A Smooth Model of Decision Making UnderAmbiguity1

Peter Klibano¤MEDS, Kellogg School of Management, Northwestern University

peterk@kellogg.northwestern.edu

Massimo MarinacciDip. di Statistica e Matematica Applicata, Università di Torino, and ICER

massimo@econ.unito.it

Sujoy MukerjiDept. of Economics, University of Oxford, and ICER

sujoy.mukerji@economics.ox.ac.uk

September 2002

1We thank N. Al-Najjar, E. Dekel, S. Grant, Y. Halevy, M. Machina, F. Maccheroni, S.Morris, R. Nau, K. Nehring, U. Segal, A. Shneyerov, J-M. Tallon, R. Vohra, P. Wakker andespecially L. Epstein and I. Gilboa, for helpful discussions and suggestions. We also thankseminar audiences at Columbia University, the RUD �02 conference, the Spring �02 MidwestEcon. Theory Conference, Northwestern�s Zell Center and CMS-EMS. Mukerji thanks MEDSat Northwestern University and ICER at the University of Torino for their hospitality duringthe visits when part of the research was completed. Mukerji gratefully acknowledges �nan-cial support from the ESRC Research Fellowship Award award R000 27 1065, and Marinaccigratefully acknowledges the �nancial support of MIUR.

Abstract

We propose and axiomatize a model of preferences over acts such that the decision makerprefers act f to act g if and only if E¹Á (E¼u ± f) ¸ E¹Á (E¼u ± g), where E is theexpectation operator, u is a vN-M utility function, Á is an increasing transformation,and ¹ is a subjective probability over the set ¦ of probability measures ¼ that thedecision maker thinks are relevant given his subjective information. A key feature of ourmodel is that it achieves a separation between ambiguity, identi�ed as a characteristicof the decision maker�s subjective information, and ambiguity attitude, a characteristicof the decision maker�s tastes. We show that attitudes towards risk are characterizedby the shape of u, as usual, while attitudes towards ambiguity are characterized by theshape of Á: We also derive Á (x) = ¡ 1

®e¡®x as the special case of constant ambiguity

aversion. Ambiguity itself is de�ned behaviorally and is shown to be characterized byproperties of the subjective set of measures ¦. This characterization of ambiguity isformally related to the de�nitions of subjective ambiguity advanced by Epstein-Zhang(2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguityattitudes. The model is also distinct from many in the literature on ambiguity in thatallows smooth, rather than kinked, indi¤erence curves. This leads to di¤erent behaviorand improved tractability, while still sharing the main features (e.g., Ellsberg�s Paradox,etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set ofmeasures may be seen as an extreme case of our model with in�nite ambiguity aversion.Two illustrative applications to portfolio choice are o¤ered.

JEL Classi�cation Numbers: D800, D810.Keywords: Ambiguity, Uncertainty, Knightian Uncertainty, Ambiguity Aversion, Un-

certainty Aversion, Ellsberg Paradox.

1 Introduction

1.1 Motivation

Consider the Ellsberg two color experiment. One urn (urn I) contains exactly 50 redand 50 black balls. A second urn (urn II) contains 100 balls divided in an unspeci�edway between red and black. An important aspect of the experiment is that the decisionmaker has di¤erent information about the stochastic fundamentals governing the outcomeof a draw from each urn. The information about one urn (urn II) is less complete, lessprecise than that for the other urn (urn I). Given the decision maker�s knowledge, allbut a single prior can be ruled out in describing the likelihood of di¤erent outcomes inone urn, whereas the information about the other urn is compatible with more than oneprior. The imprecision induces an uncertainty or ambiguity about the ex ante evaluationof bets whose payo¤s depend on the outcome of the draw from urn II. Some decisionmakers may be, and as is well known most are in the context of Ellsberg�s example,averse to the ambiguity: they worry that they may take the �wrong� decision ex antebecause they have a relatively vague idea as to what the true probability is and are proneto make a �wrong� evaluation. The related, more general intuition goes as follows: ifa decision maker perceives that, ceteris paribus, he is more certain of the evaluation ofone act as compared to that for another act then he prefers the former. The uncertaintyor ambiguity about the evaluation of an act would, in general, depend on the detailsof the decision maker�s subjective information: the di¤erent probabilities he thinks areconsistent with his knowledge and the weight of evidence with respect to each suchprobability. In general, di¤erent individuals will maintain di¤erent degrees of aversion tothe ambiguity, and indeed, some may well have a preference for it or be neutral towardsit. In this paper, we axiomatize a model of preferences which has a representation thatidenti�es all the aspects of the decision maker�s subjective information mentioned aboveas well as separating out his attitude toward ambiguity.

It is not hard to think of �real world� instances wherein a decision maker�s informationis ambiguous in the sense of being consistent with multiple probabilities on the state spacerelevant to the decision at hand. One example is expert opinion. A manager considering aparticular venture may, on seeking assessments of the probability of success of the venturefrom a set of experts, �nd that di¤erent experts report di¤erent numbers. Possibly therewould be a further aspect to the manager�s information: some prior belief about thecredibility of each expert (formed from previous experience, say). As a second example,think of a decision maker applying a model to forecast the realization of an observablevariable. The model maps a parametric con�guration, whose value is not precisely knownto the decision maker, to a probability distribution over a set of possible realizations of theobservable variable. Data on past realizations of the observable variable would allow thedecision maker to update his prior on the possible values of the parameters but is unlikelyto allow the decision maker to pin down the true values. Hence, the decision maker willtypically be left with a set of probability distributions governing the future realizations ofthe observable variable, along with some belief (a posterior updated from a prior possiblyformed on the basis of loose intuition) over the relevant probabilities. This motivationis similar to that in some of the literature on robustness and model misspeci�cationor �model uncertainty� (Hansen and Tallarini (1999), Hansen, Turmuhambetova, and

1

Williams (2001)).Given the plausibility of the ambiguity of relevant information and the fact that

non-neutrality of attitude to such ambiguity is quite common (at least, so far as it isevident in Ellsberg type experiments) it is natural to ask what e¤ect this would have oneconomically relevant behavior. To investigate the issue in a particular context one wouldtake a model describing behavior relevant to the context and pose questions involvingcomparative statics of the following kind. Suppose we allow the agents� beliefs to re�ectambiguity and consider, as a benchmark, equilibrium behavior in the model if all agentswere neutral to the ambiguity, say, subjective expected utility (SEU) maximizers. Next,without perturbing the information structure any further, we may want to ask how theequilibrium would change if ambiguity aversion were to replace ambiguity neutrality,both in the case where the aversion is uniform across agents and, more generally, if theambiguity attitude varied in intensity across the agents. Another comparative staticsexercise might hold the taste parameters such as ambiguity attitudes �xed and ask howthe equilibrium is a¤ected if the ambiguity of information varied across agents. The pointis, working out such comparative statics �properly� requires a model which separates theagents� information from his tastes and parametrizes each of these in a tractable way.We argue below that the models of ambiguity aversion available in the literature do notmake this separation in a way that is clear and complete enough.

Consider, for instance, the model of maxmin expected utility (MEU) preferences(Gilboa and Schmeidler (1989)), one of the more popular models incorporating ambi-guity aversion. These preferences are represented by the following functional:

V (f) = min¼2¦

E¼ (u ± f) ,

where f is an act and ¦ is a set of probability measures: The model does not, in general,impose a separation of information and ambiguity attitudes. In general, the set ¦ maynot be interpreted as being completely characterized by the decision maker�s information.It represents information intertwined with ambiguity attitude in an inseparable way. Acommon comparative statics exercise that uses this model to reveal the e¤ect of ambiguityaversion proceeds by comparing the behavior corresponding to one set of priors ¦ withthe behavior corresponding to a more inclusive set ¦0, i.e., ¦ ½ ¦0, while controlling forthe risk attitudes by holding u �xed. The larger set of priors could re�ect di¤erencesin information, di¤erences in ambiguity attitude, or both. This is problematic becausewe are unable to see if the answers di¤er because the ambiguity in the environment ischanging or because of a change in attitude towards ambiguity. For instance, considera principal-agent model with moral hazard which is �standard� in every respect exceptthat the players� beliefs about the contractible signal (e.g., output) re�ects ambiguous.In such a model it is natural to pose the question, �How does the optimal contract changeif the agent becomes more ambiguity averse than the principal?� Suppose we require,as in the standard model, that the comparative statics exercise respect the assumptionthat the principal and agent have the same information about the contractible signal.Such an exercise would be very hard to execute, if not impossible, in the MEU model.The problem is that there are no known results which would tell us how we could alterthe set of priors in the representation of the agent�s preferences in a way that wouldincorporate a change in ambiguity attitude while at the same time maintaining that the

2

agent�s priors and the set of priors corresponding to the principal incorporate the sameinformation about the contractible signal.

