Moral hazard and conditional preferences

Journal of Mathematical Economics 31 Ž1999. 159–181

Moral hazard and conditional preferences Jacques H. Dreze ` ) , Aldo Rustichini CORE UniÕersite´ Catholique de LouÕain, LouÕain, Belgium Received 1 October 1997

Abstract Conditional Expected Utility Theory ŽCEUT. provides an axiomatic foundation for a theory of decision under uncertainty which allows agents to choose which event in a given partition of the state space will occur. Here, we provide an alternative axiomatization of this situation. In this framework, CEUT may be seen as a special case of Decision theory with moral hazard at the same time it is possible to describe the conditions in which this theory is free from possible difficulties. They turn out to be very strong: they imply that either the decision-maker has full control of the events, or he is a standard expected utility maximizer. We also provide an axiomatic justification of the partition structure. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Decision theory; Conditional expected utility theory; Moral hazard

1. Introduction and summary Ž1. Decision theory in the ‘classical’ tradition of Ramsey Ž1926., De Finetti Ž1937. and Savage Ž1954. rests upon observable preferences A among acts f, g, . . . defined as functions from a set S of states into a set C of consequences. For any given subset B of S, an ‘event’, the conditional preference A B is derived from the observable preferences through the definition: f A B g iff f X A g X whenever f X Ž s . s f Ž s . , g X Ž s . s g Ž s . , for all s g B, and f X Ž s . s g X Ž s . for all s g S _ B. This definition is accompanied by a consistency axiom, asserting that A B is a weak order ŽSure Thing Principle.. A variant of this classical theory, called )

Corresponding author.

0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 6 5 - 7

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Conditional Expected Utility Theory ŽCEUT. and due primarily to Fishburn Ž1964, 1973., Pfanzagl Ž1967, 1968. and Luce and Krantz Ž1971., rests upon a primitive notion of preference among conditional decisions f A , g B , . . . defined as functions from the events A, B, . . . into C. This primitive relation f A A g B is directly assumed to be a weak order; the compatibility of that order with Eq. Ž1.1. is part of the axiom system. The aim of the present paper is to spell out the implications of such compatibility, or logical consistency, between the two concepts of conditional preferences. For brevity, we refer to them as ‘derived’ and ‘primitive’, respectively. We are not aware of previous work with a similar aim. 1 Ž2. In order to put our work in proper perspective, we must refer to two extensions of the classical theory which have also been addressed by CEUT, namely State-Dependent Preferences and Moral Hazard. Unlike its primitive counterpart, the derived definition ŽEq. Ž1.1.. does not operate across events Žit does not apply to conditional decisions with different domains.. A further assumption of ‘state-independent preferences’ takes a step in that direction by stipulating that, for all pairs of elements s, sX g S. f X A s g iff f X A sX g X whenever f X Ž sX . s f Ž s . , g X Ž sX . s g Ž s . .

Ž 1.2 .

This assumption requires the association of identical consequences with alternative states, for otherwise f X Ž sX . s f Ž s . becomes meaningless. Several authors Že.g., Dreze, ` 1987; Fishburn, 1973; Karni et al., 1983. regard such a structural assumption as exceedingly restrictive: the case where the decision maker is alive under s but dead under sX is a standard illustration. The CEUT work of Fishburn aims in particular at avoiding that restrictive structural assumption. Yet, the generalisation is limited, because it is assumed that ‘equally desirable’ Žthough not ‘identical’. consequences can always be associated with alternative states. With state-independent preferences, the classical approach to decision theory leads to the representation: f A g iff

Ý s Ž s . uŽ f Ž s . . F Ý s Ž s . uŽ g Ž s . . sgS

Ž 1.3 .

sgS

in terms of a probability measure s on S and a state-independent utility function u on C, where u is defined up to a positive linear transformation. In order to extend the classical approach to encompass state-dependent preferences, let B denote a set of prizes which can be associated freely with alternative states. An act is then a function from S into B, and a consequence is naturally defined as a pair Ž s, b ., or Ž s, f Ž s .. in state s under the act f. 1 Skiadas Ž1991. also obtains from assumptions about primitive conditional preferences a representation of preferences among unconditional acts. But he does not rely explicitly on derived conditional preferences.

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Using probability mixtures of prizes, one can elicit from derived preferences the conditional utilities uŽ s, b . and combine them in the representation theorem. f A g iff

Ý s Ž s . u Ž s, f Ž s . . F Ý s Ž s . u Ž s, g Ž s . . sgS

Ž 1.4 .

sgS

But one cannot identify separately the subjective probabilities s Ž s . and the units of scale of the conditional utilities uŽ s,P .. For an arbitrary but strictly positive probability on s, the preferences represented by Ss g S s Ž s . uŽ s, g Ž s .. are also represented by Ss g S p Ž s . Õ Ž s, g Ž s .., where: s Ž s. Õ Ž s,b . s c u Ž s,b . q d s ,c ) 0. Ž 1.5 . p Ž s. Primitive preferences over conditional decisions cut through that difficulty by assuming that two consequences like Ž s, b . and Ž sX , bX . are directly comparable. Preferences among probability mixtures of these are then represented by a function uŽ s, b . defined up to a positive linear transformation. This rules out nonlinear Žin s . transformations like Eq. Ž1.5.. As explained more precisely below, primitive conditional preferences formally reduce the case of state-dependent preferences to the simpler case of state-independent preferences. Ž3. In the same way that consequences cannot always be defined independently of the state, similarly many decision situations involve outcomes contingent on occurrences influenced jointly by the state of nature and the behavior of the decision maker. The death of the decision maker again illustrates such an occurrence. Such situations are said to involve ‘moral hazard’. ŽIn much of the contemporary literature, the term ‘moral hazard’ refers to situations where the probability of observable events can be manipulated at some unobservable cost or effort. In this respect, our definition differs from the standard one.. The presence of moral hazard leads to violations of the assumption that derived conditional preferences are a weak order ŽSure Thing Principle.. This is most easily seen through a simple example. Let S s Ž s1 , s2 , s3 , s4 ., and let  s1 , s2 4 denote the event ‘you attend a concert tonight’, while  s1 , s34 is the event ‘it rains tonight’. Consider then four acts with payoffs as described in Table 1. One would not be surprised to observe the pairwise orderings: f % g, f X ; g X , in apparent violation of the Sure Thing Principle. The rationale is that ‘you’ would go to the concert if the relevant acts are f and g; but ‘you’ would not go if the relevant acts are f X and g X . Table 1

f g X f X g

S1

S2

S3

S4

100 0 100 0

100 0 100 0

y1000 y1000 5000 5000

y1000 y1000 5000 5000

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The CEUT work of Luce and Krantz Ž1971. aims in particular at allowing for this type of moral hazard. Their approach would validate the pairwise ordering of the previous paragraph as reflecting the following primitive preferences among conditional decisions: fXs 3 , s4 4 ; gX s 3 , s4 4 % f s1 , s 2 4 ; fXs1 , s 2 4 % g s1 , s 2 4 ; gX s1 , s 2 4 % f s 3 , s4 4 ; g s 3 , s4 4 .

