JÜRGEN EICHBERGER and DAVID KELSEY

E-CAPACITIES AND THE ELLSBERG PARADOX

ABSTRACT. Ellsberg’s (1961) famous paradox shows that decision-makers giveevents with ‘known’ probabilities a higher weight in their outcome evaluation. Inthe same article, Ellsberg suggests a preference representation which has intuitiveappeal but lacks an axiomatic foundation. Schmeidler (1989) and Gilboa (1987)provide an axiomatisation for expected utility with non-additive probabilities.This paper introduces E-capacities as a representation of beliefs which incor-porates objective information about the probability of events. It can be shownthat the Choquet integral of an E-capacity is the Ellsberg representation. Thepaper further explores properties of this representation of beliefs and providesan axiomatisation for them.

KEY WORDS: Ellsberg paradox, Uncertainty aversion, Choquet integral, Non-additive probabilities

INTRODUCTION

For more than forty years Savage’s (1954) expected utility theorywith subjective probabilities has dominated models of decision mak-ing under uncertainty in economics and statistics. His seven axiomson preferences over uncertain acts characterize

• an expected utility function based on• a uniquesubjective(additive) probability distribution.

If one accepts Savage’s axioms, one can represent beliefs of a de-cision maker by a probability distribution over unknown states andrisk preferences by a utility function over outcomes. Hence, it ispossible to apply all rules of probability theory to this belief rep-resentation. In particular, one can update beliefs on new informa-tion using Bayes’ rule. Savage’s theory therefore provides normative‘Foundations of Statistics’, as the title of his book suggests.

Ellsberg (1961) challenged Savage’s (1954) view that individu-als’ beliefs about unknown states of nature can be adequately repre-sented by additive probability distributions. In order to distinguishbetween risk and uncertainty, Ellsberg (1961) suggested the follow-

Theory and Decision46: 107–140, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

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108 JÜRGEN EICHBERGER AND DAVID KELSEY

ing test. Consider an urn which contains 30 red balls and 60 ballsthat are either black or yellow. Four lotteries, given in the tablebelow, were presented to a large number of people.

Red Black Yellow

30 60

Lottery A $100 $0 $0

Lottery B $0 $100 $0

Lottery C $100 $0 $100

Lottery D $0 $100 $100

Decision makers first had to choose between lottery A and lottery Band then between lottery C and D. Faced with the first choice, mostpeople preferred lottery A over lottery B. When choosing betweenlottery D and C, however, lottery D was usually the preferred option.These decisions are blatantly inconsistent with Savage’ssure-thingprinciple. Moreover, beliefs of decision makers with such prefer-ences cannot be represented by an additive probability distribution:

• Since A is preferred to B, one is led to believe that the decisionmaker considers it less likely that a black ball is drawn than ared one, hence

Prob {black}< Prob {red} = 1/3.

• On the other hand, given that D is chosen rather than C, oneconcludes that the decision-maker believes that red or yellowis more likely than black or yellow, i.e.,

Prob {red or yellow}< Prob {black or yellow}.

If probabilities are additive then

1/3= Prob {red}< Prob {black}

follows. This contradicts the first assessment and therefore the exis-tence of a subjective probability distribution that represents beliefsof such a decision maker. Thus, the Ellsberg paradox challenges notonly Savage’s theory but any theory of decision making under uncer-tainty based onprobabilistic sophisticationin the sense of Machinaand Schmeidler (1992).

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E-CAPACITIES AND THE ELLSBERG PARADOX 109

The Ellsberg paradox suggests that decision makers give eventswith ‘known’ probabilities a higher weight in their outcome eval-uation. In response to this evidence, Ellsberg (1961) proposed arepresentation of preferences over uncertain acts which resolves theparadox. Though intuitively appealing, this representation lacks anaxiomatic foundation. In addition, the Ellsberg representation doesnot distinguish risk preferences from beliefs.

Recent studies by Schmeidler (1989) and Gilboa (1987) providea system of axioms characterizing expected utility with non-additiveprobabilities. This approach maintains the distinction between riskpreferences and beliefs. Representing a decision maker’s beliefs bynon-additive probabilities, one can easily accommodate the Ellsbergparadox. For example, suppose a decision maker reduces her prob-ability assessment for events with unknown probabilities by a smallamountε due to uncertainty:

Prob {black}= (1/3)− ε,Prob {red or yellow}= (2/3)− ε.

These beliefs would justify choosing lottery A over B and lottery Dover C, but such beliefs would no longer be additive.

In Schmeidler’s and Gilboa’s approach, preferences are repre-sented by a Choquet integral of expected utility with respect to ageneralcapacity, i.e., a not-necessarily-additive probability. For anystrictly increasing utility function over wealth, there are many dif-ferent capacities resolving the Ellsberg paradox. For economic andgame-theoretic applications, however, it appears desirable to de-termine a class of capacities that can accommodate the Ellsbergparadox but remain otherwise close to additive probabilities.

In particular, the following two consequences of a representa-tion of beliefs by non-additive probabilities may cause problems inapplications.

1. Familiar rules of probability theory may no longer apply forcapacities.For example, Dow and Werlang (1994) show that thesupport of a capacity may not be unique. For additive probabilities,the support may be defined either asthe smallest subset of the statespace with probability1 or the smallest subset with a complementof probability0. For non-additive probabilities, however, these two

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110 JÜRGEN EICHBERGER AND DAVID KELSEY

definitions are no longer equivalent. Moreover, whichever defini-tion one may choose, in general, the support of a capacity will notbe unique. Haller (1996) and Marinacci (1996) therefore suggestanother notion of a support for a capacity which is unique. Fortheir definition, however, the complement of the support may havepositive measure, and many capacities will have the empty set assupport.

Updating beliefs on new information is also no longer as straight-forward for capacities as it is for additive probabilities, since Bayes’rule is not the only update rule for non-additive probabilities. Gilboaand Schmeidler (1993) have axiomatised a number of update rules.For belief functions, a special class of capacities, they show thatthe Dempster–Shafer rule for updating capacities coincides with themaximum likelihood update. However, in contrast to the Bayesianupdate rule for additive probabilities, most update rules for non-additive probabilities, including the Dempster–Shafer rule, are notdynamically consistent. Indeed, Eichberger and Kelsey (1996a)show1 that preferences that are represented by a Choquet integralwith respect to a non-additive probability will have dynamicallyconsistent updates that cannot be represented by a Choquet integralwith respect to a single updated capacity. In economic and game-theoretic applications, it would be difficult to analyze dynamic de-cision making if the form of representation were not maintainedduring updating.

2. Consistency requirements in equilibrium concepts need to be re-considered.In traditional game theory, a Nash equilibrium ensuresconsistency between beliefs and actual behaviour by identifyingmixed strategies that players choose and beliefs that other playershold about their opponents’ behaviour. If beliefs are represented bynon-additive probabilities, this consistency requirement is no longerfeasible. Dow and Werlang (1994) therefore propose an equilib-rium concept for two-player games when players’ beliefs are non-additive.2 Eichberger and Kelsey (1997a) extend their equilibriumnotion to then-player case.

Representing beliefs by general non-additive probabilities weak-ens the restrictions imposed on behaviour by equilibrium consis-tency requirements. Indeed, as Eichberger and Kelsey (1997a) show,

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E-CAPACITIES AND THE ELLSBERG PARADOX 111

there is little behaviour that can be ruled out in equilibrium, if gen-eral capacities are allowed to represent beliefs of the players. Thus,it appears reasonable and desirable to impose further restrictions onthe class of non-additive probabilities that one applies to economicand game-theoretic models.

