g is r e p r e s e n t e d i n o u r t h e o r y b y t h e s e l e c t i o n o f a s u b s e t~ i p 0 o f ~ . A n a g e n t w h o t a k e s a l l e p i s t e m i c a l l y p o s s i b l e m e a s u r e s i n t o c o n s i d e r a t i o nt a k e s n o ' e p i s t e m i c ' r i s k a t a l l . I f i t i s a s s u m e d t h a t p i s a s e c o n d o r d e r p r o b a b i l i t yd i s t r i b u t i o n ( c f . n o t e 1 3 a n d [ 3 3 ] ) , w e s u g g e s t t h e f o l l o w i n g m e a s u r e o f t h e e p i s t e m i cr i s k R t a k e n b y a n a g e n t i n a d e c i s i o n s i t u a t i o n :

R ( ~ /p ~ ) = 1 - p(~]po)/p(~9),w he r e p ( ~ ) i s e qua l t o Y ~ P e ~ p( P ) a nd s im i l a r f o r p ( ~ /p0 ) . A s i s e a s i l y s e e n , M is s Ju l i ew i l l t a k e a r a t h e r g r e a t e p i s t e m i e r i s k i f s h e a c t s a s a s t r i c t B a y e s i a n i n m a t c h B . B u tw h e t h e r s h e w i ll d o s o o r n o t i s d e p e n d e n t o n h e r r i s k p r e f e re n c e s .

F o r a c r i t ic i s m o f t h e t r a d i t io n a l r i s k c o n c e p t a n d f o r a d i s c u s s i o n o f o t h e r s o l u t i o n ss e e S a h l i n [3 4] a n d H a n s s o n [ 20 ].~9 Cf . G hrd en for s [13], p . 169.

T h e r e s u l t s o f t h e m a j o r i t y o f s u b j e c t s o f th e G o l d s m i t h - S a h l i n e x p e r i m e n t s s u p p o r tt h e t h e s i s t h a t t h e d e g r e e o f e p i s t e m i c r e l i a b i l i t y o f t h e p r o b a b i l i t y e s t i m a t e s i s a ni m p o r t a n t f a c t o r w h e n c h o o s i n g l o t t e r y t ic k e t s, b u t , i t s h o u l d b e a d m i t t e d t h a t n o t a ll o ft h e s e r e s u l t s c a n b e e x p l a i n e d b y t h e M M E U c r it e ri o n . T h e m a i n r e a s o n f o r t h i s is , i no u r o p i n i o n , t h a t t h e a g e n t ' s values a r e n o t c o m p l e t e l y d e s c r i b e d b y a u t i l i t y m e a s u r e .I n t h i s p a p e r w e h a v e c o n c e n t r a t e d o n t h e e p i s t e m i c a s p e c t s o f d e c i s i o n m a k i n g a n du s e d t h e t r a d i t i o n a l w a y , i . e . u t i l i t y m e a s u r e s , t o r e p r e s e n t t h e v a l u e s o f t h e d e c i s i o nm a k e r . W e b e l i e v e t h a t t h i s p a r t o f t h e t r a d i ti o n a l B a y e s i a n d e c i s io n t h e o r y s h o u l d b em o d i f i e d as w e l l , p e r h a p s b y i n c l u d i n g a ' l e v e l o f a s p i r a t i o n ' , b u t s u c h a n e x t e n s i o n l i e sb e y o n d t h e s c o p e o f t h is p a p e r .2, S u c h a d i s t r i b u t i o n m a y , i n m a n y c a s e s , a s s i g n t h e p r o b a b i l i t y 1 t o o n e o f t h e s t a t e s ,b u t w e d o n o t a s s u m e t h a t i t a l w a y s w i l l . W e i n t e r p r e t ' f u l l i n f o r m a t i o n ' i n a p r a g m a t i cw a y , m e a n i n g s o m e t h i n g l ik e ' h a v i n g a s m u c h i n f o r m a t i o n a s i s p r a c t i c a l l y p o s s i b l e ' , so ,e v e n i f t h e w o r l d i s d e t e r m i n i s t i c , h a v i n g f u l l i n f o r m a t i o n d o e s n o t e n t a i l t h a t o n ek n o w s w h i c h i s t h e t r u e s t a t e o f n a t u r e .z2 F e l l n e r [9 ] c o n s i d e r s p r o b l e m s s i m i l a r t o E l l s b e r g ' s p a r a d o x .~ H o w e v e r , i f t h e a g e n t h a s s o m e i n f o r m a t i o n t h a t h e j u d g e s r e l e v a n t f o r t h ed i s t r i b u t i o n o f t h e b l a c k a n d y e l l o w b a l l s , t h e n t h e e p i s t e m i c r e l i a b i l i t y o f t h e d i s -t r i b u t i o n s i n ~ m a y b e q u i t e d if f e re n t . H e m a y , f o r e x a m p l e , b e l i e v e t h a t t h ed i s t r i b u t ion s i n ~ [ p0 c lu s t e r a r ou nd e . g . ( 1 /3 , 1 /2 , 1 /6 ). T he n the M M E U c r i t e r ion w i l lr e c o m m e n d t h a t a 2 b e p r e f e r r e d t o a l a n d t h a t a 4 b e p r e f e r r e d t o a 3. T h i s r e c o m -m e n d a t i o n d o e s n o t c o n f l i c t w i t h S a v a g e ' s s u r e - t h i n g p r i n c i p l e e i t h e r ( c f. [ 14 ]) .24 I t s h o u l d b e n o t e d t h a t e v e n i f ~ , t h e s e t o f a l l e p i s t e m i c a l l y p o s s i b l e p r o b a b i l i t ym e a s u r e s , i s c o n v e x , t h e s e t ~ / p 0 s e l e c t e d w i t h t h e a i d o f t h e d e s i r e d l e v e l o f e p i s te m i cr e l i a b i l i t y n e e d n o t b e c o n v e x . F o r e x a m p l e , i n t h e r e p r e s e n t a t i o n o f t h e e p i s t e m i c