Ž 1.6 .

Primitive preferences among conditional decisions are entirely meaningful in this illustration, in particular because ‘attending the concert’ or ‘not attending the concert’ is naturally within the control of he decision-maker, and is also an observable event. Thus, the primitive preferences reflect material behavior. This is also a situation where state-dependent preferences are revealed by decisions. For instance, the indifference between f X and g X reveals that a ‘bribe’ of 5000 y 100 might induce ‘you’ not to attend the concert. Ž4. Similar ideas underlie the extension of the classical approach to moral hazard Žand state-dependent preferences. in the work of Dreze ` Ž1987.. This work takes as starting point an alternative formulation of the Savage theory due to Anscombe and Aumann Ž1963.. In that formulation, the Sure Thing Principle is embedded in a broader assumption, called ‘Reversal of Order’ ŽRO., which stipulates the following: if the toss of a coin is to decide which of two acts obtains, it is immaterial to the decision-maker that the coin be tossed and the outcome known before or after observing the true state. In the example of Table 1, if the toss of the coin were to decide whether the relevant act is f or g Žrespectively, f X or g X ., ‘you’ would not care whether the coin is tossed today or tomorrow, because ‘you’ would in any case attend Žrespectively not attend. the concert. On the other hand, if the toss were to decide between f and f X , the early information would be desired, since ‘you’ would attend the concert if f, and not attend if f X . RO is then violated. To encompass moral hazard, Dreze ` replaces RO by an assumption of Non-Negative Value of Information: early information never hurts, but sometimes helps, in the single person context. This weaker assumption accommodates violations of the Sure Thing Principle, like f % g but f X ; g X in Table 1. The theory developed by Dreze ` leads to a representation theorem asserting the existence of a closed convex set S of probability measures on S such that: f A g iff max Ý s Ž s . u Ž s, f Ž s . . Fmax Ý s Ž s . u Ž s, g Ž s . . sg S sgS

s g S sgS

Ž 1.7 .

Ž5. The CEUT work of Luce and Krantz Ž1971. has a further specificity. It assumes that events subject to moral hazard are identified a priori. Let accordingly M s Ž A1 , . . . , A m . be a fixed partition of S with M the algebra generated by M. The idea is that an element A k of the partition will occur, either as a choice of the decision maker, or under the joint influence of the decision maker and nature. Conditionally on A k , the occurrence of a state s Žout of the possible states in A k . is nature’s choice. In Table 1, for instance, M s Ž A1 , A 2 ., A1 s  s1 , s2 4 , A 2 s  s3 ,

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s4 4 . In the framework used by Dreze, ` the existence of such a partition M is revealed ex post by properties of the set S . Thus, in the example of Table 1, there s1 exists a t in w0, 1x, the probability of rain, and S s  s g D 4 : s 1 q s 3 s t Ž s1 q s2 . s s3 4 . Ž s 3 q s4 . A more interesting question is whether the existence of such a partition, associated with the specific events, can be postulated a priori. 2 We provide an affirmative answer to that question in Section 7. Not surprisingly a specific axiom ŽMO. is needed to guarantee that the interplay of the choices of nature and of the decision maker respectively is captured by a partition. This is reminiscent of the assumption typically made in models of differential information, where one agent’s information structure is described by a partition. For ease of exposition, we consider first Žuntil Section 7. the simpler case where a partition M s Ž A1 , A 2 , . . . , A m . defining the events subject to moral hazard is given a priori. Ž6. In light of this brief reminder, we can summarize the contents of the present paper. We follow Luce and Krantz Ž1971. in assuming the existence of primitive preferences over elementary conditional acts, that is acts conditional on elements A k of M, as well as over unconditional acts, and over probability mixtures of all these acts. We assume that these preferences satisfy standard axioms—weak order, independence, continuity—so that they admit a representation in terms of the ‘utilities’ V Ž f, A k ., V Ž f ., defined jointly up to a common positive linear transformation. From then on, we describe all acts by these utilities. In particular, we do not spell out the representation of V Ž f, A k . in terms of a subjectively expected utility of the consequences associated with states s in A k . That would be a straightforward application of standard decision theory. We then study the consistency between the derived and the primitive approach in two steps. In the first stage ŽSection 4., derived preferences conditional on elements A k of M are assumed well defined. A consistency postulate asserts that, for elementary conditional decisions with identical domains, the primitive and derived orders are compatible. Under suitable additional postulates, the representation theorem ŽEq. Ž1.7.. is validated, in our theorem 4.12. The two approaches are, thus, fully reconciled. In addition, the primitive approach always leads to full identification of u and S . The identification problem is, thus, solved—or perhaps assumed away—by admitting as primitive the preferences among conditional decisions with different domains, which are not verifiable through material behavior when the relevant states are not objects of choice. Also, we free the analysis of the restrictive assumption present in all of the work under review—namely the assumption that, if A and g B are given, there exists f A such that f A ; g B ; see axioms 9.i of Luce and Krantz Ž1971., axiom ŽA.6. of Fishburn Ž1973., or the related Žthough distinct. assumption 8.2 of Dreze `

2

We thank an anonymous referee for urging us to deal with that question.

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Ž1987.. Again, this appears implausible if the decision-maker is alive under A, dead under B. Referring to the representation ŽEq. Ž1.7.., these assumptions would imply that for all s, sX g S, there exist b,bX g B such that uŽ s, b . s uŽ sX , bX ., i.e., there exist constant utility acts. We replace this extreme assumption by the much weaker one that there exist acts with constant differences in utility. This almost innocuous assumption boils down to eliminating null states, which correspond to generalized conditional indifference. In a second stage ŽSection 5. derived preferences conditional on an arbitrary event B in M are assumed well defined. This is also a prerequisite for primitive preferences among general conditional acts f B to be meaningful. Our theorem 5.2 establishes that, when there are at least three elements in M , the set S in Eq. Ž1.7. is either a singleton or the entire simplex. Again, the reconciliation between the derived and the primitive approach is complete: but it is shown that primitive preferences among arbitrary conditional decisions are meaningful, and compatible with derived preferences, under extreme circumstances alone-either full control of events Ž S is the entire simplex. or absence of moral hazard. Finally ŽSection 6., we show that CEUT with full control of events can be derived from a single transparent assumption, and in Section 7, we axiomatize the partition. The main proofs are collected in Appendix A.