It may seem that the problem of confounding information and taste can be �nessedif we take the special case where the set ¦ is interpreted as the set of probabilities thedecision maker thinks is relevant. Then, per force, we �x the decision maker�s ambiguityattitude to the extreme case where he only worries about the worst case scenario givenhis information. This interpretation imposes at least two kinds of limitations on the com-parative static exercises. One, we cannot allow for heterogeneity of tastes (i.e., ambiguityattitudes) among agents in the model. Two, we have no way of knowing whether theresults of the comparative static exercise are true only for the extreme attitude or extendmore generally to intermediate degrees of ambiguity aversion. The two problems mayseemingly be addressed by a generalization of the maxmin functional to the so called®-maxmin form(®-MEU):1

V̂ (f) = ®max¼2¦

E¼ (u ± f) + (1¡ ®)min¼2¦

E¼ (u ± f) ,

where the ambiguity attitude may be varied parametrically by varying ®. There is anotherlimitation of the maxmin model that is not resolved by the ®-maxmin generalization. Thelimitation is, once again, particular to the case where we interpret ¦ simply as the setof probabilities that the decision maker thinks is relevant. This interpretation of thefunctional implies that the preferences do not take into account any information thatthe decision maker may have about plausibility of each ¼ in ¦. The functional does notsmoothly aggregate the information concerning how the act performs under each possible¼ but only looks at the extremal performance values (the worst under maxmin, the bestand the worst under ®-maxmin). Taking ¦ to be the set of probabilities subjectivelythought as possible, the choice rule generated by the functional looks almost pathologicaland even, in a way, contrary to the general intuition following from Ellsberg type behavior.For instance, take two acts f and g which share the same extremal valuations (i.e.,max¼2¦E¼ (u ± f) = max¼2¦E¼ (u ± g) and min¼2¦ E¼ (u ± f) = min¼2¦E¼ (u ± g)) butfor �almost all� probabilities in ¦, E¼ (u ± f) > E¼ (u ± g) : The choice rule would rank theacts equally. Relatedly, take two acts f and g which share the same extremal valuationsbut suppose valuations for f are far less spread out than g�s. Again, f and g wouldbe ranked equally, which is somewhat contrary to the general Ellsberg intuition thatambiguity averse decision makers have preference for acts whose performance is morestable, more robust to the possible variation in probabilities. The following exampleserves to illustrate the point.

Example 1 Consider two acts f and g whose utility payo¤s in states 1, 2, 3 and 4 areas given in the following table:

State 1 State 2 State 3 State 4u ± f 1 2 3 4u ± g 1 1 4 4

Figure 1, depicts the two acts.

1This form has recently been axiomatized in Ghirardato, Maccheroni, and Marinacci (2002). A similarfunctional was proposed previously in Ja¤ray (1995).

3

Suppose a decision maker thinks the four probabilities, ¼1, ¼2, ¼3, ¼4, described inthe table below, are possible/relevant.

State 1 State 2 State 3 State 4¼1 .7 .1 .1 .1¼2 .1 .7 .1 .1¼3 .1 .1 .7 .1¼4 .1 .1 .1 .7

The points to note about the possible ¼i�s are that each favors one state, is uniform acrossthe other states, and every state is favored by some ¼i. The �linear� act f is more robustto variation in probabilities than is the �zigzag� act g. The latter, which concentratesits payo¤s on states 3 and 4, does very well under under ¼3, ¼4 but rather poorly under¼1, ¼2. Notice, though, the best and worst expected utility evaluations are the same forthe two acts, hence any ®-maxmin functional will evaluate the two acts identically. Thefunctional is insensitive to the unrobustness of g relative to f . The insensitivity arisesbecause of the �non-smooth� aggregation of the information that results from exclusiveconcentration on extremal values. The point is made more vividly if one considers a prioron the ¼i�s, say ¹, as an example of a decision maker�s information about the relevanceof each ¼i, and use the prior to aggregate the information smoothly by looking at theinduced distribution on the expected utility values generated by each act. Given f and¹, ¹f is generated in the following way: ¹f (E¼i (u ± f)) = ¹ (¼i). Figures 2 and 3 showthe distributions corresponding to ¹ being the uniform prior. As is evident, ¹g is a meanpreserving spread of ¹f . ¥

Preferences axiomatized in this paper are represented by the functional form

V (f) =

Zi2I

Á (E¼i (u ± f)) d¹:

Where, f is a real valued act de�ned on a state space S; u is a vN-M utility function; ¼i,indexed by i 2 [0; 1], is a probability measure on S; ¹ is a probability measure on the indexspace [0; 1] with I µ [0; 1] as its support; Á is a map from reals to reals. Information thatthe decision maker subjectively considers relevant to the decision problem is summarizedby the set ¦ ´ f¼i j i 2 Ig and ¹. There is subjective uncertainty about what the �true�probability on S is; ¹measures the subjective relevance of each element of ¦ as the �true�probability.2 Proceeding from a behavioral notion of ambiguity it will be shown that ifan event E µ S is identi�ed as ambiguous then the probabilities considered relevant bythe decision maker di¤er in the measure they assign to E. While u characterizes attitudetoward risk, ambiguity attitude is shown to be captured by Á. In particular, a concave Ácharacterizes ambiguity aversion or, equivalently, an aversion to mean preserving spreadsin ¹f , where ¹f is the distribution induced by ¹ on the expected utility values generatedby f and the elements of ¦, as shown in Example 1. This preference model does not, in

2The set ¦ may be due to �model uncetainty�. For instance, suppose the decision maker has in minda model such that a particular parametric con�guration of the model (where parameter con�gurationsare indexed by i) implies a particular probability ¼i on a space of observable outcomes S. In this case,we would interpret ¹ as the decision maker�s prior over the possible parameter con�gurations.

4

0

1

2

3

4

5

1 2 3 4

Contingency #C

on

tin

gen

t p

ayo

ff (

in u

tils

)

Linear act

(f)

Zig-zag act

(g)

Figure 1: Contingent utility payo¤s of f and g

0.25 0.25 0.25 0.25

0

0.25

0.5

1.6 2.2 2.8 3.4

Expected utility

Pro

bab

ilit

y

Figure 2: ¹f : Probability distribution on expected utility generated by f

0.5

0 0

0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

1.6 2.2 2.8 3.4

Expected utility

Pro

bab

ilit

y

Figure 3: ¹g : Probability distribution on expected utility generated by g:

5

general, impose reduction between ¹ and the elements of ¦. Such reduction only occurswhen Á is linear, a situation which we identify as ambiguity neutrality and whereinthe preferences are observationally equivalent to that of a subjective expected utilitymaximizer.

1.2 Organization

The rest of the paper is organized as follows. The next subsection discusses some relatedliterature. The following section states our assumptions on preferences and derives therepresentation. Section 3 de�nes and characterizes ambiguity in this model. Section 4focuses on the characterization of comparative ambiguity attitudes in the model. Section5 contains formal results and discussion which relates the notion of ambiguity presentedin Section 3 to the two leading notions of ambiguity available in the literature. Section6 presents two illustrative portfolio choice applications of the theory developed here. Allproofs, unless otherwise noted in the text, appear in the Appendix.

1.3 Related literature

A key idea in the present paper, relaxing reduction between �rst and second order prob-abilities to accommodate ambiguity sensitive preferences, owes its inspiration to the re-search reported in Segal (1987). That paper presented a model of decision making underuncertainty which assumes a unique second order probability over a set of given �rstorder probabilities, but relaxes reduction and weights the possible �rst order probabil-ities non-linearly. Using examples, Segal observed that such a model would be �exibleenough to accommodate both Allais and Ellsberg type behavior. While Segal (1987) didnot axiomatize a general preference order incorporating ambiguity aversion, Segal (1990)developed the key idea of relaxing reduction further in the context of choice under riskand obtained a novel axiomatization of the Anticipated Utility model.

Another paper that invokes this same idea of relaxing reduction to model ambiguityaversion is Nau (2001), (revised and expanded in Nau (2002)). The paper presents anaxiomatic model of partially separable preferences where the decision maker may satisfythe independence axiom selectively within partitions of the state space whose elementshave �similar degrees of uncertainty�. The axiomatization makes no attempt to uniquelyseparate beliefs from state-dependent cardinal utilities. Indeed, a major and innovativecontribution of the paper(s) is to present an intuitive notion of ambiguity aversion in astate dependent utility framework; this goes beyond what is in the previous literature,which does not allow state dependent utility. Our paper, on the other hand, axioma-tizes and investigates a model of ambiguity aversion staying very much within the stateindependent utility framework, separating beliefs and ambiguity attitude from state inde-pendent utilities, while using Segal�s idea of relaxing reduction. The notion of ambiguityaversion characterized in our paper is not as general as Nau�s in that it is not applicableto a state dependent framework. Also, as we show in our formal results, our notion ofambiguity is closely related to the principal notions in the existing literature. A relatedpoint which underscores the more modest scope of our analysis, compared to Nau�s, isthat in our framework the only departure from SEU is ambiguity aversion/preference,whereas in Nau�s framework departure from SEU can occur in various ways.