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re l i a b i l i t y c onne c t e d wi t h ma t c h C a s de p i c t e d i n F i gu re 2 , t he c o r re spond i ng s e t ~ / p0wi l l c ons i s t o f t wo d i s c onne c t e d i n t e rva l s a t t he e nd po i n t s 0 a nd 1 . I f Le v i ' s Bx,t isi de n t if i e d wi t h ~ 9 /p0, t h is e x a mp l e show s t ha t h i s r e qu i re m e n t o f c o nve x i t y i s no t a l wa y srea l i s t ic .25 F u r t he r i n fo rma t i on on t he no t i on o f P -a dmi s s i b i l i t y i s t o be found i n C h . 6 o f Le v i[ 2 7 ] .26 Levi d i scusses th i s problem in [27] , pp . 208-210.27 I t i s a l so e a sy t o s e e t ha t a c c o rd i ng t o Le v i ' s t he o ry t he re m a y be a c ha nge i n t he s e to f op t i ma l a l t e rna t i ve s i f t wo (o r more ) s t a t e s a re c on j o i ne d . Th i s i s t he c a se i f i t i sa s sume d t ha t i f s i a nd s j a re t he t wo s t a t e s t o be c on j o i ne d , t he n , fo r a ny p roba b i l i t yme a s u re , P , t he p roba b i l i t y o f t he c on j o i ne d s t a t e is e qua l t o P ( & ) + P (s i ) a nd t he u t i li t y o ft he ou t c ome i n t he ne w s t a t e , i f a l t e rna t i ve ak i s c hose n , i s e qua l t o ( P ( s i ) 'u u +P (s~). u~i)/(P(&) + P (s~)).S uc h a c on t ra c t i on o f t he de c i s i on p rob l e m i s r e a sona b l e , i f , fo re xa m pl e , i t is r e a l i z e d t ha t t he pa r t i t i on i ng a dop t e d i n i ti a ll y wa s un ne c e s sa r i l y r e f ine d . I t i se a s y t o v e r i f y t h a t t h e s e t o f a l t e rn a t i v es w h i c h a r e o p t im a l a c c o r d i n g to M M E U i s n o ta l t e re d by suc h a c on j o i n i ng o f s t a t e s .