The following five properties appear to be especially useful ineconomic and game-theoretic applications:

• The class of non-additive probabilities should be parameter-ized such that additive probabilities correspond to a particularparameter value.

• The parameters involved should have a behavioural interpreta-tion.

• Information about the objective probability of some eventsshould be easily incorporated.

• Updating on new information should leave the class of capaci-ties invariant.

• The support of the non-additive probability should be unique.

A parameterization of capacities which allows for an economic in-terpretation is necessary to make meaningful comparative static re-sults in economic models possible. Eichberger and Kelsey’s (1997c)analysis of the provision of public goods provides an example ofhow to use a parameterized version of a capacity to derive compar-ative static properties of a model. In economic modeling one oftenassumes that agents have prior information about the probability ofsome events, e.g., a type-distribution of the opponents. Such objec-tive information has to be combined with subjective beliefs of agentsthat one wishes to model by capacities. The class of capacities usedin economic models should therefore be capable of incorporatingpartial knowledge of an agent. The last two desiderata are of a moretechnical nature and have been discussed before.

The class of non-additive probabilities that we propose in thispaper will satisfy these five desiderata. Moreover, the Choquet ex-pected utility with respect to a capacity from this class is the pref-erence representation suggested by Ellsberg (1961). For this reason,we will call non-additive probabilities from this classE(llsberg)-capacities.

In Section 1 we introduce some notation and present the rep-resentation of preferences suggested by Ellsberg (1961). Section 2

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112 JÜRGEN EICHBERGER AND DAVID KELSEY

then defines and studies the class of non-additive probabilities thatwe callE-capacities. Here, we will show that the Choquet integralof an E-capacities is the Ellsberg representation. Section 3 providesan axiomatisation of preferences where the decision maker’s be-liefs can be represented by an E-capacity, and Section 4 deals withupdating of E-capacities.

1. ELLSBERG’S REPRESENTATION

Ellsberg’s (1961) argument was cast in terms of the formal frame-work of Savage (1954). Later in this paper, however, we will presentan axiomatisation of preferences for which beliefs can be repre-sented by E-capacities. This axiomatisation uses Schmeidler’s (1989)approach, which is based on the Anscombe–Aumann (1963) frame-work. Hence, our exposition of the Ellsberg representation will usethe Anscombe–Aumann (1963) framework.

Consider a finite set of states of natureS. A subset ofS will becalled anevent. The set of outcomes (or consequences) is denotedbyX and the set of all finitely supported probability distributions onX is 1(X). An action is a functiona : S → 1(X) and the set ofactions is denoted byA(S). A preference relation� is defined onthe set of actionsA(S).

For anyλ ∈ [0, 1] and any two actionsa, b ∈ A(S), λ · a +(1− λ) · b denotes the action which associates with states ∈ S theprobability distributionλ · a(s)+ (1− λ) · b(s) ∈ 1(X).REMARK. Savage (1954) defined an action as a mappinga : S →X. Let A(S) be the set of such actions ordered by�. His axiomsimply the existence of

• a utility functionu :X→R, representing risk preferences, and• an additive probability distributionπ on S, representing be-liefs,

such thatV (a) = ∑s∈S u(a(s)) · π(s) represents the preference3

relation� on A(S). 2The axioms of Anscombe and Aumann (1963), applied to the pref-erence relation� onA(S), imply the existence of

• an expected utility functionalU : 1(X)→ R, and

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E-CAPACITIES AND THE ELLSBERG PARADOX 113

• a probability distributionπ onS,

such thatV (a) = ∑s∈S U(a(s)) · π(s) represents the preference

relation� onA(S).Note that, for alls ∈ S, a(s) is a probability distribution onX that

assigns positive probabilitya(x; s) to a finite number of outcomesin X, and

U(a(s)) :=∑x∈X

u(x) · a(x; s)

is an expected utility functional. Identifying an outcomex with theprobability distribution concentrated onx, δx , one can viewA(S) asa subset ofA(S) and apply the Anscombe–Aumann representationwith U(δx) ≡ u(x).

Ellsberg (1961, p. 657) interprets the observed deviations fromthe Savage axioms of the participants in his experiment as follows:

Responses of confessed violators [of the Savage axioms] indicate that the differ-ence is not to be found in terms of the two factors commonly used to determinea choice situation, the relative desirability of the possible pay-offs and the rela-tive likelihood of events affecting them, but in a third dimension of the problemof choice: the nature of one’s information concerning the relative likelihood ofevents. What is at issue might be called theambiguityof this information, a qualitydepending on the amount, type, reliability and ‘unanimity’ of information, andgiving rise to one’s degree of ‘confidence’ in an estimate of relative likelihoods.

Hence, a representation of preferences should include not just prob-ability assessments for events but also a measure for thedegree ofconfidencethat an individual has in regard to the assessment. Inparticular, as Ellsberg’s urn experiment shows, an individual mayfeel more confident about the probability of some events than aboutthe probabilities of the others.

Based on such considerations, Ellsberg (1961, pp. 664, 665) pro-poses the following representation for preferences onA(S). Denoteby π the additive probability distribution that the decision makerconsiders as the best estimate and let5◦ be the set of additiveprobability distributions that “his information – perceived as scanty,unreliable,ambiguous– does not permit him confidently to rule out”(Ellsberg, 1961, p. 661). Then the evaluation of an actiona ∈ A(S)

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114 JÜRGEN EICHBERGER AND DAVID KELSEY

is

V (a) := % ·∑s∈S

u(a(s)) · π(s)

+ (1− %) ·min

{∑s∈S

u(a(s)) · π ′(s)|π ′ ∈ 5◦}.

Ellsberg (1961, p. 664) calls the parameter% ‘degree of confidence’.The following slight modification of an example in Ellsberg (1961,p. 665) shows that this representation can accommodate the Ellsbergparadox.

EXAMPLE 1.1. Without loss of generality assume that the deci-sion maker is risk-neutral,u(a(s)) = a(s). Assume further that thedecision maker’s best estimate of a probability distribution for thenumber of red, black and yellow balls is (1/3, 1/3, 1/3). Given theinformation about the number of balls however, one cannot excludeany probability distribution that is a convex combination of (1/3,2/3, 0) and (1/3, 0, 2/3). Hence,

Y ◦ := {(πR, πB, πY) ∈ R3+|πR = 1/3, πB + πY = 2/3}.Using the Ellsberg representation, one easily computes:

V (A) = (100/3), V (B) = % · (100/3),

V (C) = (1+ %) · (100/3), V (D) = % · (200/3).

For % < 1, one obtains the ranking of these lotteries that was mostcommonly observed in the Ellsberg experiment. Note that even asmall deviation from additivity, i.e.,% = 1− ε for a smallε, sufficesto accommodate the Ellsberg paradox. 2The example shows that adegree of confidenceof less than onein the best additive probability assessment can induce the choicebehaviour observed in the Ellsberg paradox. On the other hand, adegree of confidence equal to one leads to behaviour consistent witha representation of beliefs by an additive probability. Ellsberg (1961,p. 668) interprets the degree of confidence as a parameter charac-terizing a decision maker’s information. By repeated sampling, forexample, a decision maker may grow more confident of his best

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E-CAPACITIES AND THE ELLSBERG PARADOX 115

additive probability assessment and behave like an expected utilitymaximizer with beliefs represented by an additive probability.