2. Basic concepts and definitions We first introduce our notation and basic concepts. S is the set of states of nature, finite. Since, as we said, our decision-maker can not only choose an act, but also influence the choice of some events, we may say that he is choosing a strategy in a game against nature. For this reason, we replace the more commonly used term ‘act’ with the term ‘game’. Formally, a game is a function g g G from the state space S to B where B is the set of probability measures over B, a finite set of prizes. So f Ž s, b . g w0,1x denotes the probability of the prize b at the state s under the game f. Note that even if S and B are finite sets, G is not; and G can be described as the set of m-tuples of elements in B, one for each state. A lottery is a probability measure over games and is denoted by g g G . Since the timing of revelation of the various random events is essential in the analysis of this paper, we emphasise that a lottery has to be understood as follows: for a given lottery g a random device is activated to determine, according to the probability g , the game which will be played, and the outcome is immediately revealed to the decision maker; after this, the true state is observed, and this together with the game previously drawn determines a probability over the set of prizes; finally a random device with this probability is activated, and the prize determined. Note that if the player-decision maker can have any influence over the probability according to which the state is determined, he can do so conditional on the

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information of the outcome of the lottery, that is conditional on knowing which game he is in fact facing; but not conditionally on the outcome of the game given s. A simple lottery is a lottery with finite support on the set G. The lottery under which two lotteries g and g X receive probability a and 1 y a , respectively is denoted by gag X . Consider now a lottery g , and imagine the game which is obtained by the same sequence of random devices that define g , with the only difference that the outcomes are revealed at the same time, after the state of nature has been observed. More formally, the game gg corresponding to a simple lottery g is defined by: gg Ž s,b . s

Ý g Ž g . g Ž s,b . ggG

for every s g S and b g B. This definition is applicable to lotteries of the form gag X , with corresponding game ggag X . The critical difference between the lottery and the game is again the time at which the outcome of random devices is revealed. This is most clearly illustrated for the case of two games f, f X which assign definite prizes to all states; i.e., such that f Ž s, b . g w0,1x for all Ž s, b . and similarly for f X . The lottery f a f X involves a single random device, the outcome of which determines whether f or f X is the relevant game. The outcome is immediately revealed to the decision maker, who can act accordingly. On the other hand, in the game g f a f X , the same random device is activated but the outcome is revealed after the state has been observed, so that no use can be made of the information which game is being played. 3. Conditional games and preferences We are now ready to introduce the main structural assumptions of the paper Žthe distinction between structural and behavioral assumptions goes back to Suppes Ž1956.; see also the work of Luce and Krantz Ž1971.. Consider the situation in which the decision maker can influence the probability according to which state is chosen, but he can do so in a very specific way: namely, by controlling or influencing the probability of the element A k in the set M s Ž A1 , . . . , A m . of events, a partition of S with algebra M . The control may be complete, so that he can actually choose the event that will occur: for instance, if A1 corresponds to the event ‘attend the concert’, and A 2 ‘not attending the concert’, then we may imagine a situation in which A1 , rather than A 2 , is entirely determined by the decision maker. This control, however, may be only partial: i.e., the decision maker can make the occurrence of an event more or less likely. For instance, ‘attending the concert’ may be subject to availability of seats or transportation. Throughout this paper, we start from primitive preferences over lotteries among games conditional on a single element A k of M, and among unconditional games.

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The set of all these games is denoted by GCŽ M ., and the set of games conditional on A k by GCŽ A k .. Such a game may be understood as ‘the game f with a commitment to implement AYk . This interpretation, clear when the decision maker has full control over the events in the partition, becomes problematic when the control is not complete. To simplify the notation, we let A 0 ' S, and we occasionally write the unconditional game f as Ž f, A 0 .. Lotteries over conditional games are probability measures over them; they are still denoted by g ; the set of lotteries over GC is denoted by G C, that over GCŽ A k . is denoted G C Ž A k ., and that over GCŽ M . is G C Ž M .. The probability assigned by a lottery g to a conditional game Ž f, A k . is denoted by g Ž f, A k .. The lottery Ž f, Bk . a Ž f, Bl . can be interpreted as the ‘game f with a commitment to implement the mixed strategy of choosing the event Bk with probability a , and the event Bl with probability 1 y a ’. More generally for any s g D m , the unit simplex in the m-dimensional space, we write Ž f, s . for the game with commitment to implement the mixed strategy s s Ž s 1 , . . . , sm .. Finally, for a given g g G C Ž Bk ., Ž gg , Bk . is a conditional game, defined in the natural way; for instance if g s Ž f, Bk . a Ž h, Bk ., then: gŽ f , B k . a Ž h , B k . Ž s,b . s a f Ž s,b . q Ž 1 y a . h Ž s,b . for any s g Bk . The primitive notion of conditional preference is a binary relation over lotteries on conditional games; g Ag X means that g is not preferred to g X . We write:

Ž f , Ak . AŽ g , A j . for any pair of conditional games. We introduce the following standard assumption on preferences on lotteries in G C Ž M .: Assumption 3.1: WIC ŽWeak Order, Independence, Continuity.. WO ŽWeak Order.: The preference order A is a weak order; that is: Ži. for all g , g X g G C Ž M ., either g Ag X or g Ag ; Žii. for all g , g X , g Y g G C Ž M ., if g $ g X and g X $ g Y then g $ g Y . I ŽIndependence.: For all g , g X , g Y g G C Ž M . and all a g w0,1x, gag Y Agag Y if and only if g Ag X . C ŽContinuity.: For all g , g X , g Y g G C Ž M ., if g $$ Ž%.g X , then there is an a g w0,1x such that gag Y $ Ž%.g X . The next theorem follows immediately from the representation of Von Neumann and Morgenstern Ž1944.. Theorem 3.2: Assume WIC on G C(M). Then there exists a real Õalued function V defined on GC(M) up to a positiÕe linear transformation, such that for the lotteries g , g X g G C(M), g Ag X if and only if:

Ý

g Ž g , Ak . V Ž g , Ak . F

ggG , kg 0,1, . . . , m 4

Ý

g X Ž g , Ak . V Ž g , Ak .

ggG , kg 0,1, . . . , m 4

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For future reference, we denote by Rk the range of the functions V ŽP, A k ., and by R the product of the sets Rk .