6

The seminal work of Kreps and Porteus (1988) is not concerned with ambiguity, butis quite related to our modeling approach in that the representation we derive has a two-stage recursive form with expected utility at each stage. This is similar to a simpli�ed,two-stage, with only last stage consumption version of their functional. Moreover, asthey relate the convexity/concavity of their �rst stage evaluation function to attitudetowards the timing of the resolution of risk, we are able to relate convexity/concavityof our �rst stage evaluation function with attitude towards ambiguity. In both cases,neutrality leads reduction to hold and preferences to be expected utility. The analogy isnot perfect, however. Aside from the obvious fact that Kreps and Porteus (1988) workswith purely objective temporal lotteries while we are in a more subjective setting, thereis at least one important structural di¤erence in the objects over which preferences areformed. While it is possible to associate a natural two-stage lottery with each act in oursetting (imagine �rst choosing a probability over the state space according to a subjectivedistribution and then imagine that probability determining the state and thus the realizedoutcome according to the act) only a restricted subset of two-stage lotteries will be soassociated. This is true both because there is only one distribution over probabilitiesover the state space and, more importantly, because the second stage lotteries inducedby an act as the probabilities over the state space vary will be strongly related due to theunderlying mapping of states to outcomes. For this reason it does not appear fruitful forus to try to directly adapt existing work on recursive preferences over two-stage lotteries toour setting. For example, Grant, Kajii, and Polak (2000) examine the degree to which theassumption of expected utility at each stage may be relaxed for recursive preferences overtwo-stage lotteries. They �nd some striking results. For instance, if the decision-makeralways prefers later resolution of risk, if both �rst and second stage preferences satisfy therank dependent model, and if second stage preferences display risk aversion then secondstage preferences must in fact be expected utility. Reasoning by analogy and drawing onsome of our results, this suggests that in our model, if the decision maker is both ambiguityaverse and risk averse, then our assumption that preferences over lotteries are expectedutility may be relaxed to rank dependence while leaving the representation unchanged(i.e., still implying that preferences over lotteries are expected utility). However, webelieve that this is not true. We are working on an example/representation showingthat non-EU rank dependence in the last stage is possible under these conditions in ourmodel. We conjecture that the Grant, Kajii, and Polak (2000) results fail to translateto our setting precisely because of the restricted set of two-stage lotteries implicit in ourframework.

Another paper modelling ambiguity attitude by indirectly relaxing the reduction as-sumption is Halevy and Feltkamp (2001). They try to rationalize ambiguity aversion byassuming that a decision maker mistakenly views his choice of an action as determin-ing payo¤s for two positively related replications of the same environment, rather thansimply for a single environment. If he is risk averse and has expected utility preferencesover a single instance then this �bundling� of problems may lead to Ellsberg type behav-ior. Nehring (2001) de�nes a class of preferences, termed �utility sophisticated�, whoseidentifying spirit is that the members of this class do not show Allais type departuresfrom expected utility though they do allow for Ellsberg type behavior. The preferencesmodeled in the present paper do satisfy this spirit though it remains to be formally veri-�ed whether the preferences actually satisfy utility sophistication as de�ned by Nehring.

7

Finally, since the relation of the model in this paper to models such as MEU and ®-MEUhas been discussed above we will not go much further here except to remark that therelationship may be thought to be analogous to that between �rst and second order riskaversion (Segal and Spivak (1990), Loomes and Segal (1994)). Models such as MEU and®-MEU display ambiguity averse behavior when the corresponding indi¤erence curves inthe utility space are kinked (behavior which may be dubbed �rst-order ambiguity aver-sion). The model in this paper focuses on incorporating ambiguity aversion even whenthe indi¤erence curves are not kinked (�second order ambiguity aversion�), though it isworth noting that ambiguity aversion with kinked indi¤erence curves does not fall outsidethe scope of this theory.

2 Axioms and representation

2.1 Preliminaries

Let ­ be a �nite set. Consider the state space S = ­ £ [0; 1). Let B denote the Borel¾-algebra on [0; 1) and de�ne § ´ 2­­B, the product sigma algebra. For the remainderof this paper, all events will be assumed to belong to § unless stated otherwise. Wedenote by f a Savage act de�ned on S, f : S ! C, where C is the set of consequences.We assume C to be an interval in R containing the interval [¡1; 1]: F is the set of allsimple acts, that is, the set of all §-measurable functions f : S ! C such that the setfx 2 R : f(s) = x for some s 2 Sg is �nite. The space [0; 1) is introduced simply tomodel a rich set of lotteries3 as a set of Savage acts.4 An act l 2 F is said to be a lotteryif l depends only on [0; 1) (i.e., for any !1; !2 2 ­ and r 2 [0; 1), l(!1; r) = l(!2; r)) andis Riemann integrable. The set of all such lotteries is L. If f 2 L and r 2 [0; 1), wesometimes write f(r) meaning f(!; r) for any ! 2 ­. Let ¸ : B ! [0; 1] be the Lebesguemeasure. Let ¼ be a probability de�ned on 2­£B such that ¼(A£B) = ¼(A£[0; 1))¸(B)for A 2 2­ and B 2 B. Any such ¼ admits a unique countably additive extension to §.The set of all such ¼�s is denoted by ¢ and, since ­ is �nite, can be indexed as f¼igi2[0;1] :

Since we wish to allow ¢ to be another domain of uncertainty for the decision makerapart from S, we model it explicitly as such. To formally identify the decision maker�ssubjective uncertainty about this domain, i.e., whether he at all regards this domain asuncertain and if so, what his subjective information and beliefs are, we look at the decisionmaker�s preferences over acts which assign consequences to elements of this domain.

De�nition 1 A second order act is any function f that associates an element of theindex set of ¢ to a consequence, i.e., f : [0; 1]! C: We denote by F the set of all secondorder acts.

Let º2 be the decision maker�s preference ordering over F. The main focus of themodel is º, a preference relation de�ned on F :

3By the phrase �a rich set of lotteries� we simply mean that, for any probability p 2 [0; 1), we mayconstruct an act which yields a consequence with that probability: While this richness is not required inthe statement of our axioms or in our representation result, it is invoked later in the paper in Theorems2 and 3.

4Our modelling of lotteries in this way and use of a product state space is similar to the �single-stage�approach of Sarin and Wakker (1992) and Sarin and Wakker (1997) to Anscombe-Aumann style models.

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2.2 Basic axioms

Next we describe three axiomatic restrictions on the preference orderings º and º2. The�rst axiom applies to the preference ordering º when restricted to the domain of lotteryacts. Preferences over the lotteries are assumed to have a von Neumann-Morgensternexpected utility representation.

Axiom 1 (Expected utility on lotteries) There exists a unique u : C! R; continu-ous, strictly increasing and normalized so that u(0) = 0 and u(1) = 1 such that, for allf; g 2 L, f º g ,

R[0;1)

u(f(r))dr ¸R[0;1)

u(g(r))dr:

The following axiom is on º2, the preferences over second order acts.5

Axiom 2 (Subjective expected utility on 2nd order acts) There exists a unique�nitely additive probability ¹ : 2[0;1] ! [0; 1] and a continuous, strictly increasing v : C !R, such that

f º2 g()

Z[0;1]

v (f (i)) d¹ ¸

Z[0;1]

v (g (i)) d¹.

Further, if there exists a J µ [0; 1] such that 0 < ¹ (J) < 1, then v is unique up to positivea¢ne transformations.

We denote by I the support of ¹, that is, the smallest closed subset of [0; 1] whosecomplement has measure zero. The subset of ¢ the decision maker subjectively considersas relevant is given by ¦ ´ f¼igi2I : Given any E µ I, we interpret ¹ (E) as the decisionmaker�s subjective assessment of the likelihood that the �true� probability lies in the setf¼i j i 2 E µ Ig; hence, ¹ may be thought of as a �second order� probability over the�rst order probabilities ¼. Note, I may well be a �nite subset of [0; 1], and further, ¦ isnot necessarily a convex set.

Notice that an act f together with a probability ¼i generates a probability distribu-tion over consequences. Next we give details for the construction of the lottery lif thatgenerates an identical probability distribution over consequences.

Notation 1 ¼if is the distribution induced on consequences by ¼i and f:More formally,

¼if (x) = ¼i¡f¡1 (x)

¢=

Xf!jf(!;r)=x for some r2[0;1)g

Zfrjf(!;r)=xg

¼i (f! £ [0; 1)g) dr

We now construct a lottery lif 2 L that generates a distribution on C identical to ¼if , asfollows: First, let fx1; :::; xng be an enumeration of the support of ¼if : Second, de�ne

lif (!; r) =

8>>><>>>:

x1 if r 2 [0; ¼if (x1))x2 if r 2 [¼if (x1) ; ¼if (x2) + ¼if (x1))...

......

xn if r 2 [Pn¡1

i=1 ¼if (xi) ;Pn

i=1 ¼if (xi) = 1)

5Each of the �rst two axioms could be replaced by more primitive assumptions on º and º2, re-spectively, which deliver the expected utility representations. Since such developments are by now wellknown and easily adapted to our setting, we do not do so here.