While making an excellent case for representing the degree ofambiguityby the parameterdegree of confidence, Ellsberg (1961)remains unclear about the precise relationship between the additiveprobability distribution that serves as a best estimate,π , and theprobability distributions contained in the set of possible probabilitydistributions5◦. In addition, to the best of our knowledge, there isno axiomatic characterization of the Ellsberg representation. Section3 of this paper will provide an axiomatic foundation for the Ellsbergrepresentation.

2. E-CAPACITIES AND THE ELLSBERG REPRESENTATION

Consider the Anscombe–Aumann framework introduced at the be-ginning of Section 1. Beliefs about events will be represented bynon-additive probabilitiesor capacities. A capacityassigns proba-bility weights to eventsE ⊆ S that arenon-additive.

DEFINITION 2.1. CapacityA capacity onS is a real-valued func-tion ν on the subsets ofS which satisfies the following properties:

(i) A ⊆ B implies ν(A) 6 ν(B), monotonicity

(ii) ν(∅) = 0 and ν(S) = 1. normalization.

The capacity is calledconvexif, for all eventsA andB,

ν(A)+ ν(B) 6 ν(A ∪ B)+ ν(A ∩ B)holds.

A general capacity is a monotone measure that is normalized to 1on the full set and 0 on the empty set. Otherwise, a capacity haslittle structure. In particular, the sum of the capacities of two subsetsmay exceed 1. Or, the sum of the capacities of two subsets maybe strictly less than the capacity of the union of these sets. Thefollowing example provides some illustrations.

EXAMPLE 2.1. Capacities

(i) S = {s1, s2, s3}, ν({si}) = 1/2 for i = 1,2,3;ν({si, sj }) = 2/3 for i, j = 1,2,3, i 6= j.

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116 JÜRGEN EICHBERGER AND DAVID KELSEY

(ii) S = {s1, s2, s3}, ν({si}) = 0 for i = 1,2,3;ν({si , sj }) = 1/3 for i, j = 1,2,3, i 6= j.

(iii) S={s1, s2, s3}, ν({si})=1/4 ν({s2})=1/3, ν({s3})=1/2;ν({si, sj }) = 1 for i, j = 1,2,3, i 6= j.

It is easy to check that only capacity (ii) is convex. 2Convexity of capacities is a property suggested by the Ellsberg para-dox where the belief of the unambiguous event of drawing a blackor a yellow ball strictly exceeded the sum of the beliefs for theambiguous events that a black ball is drawn or that a yellow ballis drawn.

A special class of capacities, calledbelief functions, generalizesthe notion of a convex capacity. Non-additive belief functions willgive probability weight to events that cannot be completely dividedup between its subsets. Belief functions have been studied by Shafer(1990), Jaffray (1992, 1989) and others. Furthermore, Ghirardato(1994) and Mukerji (1997a) provide epistemic justifications for be-lief functions.

DEFINITION 2.2. Belief function.A capacity is a belief functionif, for all k = 1, 2, . . . ,

ν

(k⋃i=1

Ai

)>

∑I⊆{1,2,... ,k}

(−1)|I |+1 · ν(⋂i∈I

Ai

)for all Ai ⊆ S.

Additive probabilities are belief functions. Furthermore, Denneberg(1995) shows that all convex capacities that associate only 0 and1 with events are belief functions. Capacity (ii) in Example 2.1 isa belief function. Note that a belief function is always a convexcapacity.

Example 2.1 also illustrates that the support of a capacity is nolonger such a straightforward notion as in the case of an additiveprobability. For additive probabilities, one may define the supportas the smallest set with probability 1 or as the smallest set with acomplement of probability 0. For non-additive probabilities, how-ever, it is no longer true that the complement of the smallest set

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E-CAPACITIES AND THE ELLSBERG PARADOX 117

with probability 1 has probability zero (see Example 2.1 (iii)). Dowand Werlang (1994) provide a thorough discussion of this issue andsuggest the smallest set with a complement of measure 0 as the moreappropriate notion for the support of a capacity. We follow theirargument, because, for the class of capacities that we are going tosuggest, the set of all states is the only set with full measure.

DEFINITION 2.3. Support of a capacity. Thesupport of a capacityν is a setE ⊆ S such that

(i) ν(S\E) = 0, and

(ii) ν(F ) > 0 for allF that strictly containS\E.Example 2.1 (ii) shows that the support of a capacity need not beunique. In this case, all sets{si, sj }, for i, j = 1, 2, 3 andi 6= j ,satisfy the definition of a support.

In order to obtain a concept of a weighted sum for a real-valuedfunction onS, we introduce the following notation.

NOTATION. For any functionf : S → R, let

f 1 := max{f (s)|s ∈ S} and

Q1 := argmax{f (s)|s ∈ S}.Let T1 = Q1 and, fork > 2, define iteratively,Tk := Tk−1 ∪Qk

f k := max{f (s)|s ∈ S\Tk−1} and

Qk := argmax{f (s)|s ∈ S\Tk−1}.SinceS is finite, there must ber such thatTr = S. 2With the conventionQ0 = ∅, one can now define the Choquetintegral as a weighted average of a functionf .

DEFINITION 2.4. Choquet integral. The Choquet integralon fwith respect to the capacityν is:∫

f dν :=r∑k=1

f k · [ν(Tk)− ν(Tk−1)].

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118 JÜRGEN EICHBERGER AND DAVID KELSEY

Schmeidler (1989) suggests a set of axioms that imply a representa-tion of the preference ordering� onA(S) by the Choquet integralof an expected utility functionU : 1(X) → R with respect to anon-additive measureν, V (a) = ∫ U ◦a dν. In Schmeidler’s (1989)axiomatisation,ν is unique.

E-capacitiesrepresent a simple version of a capacity that allows oneto include information about ‘known’ probabilities of events.

Let {E1, . . . , En} be a partition ofS with ‘known’ probabilitiesp(Ei),

∑ni=1 p(Ei) = 1. Events of this partition are calledunam-

biguous. In the Ellsberg example, it was known that there were 30red and 60 black and yellow balls in the urn. The probabilities forthe events {red} and {black or yellow} are therefore known to be1/3 and 2/3, respectively. Note that these two sets from a partitionof the state space in the Ellsberg example. All other partitions areambiguous.

The following definition characterizes additive probability distri-butions that areinformation consistent. Denote the set of all additiveprobability distributions onS by1(S).

DEFINITION 2.5. Information consistent probabilities. The set ofinformation consistent additive probabilitiesis:

5(p) :=π ′ ∈1(S)|∑

s∈Eiπ ′(s)=p(Ei), i=1, . . . , n

.Finally, for i = 1, . . . , n, let

βi(A) ={

1 if Ei ⊆ A0 otherwise

be a function that indicates events that contain an unambiguousevent. Note that this function can be viewed as a capacity onS,called anelementary belief functionin the literature. It is straight-forward to show thatβi is indeed a belief function.

Given an information consistent probability distribution, the as-sessment of the decision maker, we can define an E-capacity asfollows.