4. Elementary conditional preferences Ž1. In the previous sections, we have defined primitive preferences respectively for elementary conditional games and for unconditional games, and obtained a single theorem 3.2 representing both orders; so we can assign to a game g an unconditional value V Ž g . and a set of m conditional values V Ž g, A k .. In this section we define preferences conditional on an event in M on the basis of the preferences on unconditional games, as is traditionally done in the theory of games against nature. These, however, are derived preferences, and must be related to the conditional preferences as primitive objects. In the following step we shall impose restrictions that insure the consistency of the two. Then, on the basis of this consistency condition we formulate and prove our main representation theorem 4.12. In this theorem, the utility of the decision-maker may be event dependent. It is worth noting, however, that in our setup the event dependency does not introduce a major difficulty. Decision theory with moral hazard is relatively straightforward in the case of state-independent preferences. However, state-independent preferences are particularly hard to defend for events which are subject to moral hazard, so it is desirable to eliminate that restriction. Our task is simplified by the fact that the introduction of Conditional Preferences across events as a primitive notion brings the case of state-dependent preferences back into the framework of state-independent preferences: the conditional values of theorem 3.2 provide state-independent real numbers V Ž f, A k . for every game which can then be used as a basis for the representation. Thus, the proof of theorem 4.12 follows closely the proof of theorem 6.1 in the work of Dreze ` Ž1987. which covers the case of state-independent preferences. We now define a consistency condition among the two preferences. Intuitively we require that a game g is preferred to f conditionally on an event B if and only if it is unconditionally preferred to f when f and g are equal outside B. Definition 4.1: ŽConditional Preferences.. For any B g M and any f, g g GCŽ B ., we say that: Ži. g $ B f Ž f is preferred to g given B . if and only if there exist two games h, hX with h s f, hX s g on B, h s hX on the complement of B, and hX $ h. Žii. f ; B if and only if their neither g $ B f nor f $ B g. Although the next assumption will be implied by the stronger assumption 4.3 introduced later, we state it separately for clarity and later comparison.

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Assumption 4.2: ECP ŽElementary Conditional Preferences are Well Defined.. For every k, if g $ A k f, then it is not the case that f $ A k g. It is useful to compare the assumption ECP with the Sure Thing Principle Ži.e., the Postulate P 2 in the work of Savage, 1954. The condition g $ Ak f does not exclude the existence of a pair h, hX as in 4.1 Ži. such that h ; hX : this would happen if the game h in Ack , the complement of A k , is attractive enough to induce the decision-maker to choose Ack , and he can do so. Also, ECP only applies to elementary events A k , not to general events B. So the condition ECP is precisely the weakening of the Sure Thing Principle which is necessary to allow for moral hazard. Now we introduce the assumption that imposes consistency between the conditional preferences that we have just defined and the primitive notion of preferences over conditional games. Assumption 4.3: CPŽ M . ŽConsistency of Elementary Conditional Preferences.. For every A k g M, if Ž g, A k .AŽ f, A k ., then it is not the case that f $ A k g. This is a weakening of axiom 4 of Luce and Krantz Ž1971. which states: if B l C s Ø and f s g on C, then if and only if f A B j C g. In assumption 4.3, the set B is an element of M, the set C is B c, and the condition is weakened to the implication in one direction only. Clearly, if g $ Ak f then by CPŽ M .Ž g, A k . $ Ž f, A k ., and then again by CPŽ M . it is not the case that f $ Ak g ; so assumption CPŽ M . implies ECP. Under CPŽ M ., the preference conditional on any A k g M is a weak order. The converse of the implication does not hold: if f $ Ak g then Ž f, A k . $ Ž g, A k ., but f ; Ak g , and Ž f, A k . $ Ž g, A k . are compatible. A first consequence of assumption CPŽ M . is that if a game f is preferred to g conditionally on every event in M, then it is unconditionally preferred to g. Lemma 4.4: If CP(M) holds, and (g, A k )A (f, A k ) for all A k g M, then g A f. k Proof: For each k s 1, . . . , m define a game f k as follows: f k s g on D is1 Ai, m f k s f on D iskq1 A i . Then by the assumption CPŽ M ., for every k s 1, PPP , m we have f k A f ky1 i.e., g s f m A f my1 A PPP A f 1 A f.

Ž2. To avoid uninteresting cases, we shall assume that there exist at least two such games, which are not indifferent. This includes an assumption of nondegener-

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acy Žno generalised indifference.. In addition, it imposes that no event A k is null, where: Definition 4.5: ŽNull Event.. An event A k is null if and only if f ; A k g for every pair f, g of acts. This last aspect is a harmless simplification and can be relaxed at some technical cost. These two properties together are equivalent to the assumption that, for every k, the range Rk of conditional values has a non-empty interior. Assumption 4.6: NCI ŽNo Conditional Indifference.. There exist two games g 0 , g 1 such that g 0 $ A k g 1 for every A k g M. We normalize the function V to V Ž g 1 . s 1 and V Ž g 0 . s 0. Under event-dependent preferences, two conditional games which assign the same prize respectively to all the states in A k and to all the states in A l , need not be indifferent: f Ž s, b . s b g w0, 1x for every prize b and for all s g A k j A l does not imply Ž f, A k . ; Ž f, A l ., i.e., V Ž f, A k . s V Ž f, A l .. ŽThink of the case where the decisionmaker is alive under A k and dead under A l .. Still, there may Žor may not. exist games f such that V Ž f, A k . s V Ž f, A l .. If a game f has the property that V Ž f, A k . s V Ž f, A l . for all l s 1, . . . , m, it would be natural to impose that V Ž f . s V Ž f, A k .. This would also follow from axiom 3 of Luce and Krantz Ž1971.; it is part of the structure of the model of Anscombe and Aumann Ž1963., where it follows from state-independent preferences. Although natural, such an assumption is useless if there does not exist any game with the stated property. A less stringent requirement than the existence of constant value games is readily stated in terms of differences rather than levels of conditional values. Assume that, for all k, V Ž f, A k . is in the interior of the range Rk . Because ranges are convex, there will exist another game g such that V Ž f, A k . y V Ž g, A k . s d / 0, for a real number d and for all k. A natural extension of the condition discussed in the previous paragraph would impose V Ž f . y V Ž g . s d. This will be part of the next assumption. By the same reasoning, if V Ž g, A k . s cV Ž f, A k . for some real number c / 1 and for all k, it is natural to impose V Ž g . s cV Ž f .. We assume both properties as: Assumption 4.7: SRV ŽScalarly Related Values.. Let a and b be in w0,1x; Ži. if Ž f, A k . a g 0 ; Ž g, A k . for every A k g M, then f a g 0 ; g; Žii. if Ž f, A k . a g 0 ; Ž g, A k . b g 1 for every A k g M, then f a g 0 ; g b g 1. In terms of the representation, the assumption says that for every pair of games g and f, if a V Ž f, A k . s V Ž g, A k . for every k, then a V Ž f . s V Ž g .; and if b V Ž f, A k . s b V Ž g, A k . q Ž1 y b . for every k, then b V Ž f . s b V Ž g . q Ž1 y b .. Assumption 4.7 is the logically harmless complication brought in by event-dependent preferences; it is a much weaker assumption than the existence of games with constant conditional values across events i.e., an assumption of overlapping ranges. It implies that lemma 4.4 holds in a stronger form.

