FOUNDATIONS OF BAYESIAN THEORY

Edi Karni¤

Department of EconomicsJohns Hopkins University

January 9, 2003

Abstract

This paper presents a new axiomatic subjective expected utilitymodel of Bayesian decision making under uncertainty with state-dependentpreferences and moral hazard. The theory provides choice-theoreticfoundations for the existance of prior probabilities representing deci-sion makers’ beliefs about the likely realization of events and for theupdating of these probabilities according to Bayes’ rule.

1 Introduction

Subjective expected utility theory is nowadays the standard economic modelof individual decision making under uncertainty and the choice-theoreticfoundation of the Bayesian statistics. Because of its fundamental impor-tance, the model was subjected, over the years, to careful scrutiny as aresult of which three unsatisfactory features were identi…ed:

² The theory ascribes to decision makers probabilities that do not nec-essary represent their beliefs.

² The theory does not imply the updating of subjective probabilities byBayes’ rule.

² The theory requires that preferences be state independent and theprobabilities be independent of the action taken which signi…cantlylimits the scope of its applications.

¤I am grateful to Eddie Dekel, Jacques Dreze, Peter Fishburn, Simon Grant, Faruk Gul,George Mailath, Zvi Safra, David Schmeidler, and Peter Wakker and three anonymousreferees for their valuable comments.

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In this paper I present an alternative axiomatic subjective expected util-ity model that does not su¤er from any of these drawbacks. In this theorythe probabilities ascribed to decision makers represent their beliefs, the prob-abilities are necessarily updated using Bayes’ rule, and the theory appliesto decisions making situations in which preferences are state-dependent andthe action taken a¤ects the probabilities.

1.1 On the representation of beliefs by probabilities

The search for a choice-theoretic de…nition of subjective probabilities beganwith the pioneering work of Ramsey (1931) and de Finetti (1937) and at-tained its de…nitive formulation in the work of Savage (1954). Ultimately,however, this quest failed to achieve its goal: the representation, by a prob-ability measure, of decision makers’ beliefs regarding the likely realization ofevents. The de…nitions of subjective probabilities in these and later worksinvoke a convention that is neither part of nor implied by the underlyingaxioms, namely, that the utility functions are state independent. Whereasstate-independent preferences are implied by the axiomatic structure, state-independent utility functions are not the only utility functions consistentwith the axioms. In fact, there are in…nitely many combinations of state-dependent utilities and arbitrary probability measures that are consistentwith the axioms. Consequently, the curvature of the utility functions (andthe ranking of “objective” lotteries, if such lotteries exist, as, for example,in Anscombe and Aumann [1963]) must be independent of the underlyingstates but the utility functions themselves may be positive linear transfor-mations of each other.1 In many situations this convention is untenable.For instance, it requires that the value attributed to possessing a fur coat beindependent of the temperature. Moreover, if a decision maker’s valuationsof outcomes are not independent of the underlying states, the impositionof state-independent utility functions means that, even when the decisionmaker’s beliefs (that is, a binary relation on the set of events that have theinterpretation “more likely of being realized than,”) are consistent with arepresentation by a probability measure, they may be inconsistent with thesubjective probabilities ascribed to the decision maker by the theory.

Choices among alternative courses of action, or acts, reveal the deci-sion maker’s marginal rates of substitution of outcomes across states. Thesetrade-o¤s confound subjective probabilities and marginal utilities and aretoo coarse to allow a meaningful separation of the two. Misconstrued separa-

1See the discussion in Dreze (1987); Schervish, Seidenfeldt, and Kadane (1990); Karni(1993), (1996), (2001); and Karni and Schmeidler (1993).

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tion of probabilities and utilities may result in inconsistencies between verbalexpression of preferences and observed choice behavior.2 More importantly,however, as the following example shows, in the context of principal-agentproblems if the principal ascribes to the agent incorrect subjective proba-bilities and utilities, he may fail to induce the agent to choose the optimalaction.

Example: A gambler (the principal) bets that a certain boxer will wina boxing match and then pays o¤ the other boxer (the agent) to throwthe match.3 To render this scenario concrete, assume that the winner’stake is a title and a prize of $2; 000 and the loser’s take is $1000: The two…ghters, Abe and Ben, are equally matched and the two possible resultsof the match are, A; Abe wins and B, Ben wins. Assume that both thegambler and Abe are expected utility-maximizing Bayesian decision makerswhose preferences over income are state independent (that is, they displaythe same attitude toward risk regardless of who wins the …ght), that eachof them believes that A and B are equally likely events, and that thesebeliefs are private information. The beliefs of both the gambler and Abe arerepresented by the uniform probability distribution ¼ (A) = ¼ (B) = 1=2;but the gambler does not know this and must infer Abe’s probabilities fromhis observed choice behavior. Assume that the gambler is risk neutral and hisutility function is state independent and that Abe is risk averse and that hecares about winning the title. Speci…cally let Abe’s valuations of the payo¤,w, be depicted by state-dependent utility functions uA (w) = 2

pw; and

uB (w) =pw. That is, winning the …ght makes the payo¤ more worthwhile.

The gambler ascribes to Abe utilities and (prior) subjective probabilitiesimplied by the subjective expected utility model. In other words, if the two…ghters indeed “give all they’ve got,” as far as the gambler can infer fromAbe’s choice behavior, the boxer’s preferences are represented by:

p (A)pwA + p (B)

pwB; (1)

where p (A) = 2¼ (A) =(2¼ (A) + ¼ (B)) = 2=3; p (B) = ¼ (B) =(2¼ (A) +¼ (B)) = 1=3.4

2See example and discussion in Karni (1996).3 I use an example of a boxing match to make the story more “realistic”. A reader

concerned that a …ght is a game involving two players rather than a game against nature,is welcome to substitute an archery contest (Dreze (1987)) for the …ght. The main pro-tagonists are the archer whose next shot will determines whether he wins or lose, and agambler staking money on the outcome. Aiming amiss is equivalent to throwing the …ght.

4 In the present scenario if the payo¤ in the losing state is su¢ciently large Abe willthrow the …ght. This is a violation of Savage’ Sure Thing Principle. However, if the mon-

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The outcome of the …ght depends on the “e¤ort” of the boxers. WhileAbe cannot be sure of winning the …ght even if he tries, he can ensure that heloses. He may take one of two courses of action: “…ght to win,” f; with thepossible results given by the event fA;Bg or “throw the …ght,” t: with thepossible result described by the event fBg: Conditional on these actions,Abe thinks that the probabilities of his winning are ¼ (A j f) = 1=2 and¼ (A j t) = 0: The gambler’s perception of Abe’s conditional probabilities ofwinning are p (A j f) = 2=3 and p (A j t)) = 0:

Suppose that the odds of each …ghter winning the bout are even andthat the gambler bets $x on Ben. Suppose that the gambler o¤ers Abe abribe, $b; to throw the …ght. The gambler’s problem is then to choose thesmallest payment b ¸ 0 that satis…es the incentive compatibility constraint

p (B j t)pb ¸ p (A j f)

p2000 + p (B j f)p1000 + b: (2)

It is easy to verify that the gambler thinks that any amount of moneyb > 1000 is enough to persuade Abe to throw the …ght and will o¤er himthe smallest possible sum over $1000: It is also clear that Abe is willing totake the money since, by taking it, he increases his expected utility whetheror not he actually throws the …ght (that is, the participation constraint issatis…ed). Consider next what Abe does once he accepts the bribe. Abe’sbeliefs and valuations imply that throwing the …ght yields

p1000 + b with

certainty, whereas …ghting for real entails an expected utility of

¼ (A j f) 2p2000 + ¼ (B j f)p1000 + b:

But ¼ (A j f) = ¼ (B j f), hence, for all b < 7000,

¼ (A j f) 2p2000 + ¼ (B j f)p1000 + b > p1000 + b:

Abe will take the money and …ght for real. Because he ascribed to theagent subjective probabilities that do not represent the agent’s beliefs, theprincipal designed a contract that induced the agent to chose an action thatwas not in the principal’s best interest. This problem is endemic and raisesserious doubts about the validity and even the meaning of the common priorassumption that is often invoked in the analysis of principal-agent problems.

etary payo¤s in the range [1000; 4000) all the axioms underlying the subjective expectedutility model are satis…ed. In other words, the subjective expected utility model appliespiecewise event though Abe is a subjective expected utility maximizer.

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1.2 On Bayesian updating, state-dependent preferences, andmoral hazard

From the point of view of Bayesian statistics, to which subjective expectedutility theory is supposed to provide a choice-theoretic foundation of priorprobabilities, the failure to obtain a correct representation of beliefs by prob-abilities is catastrophic. Moreover, whereas subjective expected utility the-ory is consistent with the updating of the subjective probabilities accordingto Bayes’ rule it does not imply it. In other words subjective probabilitiesare not required to be reasonable in any sense except of internal consistency,and subjective expected utility theory does not entail any conclusion aboutthe relation between decision makers’ beliefs and empirical distributions rep-resenting the relative frequencies produced by repeated trials.5 The notionthat reasonable decision makers must agree on the probabilities of outcomesthat, in repeated trials, produces stable, long-run frequency distribution isbased on a leap of faith and is not implied by subjective expected utility the-ory.6 In view of these observations it is natural to de…ne subjective expectedutility maximizing decision makers as Bayesian if, in addition to being sub-jective expected utility theory maximizers they also update of their priorbeliefs according to Bayes’ rule.

A last, well known, point criticism of subjective expected utility theory isits requirement that the preferences be state-independent and the event thatobtain be independent of the action taken. This imposes sever limitationson its possible applications. For example, the theory is inappropriate forthe analysis of the demand for health or life insurance as well as principal-agent relationships in the presence of moral-hazard problem. The last pointis particularly disturbing since, as the boxing match example illustrates, adecision maker may be Bayesian and yet does not abide by the Sure ThingPrinciple.

5Ghirardato (2002) is one exception. A more detailed discussion of this point is pro-vided in Kyburg (1968).

6The notion of conditional preferences on acts is well de…ned in subjective expectedutility theory. These conditional preferences are sometimes interpreted as the updatedpreferences. However, this interpretation, appealing as it may sound, is not implied bythe axioms. In other words, the axioms do not imply that if a decision maker receivesinformation that makes him believe that a certain event is null, he must update his priorprobabilities for the subevents in complemetary event equiproportionally.

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1.3 Preferences on conditional acts

The failure of the choice-theoretic models to quantify decision makers’ be-liefs by a probability measure is due to the restrictive nature of preferencerelations de…ned solely on acts (that is, on the set of functions from the set ofstates of nature to the set of consequences). The extension of the choice setto include conditional acts allows the expression of preferences that makes itpossible to separate utilities from probabilities in a more satisfactory man-ner. The idea of extending the choice set to include conditional acts is notnew. (Preferences on conditional acts were studied in P‡anzagl [1968], Luceand Krantz [1971], Fishburn [1973], and in Drèze and Rustichini [1999]. Ireview these contributions in Section 5.2 and contrast them with the ap-proach advanced here.) However, a few words on the meaning of preferencerelations on conditional acts are in order.