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Let ±x denote the constant act with consequence x 2 C. Let cf (¼i) denote the certaintyequivalent of the lottery lif , i.e., ±cf (¼i) » lif :

Observe that continuity of u (Axiom 1) and the assumption that lotteries have �nitesupport guarantee that the certainty equivalent exists for any lottery. We also assumed(in Axiom 1) that u is increasing; this assumption guarantees that the certainty equivalentis unique. Axiom 1 also implies that the order of the enumeration of the support of ¼ifin the construction of lif does not matter for º. Additionally, the fact that each ¼irestricted to ­£B is the Lebesgue measure implies that the only property of the chosenevents in the construction of lif that matters for º is their length.

Our �nal basic axiom requires the preference ordering of primary interest, º, to beconsistent with Axioms 1 and 2 in a certain way. Essentially, the axiom requires thedecision maker to view act f 2 F as equivalent to an act f2 2 F which yields theconsequence cf (¼i) whenever ¼i is the �true� probability law. Thus f is associated witha vector of several certainty equivalents, each certainty equivalent corresponding to aparticular possible probability. Since this vector of certainty equivalents is an act in F

it is reasonable to ask that its evaluation be consistent with Axiom 2. Since f togetherwith a possible probability ¼i generates a distribution over consequences identical to thatgenerated by lif , it is reasonable to require the certainty equivalent of f to be same asthe certainty equivalent of lif .

De�nition 2 Given f 2 F , f2 2 F denotes a second order act associated with f;de�ned as follows

f2 (i) = cf (¼i) ,

for all i 2 [0; 1].

Axiom 3 (Consistency with preferences over associated 2nd order acts) Givenf; g 2 F and f2; g2 2 F,

f º g , f2 º2 g2:

Remark 1 Let ±2c denote a second order act such that ±2c (i) = c; for all i 2 [0; 1]. Notice,

since both v and u are strictly increasing,

±2c º2 ±2c0 , ±c º ±c0 :

These three axioms are basic to our model in that they are all that we invoke toobtain our representation result. Theorem 1, below, shows that given these axioms,º is represented by a functional which is an �expected utility over expected utilities�.Evaluation of f 2 F proceeds in two stages: �rst, compute all possible expected utilitiesof f , each expected utility corresponding to a ¼i in the support of ¹; next, computethe expectation (with respect to the measure ¹) of the expected utilities obtained inthe �rst stage, each expected utility transformed by the index Á. As will be shownin subsequent analysis, this representation allows a clear decomposition of the decisionmaker�s tastes and beliefs: u determines risk attitude, Á determines ambiguity attitude,and ¹ determines the subjective ambiguity of information. The representing functionalis also invariant to positive a¢ne transforms of the vN-M utility index that applies tothe lotteries. That is, when u is translated by a positive a¢ne transformation to u0, theclass of associated Á0 is simply the class of Á (:) with domain shifted by the positive a¢netransformation. Let U denote the range fu (x) : x 2 Cg of the utility function u:

10

Theorem 1 Given Axioms 1, 2 and 3, there exists a continuous, strictly increasingÁ : U ! R such that º is represented by the preference functional V : F ! R givenby

V (f) =

ZI

Á

"X!

Z[0;1)

u (f (!; r))¼i (f! £ [0; 1)g) dr

#d¹ (1)

´ E¹Á (E¼iu ± f)

Furthermore, if there exists a J µ [0; 1] with 0 < ¹ (J) < 1, then, given u, the functionÁ is unique up to positive a¢ne transformations. Moreover, if ~u = ®u + ¯, ® > 0, thenthe associated ~Á is such that ~Á(®y + ¯) = Á (y), where y 2 U .

Hereafter when we write a preference relation º, we assume that it satis�es theconditions in Theorem 1. Note that Theorem 1 can be viewed as a part of a morecomprehensive representation result (reported in the Appendix as Theorem 6) for thetwo orderings º and º2 in which Axioms 1, 2 and 3 are both necessary and su¢cient.

2.3 Axioms characterizing ambiguity attitude

Next we show that ambiguity attitude is characterized by the properties of Á: Just as riskattitude in the classical theory is assumed to be a property of the individual and not oftheir environment, we would like to say the same about ambiguity attitude. In particular,in the context of our theory, this will require that the derived Á does not depend on I,the support of ¹.

To state this formally in our setting, we consider the family fºIgIµ[0;1] of preferencerelations characterizing each decision maker, wherein there is a preference relation cor-responding to each possible support I, that is, to each possible state of information hemay have about which probabilities ¼ are relevant to his decision problem. In readingthe axiom, recall that supports are, by de�nition, closed subsets of [0; 1].

Axiom 4 (Separation of tastes and beliefs ) For all closed subsets I µ [0; 1], eachpreference ºI admits the representation given by Eq. (1) with the same functions Á andu, and with a probability ¹ (written ¹I) having support I.

Axiom 4 says that as the support of a decision maker�s subjective belief varies (say, dueto conditioning on di¤erent information), the decision maker�s preferences over lotteries,u, and attitude towards risk on the space ¢ of probabilities, v, remain unchanged.

Notice that there is no restriction on the decision maker�s belief associated with eachºI , besides that of having support I. Though we do not need to assume it for ourresults, a natural possibility is that all such beliefs be connected via conditioning fromsome �original� common belief.

Recall that an act f together with a probability ¼i, gives a distribution on conse-quences, ¼if : Each such distribution is naturally associated with a lottery lif 2 L thatyields each consequence c in the support of ¼if on a subset of [0; 1) with length equalto ¼if (c). Each such lottery has a certainty equivalent cf (¼i). By Axiom 2, we have ameasure ¹ on the set I indexing the set of probabilities ¦. Fixing an act f , ¹ may thenbe used to induce a measure ¹f on fu (cf (¼i))gi2I , the expected utility values generated

11

by f corresponding to the di¤erent ¼�s in ¦ (using the utility function from Axiom 1).This is done in De�nition 3, as follows.

De�nition 3 Given f 2 F ; the induced distribution ¹f on U is as follows:

¹f (u (x)) ´ ¹³¡f2¢¡1

(x)´; x 2 C:

Given an act f , the derived (subjective) probability distribution over (expected) util-ities, ¹f , smoothly aggregates the information the decision maker has about the relevant¼�s and how each such ¼ evaluates f , without imposing reduction between ¹ and the ¼�s.In this framework the induced distribution ¹f represents the decision maker�s subjectiveuncertainty about the �right�(ex ante) evaluation of an act. The greater the spread in¹f the greater the uncertainty about the ex ante evaluation. In our model it is thisuncertainty through which ambiguity of the environment may a¤ect behavior: ambigu-ity aversion is an aversion to the uncertainty about ex ante evaluations. Analogous torisk aversion, aversion to this uncertainty is taken to be the same as disliking a meanpreserving spread in ¹f . First we introduce notation for denoting the mean of ¹f .

Notation 2 Let e¡¹f¢´RUxd¹f . Notice u

¡1¡e¡¹f¢¢2 C:

Recalling the notation for constant acts, we see ±u¡1(e(¹f)) is simply the constant act

which yields the same utility as the mean of ¹f . Equivalently, ¹±u¡1(e(¹f))

is the degen-

erate distribution on e¡¹f¢. The de�nition below formally describes the notion that a

given decision maker displays ambiguity aversion if each member of the family fºIgIµ[0;1]shows a preference for an act, ±

u¡1(e(¹f)), that induces the degenerate distribution on the

mean of ¹f , ¹±u¡1(e(¹f))

, over the act f that induces ¹f .

De�nition 4 A decision maker displays smooth ambiguity aversion if

±u¡1(e(¹f))

ºI f

for all f 2 F and all closed subsets I µ [0; 1].6

In a similar way, we can de�ne smooth ambiguity love and neutrality. The �rst propo-sition, below, shows that smooth ambiguity aversion is characterized in the representingfunctional by the concavity of Á. A result characterizing smooth ambiguity love by con-vexity of Á follows from the same argument. Similarly, smooth ambiguity neutrality ischaracterized by Á linear. It is worth noting that a straightforward adaptation of theproof of the analogous result in risk theory does not su¢ce here. The reason is that theneeded diversity of associated second order acts is not guaranteed in general.

6This de�nition is actually stronger than we need for our later results. It is enough that the indicatedpreference hold for (in addition to the original preference, º) some ºIwhere I contains exactly two pointswhose corresponding measures have disjoint support. While we stick with the stronger de�nition for easeof statement, the observation here indicates that many fewer preference relations need to be considered(two rather than an in�nite number) than the stronger version would lead one to think.

12

Proposition 1 Suppose the decision maker�s family of preferences fºIgIµ[0;1] satis�esAxioms 1-4. Then, the following conditions are equivalent:

(i) the function Á : U ! R is concave;

(ii) the decision maker displays smooth ambiguity aversion.

The proposition has the following corollary (whose simple proof is omitted) whichshows that the usual reduction (between ¹ and ¼i) applies whenever ambiguity neutralityholds. In that case we are back to subjective expected utility. An ambiguity neutraldecision maker, though informed of the multiplicity of ¼�s, is indi¤erent to the spreadin the ex ante evaluation of an act caused by this multiplicity; he only cares about the�expected prior� º.