Le v i d i s c us se s t he se m a t t e r s on pp . 161-162 i n [27 ] . He no t e s t ha t on e c a n ob t a i nd i f f e re n t c la s se s o f S -a dm i s s i b l e a l t e rna t ive s b y us i ng d i f f e re n t pa r t i t ions o f t he s t a t e s,bu t he c on t e nds t ha t t he a dop t i on o f a me t hod fo r f i x i ng s e c u r i t y l e ve l s i n de t e rmi n i ngS -a dmi s s i b i l i t y i s a mora l o r po l i t i c a l va l ue j udge me n t d i s t i nc t f rom a p r i nc i p l e o fra t i ona l c ho i c e . W e do no t ha ve a ny suc h p rob l e ms , s i nc e we do no t ob t a i n d i f f e re n tre su l t s whe n c on j o i n i ng s t a t e s a s a bove .

R E F E R E N C E S[1] B a r -Hi l l e l , M. : 1979 , ' The P a ra d ox o f Ide a l Ev i de nc e a nd t he C onc e p t o f R e l e -

va nc e ' , a t r a ns l a t i on o f a pa pe r a ppe a r i ng i n I Y Y U N , a H e b r e w P h i l o s o p h i c a lQuar te r ly , 27 , 1976-1977.

[2] B e c ke r , S . W . a nd B rown son , F . O . " 1964 , ' W h a t P r i c e Am bi gu i t y? Or the R o l e o fA m b i g u i t y in D e c i s io n M a k i n g ', Journal of Political Eco no m y 72, 62-73.[3] B e rns t e i n , A . R . a nd W a t t e nbe rg , F . : 1969 , ' Non -S t a nd a rd Me a su re T he o ry ' , inW . A . J . L u x e m b u r g , e d . , Appl icat ions o f Model Theory of Algebra, A nalysis andProbability, Hol t , R i ne rha r t & W i ns t on , Ne w York , 171 -185 .[4] Carnap , R . : 195 0, Log ical Foundations of Probability, U n i v e r s i t y o f C h i c a g o

P re s s , C h i c a go .[5 ] De mps t e r , A . P . : 1967 , ' Uppe r a nd Lowe r P roba b i l i t i e s Induc e d by a Mul t i va l ue dM a p p i n g ' , Ann. Math. S tat . 38, 325-339.[6] Edm a n , M . : 1973 , ' Add i n g Inde p e nde n t P i e c e s o f Ev i d e nc e ' , i n Modality, Moralityand Other Problems of Sense and Non sense; Essa ys Dedicated to Srren Hal ld~n,C .W .K. Gl e e rup , Lund , 180 -191 .[7] Ek e l6f , P .O . : 1977, Ri~tteg?mg,P . A . N o r s t e d t & S 6 n e r , S t o c k h o l m .

[8] E l l sbe rg , D . : 1961, ' R i sk , Ambi gu i t y , a nd t he S a va ge A x i om s ' , Quarterly Journal ofE c o n o m i c s 75, 643-669.[9] F e l l ne r , W . : 1961 , ' D i s t o r t i on o f S ub j e c t i v e P roba b i l i ti e s a s a R e a c t i on t o U n-c e r t a i n t y ' , Quarterly Journal o f Econ om ics 75, 670-689.