Conditional acts represent alternative courses of action when the deci-sion maker knows that a particular event obtains. One way of interpretingconditional acts is to regard them as hypothetical entities and to treat pref-erences on them as thought experiments that may be deliberately invoked bydecision makers when trying to clarify to themselves, or articulate to others,the reasoning underlying their actual choice behavior. Savage (1954) usesthis interpretation to justify his celebrated sure thing principle. To motivatethis principle, he gives the following example (italics are mine):

A businessman contemplates buying a certain piece of property.He considers the outcome of the next presidential election rel-evant to the attractiveness of the purchase. So, to clarify thematter to himself, he asks whether he would buy it if he knewthat the Republican candidate were going to win, and decidesthat he would do so. Similarly, he considers whether he wouldbuy if he knew that the Democratic candidate would win, andagain …nds that he would do so. Seeing that he would buy ineither event, he decides that he should buy. (Savage [1972] p.21)

The businessman compares the act of buying conditional on the eventthat the Republican candidate wins and the act of not buying conditional onthe same event. He then proceeds to compare the same two acts conditionalon the complementary event (a slightly di¤erent interpretation is given inGrant, Kajii and Polak [2000]). According to this description, an act condi-tional on an event corresponds to a subset of unconditional acts that agreeon that event. The comparison between two acts conditional on any given

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event does not require any restriction on the values the two acts assumeoutside of the conditioning event. If preferences among conditional acts areto be expressed in terms of unconditional acts, it requires the comparisonof subsets of unconditional acts that agree with the conditional acts on theconditioning event. Yet, when Savage formalized this concept, he took adi¤erent approach which he expressed as follows: (italics and expressions inparentheses are mine)

What technical interpretation can be attached to the idea that(the act) f would be preferred to (the act) g, if (the event) Bwere known to obtain? Under any reasonable interpretation, thematter would seem not to depend on the values f and g assumeat states outside of B. There is, then no loss of generality insupposing that f and g agree with each other except in B:(Savage [1972] p. 22)

In what follows I take conditional acts to correspond to Savage’s originaldescription. Interpreting preferences among conditional acts as a thoughtexperiment renders the axiomatic model presented in Sections 2 and 3 a nor-mative theory.7 Alternatively, conditional acts may be interpreted as con-ditional state-contingent payo¤s implemented by actions of decision makersthat restrict the set of states that might obtain. The control over eventsenvisioned here is the theoretical counterpart of what in reality is one es-sential ingredient of the moral hazard problem (the other ingredient beingthe unobservability of the action taken by the decision maker). Accordingto this interpretation, the axiomatic structure articulated in Sections 2 and3 may be regarded as a positive theory of decision making under uncer-tainty with moral hazard and costless actions, whereas the model in Section4 constitutes a positive theory of decision making under uncertainty withmoral hazard and costly actions. Moreover, because it involves choice amongalternative action-act pairs and, thus, entails “hypotheses about empiricaldata which could conceivably be refuted, if only under ideal conditions”(Samuelson [1947] p. 4) this approach renders the de…nition of probabilitiescompatible with the revealed preference methodology and hence constitutesa behavioral foundations of Bayesian theory.

Unlike some theories of state-dependent preferences with or withoutmoral hazard, the Bayesian decision theory advanced in this paper doesnot invoke objective probabilities for its formulation. As in Savage’s (1954)

7Further comments on the methodolgical issue raised by the use of conditional actsappear in section 5.?

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theory, probabilities do not enter as unde…ned (primitive) ingredients of themodel, appearing instead as a derived concept. Moreover, this theory doesnot rely on the use of or require the availability of all constant acts, therebyavoiding a problematic aspect of Savage’s model.

A di¤erent issue concerning the uniqueness of subjective probabilitiesin the choice-theoretic approach has to do with the valuation of outcomes.Generally speaking, if a decision maker believes that some state may obtainbut all the outcomes in this state are equally preferred his choice behavioris no di¤erent than if he believes it is virtually impossible that this stateobtains. The choice-theoretic model assigns such a state probability zero.Hence the interpretation of the probabilities as representation of beliefs isbased on the implicit and unveri…able assumption that in every state someoutcomes are better than others. If this assumption is not warranted, theprocedure may result in misrepresentation of beliefs. The theory advancedhere circumvents this problem by permitting decision makers to express theirpreferences among acts conditional on singleton events, thereby allowingdirect veri…cation of whether or not they are indi¤erent among all outcomesin a give state (see Karni, Schmeidler and Vind [1983].)

In the next section I describe the analytical framework and derive somepreliminary results. The main results are presented in Section 3. The impli-cations for the theory of moral hazard are examined in Section 4. Furtherdiscussion and review of the relevant literature appears in Section 5. Proofsare provided in the Appendix.

2 Subjective Expected Utility Theory

2.1 The analytical framework

Let S = f1; :::; ng, 3 · n <1; be a set of states of nature one and only oneof which is the true state. Nonempty subsets of S are events. Let E 0 denotethe set of all events. When the true state belongs to the event E; we say thatE obtains. Uncertainty is the lack of knowledge regarding which state is thetrue state. For each s 2 S, let Xs be a connected separable topological spacewhose elements are outcomes that are feasible in s. Unconditional acts arean n¡tuples x = (x1; :::; xn); where xs 2 Xs; representing possible courses ofaction. The set of all unconditional acts is the product setX := X1£:::£Xn.Note that the feasible sets of outcomes do not have to be the same acrossstates. This is a signi…cant departure from Savage’s (1954) model in whichthe set of acts includes all the constant acts and, consequently, requires thatthe feasible outcomes be the same in every state. Let ' be a preference

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relation on X and denote by ¼ the symmetric part of ' :For each s; denote by (x¡s; y) the act obtained from x by replacing its

s¡ th coordinate with y 2 Xs. Given a preference relation < on X; a states is null if (x¡s; y) ¼ (x¡s; z) for all y and z in Xs; otherwise it is nonnull.Denote by xEy the act that coincides with x on E and with y on S ¡ E(that is, (xEz)s = xs if s 2 E and (xEz)s = zs if s 2 S¡E:) Then an eventE 2 E 0 is null if xEy ¼ xEz for all y; z 2 X: Denote by E the subset of E 0that consists of all the nonnull events. Henceforth I assume that S containsat least three nonnull states.

Given E 2 E a conditional act, xE; is the element of XE :=Qs2E Xs:

For each E 2 E assume that XE is endowed with the product topology. LetX = [E2EXE denote the set of all conditional acts and assume that it isendowed with the topology whose open sets are the unions of the open setsin the product spaces XE; E 2 E:

Decision makers are characterized by a preference relation < on X where<=' on X I assume throughout that < is a continuous weak order. For-mally, < is a complete and transitive binary relation on X such that thesets fxE 2 X j xE < yE0g and fxE 2 X j yE0 < xEg are closed for allyE02X. The interpretation of the statement xB< yA requires some expla-nation. Taken literally it means that if the decision maker could choosebetween the course of action represented by the act x and the force theevent B and the course of action represented by the act y and force theevent A; he would either choose the …rst or be indi¤erent between the two.In view of the prevalence of moral hazard problems in economics, it is con-ceivable that the decision maker could take actions as a result of whichcertain nonnull events would obtain. In general these actions are costly. Tointroduce the main ideas in a way that will make them more transparent Iassume provisionally that the actions are costless and postpone the devel-opment of a full ‡edge model of choice among actions-acts pairs in whichdistinct actions entail di¤erent costs to Section 4. Thus, for the moment, Isuppress the actions and assume that the preference relation < representschoice behavior among conditional acts in the aforementioned sense. Thestrict preference relation  and the indi¤erence relation » are de…ned asusual and have the usual interpretation.

An act x¤ 2 X is an unconditional constant-valuation act if x¤fsg » x¤ftgfor all nonnull s; t 2 S:8 (Similarly, x¤E 2 XE is a conditional constant valu-

8The idea of constant valuation acts is similar to Drèze’s (1987) notion of “omnipotent”acts. In its present form it was used Karni (1993a). A similar concept appears in Skiadas(1997).

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ation act if x¤fsg » x¤ftg for all s; t 2 E:) Note that the relation x¤fsg » x¤ftgmeans that, faced with the choice between the outcome-state pairs (x¤s; s)and (x¤t ; t) the decision maker is indi¤erent between them. Constant valua-tion acts are analogous to constant acts in Savage (1954). However, unlikeSavage, who assumes implicitly that constant acts are constant-valuationacts, the present analysis recognizes that the same outcome may be assigneddistinct values in di¤erent states. Moreover, whereas Savage requires theexistence of all constant acts, I require only the existence of some constant-valuation acts. I assume that there exist constant valuation acts x and xsuch x  x:

A decision maker’s beliefs are represented by a binary relation, D; onE that has the following interpretation: For all T;Q ½ S, T D Q meansthat the decision maker considers the event T as at least as likely to obtainas the event Q: Following Ramsey (1931), it is now commonplace to infera decision maker’s beliefs from his willingness to bet on di¤erent events.However, considering the fact that the outcome valuations may be statedependent, care must taken in de…ning bets. Let D be de…ned as follows:For all constant valuation acts, x¤¤;x¤2 X, satisfying x¤¤Â x¤; and for allT;Q 2 E , T D Q if x¤¤T x¤< x¤¤Q x¤.

2.2 Preferences on conditional acts and their representation

For any given event, E; assume that the preference between any two actsconditional on E is independent of outcomes in states to which the two actsassign the same outcomes. This assumption is analogous to Savage’s (1954)Sure Thing Principle (see Wakker [1989] for more details). Formally,

(A.1) Conditional Coordinate Independence - For all x;y 2 X,E 2 E, s 2 S and w; z 2 Xs; (x¡s; z)E < (y¡s; z)E if and only if(x¡s; w)E < (y¡s; w)E :

The second axiom links the preferences on distinct conditional acts. It as-serts that, for any conditional constant valuation act x¤E, all the conditionalvaluation acts x¤G such that G ½ E are equally preferred. In particular,x¤E » x¤ for all E 2 E. Formally,

(A.2) Consequentialism - If x¤E is a conditional constant valuation actthen x¤E » x¤G for all nonnull G ½ E:

The logic underlying (A.2) is that ultimately the outcome of a decisionis a state-outcome pair (s; x¤s) : With constant valuation acts the ultimate

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outcome is of the same valued regardless which event obtains. Indi¤erenceamong all conditional constant valuation acts is form of consequentialism.It means that the decision maker is solely concerned with the ultimate out-come.

A function F : X ! R is said to be additive valued if there exist func-tions fE (¢; t) : Xt ! R, for all E 2 E and t 2 E; such that F (xE) =Ps2E fE (xs; s) : The functions fE (¢; s) are called additive-valued functions.

The following result gives necessary and su¢cient conditions for the exis-tence of an additive-valued representation of < and establishes its unique-ness.

Theorem 1 Suppose that there are at least three nonnull states. Then thefollowing conditions are equivalent:

(i) The relation < is a continuous weak-order on X satisfying (A.1) and(A.2).

(ii) There exist continuous functions fwE (¢; s) : Xs ! R j E 2 E; s 2 Egsuch that for all xE;yA2X;

xE< yA ,Xs2E

wE (xs; s) ¸Xs2A

wA (ys; s) :

and, for every conditional constant valuation act x¤E 2 X and everynonnull event G ½ E,X

s2GwG (x

¤s; s) =

Xs2E

wE (x¤s; s) :

Moreover, if fwE (¢; s) j E 2 E; s 2 Eg is another set of continuousfunctions that represent < in the sense of (ii) ; then, for all E 2 E ands 2 E; wE (¢; s) = ¯wE (¢; s) + °E (s) ; where ¯ > 0 and

Ps2E °E (s) = C:

These functions are constant if and only if s is null.

The proof of Theorem 1 is given in the Appendix.Remark 1: If there are only two nonnull states, Theorem 1 holds if <

satis…es the following hexagon condition (see Wakker [1989, Ch. III ]):

Hexagon condition - Let s and r be the only two nonnull states. Then,for all xs; ys; zs in Xs, xr; yr; zr in Xr; and x 2 X; if ((x¡s; xs)¡r; yr) »((x¡s; ys)¡r; xr) and ((x¡s; zs)¡r; xr) » ((x¡s; ys)¡r; yr) » ((x¡s; xs)¡r; zr)then ((x¡s; ys)¡r; zr) » ((x¡s; zs)¡r; yr).

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3 Subjective Expected Utility Theory

3.1 Coherence

Bayesian decision makers update their beliefs regarding the likely realizationof events independently of their valuation of outcomes in di¤erent states.To capture this idea, I assume that the preference relation < re‡ects thesame valuation of state-outcome pairs regardless of the conditioning event.Formally, let E1 the collection of all nonsingleton events in E, then

(A.3) Coherence - For all w;x;y; z 2 X, nonnull E 2 E1, s 2 E; andas; bs; cs; ds 2 Xs, if (x¡s; as) < (y¡s; bs), (y¡s; cs) < (x¡s; ds); and(z¡s; bs)E < (w¡s; as)E then (z¡s; cs)E < (w¡s; ds)E :

Axiom (A.3) is an adaptation of Wakker’s (1987) cardinal consistencyaxiom. (Wakker [1989] discusses the earlier literature on the idea underlyingcardinal consistency.) To grasp the meaning of this axiom, think of thepreferences (x¡s; as) < (y¡s; bs) and (y¡s; cs) < (x¡s; ds) as indicating thatthe “intensity” of the prior preference for cs over ds is su¢ciently greaterthan that of as over bs as to reverse the order of preference between theother coordinates of x and y: Coherence requires that these intensities arenot contradicted by the conditioning of the acts on nonnull events.