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Lemma 4.8: If CP(M), SRV hold, and (g, A k ) $ (f, A k ) for all A k g M, then g $ f. The next assumption is the weakening of RO that we had announced. In a single person problem, earlier information is never harmful, since it can be discarded. Differently from what RO implies, however, information might be useful, because it can be used in the choice of the strategy. Assumption 4.9: VI ŽValue of Information.. For every g g G , gg Ag . It may happen that information is useless, even if the player does have the power to affect the relative probability of events. Reasoning intuitively for a moment, we can think of an instance in which a lottery is a mixture of two games for which there is a common optimal strategy. In this case, the information which of these two games he is facing will be of no use to the player. More generally, it is possible to define a relation among games which makes this notion of useless information precise. Definition 4.10: ŽEquipotence.. Two games f and h are said to be equipotent, and we write fEh, if and only if f a h ; g f a h for all a g w0,1x. The equipotence relation is clearly reflexive and symmetric, but is not transitive. We said that two games are equipotent if there is a strategy, i.e., a probability measure, available to the player and optimal for both games. Now note that if two games have values which are positive linear transformations of each other, then the sets of optimal strategies coincide. Hence, in particular the two games are equipotent. We assume this as: Assumption 4.11: SRE ŽScalarly Related Games are Equipotent.. Let a and b be in w0, 1x; Ži. if Ž f, A k . a g 0 ; Ž g, A k . for every A k g M, then f Eg; Žii. if Ž f, A k . b g 0 ; Ž g, A k . b g 1 for every A k g M, then f Eg. In terms of the representation, the assumption says that for every pair of games g and f, if V Ž g, A k . s cV Ž f, A k . for every k with a c ) 0, or V Ž g, A k . s V Ž f, A k . q d for every k, then f Eg. Ž4. We can now state and prove the main representation theorem. Theorem 4.12: Assume WIC on G C(M), and CP(M), NCI, SRV, VI, SRE. Then there exists a closed and conÕex set Sc : Dm of probability measures on M such that for all f g G: m

V Ž f . s max

Ý s Ž Ak . V Ž f , Ak .

sg S cks1

Ž 4.8 .

Furthermore, there is a unique minimal set S with these properties. If there is a point Õ g int R such that Õi s Õj for every i, j, then the set Sc is unique. To be precise, we say that S is a minimal set if it is closed, convex, represents the order in Eq. Ž4.8., and there is no subset of it with the same properties. There is a good reason for focusing on minimal sets: given any other superset S X , and

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any strategy in S X _ S , the strategy s might never be observed to be chosen by the decision-maker. The non-uniqueness problem may arise when the ranges Rk do not overlap. An example will clarify this point. Take m s 2, R s R1 = R2 such that min R1 ) max R2 . If V Ž f . s pV Ž f, A1 . q Ž1 y p .V Ž f, A 2 ., then clearly the set Ž p, 1 y p .4 gives the representation in theorem 4.12, but so also does any other set which is the convex combination of the points Ž q, 1 y q . and Ž p, 1 y p ., when q - p. But p alone is ever observed. Theorem 4.12 provides a full reconciliation of decision theory in the ‘classical’ tradition with CEUT. With state-dependent preferences theorem 4.12 extends the classical results by adding a uniqueness property, which otherwise fails. Compare for this theorem 8.2 of Dreze ` Ž1987. where the uniqueness of the minimal set fails, unless there exists a Õ with Õi s Õj for every i, j. The additional property reflects both the analytical strength and behavioral weakness of the primitive definition of conditional preferences: Ž f, A k . K Ž g, A l . is not materially observable, yet solves the identification problem of subjective probabilities. We conclude this section with three remarks relating our results to the existing literature. Ži. One could prove theorem 4.12 from a weaker set of assumptions, where: Ža. WIC characterises the preferences among unconditional lotteries in G , but not among conditional lotteries in G C Ž M .; Žb. ECP is used instead of CPŽ M .; Žc. an additional assumption of independence and continuity of conditional preferences Žgiven A k , for k s 1, . . . , m. is used to elicit V Ž f, A k . from ECP, without ever resorting to conditional preferences as a primitive; the assumption ICC of Dreze ` Ž1987. plays precisely that role Žsee also Section 7.. The theorem would thus follow from a weaker set of assumptions. It would also obviate the need to identify a priori the events subject to moral hazard. In comparison with Dreze ` Ž1987., the generalisation is also significant, since the concept of omnipotent games Žgames g such that f Eg for every f g G . would be dispensed with. This is the generalisation discussed in Section 9.6, and illustrated in Appendix C of the work of Dreze ` Ž1987.. We do not elaborate on this remark here, for lack of space, but cannot refrain from mentioning that it fills Dreze ` with delight! Žii. It may be helpful for the reader to compare the result of theorem 4.12 with the analogous result that can be obtained when there is no moral hazard. Formally, the absence of moral hazard is equivalent to the assumption of RO. Assumption 4.13: RO ŽReversal of Order.. For every g g G , gg ; g . Note that if RO holds then every pair of games is a pair of equipotent games, so that assumption SRE is obviously redundant. The representation that we are after has in this case the form: m

VŽ f . s

Ý s Ž Ak . V Ž f , Ak . ks1

i.e., S s  s 4 in this case.

Ž 4.9 .

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A corollary, easily obtained from theorem 4.12 above, states that the representation in Eq. Ž4.9. holds, if WIC, CPŽ M ., NCI, RO, SRV hold. This corollary is the exact correspondent of the Anscombe and Aumann Ž1963. theory. Žiii. A different possible weakening of the assumption of RO is the assumption of Non-Positive Value of Information, to the effect that for every g g G , gg K g . This leads by the same arguments as in theorem 4.12 to a representation of the order over games as the minimisation of the expected value of a game, over a convex set S of strategies. This is the representation of Gilboa and Schmeidler Ž1989.. The rationale is altogether different.

5. When is CEUT consistent? One of the main differences between our analysis so far and that of Luce and Krantz Ž1971. has been that we do not assume that conditional preferences are well defined on the entire algebra M , but only on the set M. In this restricted case, we have derived a nontrivial representation of the order, which allows for moral hazard. Suppose that in the spirit of the theory of CEUT we extend the definition of conditional preferences to all the events in the algebra M . As it turns out, it is sufficient to that end that we extend ECP to general, as opposed to elementary, events. Our next theorem shows that this extension is possible only in very special cases: either there is no moral hazard at all, or there is full control over the events in M. As we announced, we extend the assumption 4.2 to hold for all events in M .