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E-CAPACITIES AND THE ELLSBERG PARADOX 119

DEFINITION 2.6. E-capacity/assessment/degree of confidence. TheE-capacityν(π, %) based onassessmentπ ∈ 5(p) with degree ofconfidence% ∈ [0, 1] is defined by

ν(A|π, %) :=n∑i=1

[% · π(A ∩Ei)+ (1− %) · p(Ei) · βi(A)]

for all A ⊆ S.

An E-capacity can be viewed as a convex combination of the addi-tive probability distributionπ and the capacityµ, defined as

µ(A) :=n∑i=1

p(Ei) · βi(A), for all A ⊆ S.

Once again, it is easy to check thatµ satisfies all the conditions of acapacity.

Additive probability distributions appear as special cases of E-capacities:

(i) if the degree of confidence% equals one, and(ii) if the information partition contains only single element sets.

In the first case there is ambiguity about events but the individualfeels subjectively certain that the assessmentπ is correct. In the sec-ond case, there are no ambiguous events, i.e., there is no Knightianuncertainty and the degree of confidence does not matter.

Several important special cases of E-capacities have already beenapplied to model uncertainty in economics and game theory. Dowand Werlang (1992) usesimple capacitiesto derive their result onportfolio choice. Mukerji (1997b) uses E-capacities to investigateoptimal contracts under uncertainty and Eichberger and Kelsey(1997b) apply E-capacities to model signaling games.

A simple capacityis an E-capacity with an information partition{∅, S},

ν(A|π, %) ={% · π(A) for all A ⊂ S1 forA = S.

It can be viewed as a contraction of an additive probability distri-bution, except for the full set. Given the information partition of a

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120 JÜRGEN EICHBERGER AND DAVID KELSEY

simple capacity, the set of information consistent assessments is theset of all additive probability distributions onS,1(S).

Simple capacities are especially useful for modeling uncertainty-averse behaviour in games where players have no prior informationabout the probability of events. In games, the degree of confidence%

can be taken as an exogenous characteristic of a player. The informa-tion consistent assessmentπ can then be determined endogenouslyin equilibrium. This allows one to interpret% as a player’s degreeof confidence in the game-theoretic equilibrium concept. Simple ca-pacities have the additional advantage that one can define ‘indepen-dent’ beliefs using the additive probability assessmentπ . Eichbergerand Kelsey (1997a) study games in which players’ uncertainty ismodeled by simple capacities.

A second important application of E-capacities occurs in situa-tions where a decision maker has a payoff function that dependsjointly on a variable for which objective probabilities are knownand on a variable with complete uncertainty. In such a case, the statespace can be written as a Cartesian productS = R × T with a‘known’ additive probability for events inT , sayq. The informa-tion partition in this case consists of the set{R × {t}|t ∈ T } withprobabilitiesp(R × {t}) := q(t). Such situations arise, for exam-ple, in signaling games where players have information about theiropponents’ types in form of an additive probability distribution butface uncertainty in regard to their opponents’ behaviour. Signalinggames have been studied in Eichberger and Kelsey (1997b).

The following general properties of E-capacities are useful ineconomic and game-theoretic applications.

LEMMA 2.1. E-capacities are belief functions.Proof.Since additive probabilities are belief functions, one has

π

(k⋃i=1

Ai

)>

∑I⊆{1,2,... ,k}

(−1)|I |+1 · π(⋂i∈I

Ai

)for all Ai ⊆ S

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E-CAPACITIES AND THE ELLSBERG PARADOX 121

and, sinceβj are elementary belief functions,

βj

(k⋃i=1

Ai

)>

∑I⊆{1,2,... ,k}

(−1)|I |+1 · βj(⋂i∈I

Ai

)for all Ai ⊆ S

for all j = 1, . . . , n.Straightforward calculation confirms for arbitraryAi ⊆ S,

ν

(k⋃i=1

Ai

)−

∑I⊆{1,2,... ,k}

(−1)|I |+1 · ν(⋂i∈IAi

)

=% · π ( k⋃

i=1

Ai

)+(1−%) ·

n∑j=1

p(Ej ) · βj(

k⋃i=1

Ai

)−−

∑I⊆{1,2,... ,k}

(−1)|I |+1 ·[% · π

(⋂i∈I

Ai

)+

+ (1− %) ·n∑j=1

p(Ej ) · βj(⋂i∈IAi

)= % ·

π ( k⋃i=1

Ai

)+

∑I⊆{1,2,... ,k}

(−1)|I |+1 · π(⋂i∈I

Ai

)−− (1− %) ·

n∑j=1

p(Ej) ·[βj

(k⋃i=1

Ai

)−

−∑

I⊆{1,2,... ,k}(−1)|I |+1 · βj

(⋂i∈I

Ai

)> 0.

2An immediate implication of Lemma 2.1 is that E-capacities areconvex.

LEMMA 2.2. For % > 0, E-capacities have a unique support.

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122 JÜRGEN EICHBERGER AND DAVID KELSEY

Proof.We will show that, for allA ⊆ S, ν(A) = 0 if and only ifπ(A) = 0. Hence, the support ofν equals the support of the additiveprobabilityπ . Uniqueness of the support ofν then follows from theuniqueness of the support of an additive probability.

Omitting the argumentsπ and%, it follows from the definitionof an E-capacity that, for any subsetA ⊂ S, ν(A) = 0 impliesπ(A) = 0. For someA ⊆ S, suppose thatπ(A) = 0. Hence,π(A ∩ Ei) = 0 for all i = 1, . . . , n. Sinceπ is an informationconsistent probability distribution,p(Ei) = 0 for A ∩ Ei 6= ∅andEi ⊆ A. Otherwise,βi(A) = 0. Thus,π(A) = 0 impliesν(A) = 0. 2Schmeidler’s (1989) result shows that preferences on the set of ac-tionsA(S) can be represented by an expected utility functional withrespect to a capacity. If beliefs are represented by an E-capacity,then the associated expected utility (the Choquet integral) has aparticularly simple form.

DEFINITION 2.7. Minimal expected utility on unambiguous events.The minimal expected utilityof action aon an unambiguous eventEi is defined asmi(a) := min{U(a(s))|s ∈ Ei}.The following lemma is needed in the proof of the main proposi-tion in this section. It is of more general interest, however, since itsimplifies many computations in applied work.

LEMMA 2.3. For any actiona ∈ A(S), the Choquet integral hasthe following form:∫

U ◦ a dβi = mi(a).

Proof. Denote the composition of the functionsU anda by f ,f := U ◦ a. Applying the definition of a Choquet integral, we have∫

f dβi =r∑k=1

f k · [βi(Tk)− βi(Tk−1)].

Recall thatTk = {s ∈ S|f (s) > f k}, k = 1, . . . , r. f 1 > f 2 >

· · · > f r implies thatT1 ⊂ T2 ⊂ · · · ⊂ Tr ≡ S. Hence, there must

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E-CAPACITIES AND THE ELLSBERG PARADOX 123

be exactly onek such thatEi is a subset ofTk but not ofTk−1. Thisimpliesβi(Tk) = 1 for all k > k andβi(Tk) = 0 for all k < k. Thus,we have

∫f dβi = f k. SinceEi is a subset ofTk, f (s) > f k for all

s ∈ Ei follows. Hence,f k 6 min{f (s)|s ∈ Ei}. SinceEi is not asubset ofTk−1, there must bes ∈ Ei such thatf k−1 > f (s). Hence,

f k = min{f (s)|s ∈ Ei}. 2The following characterization of the Choquet integral for an E-

capacity is now easily proved.