If axiom (A.3) is added to part (i) of Theorem 1 then the functionsfwE (¢; s) j s 2 E;E 2 Eg are positive a¢ne or constant transformationsof one another (see Lemma 8 in the Appendix). Consequently, if a state isnull it must remain so when included in the conditioning events (that is, forevery E 2 E, if s is a null state and s 2 E then (x¡s; y)E » (x¡s; z)E for ally; z 2 Xs.) The next theorem shows that the subjective prior probability ofnull states is zero. Hence the implication is that the conditional probabilityof such states cannot become positive in view of new information.

3.2 Subjective expected utility representation of state-dependentpreferences

The next theorem establishes the main result: that there exists a uniquesubjective probability distribution on the set of states representing the de-cision maker’s prior beliefs, unique posterior probabilities obtained from thegiven prior by Bayes’ rule, state-dependent real-valued utility functions onthe respective sets of outcomes representing the decision maker’s valuations,and subjective expected utility representations of his conditional and uncon-ditional preferences. Implicit in this result is the notion that the decision

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maker may choose among act-event pairs, but once the event is chosen, theprobabilities assigned to states belonging to the chosen event must increaseproportionally.

Theorem 2 Suppose that there are at least three nonnull states. Then:

a. The following two conditions are equivalent:

(i) The relation < is a continuous weak-order on X satisfying (A.1), (A.2)and (A.3).

(ii) There exists a probability measure ¼ on S and an array of continuousfunctions fus : Xs ! Rgs2S such that, for all x;y 2 X;

x < y ,Xs2S

¼ (s)us (xs) ¸Xs2S

¼ (s)us (ys) ;

for every conditional constant valuation act; x¤E; us (x¤s) = ut (x

¤t ) for

all s; t 2 E; and, for all xE;yA 2 X,xE < yA ,

Xs2E

¼ (s j E)us (xs) ¸Xs2A

¼ (s j A)us (ys) ;

where, for all B 2 E, ¼ (s j B) = ¼ (s) =Pt2B ¼ (t) is the probability ofstate s conditional on the event B:

b. The utility functions fusgs2S are cardinally measurable fully-comparable.(That is, if fusgs2S is another array of functions representing < inthe sense of (ii) then us = ¯us + ®; ¯ > 0; for all s 2 S:)

c. ¼ is unique and ¼ (s) = 0 if and only if s is null.

The proof of Theorem 2 is given in the Appendix.

3.3 Subjective expected utility representation of state-independentpreferences

State-independent preferences are a special case of the theory of the preced-ing sections. To study this case, assume, without essential loss of generality,that the same outcomes are feasible in all states (i.e., X1 = ::: = Xn = X).To help keep this in mind I denote X by Xn: Intuitively speaking, state-independent preferences requires that the “intensity” of the preferences bethe same across states. To formalize this idea I invoke the condition ofcardinal coordinate independence of Wakker (1989, Ch. IV).

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(A.4) Cardinal Coordinate Independence - For all x;y; z;w 2 Xn;nonnull s; t 2 S; and a; b; c; d 2 X; if (y¡s; b) < (x¡s; a), (x¡s; c) <(y¡s; d) and (z¡t; a) < (w¡t; b) then (z¡t; c) < (w¡t; d):

The interpretation of cardinal coordinate independence is analogous tothat of coherence. The relations (y¡s; b) < (x¡s; a) and (x¡s; c) < (y¡s; d)indicate that the “intensity” of the preference for c over d in state s issu¢ciently greater than that of b over a as to reverse the order of preferencebetween the other coordinates of x and y: State independence requires thatthese intensities are not contradicted by the preferences between the sameoutcomes in any other state t.

The next lemma gives necessary and su¢cient conditions for the state-dependent utility functions to be a¢ne transformations of one another.

Lemma 3 Let < be a continuous weak order on Xn. Then the followingconditions are equivalent:

(i) < satis…es (A.4).

(ii) There exist u : X ! R and positive a¢ne or constant functions 's :u (X)! R for all s 2 S such that, for all x;y 2 X:n;

x < y ,nXs=1

's ± u (xs) ¸nXs=1

's ± u (ys) :

The proof of Lemma 3 follows immediately from Wakker’s (1989) Theo-rem IV.2.7 and is omitted.9

In general, even if the preference relation has an expected utility repre-sentation, state-independence preferences does not imply state-independentutility functions. However, if the utility functions are not the same acrossstates then, by Lemma 3, they must be positive a¢ne transformations ofone another (i.e., for all s 2 S and x 2 Xs, us (x) := ¾su (x) + »s; where¾s > 0). In other words, the dependence of the evaluation of an outcomeon the underlying states is quanti…able by the multiplicative coe¢cients ¾sand the additive constants »s: Note that if 's is a constant function, then sis null. The next theorem captures this fact and is analogous to Theorem 2.

9 If the assumption Xs = Xt does not hold, then the utility functions of nonnull statesare positive a¢ne transformations of one another over the outcomes that are in the inter-section of the sets of feasible outcomes.

14

Theorem 4 Suppose that there are at least three nonnull states. Then:

a. The following conditions are equivalent:

(i) < is a continuous weak-order on X satisfying (A.2), (A.3), and (A.4).

(ii) There exists a probability measure ¼ on S, a continuous nonconstantfunction u : X ! R, and for all s 2 S , there are numbers ¾s > 0and »s such that, for all x;y 2Xn;

x < y ,Xs2S

¼ (s)¾su (xs) ¸Xs2S

¼ (s)¾su (ys) ;

for every conditional constant valuation act; x¤E; ¾su (x¤s)+»s = ¾tu (x

¤t )+

»t; for all s; t 2 E; and, for every xE ;yA 2 X,

xE< yA ,Xs2E

¼ (s j E) [¾su (xs) + »s] ¸Xs2A

¼ (s j A) [¾su (ys) + »s] ;

where, for all B 2 E; ¼ (s j B) = ¼ (s) =Pt2B ¼ (t) is the probability ofthe state s conditional on the event B:

b. The triplet (u; ¾s; »s) is unique. (That is, if (v; ³s; ¿s) represent thepreference relation as in (ii) then v = ¯u + ® and, for all s 2 S;³s = ¾s=¯, and ¿s = »s ¡ ®³s:)

c. ¼ is unique and ¼ (s) = 0 if and only if s is null.

The proof of Theorem 4 is similar to that of Theorem 2 and is outlinedin the Appendix.

The de…nitions of subjective probabilities in Theorems 2 and 4 representthe decision makers’ prior beliefs. Letting the probability of an event E begiven by ¼ (E) =

Ps2E ¼ (s), these de…nitions imply that for all T;Q ½ S,

T D Q , ¼ (T ) ¸ ¼ (Q) :

Moreover, for every given event E 2 E the posterior beliefs, DE; are repre-sented by the conditional probabilities ¼(¢ j E): These are the only repre-sentations of the prior and posterior beliefs of Bayesian decision makers byprobabilities.

15

4 Subjective Expected Utility Theory with MoralHazard

4.1 The analytical framework

The analysis of schemes designed to mitigate the welfare loss associated withmoral hazard have been a focal issue in economic theory in the past 30 years.Less attention has been devoted to examining the analytical underpinningof the theory, which still lacks satisfactory foundations. In this section,I build on the model of the preceding sections to address this issue. In sodoing I also develop a choice-theoretic model that encompasses the precedinganalysis and leads to a choice-theoretic de…nition of subjective probabilitiesthat represent decision makers’ beliefs. The approach taken here is basedon the idea that the moral hazard problem arises when decision makers canto a¤ect, by unobservable actions, the event that obtains.

Let A be a topological space of feasible actions and let F be a mappingof A onto E. (In general the set of actions is abstract. In speci…c situationsit may be more structured. For example, if actions correspond to levels ofe¤ort thenA may be a compact interval in the real line and the topology themetric topology. The same interpretation applies if actions correspond tomonetary expenditure.) De…ne an induced mapping, F : A£X! A£X byF (a;x) =

¡a;xF (a)

¢: A default action is an action, a0; such that F

¡a0¢= S

and, consequently, F (a0;x) =¡a0;x

¢: Let A0 be a nonempty set of default

actions and assumed that A0 ½ A: Using the default actions de…ne nullstates the set X as in Section 2.1. Assume further that A£X is endowedwith the product topology.

Decision makers are characterized by preference relations, <; on A :=f¡a;xF (a)¢ j a 2 A;xF (a) 2 Xg. The interpretation of ¡a;xF (a)¢ < ¡b;yF (b)¢is that the decision maker is better o¤ with the alternative

¡a;xF (a)

¢that

involves taking the action a when facing the payo¤ depicted by the act xthan with the alternative

¡b;yF (b)

¢involving taking the action b when the

payo¤ is depicted by the act y: I assume throughout that < is a continuousweak order and denote by  and » the asymmetric and the symmetric partsof <; respectively.

If, at a given x; imposing a su¢ciently severe penalty on the eventS ¡ F (a) induces a decision maker to take an action, a, to make sure thatF (a) obtains the action a is implementable. Formally, an action a is im-plementable at

¡a0;x

¢2 A0£X if there exist z¡a;¡a0;x

¢¢ 2 X such that¡a0;xF (a)z

¡a;¡a0;x

¢¢¢ » ¡a;xF (a)¢ :16

4.2 Axioms and preliminary results

The preference relation < is assumed to satisfy conditional coordinate inde-pendence, consequentialism, and coherence properly modi…ed to accommo-date the extended framework. The …rst axiom is analogous to conditionalcoordinate independence and, like it, requires that the preference betweenany two acts conditional on the decision maker taking the action a is indepen-dent of outcomes in states to which the two acts assign the same outcomes.In addition, it requires that the ranking of all the acts conditional on a givenevent be independent of the action that induces that event.

(A.1’) For all x;y 2X, a 2 A; s 2 S; and w; z 2 Xs;³a; (x¡s; z)F (a)

´<³a; (y¡s; z)F (a)

´if and only if

³a; (x¡s; w)F (a)

´<³a; (y¡s; w)F (a)

´; and for all E 2 E,

a; b 2 F¡1 (E) ; (a;xE) < (a;yE) if and only if (b;xE) < (b;yE) :The fact that actions must be taken in order to induce a conditioning

event implies that direct comparisons of state-outcome pairs is no longerpossible and, consequently, constant valuation acts may no longer be ob-servable. Constant valuation acts served a dual purpose in the precedinganalysis: they provided a link among preferences on conditional acts andthe mean to identify the decision maker’s utility function. To link togetherthe preference on conditional acts I assume that there exist a default actiona0 2 A0 and an act x 2 X such that every action in A is implementable at¡a0; x

¢.

The following result is analogous to Theorem 1.

Theorem 5 If there are at least three nonnull states then the followingconditions are equivalent:

(i) < is a continuous weak-order on A satisfying (A.1’) and there exist¡a0; x

¢ 2 A0 £X such that all actions are implementable at¡a0; x

¢.

(ii) There exist continuous functions fwF (a) (¢; s) : Xs ! R j a 2 A; s 2F (a)g and a function v : A!R such that, for all ¡a;xF (a)¢ ; ¡b;yF (b)¢ 2A;¡a;xF (a)

¢<¡b;yF (b)

¢if and only ifX

s2F (a)wF (a) (xs; s) + v (a) ¸

Xs2F (b)

wF (b) (ys; s) + v (b) (3)

and, for all a 2A,wF (a) (xs; s) = 0; v (a) = ¡

Xs2S¡F (a)

wS¡z¡a0; x

¢s; s¢: (4)

17

Moreover, if fw0F (a) (¢; s) : Xs ! R j a 2 A; s 2 F (a)g and v isanother set of continuous functions that represent < in the sense ofequation (3) then, for all E 2 E and s 2 E; w0F (a) (¢; s) = ¯wF (a) (¢; s)+°F (a) (s) ; and v = ¯v + °; where ¯ > 0 and

Ps2F (a) °F (a) (s) = C:

These functions are constant if and only if s is null.