Assumption 5.1: GCP ŽGeneral Conditional Preferences are Well Defined.. For every B g M , if g $ B f, then it is not the case that f $ B g. Clearly, ECP is implied by GCP. One may observe immediately that if M has only two elements, the assumption GCP is already implied by ECP. We shall discuss the case m s 2 immediately after the next theorem 5.2. When m G 3, however, from GCP we can conclude that S is either the simplex or a singleton. Before proceeding with the formal statement and the proof, let us pause to consider the simple reason behind this result. Take the simple case of M s  A1 , A 2 , A 34 , let B s A1 j A 2 , and consider how the decision-maker is deciding whether, for any two games f and g, f $ B g. According to the representation theorem, he is considering the inequality max  s 1V Ž f , A1 . q s 2 V Ž f , A 2 . q s 3V Ž h, A 3 . 4 Fmax  s 1V Ž g , A1 . sg S

qs 2 V Ž g , A 2 . q s 3V Ž h, A 3 . 4 .

sg S

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His answer will clearly depend on the game h, unless the set S has a very special form. It is easy to see that the only sets S for which the inequality is independent of h are the triangles in the simplex with vertices Ž0, 0, 1., ŽŽ a, 1 y a, 0., Ž b, 1 y b, 0.. Since this must hold simultaneously for Ž A1 , A 2 ., Ž A1 , A 3 . and Ž A 2 , A 3 ., the result follows. The reasoning does not hold in the case m s 2. In this case, to decide whether f $ A 1 g the decision-maker has to decide whether: max  s 1V Ž f , A1 . q s 2 V Ž h, A 2 . 4 F max  s 1V Ž g , A1 . q s 2 V Ž h, A 2 . 4 sg S

se S

Now some simple algebra shows that for any set S : D 2 the above inequality holds if and only if V Ž f, A1 . F V Ž g, A1 ., independently of h. So conditional preferences are well defined on elements of M Žor, which is the same thing in the present case, of M . even if S is a proper subset of D 2 . Now the theorem. Theorem 5.2: Assume WIC on G C(M), and CP(M), SRV, VI, GCP and m G 3. Then the set S of theorem 4.12 is either S s D m or S s {s }. If one wants now to exclude that S is a singleton, it is enough to introduce explicitly the condition that RO is not satisfied by the preference order oÕer G . ConÕersely, RO rules out S s D m . Note also that S s D m implies that there exists an f with V(f, A k ) s V(f, A l ) for all k, l s 1, . . . ,m. Theorem 5.2 proÕides a full reconciliation of ‘classical’ decision theory with CEUT. When S s D m , the basis of the reconciliation is that primitiÕe conditional preferences lend themselÕes to material Õerification, since the preference (f, A) % (g, B) is confirmed by the implementation of A. It could also be argued that CEUT does not add anything to ‘classical’ decision theory in that case. We show in the next section that it can lead to a drastic simplification of the axiomatics.

6. Full control of events In this section, we present a simpler, perhaps more direct approach. Suppose that the decision-maker can freely and completely choose any of the events in M, and therefore, also any event in M . A very simple axiom suffices to characterise this situation. Take any two disjoint events A and B in M , and suppose the decision-maker prefers Ž g, B . to Ž f, A.. Then he must be indifferent between Ž g, B . and the game conditional on A j B that is equal to f on A and to g on B. In fact, if he can determine for sure that B is the case, he will do so if he is facing either of these two games, among which he should be indifferent. Formally we introduce: Assumption 6.1: FC ŽFull Control of Events.. Let A, B g M , A l B s 0. If Ž f, A.AŽ g, B ., and h s f on A, h s g on B, then Ž h, A j B . ; Ž g, B ..

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This assumption is a modified version of the Axiom 3 in the work of Luce and Krantz Ž1971., which assumes Ž f, A. ; Ž g, B .. Technically, our assumption is stronger since it uses a weaker condition for the same conclusion. Substantively, however, the two versions may be considered equivalent. The following representation theorem seems to correspond precisely to the situation in which a decisionmaker has complete control of the events in M. Theorem 6.2: Assume WIC on G C(M) and FC; then for any game f, and B g M : V Ž f , B . s max V Ž f , A k . . A k:B

Ž 6.10 .

Proof: Any B g M is a finite union of l elements A k of M; the proof is by induction on l, where B s D lks 1 A k . Let without loss of generality V Ž f, A1 . G V Ž f, A 2 . G V Ž f, A k . for k / 1,2. For l s 1, the statement is trivial. We now assume it holds for l y 1: V Ž f, B _ A1 . s V Ž f, A 2 .. By assumption FC, V Ž f, A1 . G V Ž f, A 2 . s V Ž f, B _ A1 . implies V Ž f, A1 . s V Ž f, A1 j Ž B _ A1 .. s V Ž f, B ., which proves our claim. In particular, for B s S: m

V Ž f . s maxm Ý s Ž A k . V Ž f , A k . sg D ks1

Ž 6.11 .

7. Axiomatic derivation of the partition The decision problem described in Section 3 takes the partition M as given; this partition is an objective feature, as much as the state space S itself. It may be objected that this status is unwarranted, since after all the partition is more naturally interpreted Žmuch like the strategies. as a subjective feature: the set of events that the decision-maker believes he can choose. As such, it is relevant only if it can be elicited through the observation of his choices. In this section, we show that this alternative approach is possible, and briefly present a suitable axiomatic structure. Let S be the set of all the subsets of S. Given an order on unconditional games G, we extend the definition 4.1 from M to S . Definition 7.1: ŽConditional Preferences.. For any C g S and any f g g G we say that: Ži. g $ C f Ž f is preferred to g given C . if and only if there exist two games h, hX with h s f, hX s g on C, h s hX on the complement of C, and hX $ h. Žii. f ; C g if and only if neither g $ C f nor f $ C g.

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We also say that: Definition 7.2: For a given C ; S, the order A C is well defined if for every pair of games f, g, if f $ c g then it is not the case that g A c f. When it is well defined, this is a weak order Žthat is, it satisfies W .. It induces naturally a weak order on the quotient space GrC, where two games are equivalent if and only if they are equal on the complement of C; or equivalently it induces an order on the games restricted to the set C, i.e., on functions from C to B. We first isolate the set of subsets of S for which conditional preferences are well defined. Definition 7.3: D is the set of C g S such that A C is well defined. The set D is nonempty because S g D. Now two main assumptions: we require continuity of the conditional preferences. Assumption 7.4: CCPŽC. ŽContinuity of Conditional Preferences.. For all f, g, h if f $ C g $ C h, then there exists an a g w0, 1x such that g f a h ; C g. For an element C g D we introduce a weak form of the assumption of RO for the order A C . Assumption 7.5: ROŽC. ŽReversal of Order.. If f ; C g, then: Ž f a g ,C ;h,C c . ; Ž g f a g ,C ;h,C c . A refinement of D is the following: Definition 7.6: R is the set of C g D such that g C satisfies ROŽ C .. An example might help to clarify the situation. Let S s  1, 2, 3, 44 . We enter for a moment in the mind of the decision-maker and discover that he believes he can choose the probability p of the event  44 vs. the event  1, 2, 34 in an interval w a, b x, with 0 - a - b - 1. Conditional on the event  1, 2, 34 he can choose the probability q of  1, 24 in the interval w0, 1x. He believes, however, that he has no say on the relative probability of 1 vs. 2, probability fixed to r: this last choice is entirely up to nature. With the benefit of our mind-reading, we can now see what his preferences are. Clearly  1, 24 ,  34 ,  44 are contained in D : once he knows that  1, 24 will happen, for instance, his ability to influence the choice of the state is useless. Also,  1, 2, 34 is an element of D. In fact his choice of p is independent of his choice of q; so for a given vector ŽfŽ1., fŽ2., fŽ3.., he will choose q to maximize his expected utility given the event  1, 2, 34 ; and this value is unaffected by the prizes given by the function h extending f to the event  44 . However,  1, 34 is not in D : so this family is not closed under the subset operation. Finally, note that RO  1, 2, 34 is not satisfied.