Corollary 1 Under Axioms 1-4, the following properties are equivalent:

(i) the decision maker is smoothly ambiguity neutral;

(ii) Á is linear;

(iii) V (f) =RSu (f (s)) dº, where º (E) =

RI¼i (E) d¹ for all E 2 §.

Our �nal axiom imposes a (behavioral) restriction on the preference order so thatits ambiguity attitude is �well behaved�. This good behavior will be useful in the nextsection when we discuss ambiguous events and acts. In words, the restriction is that if apreference is not neutral to ambiguity then there exists at least one interval in U whichsatis�es the following property. Consider the set of acts for which the set the expectedutilities corresponding to each act lies within the given interval. We require that thedecision maker displays either strict ambiguity aversion, or strict ambiguity love, but notboth, for all such acts. What is ruled out (and this is the essence of the restriction)is the possibility that the decision maker�s ambiguity attitude �its between ambiguityaversion and ambiguity love, continuously from one point to the next, over the entirerange of U . Note, it is entirely permissible that there be several intervals, over some ofwhich the decision maker is ambiguity averse while over others he is ambiguity loving.Furthermore, outside a chosen interval, ambiguity attitude may be as inconsistent asdesired. The statement of the axiom is immediately followed by a proposition whichgives an equivalent characterization in terms of Á:

Axiom 5 (Consistent ambiguity attitude over some interval) The decision maker�sfamily of preferences fºIgIµ[0;1] satis�es at least one of the following three conditions:

(i) smooth ambiguity neutrality,

(ii) there exists an open interval J µ U such that smooth ambiguity aversion holdsstrictly when limited to all f 2 F for which supp

¡¹f¢is a non-singleton subset of

J,

(iii) there exists an open interval K µ U such that smooth ambiguity love holds strictlywhen limited to all f 2 F for which supp

¡¹f¢is a non-singleton subset of K.

13

Proposition 2 Under Axioms 1-4, we have:

1. Axiom 5 (i) holds if and only if Á linear;

2. Axiom 5 (ii) holds if and only if Á strictly concave on some open interval J µ U ;

3. Axiom 5 (iii) holds if and only if Á strictly convex on some open interval K µ U .

The following lemma and remark shows that if Áwere twice continuously di¤erentiable,as it is likely to be in any application, then Axiom 5 is actually implied by the other axiomsand is not an additional assumption.

Lemma 1 Suppose Á is twice continuously di¤erentiable. If Á is not linear, then Á iseither strictly concave or convex over some open interval.

Remark 2 It follows immediately from Proposition 2 and Lemma 1 that under twicecontinuous di¤erentiability of Á, Axioms 1�4 imply Axiom 5.

It is perhaps worth noting that there exist functions on [0; 1] that are strictly increas-ing, continuous and almost everywhere di¤erentiable for which Lemma 1 does not hold.It can be shown that this is the case, for example, of the function presented in Takacs(1978).

3 Ambiguity: ambiguous events and ambiguous acts

We have mentioned that an attractive feature of our model is that it allows one to separateambiguity from ambiguity attitude. In this section we concentrate on the ambiguity part.First, we propose a preference based de�nition of ambiguity. We then show that thisnotion of ambiguity has a particularly simple characterization in our model and go onto present results to demonstrate appropriateness of the de�nition in the context of ourmodel.

What makes an event ambiguous or unambiguous by our de�nition rests on a test ofbehavior, with respect to bets on the event, inspired by the Ellsberg 2-color experiment(Ellsberg (1961)). The role corresponding to bets on the draw from the urn with theknown mixture of balls is played here by bets on events in ­£ B. To �x ideas, it mighthelp to think of two distinct, complementary pairs of events in ­ £ B, fB;Bcg andnB̂; B̂c

o, as analogous to events corresponding to two di¤erent (known) mixtures. An

event E 2 § is deemed ambiguous if, for instance, as in the modal behavior observed inthe Ellsberg experiment, betting on E is preferred to betting on some event B in ­£B,and betting on Ec is also preferred to betting on Bc: The proposition following thede�nition shows an equivalent, shorter, form of the de�nition which essentially followsfrom the continuity of Á, which is assured in our model given the �rst three axioms.This equivalent form, though it lacks immediate intuitive identi�cation with the Ellsbergexperiment, adds clarity to our understanding of what makes an event unambiguous: anevent is unambiguous if it is possible to calibrate the likelihood of the event with respectto events in ­£ B.

14

Notation 3 If x; y 2 X and A 2 §, xAy denotes the binary act which pays x if s 2 Aand y otherwise.

De�nition 5 An event E 2 § is unambiguous if, for each event B 2 ­ £ B, andfor each x; y 2 X such that ±x  ±y; either, [xEy  xBy and yEx Á yBx] or,[xEy Á xBy and yEx  yBx] or [xEy » xBy and yEx » yBx] : An event is ambiguous

if it is not unambiguous.

Proposition 3 Assume º satis�es the conditions in Theorem 1. An event E 2 § isunambiguous if and only if for each x and y with ±x  ±y,

xEy » xBy () yEx » yBx: (2)

whenever B 2 ­£ B.

If º satis�es Axioms 1 through 3 then all events in ­£B are unambiguous. Given anyparticular preference relation, it of course may be checked using our de�nition whether anevent in ­£B is unambiguous. We also observe here that the role of B in our de�nitionmay be played equally well by some other rich set of events over which preferences displaya likelihood relation representable by a probability measure. Furthermore, the productstructure of our state space also does not play an essential role in formulating such ade�nition. In general, replace ­£ B with the desired alternative set.

The next theorem relates ambiguity of an event to event probabilities in our repre-sentation.

Theorem 2 Assume º satis�es the conditions in Theorem 1. If the event E is ambiguousaccording to De�nition 5, then there exist ¹-non-null sets I 0 µ I and I 00 µ I and ° 2(0; 1), such that ¼i(E) < ° for all i 2 I 0 and ¼j(E) > ° for all j 2 I 00. If the event E isunambiguous according to De�nition 5 then, provided º satis�es Axioms 4 and 5 and isnot smoothly ambiguity neutral, there exists a ° 2 [0; 1] such that ¼i(E) = ° for ¹ almostall i.

Thus, in our model, if there is agreement about an event�s probability then thatevent is unambiguous. Furthermore, if º has some range over which it is either strictlysmoothly ambiguity averse or strictly smoothly ambiguity loving then disagreement aboutan event�s probability implies that the event is ambiguous. Note, in the case where Iis �nite (i.e., the support of ¹ is �nite, so that ¦ = f¼igi2I contains a �nite number ofprobabilities) the meaning of disagreement about an event�s probability in the theoremabove simpli�es to: there exists i; j 2 I such that ¼i(E) 6= ¼j(E).

To understand why conditions are needed for one direction of the theorem think of thecase of ambiguity neutrality i.e., Á linear. Recall that in this case, even if the measures in¦ disagree on the probability of an event, the decision maker behaves as if he assigns thatevent its ¹-average probability. Recall that Lemma 1 and Remark 2 showed that underconditions likely to be assumed in any application (twice continuous di¤erentiability ofthe function Á and Axiom 4) ambiguity neutrality is the only case where there will fail tobe a range of strict ambiguity aversion (or love) and so the only case where disagreementabout an event�s probability will not imply that the event is ambiguous.

15

We say that a collection of events A µ § is a ¸-system if (i) S 2 A; (ii) A 2 A )Ac 2 A; and (iii) A1; A2 2 A and A1 \ A2 = ? ) A1 [ A2 2 A. This could be calleda �nite ¸-system since (iii) requires only closure under �nite disjoint unions rather thanthe closure under countable disjoint unions required in e.g., Billingsley (1986), p. 36.It has been widely argued in the literature (e.g., Epstein and Zhang (2001), Ghirardatoand Marinacci (2002), Zhang (2002)) that the maximal collection of unambiguous eventsforms a ¸-system and not necessarily an algebra. Hence we include the following corollaryto Theorem 2 which observes that in our model, given our de�nition of ambiguous events,the collection of all unambiguous events does form a ¸-system.

Corollary 2 Suppose º satis�es Axioms 1 through 5. Let ¤ µ § be the collection of allunambiguous events in §. Then ¤ is a (�nite) ¸-system.

Next, we identify an unambiguous act as an act which is measurable with respect to¤, the collection of all unambiguous events in §. Notice, if an act f is unambiguous thenthe induced distribution ¹f is degenerate.

De�nition 6 An act f 2 F is an unambiguous act if it is measurable with respect to¤: Let H be the set of all unambiguous acts.

To gain further insight into our de�nition of an ambiguous event and its appropri-ateness in the context of our model, we introduce the following concept combining thestandard notion of a qualitative probability with an additional necessary condition for theexistence of a probability and likelihood revealed through bets on events. (Conditions(i) through (iv) in the de�nition de�ne a qualitative probability, condition (v) is alsonecessary for the existence of a representing probability and condition (vi) connects thisqualitative probability with º over bets.)