PROPOSITION 2.1.Expected utility with an E-capacity. The ex-pected utility of an action a with respect to the E-capacityν(π, %)is

V (a|π, %) := % ·[∑s∈S

π(s) · U(a(s))]

+ (1− %) ·[

n∑i=1

p(Ei) ·mi(a)].

Proof. An E-capacityν(π, %) can be written as a linear com-bination of the additive probabilityπ and the capacitiesβi , i =1, . . . , n:

ν(π, %) := % · π +n∑i=1

(1− %) · p(Ei) · βi.

For positive coefficients the Choquet integral is linear in the capac-ity. Hence,

∫U ◦ a dν(π, %) = % · U ◦ a dπ + ∑n

i=1 (1 − %) ·p(Ei) ·

∫U ◦ a dβi . Sinceπ is additive, we have

∫U ◦ a dπ =∑

s∈S π(s) ·U(a(s)) and, from Lemma 2.3,∫U ◦a dβi = mi(a).2

Proposition 2.1 shows that the expected utility of an action withrespect to an E-capacity is a convex combination of the expectedutility with respect to the additive probability distributionπ and aterm which is easily recognized as the minimum expected utility

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124 JÜRGEN EICHBERGER AND DAVID KELSEY

over all information consistent probability distributions,n∑i=1

p(Ei) ·mi(a)=min

{∑s∈S

U(a(s)) · π ′(s)|π ′ ∈5(p)}.

Hence, the expected utility of actiona with respect to the E-capacityν(π, %), V (a|π, %), is the Ellsberg representation discussed in Sec-tion 1. Example 1.1 shows that the Ellsberg paradox can be accom-modated by preferences that are represented by a Choquet integralwith respect to an E-capacity. Moreover, the intuitive interpretation,put forward so convincingly in Ellsberg (1961), applies directly tothis approach.

3. AXIOMATISATION OF E-CAPACITIES

In this section, we will provide an axiomatisation of E-capacitiesin the Anscombe–Aumann framework used in Schmeidler (1989).Since E-capacities are a special type of capacity, it will come asno surprise that most axioms of the Schmeidler approach will bemaintained. In contrast to the latter paper however, we require theindependence axiom to hold for a larger set of acts which strictlyincludes comonotonic acts. Since we strengthen Schmeidler’s ax-ioms we are able to make stronger predictions about the form ofpreferences.

To simplify notation,pS will denote the action with a constantoutcome probability distributionp ∈ 1(X). The following axiomsare well-known and discussed in Schmeidler (1989).

AXIOM 3.1. Weak order. For alla, b, c ∈ A(S),(i) a � b or b � a;(ii) a � b and b � c implies a � c.

The preference relation� onA(S) induces a preference order on theset of probability distributions onX,1(X), denoted by� as well:

for all f, g ∈ 1(X), f � g if and only if fS � gS.AXIOM 3.2. Continuity. For alla, b, c ∈ A(S)with a � b andb �c, there existα ∈ (0, 1) andβ ∈ (0, 1) such thatαa+ (1− α)c � bandb � βa + (1− β)c.

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E-CAPACITIES AND THE ELLSBERG PARADOX 125

AXIOM 3.3. Weak dominance. For alla, b ∈ A(S),if a(s) � b(s) for all s ∈ S, then a � b.

AXIOM 3.4. Uncertainty aversion.For alla, b ∈ A(S) and allλ ∈[0, 1],

a � b implies λa + (1− λ)b � b.

Forming a convex combination over two acts makes the lotteries ineach state less extreme. Though it does not change the uncertaintyabout the states, it reduces the importance of uncertainty. Hence,for an uncertainty averse decision maker, a convex combination oftwo acts should be at least as good as the worse of the two actsfrom which it is formed. This is the notion of uncertainty aversionof Axiom 3.4.

The following two axioms will replace the co-monotonicity ax-iom of Schmeidler (1989). Note that E-capacities are characterizedby the fact that they are additive on the unambiguous events andthat preferences on actions which have the same minimum valueon each element of the unambiguous partition can be representedby an additive probability distribution. The following two axiomswill therefore require the independence axiom for these two typesof actions.

DEFINITION 3.1. An actiona ∈ A(S) is said to bemeasurablewith respect to the (unambiguous) partitionE1, . . . , En) if thereexists noEi, i = 1, . . . , n such thata(s) 6= a(s ′) for somes, s ′ ∈Ei. LetAM(S) denote the set of actions which are measurable withrespect to the unambiguous partition.

AXIOM 3.5. Unambiguous independence.For alla, b, c ∈ AM(S)and allλ ∈ [0, 1],

a � b implies λa + (1− λ)c � λb + (1− λ)c.

Unambiguous independence captures the notion that deviations fromexpected utility theory with additive probabilities are due to thepresence of ambiguity, by allowing the independence axiom to apply

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126 JÜRGEN EICHBERGER AND DAVID KELSEY

to actions inAM(S). It seems not unreasonable to assume that thereis no ambiguity involved in the evaluation of actions that yield aconstant outcome on each element of the unambiguous partition.Thus, Axiom 3.5 requires preferences for known probabilities tosatisfy the expected utility hypothesis.

AXIOM 3.6. Extremal independence. For all a, b, c ∈ A(S) suchthat, for i = 1, . . . , n, there existσi ∈ Ei with a(si) � a(σi),b(si) � b(σi), c(si) � c(σi) for all si ∈ Ei, and allλ ∈ [0, 1]

a � b implies λa + (1− λ)c � λb + (1− λ)c.Axiom 3.6 is stronger than comonotonic independence, because itapplies the independence axiom to a larger class of acts. Essentiallyit says that the usual independence axiom applies to actions whoseminima on the unambiguous events occur in the same states. Onereason for preferring a convex combination of two actions to eitherof the original actions is that the minimum of the convex combina-tion may be higher. This is illustrated by the following example.

EXAMPLE 3.1. Consider the following table with two actions andthe convex combination of these actions:

s1 s2 s3 mina 1 2 0 0

b 2 0 1 012a + 1

2b 1.5 1 0.5 0.5 2This cannot happen if the minima of the original actions occur inthe same state. Hence it seems more reasonable to apply the inde-pendence axiom in the latter case.

In the example above, the only unambiguous event was the wholespace. Suppose now there is a partition ofS, (E1, . . . , En), whichconsists of unambiguous events. Consider three actionsa, b andc such that for eachEi, i = 1, . . . , n, the minima ofa, b and coccur in the same state. The above argument should then be ex-tended to these actions because it applies conditional on any of theseunambiguous eventsEi occurring. This is what Axiom 3.6 requires.

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E-CAPACITIES AND THE ELLSBERG PARADOX 127

PROPOSITION 3.1.Assume that there are at least four states. Sup-pose that the preference relation� on A(S) satisfies Axiom 3.1(weak order), Axiom 3.2(continuity), Axiom 3.3(weak dominance),Axiom 3.4(uncertainty aversion), Axiom 3.5(unambiguous indepen-dence)and Axiom 3.6(extremal independence), then there exists anadditive probability distributionπ onS, real numbers%1, . . . , %n ∈[0, 1] and an expected utility functionU : 1(X)→ R such that

n∑i=1

∑s∈Ei

%i · π(s) · U(a(s))+ (1− %i) · p(Ei) ·mi(a)

represents these preferences.