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Definition 7.7: For every s g S, P Ž s . is the set of maximal elements A in R Žthe order is induced by the inclusion. such that s g A. Suppose now that A and B are both in P Ž s .. This corresponds to a situation in which for every s1 g A, s1 / s, the decision-maker cannot affect the relative probability of s1 vs. s; similarly for any s2 g B. But then he cannot affect the relative probability of any sX g A j B vs. s, nor of s1 vs. s2 . We record formally: Assumption 7.8: MO ŽMonotonicity.. If A g R, B g R, A l B / 0, then A j B g R. From the monotonicity assumption we derive immediately that P Ž s . is a single set. It is clear that the correspondence P is reflexive by construction, and symmetric and transitive: this follows from an easy argument using the assumption MO. Proposition 7.9: Assume MO; then there exists a partition M s (A 1 , . . . A m ) of S such that: (i) A A is well defined for eÕery A k g M; k (ii) CCP(A k ) and RO(A k ) hold for eÕery A k g M; (iii) M is the coarsest partition with the two properties aboÕe. In the example the partition with elements {1},{2},{3},{4} is a finer partition than {1,2},{3},{4}; but this last is the only one that satisfies the point (iii) of the proposition. Acknowledgements We thank Peter Hammond, Philippe Mongin and two referees for very useful comments on a previous draft. Appendix A In this appendix, we collect the proofs of theorems 4.12 and 5.2 in the text. Proof of theorem 4.12: For a game f g G denote the vector of conditional values m zf s Ž V Ž f, A1 ., . . . ,V Ž f, A m .., or ‘ value vector’. The function F: R s P ks1 Rk ™ R, defined by F Ž x . s V Ž f . for any f such that zf s x is well defined. In fact, by lemma 4.4, if V Ž f, A k . s V Ž g, A k . for all k, then f ; g; so if f X is any other game such that zXf s x, then V Ž f . s V Ž f X .. From the assumption VI, F is a convex function; from assumption SRV it is homogeneous of degree one; also, F is continuous. Extend now F to a function from R m to R j q ` by first extending its domain of definition, by homogeneity of degree one, to the smallest convex cone containing the set R; and then extend its domain to the entire space by setting F equal to q` on the complement of this cone. Note that F, once extended, is still

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homogeneous of degree one and convex; therefore, by Corollary 13.2.1 of Rockafellar Ž1970., it is the support function of a uniquely determined closed and convex set:

S ) s  s g R m :for all z g R , s P z F F Ž n . 4 Note that F Ž Õ . s sups g S ) s P Õ. Define now S ' S ) l D m . We claim that for all f g G there is a s g S such that F Ž zf . s s P zf . Consider first any g such that zg is in the interior of the set R: such a game exists by NCI. Hence, for some f g G and some d ) 0 we have: V Ž f , Ak . y V Ž g , Ak . s d for every k. Since by assumption SRE the two games f and g are equipotent, then: F Ž a zf q Ž 1 y a . zg . s a F Ž zf . q Ž 1 y a . F Ž zg . for every a g w 0,1 x

Ž 8.12 . This implies that if a sequence  sn4 ; S ) is such that sn P Ž a zf q Ž1 y a . zg . ™ F Ž a zf q Ž1 y a . zg . then also: sn P zg ™ F Ž zg .. We consider first the simple case in which the supremum in the definition of the function F is achieved; i.e., there is a sg such that: V Ž g . s sg P zg ,V Ž f . s sg P zf s sg P zg q d sg P e

Ž 8.13 .

where e is the vector with each component equal to 1. From assumption SRV, V Ž f . y V Ž g . s d, hence, from Eq. Ž8.13., sg P e s 1. Also for every k: V Ž f . G V Ž Ž f , A k . . , Ž g , Ack . . G sg P zg q d sg Ž A k . s V Ž g . q d sg Ž A k . where the first inequality follows from the lemma 4.4. Hence:

sg Ž A k . F

V Ž f . yV Ž g .

d for every k. Finally for every k:

s1

V Ž f . G V Ž Ž g , A k . . , Ž f , Ack . . G sg P zf y d sg Ž A k . s V Ž f . y d sg Ž A k . and therefore sg Ž A k . G 0 for every k. To show that a sg at which the maximum is achieved exists take a sequence of ´ tending to 0, and a corresponding sequence  sg´ 4 ; S ) such that sg´ P zg s V Ž g . y ´ . An argument similar to the one we have just given can be used to show that the sequence  sge 4 is bounded; since S ) is a closed set, this proves our claim. Finally we extend the result to any g g G. Take a sequence  g n4 such that: zg n ™ zg ; sg n P zg n s F Ž zg n . , sg n g S Now a limit point sg of the sequence  sg n4 proves our claim.

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The proof has also provided an explicit construction of the set Sc , which determines it uniquely. We now prove that this Sc is the unique maximal set. Take any other candidate set S 1 , and let F1 be its support function. Clearly F G F1 Žthey are equal on the cone spanned by R, and F is defined to be identically equal to q` for any other z .. A well known convex analysis result now gives that S 1 : Sc . Now for the uniqueness of the minimal set, take any two subsets S 1 , S 2 that are closed, convex, and represent the order in Eq. Ž4.8.. It suffices to prove that S 1 l S 2 has the same properties. This set is closed and convex; but also if s g S is such that for some z g R: zs ) zs X , for everys X g S _ s Ž 8.14 . then s g S 1 l S 2 ; so S 1 l S 2 is nonempty, and contains all the points as in Eq. Ž8.14. for some z g R. Since the set of such Õ’s is dense in R, the set S 1 l S 2 represents the order in Eq. Ž4.8., and this proves our claim. Finally, for the last statement observe that the cone spanned by R is the entire space. Proof of theorem 5.2. The proof requires a few preliminary lemmas. We begin with the most important. Let B be an event in M ; then any act can be written as a pair f s Ž g, h. where g and h are the restriction of f to B and B c, respectively; and zg and zh are the vectors of values of the acts g and h in those two events. Also for any s g S , sB denotes the restriction of s to B. Lemma 8.1 Assume that for two acts (g, h) and (g, hX ),  s 4 s argmax t g S t Ž zg ,zh . and s X g argmax t g S 4t Ž zg ,zXh . Then:

sB s Ž B.

s

s BX s X Ž B.