De�nition 7 We call a binary relation ºq on a ¸-system A µ § a qualitative proba-

bility relation for º if it is (i) complete and transitive, (ii) S Âq ; , (iii) A ºq ; forA 2 A; (iv) for A;B;C 2 A; if A \ C = B \ C = ; then

A Âq B () A [ C Âq B [ C

(v) for A;B 2 A;A Âq B () Bc Âq A

c

and (vi) for A;B 2 A; A ºq B if there exist consequences x; y with ±x  ±y such that

xAy º xBy:

º is said to have probabilistic beliefs if there exists a qualitative probability relation forº on all of §.

If A is an algebra, conditions (i)-(iv) imply (v) (see e.g., Kreps (1988), p. 118).However, as observed by Zhang (1999), this is not true if A is merely a ¸-system. Since(v) is a necessary condition for the existence of a probability representing ºqon A, itmakes sense to include it here. In any event, the result below on the existence of aqualitative probability relation for º is false without condition (v).

The next result uses this preference based notion of qualitative probability relation togive an alternative characterization of ambiguous events in our setting.

16

Corollary 3 Assume º satis�es Axioms 1 through 5. Fix an event E µ §. Then E isunambiguous if and only if there exists a qualitative probability relation for º on some¸-system that is a superset of fE;Ec;­£ Bg.

Remark 3 The �if� part of the claim in Corollary 3 does not depend on º satisfyingAxioms 1 through 5.

The existence of a qualitative probability relation for º on a ¸-system containingfE;Ec;­£ Bg intuitively means that the event E can be compared in a consistent waywith the rich set of events in the Borel ¾-algebra B:

The most important class of preferences exhibiting probabilistic beliefs are the prob-abilistically sophisticated preferences of Machina and Schmeidler (1992). Besides prob-abilistic beliefs, they also require some additional conditions which we do not considerbecause they are super�uous for our purposes. Given Axioms 1 through 5, the only depar-ture from expected utility that may arise in our model is one due to ambiguity sensitivebehavior, behavior that is not probabilistically sophisticated, as formally detailed in thefollowing corollary.

Corollary 4 Assume º satis�es Axioms 1 through 4. Consider the following four prop-erties:

(i) º has probabilistic beliefs

(ii) º is probabilistically sophisticated

(iii) º has a subjective expected utility representation

(iv) For each event E, there exists a ° 2 [0; 1] such that ¼i(E) = ° for ¹-almost-all i.

Then,(iv) =) (iii) =) (ii) =) (i):

Moreover, all four properties are equivalent whenever º satis�es Axiom 5(ii) or (iii),while the �rst three are equivalent (and true) whenever º satis�es Axiom 5(i).

When Á is linear (º satis�es Axiom 5(i)), recall that reduction applies and º has aSEU representation with the subjective prior on E equal to

RI¼i (E) d¹: Hence, if Á is

linear º is SEU even if all the ¼i�s do not agree. This is meaningful since all it says isthat if the decision maker is ambiguity neutral his behavior is indistinguishable from anSEU maximizer even if the information is ambiguous.

4 Comparison of ambiguity attitudes

In this section we study di¤erences in ambiguity aversion across decision makers. Foreach decision maker we consider the entire family fºIgIµ[0;1], where each ºI correspondsto the state of information given by I. Throughout the section (with the exception ofProposition 5), we assume Axiom 4 holds, in addition to the �rst three axioms. Hence,

17

ambiguity attitudes do not depend on the support I of ¹, that is, on the particularinformation available to the decision makers.

We begin with our de�nition of what makes one preference order more ambiguityaverse than another. The idea behind it is that if two decision makers are such thatthey share the same attitude to risk and the same beliefs and yet one of the two is moreaverse to acts with uncertain consequences, then the comparatively �extra� aversion touncertainty is due to a relatively greater aversion to ambiguity.

De�nition 8 Let A and B be two decision makers whose families©ºA

I

ªIµ[0;1]

and©ºB

I

ªIµ[0;1]

share the same vN-M utility function u and the same measures ¹I over each I. We saythat decision maker A is more ambiguity averse than B if

f ºAI ±x =) f ºB

I ±x (3)

for every f 2 F , every x 2 C, and every closed subset I µ [0; 1].

We can now state our comparative result, which shows that di¤erences in ambiguityaversion across decision makers are captured by the relative concavity of their functionsÁ, thus showing that the concavity of Á plays here the role of the concavity of utilityfunctions in standard risk theory.

Theorem 3 Let A and B be two decision makers whose families©ºA

I

ªIµ[0;1]

and©ºB

I

ªIµ[0;1]

share the same vN-M utility function u and the same measures ¹I over each I. Then, decisionmaker A is more ambiguity averse than B if and only if

ÁA = h ± ÁB

for some strictly increasing and concave h : ÁB (U)! R.

Using results from standard risk theory we get the following corollary as an immediateconsequence of Theorem 3.

Corollary 5 Suppose the hypotheses of Theorem 3 hold. If ÁA and ÁB are twice contin-uously di¤erentiable, then decision maker A is more ambiguity averse than B if and onlyif, for every x 2 U ,

¡Á00A (x)

Á0A (x)¸ ¡

Á00B (x)

Á0B (x):

Analogous to risk theory, we will call the ratio

¸(x) = ¡Á00 (x)

Á0 (x)

the coe¢cient of ambiguity aversion at x 2 U .

Corollary 6 Under Axioms 1-4, Á is concave if and only if the decision maker is moreambiguity averse than an expected utility decision maker, that is, a decision maker all ofwhose associated preferences ºI are expected utility.

18

Remark 4 Corollary 6 connects our de�nition of smooth ambiguity aversion (De�nition4) to the comparative notion of ambiguity aversion in De�nition 8. It shows that theyagree, with expected utility taken as the dividing line between ambiguity aversion andambiguity loving. Corollary 4, in Section 3, shows that if Axiom 5 holds nothing wouldchange if we were to take probabilistic sophistication, rather than expected utility, as thebenchmark.

We close the section by considering the two important special cases of constant andextreme ambiguity attitudes. We begin by de�ning a behavioral notion of constant am-biguity attitude.

De�nition 9 Suppose acts f , g, f 0, g0 and k 2 R are such that, for each ! 2 ­ andr 2 [0; 1),

u(f 0(!; r)) = u(f(!; r)) + k

u(g0(!; r)) = u(g(!; r)) + k:

We say that the decision maker displays constant ambiguity attitude if, for each closedsubset I µ [0; 1],

f ºI g () f 0 ºI g0:

To see the spirit of the de�nition notice, by bumping up utility (not the raw payo¤s)in each state by a constant amount we achieve a uniform shift in the induced distributionover ex-ante evaluations, i.e.,

¹f 0 = ¹f + k and ¹g0 = ¹g + k

The intuition of constant ambiguity attitude is that the decision maker views the �am-biguity content� in ¹f and its �translation� ¹f + k to be the same. Next we show thatconstant ambiguity attitudes are characterized by a negative exponential Á. It is of someinterest to note that the proposition does not assume that Á is di¤erentiable.

Proposition 4 The decision maker displays constant ambiguity attitude if and only ifthere exists an ® 6= 0 such that, for all x 2 U , either Á (x) = x or Á (x) = ¡ 1

®e¡®x, up to

positive a¢ne transformations.

We now turn to extreme ambiguity attitudes. The Maxmin expected utility model(e.g., Gilboa and Schmeidler (1989)) with a given set of measures ¦, may be viewed asan extreme case of our model as the ambiguity aversion tends to in�nity, as shown in thefollowing proposition. It may similarly be shown that Maxmax expected utility with agiven set of measures ¦, may be viewed as the case where ambiguity aversion tends tonegative in�nity (extreme ambiguity love).

Proposition 5 Let º be an ordering on F satisfying Axioms 1, 2 and 3. Then thereexists a sequence fºng

1n=1 of orderings on F satisfying Axioms 1, 2 and 3, with º1=º,

such that

(i) all ºn share the same vN-M utility function u and the same measure ¹,

19

(ii) limn ¸n (x) = +1 and ¸n (x) ¸ ¸n¡1 (x) for all n ¸ 1 and all x 2 U .

Moreover, given any f and g in F if it holds, eventually, that f ºn g, then,

mini2I

E¼iu (f) ¸ mini2I

E¼iu (g) ;

whilemini2I

E¼iu (f) > mini2I

E¼iu (g)

implies that, eventually, f Ân g.

5 Relating to other notions of ambiguity

In this section, we compare, in the context of our model, our de�nition of ambiguitywith the de�nitions proposed in Epstein and Zhang (2001) and Ghirardato and Marinacci(2002). First, we will relate our de�nition to the behavioral notion of ambiguity developedin Epstein and Zhang (2001). Their notion of ambiguity was designed to apply to a widevariety of models of preferences.

De�nition 10 (Epstein-Zhang (2001)) An event T is unambiguous if: (a) for all dis-joint subevents A, B of T c, acts h, and outcomes x¤; x; z; z0;0

BB@x¤ if s 2 Ax if s 2 B

h (s) if s 2 T cn (A [ B)z if s 2 T

1CCA %

0BB@

x if s 2 Ax¤ if s 2 Bh (s) if s 2 T cn (A [B)z if s 2 T

1CCA

)

0BB@

x¤ if s 2 Ax if s 2 B

h (s) if s 2 T cn (A [ B)z0 if s 2 T

1CCA %

0BB@

x if s 2 Ax¤ if s 2 Bh (s) if s 2 T cn (A [B)z0 if s 2 T

1CCA ;

and (b) the condition obtained if T is everywhere replaced by T c in (a) is also satis�ed.Otherwise, T is ambiguous .