The proof of this proposition requires four lemmas and is given inthe Appendix. The Choquet integral with respect to an E-capacityderived in Section 2 is easily recognized as the special case whereall the%i are equal,%i = % for all i = 1, . . . , n.

The representation obtained in Proposition 3.1 is more generalthan the Choquet integral of an E-capacity because it allows the de-gree of confidence to vary across the unambiguous partition. This isan advantage in terms of the larger set of acts to which one can applythe representation. Since beliefs about ambiguous events can be heldwith differing degrees of confidence, one can consider cases wherea decision maker has differential information about the ambiguousstates in different unambiguous events. The representation of beliefsby an E-capacity corresponds to the special case of a common de-gree of confidence for the probability assessment of the ambiguousstates.

4. UPDATING OF E-CAPACITIES

Gilboa and Schmeidler (1993) have axiomatised various updatingrules for capacities. All of these rules equal Bayesian updating ifthe capacities are additive. They show in particular that only a verysmall class of updating rules yields updates that can be representedagain by a single capacity (Theorem 3.2). Among these theDempster–Shafer ruleapplied to belief functionsis equivalent tothe maximum likelihood principle (Theorem 3.3). As Camerer and

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128 JÜRGEN EICHBERGER AND DAVID KELSEY

Weber (1992, p. 352) note: “The equivalence of maximum like-lihood (which is widely used in classical statistics) gives a novelunderpinning to the Dempster–Shafer rule.”

E-capacities are belief functions and updating them by theDempster–Shafer rule corresponds to updating according to the max-imum likelihood principle. Dempster–Shafer updating guaranteesalready that the updated preferences can be represented again by aChoquet integral. We will show however that the Dempster–Shaferupdate of an E-capacity is itself an E-capacity. This is a particularlyuseful property in economic applications.

Denoting by¬A the complement of a setA, the Dempster–Shaferupdating rule is defined as follows.

DEFINITION 4.1. Dempster–Shafer updating.

ν(A|E) := ν((A ∩ E) ∪ (¬E))− ν(¬E)1− ν(¬E) .

Note that the Dempster–Shafer updating rule coincides with Bayesianupdating ifν is additive. For an additive capacityν, ν((A ∩ E) ∪(¬E)) = ν(A ∩ E) + ν(¬E) and ν(E) + ν(¬E) = 1. Hence,ν(A|E) := ν(A∩E)/ν(E) is the usual Bayesian updating formula.

Consider an E-capacity based on the partition{E1, . . . , En} withknown probabilitiesp:

ν(A|π, %) :=n∑i=1

[% · π(A ∩ Ei)+(1−%) · p(Ei) · βi(A)].

To simplify notation, the reference to the information-consistent as-sessmentπ and the degrees of confidence%will be suppressed in theremainder of this section. Furthermore, for any two setsA,B ⊆ S,denote by

Z(A,B) := {Ei|Ei ∩ (B\A) = ∅, Ei ∩ B 6= ∅}the (possibly empty) set of unambiguous eventsEi that intersectBbut notB\A and let

Z(A, B) :=⋃

Ei∈Z(A,B)Ei

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E-CAPACITIES AND THE ELLSBERG PARADOX 129

the union of the sets inZ(A,B). With this notation, it is easy to seethat the Dempster–Shafer updates of an E-capacity are E-capacities.

LEMMA 4.1. The Dempster–Shafer updates of an E-capacity havethe following form:

ν(A|B) = % ·∑s∈A∩B

π(s)+ (1− %) ·∑

Ei∈Z(A,B)p(Ei) (4.1)

with

π (s) := π(s)/π(B) for all s ∈ B,p(Ei) :=p(Ei)/p(Z(B, B)) for all Ei ∈ Z(B,B),

and

% := % · π(B)

% · π(B)+ (1− %) · p(Z(B, B)).

Proof. For any two eventsA,B ⊆ S, a straightforward calcula-tion shows that

ν(A ∩ B ∪ ¬B)− ν(¬B) = % ·∑s∈A∩B

π(s)

+ (1− %) · ∑Ei⊆A∩B∪¬B

p(Ei)−∑

Ei⊆¬Bp(Ei)

= % ·

∑s∈A∩B

π(s)+ (1− %) ·∑

Ei∈Z(A,B)p(Ei).

Noting that, forA = B, ν(A ∩ B ∪ ¬B) = 1, one obtains

ν(A|B) =% ·

∑s∈A∩B

π(s)+ (1− %) ·∑

Ei∈Z(A,B)p(Ei)

% ·∑s∈B

π(s)+ (1− %) ·∑

Ei∈Z(B,B)p(Ei)

.

Using the definitions for%, π(s), andp(Ei), completes the proof.2Lemma 4.1 shows that Dempster–Shafer updates of E-capacities

are again E-capacities. Hence the same representation of beliefs can

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130 JÜRGEN EICHBERGER AND DAVID KELSEY

be used after receiving new information. Note that the degree ofconfidence will be adjusted too. However, learning can only increaseambiguity. The degree of confidence remains unchanged only if thedecision maker learns that an unambiguous event has happened.This corresponds to the intuition that nothing has been learned inregard to the unambiguous events. On the other hand, if the new in-formation does not confirm an unambiguous event, then uncertaintyincreases. Reconsidering the Ellsberg example illustrates this point.

EXAMPLE 4.1.4 Reconsider the Ellsberg example (Example 2.1)and assume that the decision maker learns that the ball that hasbeen drawn isnot red. Hence, the event{B, Y }must have happened.Applying formula 4.1 yields:

% = %, π(B) = π (Y ) = 0.5, p({B, Y }) = 1, and

ν(B|{B, Y }) = ν(Y |{B, Y }) = %/2.In this case, no new information about the ambiguous events hasarrived. Hence, the degree of confidence remains unchanged,% = %,andπ is updated in the Bayesian way.

In contrast, suppose the ball drawn isnot yellow. Formula 4.1now yields:

% = % · 2

3− %, π(B) = π (R) = 0.5, p({R}) = 1

3,

p({B, Y }) = 2

3and

ν(B|{B,R}) = 2− %3− %, ν(R|{B,R}) = 1

3− % .

If the decision maker is initially uncertainty averse,% < 1, thenlearning thatY did not occur increases ambiguity:% > %. Note that,in case of updating on(B, Y ), the updated beliefs are represented bya simple capacity, since the information partition is now{(B, Y ), ∅}.On the other hand, updating on(B, R) leaves the information parti-tion unchanged. 2

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E-CAPACITIES AND THE ELLSBERG PARADOX 131

Updating of capacities on a product space

An important application of E-capacities are games of incompleteinformation where players have to form expectations about an op-ponent’s strategy and type. In this context, it is often assumed thatplayers have common knowledge of a prior distribution over types.Observing the strategy choice of the opponent provides a signal toupdate the prior beliefs about types. Eichberger and Kelsey (1997b)analyze signaling games and provide generalizations of the notionsof Bayesian Equilibrium and Perfect Bayesian Equilibrium for thecase where players’ beliefs are represented by non-additive proba-bilities.

The following example defines an E-capacity for the case of aproduct space, where there is an additive probability on one spaceand a non-additive probability on the other, and derives its Dempster–Shafer updates.