The proof is given in the Appendix.Because constant valuation acts may no longer be used to de…ne the

utilities, I introduce the concept of compensating-variations payo¤ pro-…les. Formally, denote by as an action whose image under F is fsg: Fixa =(a1; :::; an) than an act x¤ is a compensating-variations payo¤ pro…le(CVPP) if

³as;x

¤fsg´»³at;x

¤ftg´: Compensating-variations payo¤ pro…les

have the following interpretation: The decision maker’s valuation of the dif-ference in the impact on the state-outcome pairs (s;x¤s) and (t;x¤t ) on hiswell-being is equal to the di¤erence in the direct e¤ect on his well-being ofthe corresponding actions as and at that yield them. Given a as above, letX (a) = fx 2 X j ¡as;xfsg¢ » ¡at;xftg¢g be the set of CVPPs generated bya: Assume that the set of actions is su¢ciently rich so that given x¤ 2X (a)and any E;E0 2 E there exist a 2 F¡1 (E) and b 2 F¡1 (E0) such that³a;x¤F (a)

´»³b;x¤F (b)

´: Analogous to (A.2) I assume:

(A.2’) For every given a = (a1; :::; an) 2An; for all compensating-variationspayo¤ pro…les x¤, x¤¤ 2 X (a) and actions a; b 2A;

³a;x¤F (a)

´»³b;x¤F (b)

´if and only if

³a;x¤¤F (a)

´»³b;x¤¤F (b)

´:

The next axiom is analogous to coherence (A.3) and has a similar inter-pretation.

(A.3’) For all w;x;y; z 2 X, a 2 A, s 2 F (a) ; and as; bs; cs; ds 2 Xs,if¡a0; (x¡s; as)

¢<¡a0; (y¡s; bs)

¢,¡a0; (y¡s; cs)

¢<¡a0; (x¡s; ds)

¢; and³

a; (z¡s; bs)F (a)´<³a; (w¡s; as)F (a)

´then

³a; (z¡s; cs)F (a)

´<³a; (w¡s; ds)F (a)

´:

The meaning of (A.3’) is easy to grasp if the preferences¡a0; (x¡s; as)

¢<¡

a0; (y¡s; bs)¢and

¡a0; (y¡s; cs)

¢<¡a0; (x¡s; ds)

¢are taken to indicate that,

given that act of omission, a0, the “intensity” of the preference for cs overds is su¢ciently greater than that of as over bs as to reverse the order ofpreference between the other coordinates of x and y: Axiom (A.3’) requiresthat these intensities not be contradicted when some other action is taken

18

that restricts the event that obtains. The implication of this assumption isthat intensity of preferences between pairs of outcomes in any given stateis una¤ected by the action taken provided that the state may still be truefollowing that action.

4.3 Subjective expected utility with moral hazard and state-dependent preferences

The next theorem extends the main result, Theorem 2, to the case of de-cision making under uncertainty with moral hazard and state-dependentpreferences.

Theorem 6 Suppose that there are at least three nonnull states. Then:

a. The following two conditions are equivalent:

(i) The relation < is a continuous weak-order on A satisfying (A.1’),(A.2’), and (A.3’) and there exist

¡a0; x

¢ 2 A0 £ X such that allactions are implementable at

¡a0; x

¢.

(ii) There exist probability measure ¼ on S; continuous functions us : Xs !R; s 2 S; and a function v : A!R such that, for all x;y 2 X;¡

a0;x¢<¡a0;y

¢ ,Xs2S

¼ (s)us (xs) ¸Xs2S

¼ (s)us (ys) :

For every¡a;xF (a)

¢;¡b;yF (b)

¢ 2 A, ¡a;xF (a)¢ < ¡b;yF (b)¢ if and onlyifXs2F (a)

¼ (s j F (a))us (xs) + v (a) ¸Xs2F (b)

¼ (s j F (b))us (ys) + v (b) ;

where ¼ (s j F (a)) = ¼ (s) =Pt2F (a) ¼ (t) is the probability of state s

conditional on the event F (a) :

b. The utility functions fusgs2S and v are cardinally measurable fully-comparable.

c. ¼ is unique and ¼ (s) = 0 if and only if s is null.

An outline of the proof of Theorem 6 is given in the Appendix.

19

4.4 Subjective expected utility with moral hazard and state-independent preferences

The case of state-independent preferences may be treated similarly. Specif-ically, the axiom of cardinal coordinate independence may be restated asfollows:

(A.4’) For all x;y; z;w 2 Xn; nonnull s; t 2 S; and ®; ¯; °; ± 2 X; if¡a0; (y¡s; ¯)

¢<¡a0; (x¡s; ®)

¢;¡a0; (x¡s; °)

¢<¡a0; (y¡s; ±)

¢and

¡a0; (z¡t; ®)

¢<¡

a0; (w¡t; ¯)¢then

¡a0; (z¡t; °)

¢<¡a0; (w¡t; ±)

¢:

Analogous to Theorem 4, the next theorem provides necessary and suf-…cient conditions for the existence of subjective expected utility representa-tion of state-independent preferences with moral hazard:

Theorem 7 Suppose that there are at least three nonnull states and that allactions are implementable. Then:

a. The following conditions are equivalent:

(i) < is a continuous weak-order on f(a;x) j a 2 A;x 2Xng satisfying(A.2’), (A.3’) and (A.4’).

(ii) There exists a probability measure ¼ on S, a continuous, nonconstant,functions u : X ! R and v : A!R and numbers ¾s > 0; »s; s 2 Ssuch that , for all x;y 2Xn;¡a0;x

¢<¡a0;y

¢ ,Xs2S

¼ (s)¾su (xs) ¸Xs2S

¼ (s)¾su (ys) :

For all¡a;xF (a)

¢;¡b;yF (b)

¢ 2 A, ¡a;xF (a)¢ < ¡b;yF (b)¢ if and only ifXs2F (a)

¼ (s j F (a)) [¾su (xs) + »s] + v (a) ¸Xs2F (b)

¼ (s j F (b)) [¾su (ys) + »s] + v (b) ;

where ¼ (s j F (a)) = ¼ (s) =Pt2F (a) ¼ (t) is the probability of state s

conditional on the event F (a) :

b. The triplet (u; ¾s; »s) is unique. (That is, if (u0; ³s; ¿s) represent the

preference relation as in (ii) then u0 = ¯u + ® and, for all s 2 S;³s = ¾s=¯, and ¿s = »s ¡ ®³s:).

c. ¼ is unique and ¼ (s) = 0 if and only if s is null.

The proof of Theorem 7 follows immediately from the proofs of Theorems4 and 6 and is omitted.

20

4.5 The moral hazard problem

To relate the axiomatic models of the preceding sections to the litera-ture on optimal contracts in the presence of moral hazard let X =Rn andx represents state-contingent output levels. An incentive contract is a vec-tor w 2Rn satisfying ws = wt if xs = xt, for all s; t 2 S. Denote by <Pthe principal’s preference relation on X and suppose that it satis…es axioms(A.1), (A.2), and (A.3). Denote by <A the agent’s preference relation onA£X, and suppose that it satis…es axioms (A.1’), (A.2’), (A.3’). LetB ½ Abe a subset of feasible actions. Then the principal’s problem may be statedas follows: Given x 2Rn design a contract w¤ and choose an action a¤ 2 Bsuch that (x¡w¤)F (a¤) <P (x¡w)F (a) for all (a;w)2 B£Rn subject tothe incentive compatibility constraints:³

a¤;w¤F (a¤)´<A

³a;w¤F (a)

´; for all a 2 B;

and the participation constraint³a¤;w¤F (a¤)

´<A

¡c; zF (c)

¢;

where¡c; zF (c)

¢represents the agent’s best alternative course of action if he

refuses the contract.By Theorems 2 and 6, this problem may be restated as follows: Choose

(a¤;w¤) 2 B£Rn so as to maximize the principal’s objective functionXs2S

¼P (s j F (a¤))uPs (xs ¡w¤s)

subject to the participation constraint:Xs2S

¼A (s j F (a¤))uAs (w¤s) + v (a¤) ¸ v0

and the incentive compatibility constraints: for all a 2 BXs2S

¼A (s j F (a¤))uAs (w¤s) + v (a¤) ¸Xs2S

¼A (s j F (a))uAs (w¤s) + v (a) ;

where v0 =Ps2S ¼

A (s j F (c))uAs (zs) + v (c) and the superscripts P and Adenote the variables corresponding to the principal and agent, respectively.

Note that in the usual formulation of the moral hazard problem (e.g.,Shavell [1979], Holmstorm [1979]) it is assumed that the principal and the

21

agent agree on the probabilities (i.e., ¼A = ¼P ) and that the utility functionsare state independent, so that uPs = u

P and uAs = uA for all s 2 S: Neither

of these assumptions is necessary or compelling. Notice also that the formu-lation above is one way of stating the moral hazard problem. An alternativeapproach is to assume that the subjective probability on the state-space isuna¤ected by the agent’s action and that there is a “production function”that selects a random variable, that is, an act in X; as a function of theaction. Consequently, the choice of action a¤ect the probability of the out-comes, through the selection of the random variable without a¤ecting theprobabilities of the underlying states.

5 Discussion

5.1 Beliefs and probabilities

The theory developed here yields a unique subjective probability distribu-tion regardless of whether the preference relation is state independent. Theunderlying premise is that the mental processes at work in the assessment ofthe likelihood of events and the valuation of outcomes are the same whetheror not the preferences are state independent. It is reassuring, therefore,that both cases are addressed using the same approach and that the caseof state-independent preferences is merely a special instance of the moregeneral model.

A crucial aspect of the de…nition of subjective probabilities in this pa-per is the fact that they quantify the decision makers’ prior and posteriorbeliefs correctly. The following example illustrates this assertion. Con-sider a subjective expected utility - maximizing Bayesian decision makerwhose preferences are state independent. Let there be three states of na-ture, S = f1; 2; 3g: Suppose that the decision maker’s valuations of outcomesare depicted by state-dependent utility functions satisfying u1 = u2 = u; andu3 = 2u. Assume further that the decision maker believes the three statesto be equally likely. This belief is represented by the uniform probabilitydistribution p1 = p2 = p3: However, based on the observations of his choiceamong acts, the choice-theoretic models ascribe to the decision maker utili-ties and subjective probabilities so that his preferences are represented by:

q1u (x) + q2u (y) + q3u (z) ; (5)

where q1 = p1=(p1 + p2 + 2p3) = 1=4; q2 = p2=(p1 + p2 + 2p3) = 1=4; andq3 = 2p3=(p1 + p2 + 2p3) = 1=2.

22

Consider next the decision maker’s choices among acts conditional onthe events f1; 2g and f2; 3g: According to the decision maker’s beliefs theprobabilities of these events are: p (f1; 2g) = p (f2; 3g) = p (f1; 3g) = 2=3and according to the ascribed probabilities they are: q (f1; 2g) = 1=2;q (f2; 3g) = q (f1; 3g) = 3=4:

Let xf2;3g » yf1;2g and suppose that x2 = y2 then, by Theorem 4,

3

2

·1

3u (x2) +

2

3u (x3)

¸=3

2

·1

3u (y1) +

1

3u (y2)

¸: (6)

According to the ascribed probabilities, the same preference is representedby:

4

3

·1

4u (x2) +

1

2u (x3)

¸= 2

·1

4u (y1) +

1

4u (y2)

¸: (7)

But equations (6) and (7) cannot both be true. In fact, the choice-theoreticsubjective expected utility model fails to predict the behavior of an expectedutility - maximizing Bayesian decision maker even though his preferencessatisfy the underlying axiomatic structure. This failure is endemic.

In Karni (1996) I argued that a correct representation of beliefs is usefulsince it is consistent with both decision-makers’ choice behavior and theverbal expressions used to exchange information. The boxing example inthe introduction and subsequent work by Grant and Karni (2002) illustratethe kind of economic problems that may arise if agent’s beliefs and utilitiesare misconstrued. As I argue next, correct representation of beliefs is alsoimportant for normative economic analyses.