Proof. We assume that the conclusion is contradicted, and prove that assumption GCP is violated. Note first that since the two vectors s b and s b are probability X

s Ž B.

X

s Ž B.

vectors, if they are different they are not co-linear. Hence, there exists a vector j g R n, with j k s 0 if k g B c, such that: sj - s Xj Ž 8.15 . Now for any u ) 0 define an act g Ž u . such that: zg Ž u . s zg q uj , and denote s Ž u . the element in argmax t g S 4t Ž zg Ž u . , zh . which is closest to s . Note that s Ž0. s s . Now, perhaps on a subsequence, the limit of s Ž u . as u tends to zero exists, and this limit is s . The first statement is obvious from the fact that S is compact. To see why the second is true, note that:

s Ž u . Ž zg q uj ,zh . G s Ž zg q uj ,zh .

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by definition of s Ž u .; hence:

ž lim s Ž u . / Ž z ,z . G s Ž z ,z . ; g

u™0

h

g

h

and now the second claim follows from the fact that s is the unique maximizer of Ž zg , zh .. We conclude in particular that for some u ) small enough, s Ž u ) . j - 0. Now: V Ž g Ž u ) . ,h . s s Ž u ) . Ž zg ,zh . q u )s Ž u ) . j - s Ž u ) . Ž zg ,zh . F V Ž g ,h . . Also using Eq. Ž8.15., again:

Ž 8.16 .

V Ž g Ž u ) . ,h . G s X Ž zg Ž u ) . ,zhX . s sB Ž zg q u )j . q sBX c zhX ) sBX zg q sBX c zhX s V Ž g ,hX . Ž 8.17 . From Eqs. Ž8.16. and Ž8.17., we now conclude that g Ž u ) . $ B g and g $ B g Ž u ) ., contradicting GCP. By adding a constant, we can insure that every element in R has all coordinates strictly positive. Of course the normalization V Ž g i . s i, i s 0, 1 will not hold any more: but this transformation does not change the set S . For every Õ g R we let:

½

5

m Ž z . ' te S :t z s max s z ; se S

while for every t g S we let: my1 Ž t . '  ze R:te m Ž z . 4 and ny1 Ž t . '  z g my1 Ž t . :t z ) s z for every s g S _ t 4 Note that my1 Žt . is the intersection of the cone to S at t with R; and ny1 Žt . is the Žpossibly empty. set of the value vectors that have t as unique maximizer. Finally, we denote by ext Ž S . Žrespectively exp Ž S .. the set of extreme Žrespectively exposed. points of S , and for any Õ g R: F Ž z . ' maxs Õ se S

Recall that S is defined to be a minimal convex set that represents the order over acts, i.e.,: f K g iff F Ž zf . G F Ž zg . , Ž 8.18 . and there is no S X ; S , S X convex and closed, that also satisfies Eq. Ž8.18.. The conclusion of the proof proceeds as follows: we assume that S is neither the simplex D m nor a singleton s , and we derive a contradiction to the conclusion of lemma 8.1. This is accomplished by showing that two pairs Ž s , Õ . and Žt , z . of strategies and value vectors exist such that, for some pair of indices Ž i, j .: Ži. Ž si , sj . is not a multiple of Žt i , t j .; Žii. s is the unique maximizer of Õ, and t is a maximizer of z; Žiii. Ž Õi , Õj . s Ž z i , z j ..

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Lemma 8.2 Suppose S is not the simplex; then there exists a: s g exp S _ exp D m with ny1 Ž s . l int R / 0. Proof. Note first that exp S : exp D m is impossible because of the assumption NCI Žwhich excludes exp S ; exp D m . and our assumption that S is not the simplex. Next, we prove that there exists a t g ext S _ ext D m and a value vector u g int R such that: max t u) max t u. Ž 8.19 . m tgext S l Žext D m . c

tgext S lext D

Suppose in fact that Eq. Ž8.19. does not hold, then for every w g int R, and therefore for every w g R, if we denote: S X ' co Ž ext S l ext D m . we have: maxXtv s maxtv tg S

tg S

contradicting the minimality of S . Now for any neighborhood of the value vector u we can find a new value vector z and a s such that: Ži. z g ny1 Ž s . l int R, Žii. zi / zj for every pair Ž i, j ., Žiii. s g exp S _ exp D m . In fact, we can always find a z such that Ži. and Žii. above are satisfied because the set of vectors z which have a unique maximizer is dense in R. But also for some e ) 0 Žsee Eq. Ž8.19.. W Ž u. G max s g Žext S l ext D m. s u q 3e. So if Õ is chosen to be close enough to u we have: F Ž z.

GF Ž z. yg G max

s uq2g

sg Ž ext S lext D m

G

max

Ž 8.20 .

s zqg

s g Ž ext S lext D m

which shows that s f ext D m s exp D m. Lemma 8.3 Suppose S is not a singleton; then for no s g S , int R : m y 1 (s ). Proof. If the conclusion does not hold, then for every u g int R, and so for any u g R, F Ž u. s s u; this contradicts the minimality of S . Next note an obvious fact. For every u g R n, for every b g R and for every l ) 0, mŽ l u q Ž b , . . . , b .. s mŽ u.. This observation has an important consequence. The set mŽ u. is independent of the choice of two variables; and to reach our contradiction we need two vectors with at least two coordinates equal. The degrees of freedom we have will exactly compensate for this restriction. We can now conclude our proof. From the assumption that S is neither the simplex nor a singleton, and lemmata 8.2 and 8.3, we conclude that there exists a

J.H. Dreze, ` A. Rustichinir Journal of Mathematical Economics 31 (1999) 159–181

181

s g exp S _ exp D m and a vector w g bd ny1 Ž s . l int R. Accordingly, there exist z g ny1 Ž s . l int R, and u f ny1 Ž s . arbitrarily close to z. Without loss of generality, u i / u j for every i / j. Because u f ny1 Ž s ., there exists a t / s , t g mŽ u.; hence, there are two indices Ž i, j . such that Ž si , sj . / Žt i , t j .. We only have to prove that, for that pair of indices, there exists a z g int R, with mŽ z . s mŽ u., and Ž z i , z j . s Ž Õi , Õj .. For simplicity Ž i, j . s Ž1, 2.; recall that we have chosen the vector z so that zi / zj for every pair Ž i, j .. We shall construct z as z i s l u i q b for every i, with z 1 s z1 , z 2 s z2 . The last two equalities are satisfied if: z 2 y z1 z1 z 2 y z 2 z1 ls and b s . u 2 y u1 u 2 y u1 Since z1 / z2 , the limit of l and b as u tends to z are respectively 1 and 0. Hence, z is in the interior of R if u is chosen sufficiently close to z, and s / t g mŽ u. s mŽ z ..

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