EXAMPLE 4.2. Consider beliefs over a product spaceS = M×T .In the context of games of incomplete information,T could be theset of types of an opponent andM may be the set of feasible actions.Suppose the decision maker knows that the probability distributionoverT is (p(1), . . . , p(n)),

∑ni=1p(t) = 1. Beliefs over the prod-

uct spaceS = M × T will be represented by an E-capacity that isconsistent with the known marginal distribution over types.

Let Mt = M × {t}, t ∈ T , be the unambiguous partition ofS = M × T with probability distributionp(t), t ∈ T . Given aninformation-consistent additive measureπ and a degree of confi-dence%, one can define an E-capacity in the usual way:

ν(A) :=∑t∈T[% · π(A ∩Mt)+ (1− %) · p(t) · βt(A)]

for anyA ⊆ M × T . Note that, for any subset of typesB ⊂ T ,

ν(M × B) =∑t∈T[% · π((M × B) ∩Mt)

+ (1− %) · p(t) · βt(M × B)] =∑t∈B

p(t).

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132 JÜRGEN EICHBERGER AND DAVID KELSEY

Now suppose that the decision maker learns that the eventC ⊂ Mhas occurred. The Dempster–Shafer update for an eventB ⊆ T thentakes the following form.

LEMMA 4.2. Dempster–Shafer update.

ν((M × B)|(C × T )) == % ·

∑s∈C×B π(s)+ (1− %) ·

∑t∈B p(t)

% ·∑s∈C×T π(s)+ (1− %). (4.2)

Proof.Using the updating formula for an E-capacity (4.1), we get

ν((M × B)|(C × T )) = % ·∑

s∈(M×B)∩(C×T )π(s)

+ (1− %) ·∑

Mt∈Z((M×B),(C×T ))p(Mt)

= % ·∑

s∈C×Bπ(s)+ (1− %) ·

∑t∈B

p(Mt),

sinceMt∩(C×T ) = ∅ for all t ∈ T andMt∩((C×T )\(M×B)) =∅ if and only if t ∈ B. Noting thatπ (s) := π(s)/π(C × T ) for alls ∈ C×T , p(Mt) := q(t) for all t ∈ T , and% := % ·π(C×T )/[% ·π(C × T )+ (1− %)] completes the proof. 2

Updating on events of probability zero

A special feature of the Dempster–Shafer formula derived in Lemma4.2 is well worth emphasizing. Suppose that the decision makerlearns that an eventC ⊂ M has occurred that he considered im-possible,ν(C× T ) = π(C× T ) = 0. If the degree of confidence inthe assessmentπ was less than 1, then the Dempster–Shafer updateremains well-defined. The updated degree of confidence in the prob-ability assessmentπ , %, falls however to 0 and the posterior beliefcoincides with the prior distribution onT , ν(M × B|C × T ) =∑t∈B p(t). Note that this appears to be quite sensible, since the

new information invalidates the probability assessmentπ but notthe prior beliefs onT . Falling back on one’s prior knowledge, if one

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E-CAPACITIES AND THE ELLSBERG PARADOX 133

learns that a ‘theory’, the assessmentπ , was inconsistent with theactual event, corresponds to a dismissal of this theory.

In signaling games, this suggests that uncertainty-averse playerswho observe some out-of-equilibrium move dismiss their equilib-rium beliefs and act based on their prior. Eichberger and Kelsey(1997d) show that, with E-capacities that are updated in theDempster–Shafer way, the Spence education-signaling model hasthe Pareto-dominating pooling equilibrium as its only equilibriumwhen the degree of confidence is not too low. This is in stark con-trast with most equilibrium refinements which select a separatingequilibrium, relying on different assumptions in regard to out-of-equilibrium moves. Note that, in contrast to many equilibrium re-finements that are suggested in the literature, the Dempster–Shaferupdates can be based on a well-developed consistent decision the-ory. We believe that a reversion to prior beliefs if an assessment isproven to be false is a quite intuitive adjustment of beliefs, support-ing our view that E-capacities are worth studying.

5. CONCLUDING REMARKS

In this paper we have introduced the concept of anE-capacity.E-capacities provide a simple parameterized version of a capacitybased on an additive probability distribution that makes it possi-ble to include ‘known’ probabilities for a partition of unambiguousevents. The Choquet integral of an E-capacity is a weighted aver-age of the expected utility with regard to an additive probabilitydistribution and the worst expected outcome obtained in the unam-biguous events. This Choquet integral is identical to a representationof preferences over actions suggested in Ellsberg (1961).

We provide an axiomatisation of preferences for which beliefsare represented by E-capacities and show that E-capacities satisfya list of desiderata for a concept that is applicable to economicand game-theoretic problems. Special cases of E-capacities includesimple capacities, the case where no ‘known’ probabilities exist. Fi-nally, we investigate the Dempster–Shafer updates of E-capacities.It can be shown that the updates of E-capacities remain E-capacities.The degree of confidence, however, will not increase with new in-formation.

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134 JÜRGEN EICHBERGER AND DAVID KELSEY

APPENDIX

To simplify notation, we will writeEpU(a) to denote the expectedutility of actiona with respect to the additive probability distributionp, i.e.,

EpU(a) :=∑s∈S

p(s) · U(a(s)).

Proof of Proposition 3.1.The proof of this proposition requirestwo definitions and four lemmas:

DEFINITION A.1. Define6 = {(s1, ..., sn)|si ∈ Ei, i = 1, ..., n}.Forσ = (s1, . . . , sn) defineσi = si .

DEFINITION A.2. Forσ ∈ 6, let

Aσ (S) ={a ∈ A(S)|a(s) � a(σi)for all s ∈ Ei, i = 1, . . . , n}

be the set of actions for which the worst outcome in each unambigu-ous eventEi occurs atσi .

LEMMA A.1. Suppose the preference relation� satisfies Axiom3.1 (weak order), Axiom 3.2(continuity), Axiom 3.3(weak domi-nance), Axiom 3.5(unambiguous independence)and Axiom 3.6(ex-tremal independence), then there exist additive probability distribu-tionspσ onS for anyσ ∈ 6 and a utility functionU : 1(X)→ Rsuch that, for alla ∈ Aσ (S) andb ∈ Aτ(S),

a � b if and only if EpσU(a) > EpτU(b). (A.1)

Proof. On the setAσ(S) ⊆ A(S) extremal independence (Ax-iom 3.6) implies independence. Using standard arguments (see forinstance Fishburn (1970)) it can be shown that the restriction of thepreferences to the setAσ(S) may be represented in the expectedutility form, i.e., there exists an additive probability distributionpσ

on S and a utility functionUσ : 1(X) → R such that, for alla, b ∈ Aσ(S),

a � b if and only if EpσUσ (a) > EpσUσ (b).

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E-CAPACITIES AND THE ELLSBERG PARADOX 135

Suppose that all consequences are indifferent. Then a constant utilityfunction can be used to represent preferences in the form of (A.1)for anyσ, τ ∈ 6. Otherwise, there are two lotteriesf0, f1 ∈ 1(X)such thatf1 � f0. Hence, we may normalize the utility function bysettingUσ (f0) = 0 andUσ(f1) = 1. Standard arguments5 showthat with this normalizationUσ (f ) = Uτ(f ) =: U(f ) for all f ∈1(X) and allσ, τ ∈ 6.