This de�nition works by looking for conditional likelihood reversals over events inthe complement of the event being tested for ambiguity. Our de�nition instead looksfor likelihood reversals involving the event being tested for ambiguity and events in B.Note that such a strategy was not available to Epstein and Zhang since they wished theirde�nition to have power in environments that might lack an appropriately rich set ofunambiguous events.

Is our de�nition of ambiguity identical to that of Epstein and Zhang (2001) when ap-plied to our model of preferences? The answer is no. How do they di¤er? It is of interestto note at the outset, given the many discussions comparing notions of ambiguity dueto Epstein and Zhang (2001) and Ghirardato and Marinacci (2002), the reason for thedi¤erence cannot be that our de�nition confounds ambiguity with probabilistically sophis-ticated departures from expected utility. As we have shown in the previous subsection,whenever our de�nition identi�es an event as ambiguous, behavior is not probabilistically

20

sophisticated. Nevertheless, one might still expect, since we can take advantage of therich structure of our model, that any di¤erence would lie in the direction of our de�nitionclassifying more events as ambiguous than Epstein and Zhang. This is part of the story:in Example 2 we show that there may be some events that are ambiguous according toour de�nition that are not according to Epstein and Zhang.

Example 2 Let ­ = f!1; : : : ; !ng. The measure ¹ assigns probability 12to i = 0 and 1

2

to i = 1 where ¼0 and ¼1 yield marginals on ­ of

¼0 (!1) = ¸1; ¼0 (!j) =1¡ ¸1n¡ 1

; j 6= 1

and

¼1 (!1) = ¸2; ¼1 (!j) =1¡ ¸2n¡ 1

; j 6= 1;

respectively with ¸1 < ¸2. The utility function is u(x) = x (risk neutrality). The functionÁ is Á(x) = ¡e¡®x with ® > 0.

Since ¼0 (!1) = ¸1 < ¸2 = ¼1 (!1)and Á is strictly concave, Theorem 2 implies thatthe event !1 £ [0; 1) is ambiguous according to the de�nition in this paper. We nowdemonstrate that it is unambiguous according to the de�nition of Epstein and Zhang(2001) (De�nition 10). To this end, consider T = !1£[0; 1) in their de�nition. Notice thatall events in T c are assigned the same relative weights under ¼0 as under ¼1. Speci�cally,for any E µ T c; ¼0(E) = 1¡¸1

1¡¸2¼1(E): Under the speci�ed preferences, the top pair of acts

in De�nition 10 is evaluated according to

¡1

2e¡®(¸1z+¼0(A)x

¤+¼0(B)x+Rs2Tcn(A[B) h(s)¼0(s)) ¡

1

2e¡®(¸2z+¼1(A)x

¤+¼1(B)x+Rs2Tcn(A[B) h(s)¼1(s))

(4)and

¡1

2e¡®(¸1z+¼0(A)x+¼0(B)x

¤+Rs2Tcn(A[B) h(s)¼0(s)) ¡

1

2e¡®(¸2z+¼1(A)x+¼1(B)x

¤+Rs2Tcn(A[B) h(s)¼1(s))

(5)respectively. Substituting ¼0(E) = 1¡¸1

1¡¸2¼1(E) and simplifying yields (4) ¸ (5) if and only

if¼1(A)x

¤ + ¼1(B)x ¸ ¼1(A)x+ ¼1(B)x¤: (6)

The evaluation of the lower pair of acts in De�nition 10 di¤ers only by substituting z0

for z. It is not hard to show that preference in the lower pair is also determined by (6).Therefore condition (a) of the de�nition is satis�ed. Setting T = (!1£ [0; 1))c and notingthat, again, all events in T c will be assigned the same relative weights under any ¼ leadsto the conclusion that condition (b) of the de�nition holds as well. Therefore, !1£ [0; 1)is unambiguous according to De�nition 10. ¥

The intuition behind this example is that Epstein and Zhang (2001) ambiguity requiresa reversal in the relative likelihoods of two events lying in the complement of the candidateambiguous event. The preferences in the example have the property that the ambiguity onthe candidate event a¤ects the likelihoods of all events in its complement in the same way� thus relative likelihoods in the complement are unchanged. In the formal result below,

21

we show that such situations are in some sense rare in that by perturbing the beliefs ofthe decision maker even slightly one may have the ambiguity generate the di¤erences inrelative likelihoods needed for De�nition 10.

This is not the whole story however. We show in Example 3 that the Epstein andZhang de�nition may, more surprisingly, classify as ambiguous some events that areunambiguous according to our de�nition. This may appear somewhat strange, giventhat if E is unambiguous in our framework (with º satisfying Axioms 1 through 5)then preferences restricted to acts measurable with respect to the ¸-system generated byf;; E;Ec;­£ B; Sg are probabilistically sophisticated (and, in fact, are expected utility).

Example 3 Let ­ = f!1; !2; !3; !4g. The measure ¹ assigns probability 12to i = 0 and

12to i = 1 where ¼0 and ¼1 yield marginals on ­ of

¼0 (!1) = 0:5; ¼0 (!2) = 0:22; ¼0 (!3) = 0:18; ¼0 (!4) = 0:1

and¼1 (!1) = 0:5; ¼1 (!2) = 0:02; ¼1 (!3) = 0:18; ¼1 (!4) = 0:3

respectively. The utility function is u(x) = x (risk neutrality). The function Á is Á(x) =ln(x).

Since ¼0 (!1) = ¼1 (!1) = 0:5, the event !1 £ [0; 1) is unambiguous according tothe de�nition in this paper. We now demonstrate that it is ambiguous according tothe de�nition of Epstein and Zhang (2001) (De�nition 10). In their de�nition, let T =!1 £ [0; 1), A = !3 £ [0; 1), B = !4 £ [0; 1), x¤ = 1; x = 0; h(s) = 0; z = 0; and z0 = 100:Calculation shows that for the top pair of acts in their de�nition the act on the left isstrictly preferred to the one on the right, since ln(0:18) > 0:5 ¤ ln(:1) + 0:5 ¤ ln(:3), yetfor the bottom pair of acts the preference is reversed, as ln(50:18) < 0:5 ¤ ln(50:1)+0:5 ¤ln(50:3):¥

What is going on in this example? The function Á(x) = ln(x) does not re�ect aconstant ambiguity attitude. In particular, as the acts under consideration get betterand better in terms of expected utilities, the ambiguity aversion of such a decision makerdiminishes. This is in close analogy to risk theory, since ln(x) displays constant relativerisk aversion but diminishing absolute risk aversion when used as a utility function. In ourexample, A is unambiguous, while B is ambiguous but has a higher average probabilitythan A. When z = 0 the expected utilities of the acts are relatively low under all themeasures and the portion of the ln(x) function that is quite ambiguity averse is therelevant one. Here A gets favored over B. When considering z0 = 100 however, the actsget considerably higher expected utilities and a portion of the ln(x) function that is muchless concave (ambiguity averse) is relevant. In the second case, ambiguity aversion hasdiminished enough that the decision maker is now willing to favor the ambiguous-but-higher-average-probability event, B over the unambiguous one, A. Examples of this kindmay be constructed quite generally when ambiguity aversion is not constant. In such acase, changing payo¤s on any event E, no matter what its ambiguity status, may leadto a conditional likelihood reversal between two events in Ec if at least one of the twoevents in Ec is ambiguous. This occurs because changing payo¤s may change ambiguityattitude thus possibly a¤ecting the decision maker�s ranking of events. If all events in Ec

22

are unambiguous, changing ambiguity attitude cannot a¤ect the ranking of these eventsand thus a conditional likelihood reversal would not occur in this case.

This example suggests that when a rich set of events like B over which the decisionmaker has a probability is available, our approach allows one to distinguish betweenreversals due to the ambiguity of the event being tested and those due to changingambiguity attitude.

As the next theorem shows, the di¤erences identi�ed in the above two examples arein some sense the only ones separating the two de�nitions of ambiguity in our setting.The second example is dealt with by the assumption of constant ambiguity aversion.The �rst example is dealt with by showing that perturbing the beliefs of the decisionmaker by as small amount as one wishes can eliminate this type of disagreement. Weshould note upfront that the statement in part (b) of the theorem below (concerningdi¤erences in the direction of the �rst example) does not go as far as one might hope. Inparticular, the perturbation argument we develop works for events in 2­ £ [0; 1) ratherthan general events in §, assumes that ¹ has a �nite support and, even though theperturbations required are arbitrarily small, we have not been able to rule out that theymight change the ambiguity classi�cation of some compound events (i.e., events not of theform ! £ [0; 1)). This contrasts with part (a) of the theorem (concerning di¤erences inthe direction of the second example), which applies to all events and whose proof does notmake use of the �nite support assumption. Part (a) shows quite strongly that di¤erencesin the direction of the second example indeed stem from non-constant ambiguity attitude.