Aggregation of Beliefs and the Pareto Principle: Harsanyi’s (1955)aggregation theorem shows that if individuals and social preference relationssatisfy the axioms of expected utility theory of von Neumann and Morgen-stern (1944) and a Pareto indi¤erence condition, then the social preferencesmay be represented as a linear combination of individual utilities. Harsanyi’stheorem takes the probabilities of the social state-lotteries as given. If theprobabilities are subjective, then Harsanyi’s approach suggests that individ-ual utilities and probabilities be aggregated separately into social utilitiesand probabilities and then combined to obtain an expected utility repre-sentation of social preferences. Unfortunately, as noted by Hylland andZeckhauser (1979) and Mongin (1995), such an aggregation is inconsistentwith Pareto indi¤erence.

Gilboa, Samet, and Schmeidler (2001) claim that this inconsistency doesnot pose an ethical problem. They argue, convincingly, that the Pareto

23

condition is an expression of individual preferences that combines beliefs(subjective probabilities) and tastes (utilities). Hence the Pareto conditionthat requires that when all members of a society are indi¤erent betweentwo alternatives the social preferences must also be indi¤erent is compellingonly when the individual members do not hold contradictory beliefs. Inother words, without some quali…cation, the Pareto condition cannot be usedas an argument to justify social preference over alternatives about whichindividual members hold con‡icting beliefs. On the positive side, Gilboaet al. show that if the Pareto indi¤erence condition is imposed only whenthere is agreement among individuals’ beliefs, the inconsistency disappears.Hence if the individual and social preferences satisfy the axioms of subjectiveexpected utility theory, then imposing Pareto indi¤erence implies that thesocial preferences are represented by a subjective expected utility functionalwith probabilities that are an a¢ne combination of the individual subjectiveprobabilities, and a social utility function that is a linear combination of theindividual utilities.

Gilboa et al. do not distinguish between probabilities and beliefs. Infact, following the traditional practice in decision theory, they tacitly de…nebeliefs by probabilities and use these probabilities in the formulation of theirmain axiom, namely, the restricted Pareto condition. This approach opens agap between the verbal argument, which is stated, quite compellingly, usingthe language of beliefs, and the formal argument ,which is presented interms of Savage-type ascribed probabilities. What happens if beliefs are notrepresented by the ascribed probabilities? Not surprisingly, this may lead totwo types of errors. Errors of the …rst type occur when the restricted Paretocondition is not used to justify social preferences when in fact it should be.Errors of the second type occur when the restricted Pareto condition is usedto justify social preferences when it should not be.

To understand the problem, consider the following example. Let therebe two individuals, a and b; and two states of nature, 1 and 2: Suppose thatindividual tastes are captured by state-dependent utility functions de…nedon the level of wealth as follows:

StateIndividual 1 2

a w® 2w®

b w¯ w¯

Consider next the beliefs of the individuals.Case 1: Both individuals believe that state 1 is twice is likely to obtain

as state 2: Being subjective expected utility maximizers, their subjectiveprobabilities are ¼ (1) = 2=3 and ¼ (2) = 1=3: However, in the context

24

of traditional subjective expected utility theory, the representation of theindividual preferences are:

Ua ((w1; w2)) =1

2w®1 +

1

2w®2

U b ((w1; w2)) =2

3w¯1 +

1

3w¯2

(These are the probabilities and representation that would …gure in Gilboa etal. [2001].) Thus the two individuals appear to disagree on the probabilitiesand therefore, the restricted Pareto condition of Gilboa et. al. does notapply, even though, by the normative argument, it should.

Case 2: Individual a believes that state 1 is four times more likely toobtain than state 2 (i.e., ¼a (1) = 4=5 and ¼a (2) = 1=5), while individual bbelieves, as before, that state 1 is twice is likely to obtain as state 2: Thepreference of the two individuals are represented by:

Ua ((w1; w2)) =2

3w®1 +

1

3w®2 ;

and

U b ((w1; w2)) =2

3w¯1 +

1

3w¯2 :

In this case, the model of Gilboa, Samet, and Schmeidler implies that re-stricted Pareto indi¤erence should apply, even though, in fact, the individ-uals hold con‡icting beliefs. In other words, Gilboa et al. would use therestricted Pareto condition to justify social preferences even though, by theirown argument, the situation does not warrant it.

Conclusion: To avoid making errors in using the restricted Paretocondition to justify social preferences it is necessary to use the correctedprobability representations of individual beliefs.

5.2 Related literature

Luce and Krantz (1971) maintain that, in many circumstances, decisionsdelimit which events may obtain and that in such circumstances the ap-plication of Savage’s theory is cumbersome and unintuitive. They proposeinstead a theory based on choice among conditional acts that, they believe,is simpler and more natural. The critical view of the adequacy of Savage’stheory is shared by Fishburn, according to whom “although the Luce-Krantz

25

theory might seem a bit more intricate than Savage’s, it surely comes closerto making contact with the structure of actual decision situations” (Fish-burn [1973] p. 5). The analytical framework of Luce and Krantz includes aset of states (…nite or in…nite), an algebra of events, and an arbitrary set ofconsequences that is the same across events (as opposed to the structuredsets of state-dependent outcomes in this paper).

Both the Luce and Krantz model and the model presented here requirethat the preference relations be continuous weak orders satisfying a versionof the “sure thing principle.” Moreover, both models include an axiom thatcalibrates the intensity of preference (or preferential “di¤erences”). The dif-ference between the two models is the formalization of this idea. Looselyspeaking, the coherence axiom (A.2) requires that the intensity of prefer-ences between pairs of outcomes in a given state be independent of theevent in which this state occurs. Axiom 5 of Luce and Krantz requires thatf(i)A » g(i)B ; i = 1; 2; 3; 4 (a conditional act fA is a function from the subset ofstates A to the set of consequences) implies that if the preferential di¤erencebetween pairs of conditional acts f (3)A and f (4)A exceeds that of f (1)A and f (2)A ;

then the preferential di¤erence between g(3)B and g(4)B must exceed that be-

tween g(1)B and g(2)B independently of the (disjoint) conditional act to whichthey may be attached. This distinction has signi…cant implications for therepresentation. In particular, the axioms of Luce and Krantz do not implya subjective expected utility representation in which the utility function isde…ned on the set of consequences. To obtain such an event-independent rep-resentation they require, in addition, that there exist constant acts, namely,that there exist consequences whose valuations are event independent. Thisis in contrast to the present model in which the utility function, de…ned onthe state-dependent sets of outcomes, may be either state dependent or stateindependent and its existence does not require the availability of constantacts.

Fishburn (1973) proposed an alternative axiomatization of subjectiveexpected utility that combines elements of the Luce-Krantz model with ele-ments of the model of Anscombe and Aumann (1963). Fishburn’s analyticalframework includes extraneous probabilities, and his choice set consists of“objective” probability mixtures of conditional acts. Fishburn de…nes prob-ability mixtures on acts conditional on the same event and introduces aversion of the independence axiom requiring that if two mixture-acts condi-tional on one event are each indi¤erent to a corresponding act on a secondevent, then the 50-50 mixture of the …rst pair of conditional acts is indi¤erentto the same mixture of the second pair. He also requires that if a mixture-act

26

conditional on one even is weakly preferred over the same mixture-act con-ditional on another, disjoint, event, then the same mixture-act conditionalon the union of the two events is (weakly) less desirable than the former and(weakly) more desirable than the latter. With these and some additional in-nocuous conditions, Fishburn shows that there exist a subjective conditionalexpected utility representation of the preference relation on conditional actswith an event-dependent utility function that is unique up to positive lineartransformations and the unique subjective conditional probabilities.

The main di¤erence between Fishburn’s approach and the approachtaken here and by Luce and Krantz concerns the role of probabilities. Theformer relies on extraneous probabilities in the statement of the axioms,while the latter works do not invoke the notion of probabilities at the prim-itive level.

Unlike the present work, neither Luce and Krantz (1971) nor Fishburn(1973) attempted to apply their theories to the formulation of the moral-hazard problem. A decision theory with moral hazard and state-dependentpreferences was proposed by Drèze (1961, 1987). Building on the modelof Anscombe and Aumann (1963), Drèze (1987) and Drèze and Rustichini(1999) replace the formers’ assumption of reversal of order with preferencefor early resolution of the outcome of a random device used to choose amongacts (“games” in their terminology). This preference for information re‡ectsthe decision makers con…dence in their ability to in‡uence the likely real-ization of alternative events by taking appropriate actions. To exploit theirpower decision makers need to know in advance what game they are engagedin (the payo¤ associated with every state). They obtain a utility represen-tation of choice among games that is the maximum of subjective expectedutility over a closed convex set of probability distributions over a …nite statespace. Moreover, under additional assumptions there is a unique minimalset of such probability distributions.

Drèze (1987) is critical of the approach that uses preferences on condi-tional acts as primitive, even though implicit in his preference for informa-tion is the assumption that decision makers are capable of foreseeing andevaluating the merits of alternative games conditional on their “preemptive”actions. Drèze and Rustichini (1999) use the preference relation on condi-tional acts as a central ingredient of their model. Their work spells out thelink between the model of Drèze (1987) which uses the (derived) conditionalpreferences and conditional expected utility theory (e.g., of Luce and Kranz[1971]). In fact, the main novelty of their analysis is the imposition of con-sistency between the (primitive) preference relations on conditional acts andderived conditional preferences over unconditional acts. Thus the di¤erence

27

between the approaches taken by Drèze (1987) and Drèze and Rustichini(1999) and the one pursued here has less to do with the use of conditionalacts and more with the other assumptions of the model. In particular, itis noteworthy that, in general, in these works the sets of probabilities overwhich decision makers exercise choice are not the conditional probabilitiesobtained from a given prior. In fact, in the model of Dreze (1987) condition-ally on a speci…c action, Bayes’s rule holds; but across actions it need nothold since di¤erent actions may imply di¤erent relative probabilities acrossstates in the conditioning event.10 Second, even though their formulationsdepend critically on the presence of moral hazard, no (dis)utility is assignedto actions intended to change the probability distribution on the states space(Drèze and Rustichini [1999] allude to this point). Third, by avoiding us-ing probabilities as a primitive concept, the model presented here is moreappealing as a foundation of subjective probabilities.

An axiomatic model of subjective expected utility and Bayesian updat-ing using derived conditional preferences over unconditional acts was studiedby Ghirardato (2002). The framework is similar to that of Savage (1954)with the sure thing principle replaced by dynamic consistency. The lat-ter assumption connects the unconditional and conditional preferences. Inaddition, the model imposes consistency between unconditional and condi-tional preferences over constant acts, which together with Savage’s P4 implystate-independent preferences. This implies the existence of a unique priorand event-dependent posterior probability distributions connected throughBayes’ rule. Ghirardato’s model is an important extension of Savage’s work.It is di¤erent from the results presented here in some signi…cant ways. First,unlike the present results, the prior and posterior probabilities obtained byGhirardato do not necessarily represent the decision makers’ beliefs. Second,like Savage, Ghirardato’s representation does not admit state-dependentpreferences. Third, by not including the events as an ingredient of the choiceset, Ghirardato’s model is not (and was not meant to be) a framework thatcan accommodate decision making in the presence of moral hazard. Thesedivergent results are manifestations of the fundamental di¤erence betweenGhirardato’s approach and the approach pursued here, namely, the formerapproach events signify information while in the latter approach events con-stitute an ingredient of the choice variable.

Skiadas (1997, 1997a) axiomatized subjective probabilities representingdecision makers’ beliefs in a model that accommodates state-dependent pref-erences and admits non-separability (across sates) of the evaluation of acts.

10 I am grateful to Jacques Drèze for clarifying this point.

28

In Skiadas’ model acts and states are primitive concepts and the set of con-sequences is a derived concept intended to capture the subjective nature ofits elements. Preferences are de…ned on acts for any given event expressingthe decision maker’s desire for the overall consequences of an act on an eventnot knowing whether the event occurred. In fact, to express disappointmentaversion, given an event, one act may be conditionally preferred over an-other with the preferences between the same two acts if the same event wereknown to occurred is reversed. Thus conditional preferences is an expressionof anticipated feeling and do not have clear choice-theoretic interpretation.It is also worth mentioning that Skiadas’ model was not intended to nordoes it imply Bayesian updating. In fact, disappointment aversion meansthat learning that an even was realized a¤ects the decision maker’s beliefsat the same time that it a¤ects his valuations of di¤erent acts and it is notclear what restrictions need to be imposed if beliefs are to be updated usingBayes rule.