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Kyberg and H. E . Smokie r , eds . , Studies in Subjective Probabili ty, R. E. Kr ieger ,New York, 53-118.[ I1 ] F r ee l ing , A . : I 980 , 'Fuzzy Se t s and Dec i s ion Ana lys i s ' , IE E E Transact ions onSystems, Man, and Cybernet ics SMC -10, no. 1, 341-354.[12] G~trdenfors, P . : 1978, 'On th e L ogic of R eleva nce ' , Syn these 37, 351-367.[13] Gf i rdenfo r s , P . : 1979 , 'Fo reca s t s , D ec i s ions and Unce r t a in P robab i l i t i e s ' ,Erkenntnis 14, 159-181.[14] G/ i rdenfors , P . and Sahl in , N-E. : 1981, 'Dec ision Making wi th U nrel iable Prob-ab i l i ti e s ' . Pap er p r esen ted a t E igh th Re search C onfe r ence on Sub jec t ive P rob-ab i l it y , U t i l it y and Dec i s ion Making , Budapes t .[15] G oldsm ith , R. W. and S ahl in , N-E . : 1982, 'The Role of Sec ond -order Probabi l i t iesin Dec i s ion Ma king ' , i n Hum phreys , P . C . , Svenson , O . and V ar i , A . , eds . , Analys ingand Aiding Decision Processes, Nor th -Hol l and , Amste rdam.[16] Goldsm ith , R. W . and S ahl in , N-E . : 1981, 'Do Seco nd-o rder Proba bi l i t ies AffectDec i s ions? ' , f o r thcoming .[17] Goldsm ith , R. W . and S ahl in , N-E . : 1981 'Seco nd-o rder Probab i l i t ies and R isk inDec i s ion M aking ' , f o r thcoming .[18] G ood , I . J .: 1962 , 'Sub jec t ive P robab i l i t y a s a Measure o f a N on-M easurab le Se t ' ,i n E . N age l , P . S uppes and A . Tar sk i , eds . , Log ic, Methodology and Philosophy o fScience; Proe. o f the 1960 Int . Congress, Stanfo rd Univer s i ty P ress , S t an fo rd .[19] HaUd6n, S .: 1973, ' Indicie me kan ism er ' , Tidskri f t for Ret tsvi tenskap 86, 55-64.[20] Hansson , B . : 1981 , 'The Dec i s ion Game - t he C oncep tua l i sa t ion o f Ri sk andUt i l i t y ' , i n E . M orscher and R. S t r anz inger , eds ., Eth ics - Foundations, Problems,

and Appl icat fons, Wien .[21] Hodges , J . L . and Lehm ann , E . L . : 1952 , 'The U ses o f P r ev ious Exper i ence inReach ing S ta t i s ti ca l Dec i s ions ' , Annals o f Mathemat ical S tat is t ics 23, 396--407.[22] H urwicz , L : 1951 , 'Som e S pec i f ica t ion P rob lems and App l i ca t ions to Econo met r i cM o d e l s ' , Economet r i ca 19, 343-344 (abstract) .[231 Jef f rey, R. C. : 1977, 'Sav age 's Ome let ' , in F . S uppe and P . D. A squi th , eds. , P S A1976, volum e 2 , East Lansing, M ichigan, 361-371,[24] Kahn eman , D . and Tver sky , A . : 1979 , 'P rosp ec t The ory ' , Economet r i ca 47,263-292.[25] Keynes, J . M. : 1921, A Treatise on Probability, Macm i l lan , Londo n .[26] L evi , I . : 1974, 'On Ind eterm inate Proba bi l i t ies ' , Journal o f Phi losophy 71,391-418.[27] Levi, I . : 1980, The Enterprise o[ Knowledge, MIT Press , Cambr idge , Mass .[28 ] Lu te , D . and Kran tz , D . H . : 1971 , 'Cond i t iona l Exp ec ted U t i l i t y ' , Economet r i ca39, 253-271.[29] Luce, D. and Raiffa, H.: 1957, Games and Decisions, W iley , New York , 288-290 .[30] Peirce, C. S.: 1932, Collected Papers, ed . by H ar t shorne and W eiss , Be lknap P ress ,Cam br idge , Mass .[31] Popper , K. R. : 1974, The Logic of Scienti f ic Discovery, Hutch inson , London .[32] Rosenkran tz , R . D . : 1973 , 'P robab i l i t y Mag ic Un mas ked ' , Philosophy of Scienc e40, 227-233.[33] Sah l in , N-E. : 1980 , 'Genera l i zed Bayes i an D ec i s ion Mode l s ' , Working P aper.[34] Sahl in , N-E . : 1980, 'Le ve l of Aspirat io n and R isk ' , Working Paper No. 21, TheMattias Fremling Society.

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[35J Savage, L. J.: 1972, The Foundat ions of S tat is t ics , Dover, New York.[36] Smith, C. A. B.: 1961, Consistency in Statistical Inference and Decision', Journalof the Ro yal S tat is t ical Society Ser. B 23, 1-25.[37] Smith, C. A. B.: 1965, 'Personal Probabilities and Statistical Analysis', Journal ofthe Roy al S tat is t ical Society Set .