It remains to show that (A.1) holds for the whole ofA(S). Foranya, b ∈ A(S), with a � b, there existsσ, τ ∈ 6 such thata ∈Aσ (S) andb ∈ Aτ(S). Let c ∈ A(S) be a constant action such thata ∼ c. Such an action exists by continuity (Axiom 3.2) and weakdominance (Axiom 3.3). Sincec ∈ Aσ(S) ∩ Aτ(S),

EpσU(a) = EpσU(c) = EpτU(c) > EpτU(b). 2LEMMA A.2. For all Ei and all σ, τ ∈ 6, pσ (Ei) = pτ (Ei) =p(Ei).

Proof.On the setAM(S), our Axioms 3.1, 3.2, 3.3, 3.5 are iden-tical to those of Anscombe and Aumann (1963). Hence there mustbe a unique subjective probability distribution on actions that aremeasurable with respect to the partitionEi . 2LEMMA A.3. Suppose the preference relation� satisfies Axiom3.1 (weak order), Axiom 3.2(continuity), Axiom 3.3(weak domi-nance), Axiom 3.5(unambiguous independence)and Axiom 3.6(ex-tremal independence), then there exist real numbersδ1, . . . , δn andnon-negative real numbersα1, . . . , aS with∑

s∈Sαs +

n∑i=1

δi = 1

and a cardinal utility functionU : 1(X) → R such thata � b ifand only if∑

s∈Sαs · U(a(s))+

n∑i=1

δi ·mi(a) >∑s∈S

αs · U(b(s))

+n∑i=1

δi ·mi(b) (A.2)

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136 JÜRGEN EICHBERGER AND DAVID KELSEY

Proof.By Lemma A.1, for any two actsa, b ∈ A(S), there existσ, τ ∈ 6 such thata � b if and only if

∑s∈S pσ (s) · U(a(s)) >∑

s∈S pτ (s) · U(b(s)), and Lemma A.2 impliesp(Ei) = ∑s∈Ei

pσ (s) =∑s∈Ei pτ (s) for all σ, τ ∈ 6.

Consider first an elementEi which consists of a single states.By Lemma A.2,p(Ei) = pσ (s) = pτ (s) for all σ, τ ∈ 6. Hence,we can defineαs := p(Ei) for s ∈ Ei .

For anyEi with at least two states, consider the following twoactionsa, b ∈ A(S):• a constant actiona, a(s) = f for all s ∈ S, wheref ∈ 1(X);• an actionb with b(s) = f for all s ∈ S, s 6= s ′ ∈ Ei , andb(s ′) = g, whereg ∈ 1(X) with g � f .

By Lemma A.1,EpσU(a) = EpτU(a) andEpσU(b) = EpτU(b)

for all σ, τ ∈ 6 such thatσi 6= s ′ 6= τi . Hence,

EpσU(b) = EpσU(a)+ pσ (s ′) · [U(b(s ′))− U(a(s ′))]and

EpτU(b) = EpτU(a)+ pτ (s ′) · [U(b(s ′))− U(a(s ′))]imply pσ (s ′) = pτ (s ′) for all σ, τ ∈ 6 such thatσi 6= s ′ 6= τi .Hence, we can defineαs := pσ (s) for all s ∈ Ei, s 6= σi . Sinceτi 6=σi , one hasασi := pτ (σi) 6= pσ (σi). Defineδσi := pσ (σi) − ασi .Now

p(Ei) =∑s∈Ei

pσ (s) =∑s∈Ei

αs + δσi . (A.3)

Hence,δσi is independent ofσ and we may suppress the superscript.By construction, ifa ∈ Aσ(S), EpσU(a) = ∑

s∈S αs · U(a(s)) +∑ni=1 δi ·mi(a), sincepσs = αs unlesss = σi , in which casepσs =

αs + δi wheres ∈ Ei. By A.3,n∑i=1

p(Ei) =∑s∈S

αs +n∑i=1

δi = 1. 2LEMMA A.4. Suppose the preference relation� satisfies Axiom3.1 (weak order), Axiom 3.2(continuity), Axiom 3.3(weak dom-inance), Axiom 3.4(uncertainty aversion), Axiom 3.5(unambigu-ous independence)and Axiom 3.6(extremal independence), then wemay ensure thatδi > 0 for i = 1, . . . , n.

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E-CAPACITIES AND THE ELLSBERG PARADOX 137

Proof. For unambiguous eventsEi with a single elements, wehaveαs := p(Ei). Hence,δi = 0. Now considerEi such that|Ei | >2. If all consequences are indifferent then we may setδi = 0, for i =1, . . . , n, in Equation A.2 without affecting the ordinal properties ofthe preferences.

Hence, we may assume that there exist consequences which arenot indifferent. Fors ′, s ′′ ∈ Ei and somef, g ∈ 1(X), definea, b ∈A(S) as

a(s) :={f for s ∈ Ei, s 6= s ′f0 otherwise

and

b(s) :={g for s ∈ Ei, s 6= s ′′f0 otherwise.

By continuity (Axiom 3.3), we may findf, g ∈ 1(X) such thata ∼ b.

Sincea ∼ b,∑s∈Sαs ·U(a(s)) =∑s∈S αs ·U(b(s)). Defineaλ :=

λa + (1− λ)b for λ ∈ [0, 1]. If λ is sufficiently small, it followsfrom Lemma A.3 that

V (aλ) =λ ·∑s∈S

αs · U(a(s))

+ (1− λ) ·∑s∈S

αs · U(b(s))+ δi · λ · U(f )

=∑s∈S

αs · U(a(s))+ δi · λ · U(f ),

because∑s∈S αs · U(a(s)) =

∑s∈S αs · U(b(s)). By uncertainty

aversion (Axiom 3.4)aλ � a and therefore∑s∈S

αs · U(a(s))+ δi · λ · U(f ) >∑s∈S

αs · U(a(s)).

This impliesδi > 0. 2Proof of Proposition 3.1.Define

%i = 1− δi

p(Ei)

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138 JÜRGEN EICHBERGER AND DAVID KELSEY

and, fors ∈ Ei, let

πs := αs

%i.

One easily computes

∑s∈S

π(s) =n∑i=1

∑s∈Ei

αs

%i=

n∑i=1

∑s∈Ei

p(Ei)− δi1− δi/p(Ei)

=n∑i=1

p(Ei) = 1.

Substituting,δi = (1 − %i) · p(Ei) andαs = %i · π(s) into A.2completes the proof. 2

ACKNOWLEDGMENTS

This research was in part supported by the ESRC senior researchfellowship scheme, Award No. H52427502595.

NOTES

1. Epstein and Le Breton (1993) make a similar point in a somewhat differentaxiomatic context.

2. Alternative concepts have been suggested by Klibanoff (1994) and Lo (1995).3. Savage’s (1954) axioms actually imply that the state spaceS must be infi-

nite. Since this is a property of his axioms which does not carry over tothe Anscombe–Aumann framework used in this paper, we write Savage’srepresentation as a sum rather than an integral to avoid unnecessary extranotation.

4. Gilboa and Schmeidler’s (1993, p. 36) updating example is the special case ofExample 4.1 where% = 0.

5. See for instance Fishburn (1970).

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Addresses for correspondence:JÜRGEN EICHBERGER, Wirtschaftstheorie,Fachbereich 2, Universität des Saarlandes, D-66041 Saarbrücken, GermanyPhone: +49 681 302-4822; Fax: +49 681 302-4823;E-mail: j.eichberger@with.uni-sb.de

DAVID KELSEY, Department of Economics, The University of Birmingham,Birmingham, B15 2TT, United Kingdom

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