A di¤erent branch of literature is related to the coherence axiom. Thisaxiom serves the purpose of linking the preference relations on acts condi-tional on distinct events. The idea of axiomatically linking di¤erent pref-erence relations was originally used in Karni and Schmeidler (1980) andin Karni, Schmeidler, and Vind (1983) to connect the preference relationson actual and hypothetical acts in the framework of Anscombe and Au-mann (1963) and, thereby, model subjective expected utility with state-dependent preferences. Subsequently Wakker (1987) extended the work ofKarni, Schmeidler, and Vind by replacing the roulette lotteries of Anscombeand Aumann with topologically connected consequence spaces. The di¤er-ent structure of the consequence spaces requires the use of a linkage axiomdi¤erent from the one used by Karni, Schmeidler, and Vind and much closerin spirit to the coherence axiom of this paper. In both cases, however, thelinkage imposes state-by-state consistency of conditional preferences.

Karni and Mongin (2000) observed that only the model of Karni andSchmeidler (1980) leads to a de…nition of subjective probabilities that faith-fully represents the decision maker’s beliefs. Other models, including Karni,Schmeidler, and Vind (1983) and Wakker (1987), involve a choice of hypo-thetical probabilities over the states that renders the resulting subjectiveprobabilities arbitrary. This observation lends weight to the work of Grantand Karni (2000) extending the work of Karni and Schmeidler (1980) tononexpected utility preferences and to Karni (2001) extending it to theframework of Wakker (1987).

A common feature of all these contributions is the reliance on expressedpreferences among hypothetical lotteries or objective probability distribu-

29

tions on the states. The present model is di¤erent in that it relies on theuse of preferences on conditional acts. It thus circumvents the need to useprobabilities as primitive concepts.

30

APPENDIX

A. Proof of Theorem 1 - (i) ) (ii): Null states do not a¤ect thepreferences among acts. Thus, without loss of generality, when writing xEit is assumed that all the states in E are nonnull. If s is a null state setwS (¢; s) = 0: By Theorem III.4.1 of Wakker (1989) there exist additive valuefunction wS : X!R that represent < on X with jointly cardinal continuousfunctions fwS (¢; s) : Xs ! Rgs2S : (I.e., for all x;y 2 X;x < y if and onlyif wS (x) =

Ps2S wS (xs; s) ¸

Ps2S wS (ys; s) = wS (y) :) Let ¹x and x be

constant valuation acts such that ¹x  x. Using the uniqueness property ofthe jointly cardinal representation normalize fwS (¢; s)gs2S as follows: SetwS (xs; s) = 0 for all s 2 S and let

wS (¹x) =Xs2S

wS (¹xs; s) = 1. (8)

To construct functions wE (¢; s) on Xs note that, by (A.2), x » xE . SetwE (xs; s) = 0 for all s 2 E: For each s 2 E and xs 2 Xs let x0s (xs; E) ½ Xsbe de…ned by: (x¡s; xs) » (x¡s; x0s (xs; E))E : Set wE (ys; s) = wS (xs; s) forall ys 2 x0s (xs; E) : If the [xsx0s (xs;E) is a proper subset of Xs extend itas follows: For zs =2 [xsx0s (Xs;E) take x 2 X such that x » (x¡s; zs)E andde…ne wE (zs; s) =

Pt2S wS (xt; t) :

De…ne a function wE on XE by wE (yE) = wS (xEx) whenever yE »xEx: But, by (A.1), yE » xEx implies that ys 2 x0s (xs;E) :Thus wS (xEx) =Ps2E wS (xs; s) =

Ps2E wE (ys; s) : Hence wE (yE) =

Ps2E wE (ys; s) for

all yE 2 XE. On the set of conditional constant valuation acts (A.2) impliesthat wE (x¤E) = wG (x

¤G) for all E 2 E and G ½ E:

Next we show that the functions fwE (¢; s) j s 2 S;E 2 Eg constitutean additive-valued representation of < on X. Let xE ; yA 2 X and x;y 2 Xsuch that xs 2 x0s (xs;E) for all s 2 E and ys 2 x0s (ys;A) for all s 2 A: ThenxE » xEx and yA » yAx: By transitivity and the presentation, xE < yA ifand only if xEx < yAx if and only if

Pt2E wS (xt; t) ¸

Pt2S¡AwS (yt; t) :

But, by de…nition, wS (xs; s) = wE (xs; s) for all s 2 E and wS (ys; s) =wA (ys; s) for all s 2 A. Hence

Xt2E

wS (xt; t) ¸Xt2A

wS (yt; t),Xt2E

wE (xt; t) ¸Xt2A

wA (yt; t) ; (9)

31

for all E;A 2 E. Thus

xE < yA ,Xt2E

wE (xt; t) ¸Xt2A

wA (yt; t) : (10)

This completes the proof that (i)! (ii) : The proof that (ii) implies (ii)is immediate.

To prove the uniqueness of fwE (¢; s) j s 2 S;E 2 Eg note that, byWakker (1989) Observation III.6.6’, fwE (¢; s) j s 2 Sg is unique up to unit-comparable transformations. (That is, fwE (¢; s) j s 2 Sg represent < onXE in the sense of (ii) if and only if wE (¢; s) = ¯wE (¢; s) + °E (s) ; ¯ > 0for all s 2 E:) But, by (ii) ; for all E 2 E and x;y 2 X; xS » yE if and onlyif X

s2SwS (xs; s) =

Xs2E

wE (ys; s) ;

which is equivalent toXs2S

[¯wS (xs; s) + °S (s)] =Xs2E

[¯wE (ys; s) + °E (s)] :

But this implies that C =Ps2S °S (s) =

Ps2E °E (s) for all E: ¤

B. Proof of the Main Result.The following Lemma is needed for the proof of the main result.

Lemma 8 Assume that there are at least three nonnull states and that therelation < is a continuous weak-order on X: Then the following conditionsare equivalent:

(i) < satis…es (A.1), (A.2) and (A.3).

(ii) For every E 2 E there exist positive a¢ne or constant function ÁE :[s2EwS (Xs; s) ! R such that, for all and s 2 E; wE (¢; s) = ÁE ±wS (¢; s) ; where fwE (¢; s) j E 2 E, s 2 Eg are as in Theorem 1.

Proof of Lemma 8. The proof of Lemma 8 involves two stages. The…rst stage is an adaptation of the proof of Wakker’s (1987) Proposition 4.5and proves the results for E 2 E1: The second stage is new and extends theresult to nonnull singleton events.

32

Stage 1 - (i) ) (ii) : Suppose that < is a continuous weak order sat-isfying (A.1), (A.2) and (A.3). Fix E 2 E1 then, by Theorem 1, for everyt 2 E there exist w; z 2X such thatX

r2S¡ftg[wS (wr; r)¡wS (zr; r)] = ³ > 0; (11)

and xE ;yE2 XE satisfyingXr2E¡ftg

[wE (xr; r)¡wE (yr; r)] = " > 0. (12)

By continuity of the additive valued functions wE (¢; s) and the connected-ness of the sets Xs; for every ³ 2 [¡³; ³]; " 2 [¡"; "]; and t 2 E there exist¹w;¹z 2 X and ¹xE ; ¹yE2 XE such thatX

r2S¡ftg[wS ( ¹wr; r)¡wS (¹zr; r)] = ³ (13)

and Xr2E¡ftg

[wE (¹xr; r)¡wE (¹yr; r)] = ". (14)

De…ne ÁE by wE (¢; ¢) = ÁE ± wS (¢; ¢) : Then ÁE is continuous. To showthat ÁE is positive a¢ne or constant function …x t 2 E and let Wt =wS (Xt; t) : Then, by the connectedness of Xt and the continuity of wS (¢; t) ;Wt is an interval in R. Take ®; ¯; °; ± 2Wt such that ¡³ · ®¡¯ = °¡± · ³and ¡" · ÁE (®) ¡ ÁE (¯) · ": Let at; bt; ct; dt 2 Xt satisfy wS (at; t) = ®;wS (bt; t) = ¯; wS (ct; t) = ° and wS (dt; t) = ±: Take w; z 2 X such thatwS (wr; r)¡wS (zr; r) = (®¡ ¯) = (j S j ¡1) for all r 2 S: ThenX

r2S¡ftg[wS (wr; r)¡wS (zr; r)] = ®¡ ¯: (15)

By Theorem 1¡w¡t;at

¢ » ¡z¡t; bt¢ and ¡w¡t; ct¢ » ¡z¡t; dt¢ :Take xE ; yE2 XE such that wE (xr; r)¡wE (yr; r) = (ÁE (®)¡ ÁE (¯)) = (j E j ¡1)

for all r 2 E: ThenXr2E¡ftg

[wE (xr; r)¡wE (yr; r)] = ÁE (®)¡ ÁE (¯) : (16)

33

Since wE (¢; t) = ÁE ±wS (¢; t) this implies¡x¡t;at

¢E» ¡y¡t; bt¢E : Applying

(A.3) twice yields¡x¡t; ct

¢E» ¡y¡t; dt¢E : Thus

ÁE (°)¡ ÁE (±) =X

r2E¡ftg[wE (xr; r)¡wE (yr; r)] = ÁE (®)¡ ÁE (¯) :

(17)

By Wakker (1987) Lemma 4.4 this implies that ÁE is a¢ne. But ÁE isnondecreasing. (To see this, let

¡x¡t; at

¢<¡x¡t; bt

¢: But

¡x¡t; at

¢<¡

x¡t; at¢;¡x¡t; at

¢<¡x¡t; bt

¢; and

¡x¡t; at

¢E<¡x¡t; at

¢E: Thus, by (A.3)¡

x¡t; at¢E<¡x¡t; bt

¢E: The conclusion is implied by the representation of

<.) Hence ÁE is constant or positive.(ii) ) (i) : Assume that there exist positive a¢ne or constant transfor-

mations ÁE such that wE (¢; ¢) = ÁE ± wS (¢; ¢). Suppose that (x¡t; at) <(y¡t; bt), (y¡t; ct) < (x¡t; dt) and

¡z¡t; bt

¢E<¡w¡t; at

¢E: By Theorem 1,

(x¡t; at) < (y¡t; bt) if and only if

wS (at; t) +X

s2S¡ftgwS (xs; s) ¸ wS (bt; t) +

Xs2S¡ftg

wS (ys; s) (18)

and (y¡t; ct) < (x¡t; dt) if and only if

wS (dt; t) +X

s2S¡ftgwS (xs; s) · wS (ct; t) +

Xs2S¡ftg

wS (ys; s) : (19)

Hence

wS (bt; t)¡wS (at; t) ·X

s2S¡ftg[wS (xs; s)¡wS (ys; s)] · wS (ct; t)¡wS (dt; t) :

(20)

By positive a¢neness or constancy of ÁE these inequalities imply

wE (bt; t)¡wE (at; t) · wE (ct; t)¡wE (dt; t) : (21)

By Theorem 1¡z¡t; bt

¢E<¡w¡t; at

¢Eif and only ifX

s2E¡ftgwE (zs; s) +wE (bt; t) ¸

Xs2E¡ftg

wE (ws; s) +wE (at; t) : (22)

Thus

wE (bt; t)¡wE (at; t) ¸X

s2E¡ftg[wE (ws; s)¡wE (zs; s)] : (23)

34

But wE (bt; t)¡wE (at; t) · wE (ct; t)¡wE (dt; t) impliesXs2E¡ftg

wE (zs; s) +wE (ct; t) ¸X

s2E¡ftgwE (ws; s) +wE (dt; t) : (24)

Hence, by Theorem 1,¡z¡t; ct

¢E<¡w¡t; dt

¢E: This completes the proof of

stage 1.Stage 2 - (ii)! (i): For the nonnull singleton events fsg; s 2 S suppose

that

wfsg (xs; s) = ¸¡1s wS (xs; s) + ·s; 8xs 2 X (25)

where ¸s > 0: By Theorem 1, for every constant valuation act, x¤; and allnonnull s 2 S;

wfsg (x¤s; s) =Xt2S

¸twftg (x¤t ; t) =Xt2S

wS (x¤t ; t) : (26)

(Notice that, by the normalization of wS (¢; s) ; wfsg (¹xs; s) =Pt2S wS (¹xt; t) =

1: Hence, for all s; ¸s = wS (¹xs; s), ·s = 0, andPs2S ¸s = 1:) But equations

(26) implies (A.2) for all nonnull singleton events. Axiom (A.1) and (A.3)are implied trivially. Thus (ii)! (i):

Next we show that (A.1)-(A.3) imply the de…nition in equations (25).Without loss of generality assume that S = f1; 2; 3g and that all three statesare nonnull. Suppose also, without loss of generality that, for some constantvaluation act x¤;

wf1g (x¤1; 1) = µ¡11 wS (x

¤1; 1) ; wf2g (x

¤2; 2) = µ

¡22 wS (x

¤2; 2) ; wf3g (x

¤3; 3) = ¸

¡13 wS (x

¤3; 3) :

(27)

where µ1 < ¸1 and µ2 > ¸2. Let E = f2; 3g; then, since E is nonnull, theproof of stage 1 implies that there exist bE > 0 and aE such that, for allx 2 Xt and t 2 E;

wE (x; t) = bEwS (x; t) + aE : (28)

Moreover, by (A.2),

wE (¹x2; 2) +wE (¹x3; 3) = bE [wS (¹x2; 2) +wS (¹x3; 3)] + 2aE = 1: (29)

Hence bE = [wS (¹x2; 2) +wS (¹x3; 3)]¡1 and aE = 0: By (A.2), for every

constant valuation act, x¤,

bE [wS (x¤2; 2) +wS (x

¤3; 3)] = wf2g (x

¤2; 2) = wf3g (x

¤3; 3) : (30)

35

But

wS (x¤3; 3) = ¸3wf3g (x

¤3; 3) = wf3g (x

¤3; 3)wS (¹x3; 3) (31)

and

wS (x¤2; 2) = µ2wf2g (x

¤2; 2) > wf2g (x

¤2; 2)wS (¹x2; 2) = ¸2wf2g (x

¤2; 2) : (32)

Hence, using the fact that wf2g (x¤2; 2) = wf3g (x¤3; 3) ; ¸s = wS (¹xs; s) for alls;

bE [wS (x¤2; 2) +wS (x

¤3; 3)] > bE (¸2 + ¸3)wf2g (x

¤2; 2) = wf2g (x

¤2; 2) : (33)

This contradicts equation (30). Hence, for all s and xs 2 Xs; wfsg (xs; s) =¸¡1s wS (xs; s) : ¤

Proof of Theorem 2 - (a) (i)) (ii) : Suppose that < satis…es (A.1),(A.2), and (A.3): Then, by Theorem 1, for all E 2 E, t 2 E; xE2 XE andx; x0 2 Xt;

(x¡t; x)E< (x¡t; x0)E , wE (x; t) ¸ wE¡x0; t

¢: (34)

By Lemma 8 there exist bE ¸ 0 and aE such that, for all x 2 Xt and t 2 E;wE (x; t) = bEwS (x; t) + aE (35)

In particular,by the proof of stage 2 of Lemma 8, for all nonull t 2 S;wftg (x; t) = bftgwS (x; t), where bftg = wS (¹xt; t)

¡1 :For each s 2 S let us (¢) = wfsg (¢; s) and de…ne ¼ (s) = b¡1fsg = wS (¹xs; s) ;

if bfsg > 0 (i.e., if s is nonnull) and ¼ (s) = 0 otherwise: Then wS (x; s) =¼ (s)us (x) and, by Theorem 1, for all x;y 2 X;

x < y ,Xs2S

¼ (s)us (xs) ¸Xs2S

¼ (s)us (ys) : (36)

Moreover, by Theorem 1 and equations (35) ; for all xE;yA 2 X, xE < yAif and only if

bEXs2E

¼ (s)us (x) + º (E) ¸ bAXs2A

¼ (s)us (x) + º (A) ; (37)

where º (B) =j B j aB; for all B 2 E:Next use the constant valuation acts to determine the probabilities. Let

x¤ and x¤¤ be constant valuation acts and suppose that x¤¤  x¤: Denote

36

by ¹u and u the common value us (x¤¤s ) and us (x¤s) ; respectively. Then, by(A.3), x¤¤E » x¤¤A and x¤E » x¤A. Thus equations (37) imply

¹u[bEXs2E

¼ (s)¡ bAXs2A

¼ (s)] + [º (E)¡ º (A)] = 0: (38)

and

u[bEXs2E

¼ (s)¡ bAXs2A

¼ (s)] + [º (E)¡ º (A)] = 0: (39)

But ¹u > u hence equations (38) and (39) imply that bEPs2E ¼ (s) =

bAPs2A ¼ (s) and º (E) = º (A) : Let A = S then, by de…nition, bS = 1 and

º(S) = 0: SincePs2S ¼ (s) = 1 equation (39) implies that bE

Ps2E ¼ (s) = 1

and º (E) = 0 for all E 2 E. Letting ¼ (t j E) = bE¼ (t) = ¼ (t) =Ps2E ¼ (s)

and invoking equations (37) we conclude that, for all xE;yA 2 X;

xE < yA ,Xs2E

¼ (s j E)us (xs) ¸Xs2A

¼ (s j A)us (ys) :

This completes the proof that (i)) (ii) :(ii)) (i) : The fact that (ii) implies (A.1) is an immediate implication of

Theorem 1. The fact that it implies (A.3) is straightforward. To show that(ii) implies (A.2) take E 2 E1, w;x;y; z 2X, aj; bj ; cj; dj 2 Xj and supposethat (x¡j ; aj) < (y¡j; bj), (y¡j ; cj) < (x¡j; dj);

¡z¡j; bj

¢E<¡w¡j; aj

¢E:

By (ii) (x¡j ; aj) < (y¡j; bj) impliesXs2S¡fjg

¼ (s)us (xs) + ¼ (j)uj (aj) ¸X

s2S¡fjg¼ (s)us (ys) + ¼ (j)uj (bj) ;

(40)

and (y¡j; cj) < (x¡j; dj) impliesXs2S¡fjg

¼ (s)us (xs) + ¼ (j)uj (dj) ·X

s2S¡fjg¼ (s)us (ys) + ¼ (j)uj (cj) :

(41)

and¡z¡j ; bj

¢E<¡w¡j ; aj

¢EimpliesX

s2S¡fjg¼ (s j E)us (zs) + ¼ (j j E)uj (bj) ¸

Xs2S¡fjg

¼ (s j E)us (ws) + ¼ (j j E)uj (aj) :

(42)

37

But equations (40) and (41) imply that

uj (cj)¡ uj (dj) ¸ uj (bj)¡ uj (aj) : (43)

Hence equations (42) and (43) implyXs2S¡fjg

¼ (s j E)us (zs) + ¼ (j j E)uj (cj) ¸X

r2S¡fjg¼ (s j E)us (ws) + ¼ (j j E)uj (dj) :

(44)

Equation (44) and (ii) imply¡z¡j ; cj

¢E<¡w¡j ; dj

¢E: Hence (ii) implies

(A.2).(b) The uniqueness of fusgs2S follows directly from the uniqueness of

fwE (¢; s) j s 2 S;E 2 Egs2S of Theorem 1. In particular, ® = C:(c) Let ¼ and fusgs2S satisfy part (a) of Theorem2. If t 2 S is nonnull

then, by the maintained assumption (A.0), for some ¹xt; xt 2 Xt; and x 2 X;¡x¡t; ¹xt

¢ftg Â

¡x¡t; xt

¢ftg :Hence ut (¹xt)¡ut (xt) > 0. Moreover,

¡x¡t; ¹xt

¢ ¡x¡t; xt

¢implies ¼ (t) [ut (¹xt)¡ ut (xt)] > 0: Thus ¼ (t) > 0: If t is null then¡

x¡t; ¹xt¢ » ¡x¡t; xt¢ implying ¼ (t) [ut (¹xt)¡ ut (xt)] = 0: Hence ¼ (t) = 0:

To prove the uniqueness of ¼ suppose, by way of negation, that thereexists a probability measure, ¹; on S and utility functions fusgs2S thatsatisfy the representation in (a.ii), but ¹ 6= ¼ . Then there are statess; t 2 S such that ¹ (s) > ¼ (s) and ¼ (t) > ¹ (t) : Note that ¹ (s) > ¼ (s) and¼ (t) > ¹ (t) imply that s and t are nonnull. Moreover, let us (¢) = csus (¢) ;for some cs > 0: Then the representation requires that ¹ (s) = cs¼ (s) =Cfor all s 2 S; where C =Pt2S ct¼ (t) : Moreover, ¼ (s)us (¢) = C¹ (s) us (¢)for all s 2 S. Let r 2 S ¡ fs; tg be a nonnull state and consider the eventsE = fs; rg and A = ft; rg: Suppose that xE » yA: Then

¼ (s)us (xs) + ¼ (r)ur (xr)

¼ (s) + ¼ (r)=¼ (t)ut (yt) + ¼ (r)ur (yr)

¼ (t) + ¼ (r):

But ¹ (s) + ¹ (r) > ¼ (s) + ¼ (r) and ¹ (t) + ¹ (r) < ¼ (t) + ¼ (r) which,together with ¼ (s)us (¢) = C¹ (s) us (¢) for all s 2 S; imply

¹ (s) us (¢) + ¹ (r) ur (xr)¹ (s) + ¹ (r)

<¹ (t) ut (yt) + ¹ (r) ur (yr)

¹ (t) + ¹ (r):

This contradicts xE » yA. Hence ¹ = ¼: ¤

C. Proof of Theorem 4 - Theorem 4 follows from Theorem 2 with thefollowing additional speci…cations: To show that, in part (a), (i)) (ii) note

38

that (A.4) makes (A.1) super‡uous. In particular, the existence of additiverepresentation is implied by Wakker (1989) Theorem IV.2.7. Without lossof generality let 1 be a nonnull state and set u (x) = u1 (x) ; x 2 X: Then,by Lemma 3 and Theorem 2, for every nonnull s 2 S; us (x) = ¾su (x) + »s;where ¾s > 0: Hence, by Theorem 2, for all x;y 2 X;

x < y ,Xs2S

¼ (s)¾su (xs) ¸Xs2S

¼ (s)¾su (ys) : (45)

and, for every xE;yA 2 X,

xE < yA ,Xs2E

¼ (s j E) [¾su (xs) + »s] ¸Xs2A

¼ (s j A) [¾su (ys) + »s] :

(46)

The proof that (ii) implies (i) is follows immediately from the represen-tation.

The proofs of the part (b) is straight forward. The proof of part (c)follow from the proof of part (c) in Theorem 2. In particular,

¡x¡1; ¹x

¢f1g ¡

x¡1; x¢f1g :Hence u (¹x)¡u (x) > 0 and, by (A.4), for all s 2 S; (x¡s; ¹x)fsg Â

(x¡s; x)fsg : Hence us (¹x)¡ us (x) = ¾s [u (¹x)¡ u (x)] > 0: Thus, ¾s > 0 forall s 2 S: ¤

D. Proof of Theorem 5 - (i)) (ii): Suppose that < is a continuousweak order satisfying (A.1’) and that every action in A is implementable at¡a0; x

¢: By Theorem III.4.1 of Wakker (1989), for every a0 2 A0 there exist

jointly cardinal continuous additive value functions fwS¡¢; s; a0¢gs2S that

represent < on fa0g £ X: Furthermore, (A.1’) implies that, for all s 2 Sand a0 2 A0; wS

¡¢; s; a0¢ = wS (¢; s) + v ¡a0¢. Using the uniqueness of thejointly cardinal representation normalize fwS (¢; s) + v

¡a0¢gs2S as follows:

Set wS (xs; s) = 0 for all s 2 S, v¡a0¢= 0:

Fix E 2 E and a 2 F¡1 (E) and let¡a0; xEz (a)

¢ » (a; xE) : (Suchz (a) := z

¡a;¡a0; x

¢¢exist since all actions are implementable at

¡a0; x

¢.)

De…ne v (a) = ¡Ps2S¡E wS (zs (a) ; s) and set wE (xs; s) = 0 for all s 2 E:Hence (i) implies (4).

For each s 2 E and xs 2 Xs let x0s (xs; a) ½ Xs be de…ned by:¡a0;¡x¡s; xs

¢Ez (a)

¢ » ¡a; ¡x¡s; x0s (xs; a)¢E¢ :Set wE (ys; s) = wS (xs; s) for all ys 2 x0s (xs; a) ; s 2 S: (If the [x2Xsx0s (x; a) +Xs extend the correspondence x0s(¢; a) it as follows: For qs 2 Xs¡[x2Xsx0s (x;a)

39

take x 2X such that¡a0;xEz (a)

¢ » (a; (x¡s; qs)E) and de…ne wE (qs; s) =Ps2E wS (xs; s) :)Let

WF (a) (x;a) =Xs2F (a)

wF (a) (xs; s) + v (a) : (47)

Next we show that the functions fwF (a) (¢; s)+v (a) j s 2 S; a 2 Ag con-stitute an additive-valued representation of < on A. Let

¡a; ~xF (a)

¢;¡b; ~yF (b)

¢ 2A and x;y 2X such that ~xs 2 x0s (xs;a) for all s 2 F (a) and ~ys 2 x0s (ys; b)for all s 2 F (b) : Then, by transitivity of < and (A.1’), ¡a; ~xF (a)¢ » ¡a0; ¡xF (a)z (a)¢¢and

¡b; ~yF (b)

¢ » ¡a0; ¡yF (b)z (b)¢¢ : By transitivity and the presentation, ¡a; ~xF (a)¢ < ¡b; ~yF (b)¢if and only if

¡a0;¡xF (a)z (a)

¢¢<¡a0;¡yF (b)z (b)

¢¢if and only if

Pt2F (a) wS (xt; t)+

v (a) ¸Pt2F (b) wS (yt; t)+v (b) : But, by de…nition, wS (xs; s) = wF (a) (~xs; s)for all s 2 F (a) and wS (ys; s) = wF (b) (~ys; s) for all s 2 F (b). Hence¡a; ~xF (a)

¢<¡b; ~yF (b)

¢, Xt2F (a)

wF (a) (~xt; t) + v (a) ¸Xt2F (b)

wF (b) (~yt; t) + v (b) :

(48)

This completes the proof that (i) ! (ii) : The proof that (ii) impliesthat < on A is a continuous weak order satisfying (A.1’) that is immediate.To show that (ii) implies that all actions are implementable at

¡a0; x

¢note

thatXs2F (a)

wF (a) (xs; s) + v (a) =Xs2F (a)

wF (a) (xs; s)¡X

s2S¡F (a)wS (zs (a) ; s) + v

¡a0¢:

(49)

Hence (a; x) » ¡a0; xF (a)z (a)¢ and a is implementable at ¡a0; x¢ : The proofof uniqueness is similar to the proof of uniqueness in Theorem 1. ¤

E. Proof of Theorem 6The following lemma is analogous to Lemma 8.

Lemma 9 Assume that there are at least three nonnull states and that therelation < is a continuous weak-order on A: Then the following conditionsare equivalent:

(i) < is a continuous weak order satisfying (A.1’) and (A.3’) and there exist¡a0; x

¢ 2 A0 £X such that all actions are implementable at¡a0; x

¢.

40

(ii) For every a 2 A and s 2 F (a) ; wF (a) (¢; s) = ¯F (a)wS (¢; s) + ®F (a);¯F (a) ¸ 0; where the functions fwF (a) (¢; s) j a 2 A, s 2 F (a)g are asin Theorem 5.

Proof of Lemma 9 - (i)) (ii) : Suppose that (i) holds the, by Theorem5, for every t 2 S there exist w; z 2 X such thatX

r2S¡ftg[wS (wr; r)¡ wS (zr; r)] = ³ > 0;

and, for every a 2 A there exist x;y 2 X such thatXr2F (a)¡ftg

£wF (a) (xr; r)¡ wF (a) (yr; r)

¤= " > 0.

By continuity of the functions wF (a) (xs; s) and the connectedness of the setsXs; for every ³ 2 [¡³; ³]; " 2 [¡"; "]; and t 2 F (a) there exist ¹w;¹z 2 X and¡a; ¹xF (a)

¢;¡a; ¹yF (a)

¢ 2 fag£X; such thatXr2S¡ftg

[wS ( ¹wr; r)¡ wS (¹zr; r)] = ³

and Xr2F (a)¡ftg

£wF (a) (¹xr; r)¡ wF (a) (¹yr; r)

¤= ".

De…ne ÁF (a) by wF (a) (¢; ¢) = ÁF (a) ± wS (¢; ¢) : To show that ÁF (a) ispositive a¢ne or constant function, let Wt = wS (Xt; t) : Then, by theconnectedness of Xt and the continuity of wS (Xt; t) ; Wt is an intervalin R. Take ®; ¯; °; ± 2 Wt such that ¡³ · ® ¡ ¯ = ° ¡ ± · ³ and¡" · ÁF (a) (®) ¡ ÁF (a) (¯) · ": Let at; bt; ct; dt 2 Xt satisfy wS (at; t) = ®;wS (bt; t) = ¯; wS (ct; t) = ° and wS (dt; t) = ±: Take w; z 2 X such thatX

r2S¡ftg[wS (wr; r)¡ wS (zr; r)] = ®¡ ¯:

Then, by Theorem 5,¡a0;¡w¡t;at

¢¢ » ¡a0; ¡z¡t; bt¢¢ and ¡a0; ¡w¡t; ct¢¢ » ¡a0; ¡z¡t; dt¢¢ :Take

¡a; xF (a)

¢;¡a; yF (a)

¢ 2 fag£X; such thatXr2F (a)¡ftg

£wF (a) (xr; r)¡ wF (a) (yr; r)

¤= ÁF (a) (®)¡ ÁF (a) (¯) :

41

Since wF (a) (¢; t) = ÁF (a)±wS (¢; t) this implies³a;¡x¡t;at

¢F (a)

´»³a;¡y¡t; bt

¢F (a)

´:

Applying (A.3’) twice yields³a;¡x¡t; ct

¢F (a)

´»³a;¡y¡t; dt

¢F (a)

´: Thus

ÁF (a) (°)¡ ÁF (a) (±) =X

r2F (a)¡ftg

£wF (a) (xr; r)¡ wF (a) (yr; r)

¤= ÁF (a) (®)¡ ÁF (a) (¯) :

By Wakker (1987) Lemma 4.4 this implies that ÁF (a) is a¢ne. But ÁF (a) isnondecreasing. Hence it is constant or positive.

Since F may be many-to-one it is possible that there are a; b 2 A suchthat F (a) = F (b) = E: Let (a;xE) < (b;xE) then

(¯ (a;E)¡ ¯ (b;E))Xs2E

wS (xs; s) + ®F (a) ¡ ®F (b) + v (a)¡ v (b) ¸ 0:

By (3) in Theorem 5, for all y 2X;

(¯ (a;E)¡ ¯ (b; E))Xs2E

wS (ys; s) + ®F (a) ¡ ®F (b) + v (a)¡ v (b) ¸ 0:

Let xÀ y thenPs2E wS (ys; s) <

Ps2E wS (xs; s) = 0; where the last

equality follows from Theorem 5 and the de…nition of¡a0; x

¢: Hence for suf-

…ciently unfavorable payo¤ pro…le y this inequalities implies that ¯ (a;E) =¯ (b;E) = ¯ (E) = ¯F (a): Hence (i) implies (ii) :

(ii)) (i) : The proof that (ii) implies (i) follows the same argument asin the proofs of Lemma 8 and of Theorem 5. ¤

Proof of Theorem 6 - (a) (i) ) (ii) : Suppose that < satis…es theassumptions in (i) : Then, by Theorem 5, for all a 2 A; x 2 X; t 2 F (a)and x; x0 2 Xt;¡

a; (x¡t; x)F (a)¢<¡a; (x¡t; x0)F (a)

¢ , wF (a) (x; t) ¸ wF (a)¡x0; t

¢:

(50)

By Lemma 9, wF (a) (x; t) = ¯F (a)wS (x; t) + ®F (a); ¯F (a) ¸ 0. Hence, for alla 2 F¡1 (fsg) wfsg (¢; s) = ¯fsgwS (x; t) + ®fsg: For each s 2 S let us (¢) =wfsg (¢; s).and de…ne ¼ (s) = ¯¡1fsg=

Pt2S ¯

¡1ftg if ¯fsg > 0 and ¼ (s) = 0

otherwise: Then wS (x; t) = ¼ (s)£us (x)¡ ®fsg

¤Pt2S ¯

¡1ftg and, by Theorem

5, for all¡a0;x

¢;¡a0;y

¢ 2 fa0g £X;42

¡a0;x

¢<¡a0;y

¢,Xs2S

¼ (s)us (xs) ¸Xs2S

¼ (s)us (ys) : (51)

Moreover, by Theorem 5, for all¡a;xF (a)

¢;¡b;yF (b)

¢ 2 A, ¡a;xF (a)¢ < ¡b;yF (b)¢if and only if

¯F (a)Xs2F (a)

¼ (s)us (xs) + · (a) + v (a) ¸ ¯F (b)Xs2F (b)

¼ (s)us (ys) + · (b) + v (b) ;

(52)

where · (a) = ¡Ps2F (a) ¼ (s)®fsgPt2S ¯

¡1ftg+ j F (a) j ®F (a); a 2 A:

Let x¤;x¤¤ 2 X (a) be CVPP. Then us (x¤¤s ) ¡ us (x¤s) = ut (x¤¤t ) ¡

ut (x¤t ) for all s; t 2 S: By (A.3’)

³a;x¤F (a)

´»³b;x¤F (b)

´if and only if³

a;x¤¤F (a) (a)´»³b;x¤¤F (b) (b)

´: Hence, by equation (52),

¯F (a)Xs2F (a)

¼ (s) [us (x¤¤s )¡ us (x¤s)] = ¯F (b)

Xs2F (b)

¼ (s) [us (x¤¤s )¡ us (x¤s)] :

(53)

Thus

[us (x¤¤s )¡ us (x¤s)]

24¯F (a) Xs2F (a)

¼ (s)¡ ¯F (b)Xs2F (b)

¼ (s)

35 = 0: (54)

Without loss of generality assume that³a;x¤¤F (a)

´Â³a;x¤F (a)

´then [us (x¤¤s )¡ us (x¤s)] >

0: Equations (53) and (54) imply that ¯F (a)Ps2F (a) ¼ (s) = ¯F (b)

Ps2F (b) ¼ (s) :

Let b 2 A0 (i.e., b is a default action). Because ¯S = 1 =Ps2S ¼ (s) the

preceding argument implies that, for all a 2 A, ¯F (a)Ps2F (a) ¼ (s) = 1.

De…ne ¼ (s j F (a)) = ¼ (s) =Ps2F (a) ¼ (s) :Next observe that since all actions are implementable at

¡a0; x

¢, for

all a 2 A;¡a0; xF (a)z (a)

¢ » ¡a; xF (a)¢ : Invoking the fact that us (xs) =·¡a0¢= v

¡a0¢= 0; the representation in equation (52) implies: · (a) +

v (a) =Ps2S¡F (s) ¼ (s)us (zs (a)) : But v (a) =

Ps2S¡F (s) ¼ (s)us (zs (a))

hence · (a) = 0. Thus equation (52) imply that, for all¡a;xF (a)

¢;¡b;xF (b)

¢2 A£X;¡a;xF (a)

¢<¡b;xF (b)

¢if and only ifX

s2F (a)¼ (s j F (a))us (xs;a) + v (a) ¸

Xs2F (b)

¼ (s j F (b))us (xs; b) + v (b) :

(55)

43

This completes the proof that (i)) (ii) :The proof that (ii)) (i) follows by the same arguments as the proof of

the same implications in Theorem 2. The proof of parts (b) and (c) followsby the corresponding arguments in the proof of Theorem 2. ¤

44

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