De�nition 11 An event E µ ­ is null if, for all x; y; z 2 C;µx if s 2 E £ [0; 1)

y if s 2 Ec £ [0; 1)

¶»

µz if s 2 E £ [0; 1)

y if s 2 Ec £ [0; 1)

¶:

Proposition 6 An event E µ ­ is null i¤ ¼i(E £ [0; 1)) = 0, for ¹ almost all i.

Theorem 4 Assume that ­ contains at least three non-null states, that Á (x) = ¡e¡®x;® > 0; and that I is �nite.

(a) If an event E 2 § is unambiguous according to De�nition 5 then it is unambiguousaccording to the de�nition in Epstein-Zhang (2001).

(b) If an event E £ [0; 1) is ambiguous according to De�nition 5, then there existsa sequence of perturbations of ¦, denoted ¦(²n), with limn!1¦(²n) = ¦ andlimn!1 ²n = 0, such that for all n, given ¦(²n)

(i) The event E£ [0; 1) is ambiguous according to the de�nition in Epstein-Zhang(2001), and,

(ii) for each ! 2 ­; ! £ [0; 1) is ambiguous according to De�nition 5 i¤ ! £ [0; 1)was ambiguous according to De�nition 5 given ¦:

An analogous result is true with ® < 0 and Á(x) = e¡®x (constant ambiguity love).Next we relate our behavioral de�nition of ambiguity to the behavioral de�nition

proposed by Ghirardato and Marinacci (2002). The de�nition they propose applies to avery general class of preferences which includes the preferences axiomatized in the presentpaper. In their de�nition, they invoke a notion of a benchmark preference for º which isany SEU preference that is less ambiguity averse than º (in the sense of De�nition 8).

23

De�nition 12 (Ghirardato and Marinacci (2002)) For a given benchmark preference> for º the set of > ¡unambiguous acts, denoted Hgm

> , is the largest subset of Fsatisfying the following two conditions:

(A) For every f 2 Hgm> and every x 2 C,

±x > f () ±x º f and ±x 6 f () ±x ¹ f ;

(B) For every f 2 Hgm> and every g 2 F ; if fg¡1(x) : x 2 Cg µ ff¡1(x) : x 2 Cg, then

g 2 Hgm> .

Theorem 4 in the working paper version of Ghirardato and Marinacci (2002),7 showsthat Hgm

> is actually independent of the particular benchmark > and hence we maydrop the subscript and let Hgm denote the set of acts identi�ed as unambiguous in theGhirardato-Marinacci de�nition. Correspondingly, they de�ne the set of unambiguousevents ¤gm as the collection of all the �upper pre-image� sets of the unambiguous actsf 2 Hgm. (An upper pre-image set of an act f : S ! C is a set of the form

©s : ±f(s) º ±x

ªfor x 2 C.)

In the following theorem we show if º satis�es all our axioms, the collection of unam-biguous events (¤) and the collection of unambiguous acts (H) identi�ed by the de�nitionsin this paper (De�nitions 5 and 6, respectively) are identical to the collections identi�edby the corresponding de�nitions in Ghirardato and Marinacci (2002).

Theorem 5 Suppose º satis�es Axioms 1 through 5. Then, H = Hgm and ¤ = ¤gm.

While Ghirardato and Marinacci (2002) does provide a simple characterization ofambiguous acts and events for case of biseparable preference they do not have a corre-sponding result for general preferences. In particular, the class of preferences axiomatizedin the present paper is not contained in the biseparable class. Hence, the formal relation-ship described above between our de�nitions and the Ghirardato-Marinacci de�nitions ofambiguous acts and events in the context of º, does not follow from the characteriza-tion obtained in Ghirardato and Marinacci (2002). What explains the above result thatthe two de�nitions of ambiguous acts, ours and Ghirardato-Marinacci�s, coincide in ourmodel? The explanation lies in two key properties shared by the de�nitions. One is thataccording to both de�nitions, if two acts are measurable with respect to the same collec-tion of events, then if one of the acts is unambiguous so is the other; the actual payo¤s donot matter only the partition generated by the acts. This property is, for instance, notshared by the Epstein-Zhang de�nition: we saw in Example 3 simply changing payo¤s(without changing the measurability of the act) can a¤ect the ambiguity status of the un-derlying events. The other key property is that the set of unambiguous events, accordingto Ghirardato-Marinacci, is a partition such that the ranking between acts measurablewith respect to that partition is the same, whether given by the benchmark preferencerelation or whether given by the ambiguity sensitive preference relation: behavior withrespect to these events is una¤ected by ambiguity attitude. In our model there is a nat-ural benchmark corresponding to any given preference, namely the SEU relation obtained

7For convenience, we reproduce the relevant section of the theorem in the Appendix as Theorem 7.It appears just before the statement of the proof of our Theorem 5.

24

by making reduction hold (i.e., Axiom 5 (i) ) without changing any other aspect of thegiven preference. Our de�nition of ambiguity identi�es those events as unambiguous forwhich there is no disagreement about probabilities. In our model, ranking between actsmeasurable with respect to such events according to the given preference must coincidewith that corresponding to the benchmark SEU preference.

6 Portfolio Choice Examples

In this section we consider two examples of simple portfolio choice problems. The ex-amples are intended both as a simple, concrete illustration of our framework and assuggestive of the potential of our approach in applications. We focus, in particular, oncomparative statics in risk attitude and in ambiguity attitude.

The environment for the examples is as follows. The space ­ contains two elements,!1 and !2: The measure ¹ assigns probability 1

2to i = 0 and 1

2to i = 1 where ¼0 and ¼1

yield marginals on ­ of

¼0 (!1) =1

4; ¼0 (!2) =

3

4and ¼1 (!1) =

3

4; ¼1 (!2) =

1

4;

respectively. The function u is given by u(x) =

½¡e¡ax if a > 0x if a = 0

: This utility func-

tion displays constant absolute risk aversion with a as the coe¢cient of absolute risk

aversion. The function Á is given by Á(x) =

½¡e¡®x if ® > 0x if ® = 0

: This function may

be said to display constant ambiguity aversion with this terminology justi�ed by Propo-sition 4 of the previous section. ® is thus the coe¢cient of ambiguity aversion.

Table 1 illustrates the acts that will appear in our examples. Each of these acts ismeant to represent the gross payo¤ (in dollars) per dollar invested in a particular assetas a function of the state of the world.

!1 £ [0; 12) !1 £ [1

2; 1) !2 £ [0; 1

2) !2 £ [1

2; 1)

f 2 2 1 1l 3 1 3 1±1:15 1:15 1:15 1:15 1:15

Table 1. Gross $ payo¤ per $ invested for each of three assets.

Observe that f is an example of an ambiguous act, as its payo¤ depends on theambiguous events !1 £ [0; 1) and !2 £ [0; 1). l is an example of an unambiguous, butrisky, act (it is also a lottery). ±1:15 is an example of a constant act, involving neither risknor ambiguity. Thinking of these in terms of assets and asset returns, f re�ects a 100%return when the state of the world s 2 !1£ [0; 1) and 0% otherwise; l re�ects a return of200% with probability 1

2and a return of 0% with probability 1

2; and ±1:15 re�ects a sure

return of 15%.

Example 4: (Allocating $1 between a safe asset & an ambiguous asset)The classic simple example of a static portfolio choice problem is the decision of how

to allocate wealth between a safe asset and a risky asset. As is well known, an increase

25

in risk aversion (here an increase in a) leads more wealth to be invested in the safe asset.Here, the asset underlying f is not only risky but is also ambiguous. ±1:15 is the safeasset. By varying ® and a in this example, we can vary the ambiguity aversion andrisk aversion of the agent respectively. What are the comparative statics results in thisframework? Just as with a risky asset, holding ambiguity aversion (®) �xed, an increasein risk aversion (a) leads more to be invested in the safe asset. Furthermore, holding riskaversion (a) �xed, an increase in ambiguity aversion (®) leads more to be invested in thesafe asset. Table 2 gives a numerical illustration of this e¤ect when risk aversion is �xedat a = 2.

ambiguity aversion (®) amount allocated to safe asset0 0.1327

0.02 0.1329732 0.15894120 0.344943100 0.620079200 0.675267

Table 2: Optimal amount out of $1 allocated to the safe asset as

ambiguity aversion varies holding risk aversion at a = 2.

In this example, ambiguity aversion and risk aversion work in the same direction. If weview the ambiguous asset as a proxy for equities, this example suggests that if observedportfolio allocations between equities and safe assets are rationalized by risk aversion only� ignoring ambiguity aversion � then levels of risk aversion may be overestimated. Thusambiguity aversion may play a role in helping to explain the equity premium puzzle. Anumber of previous papers have noted this possible role for ambiguity aversion, includingChen and Epstein (2002), Epstein and Wang (1994). Also, work including Hansen andTallarini (1999) and Hansen, Turmuhambetova, and Williams (2001) has suggested thatmodel uncertainty plays a similar role in reinforcing risk. While the cited papers arecomplete, dynamic models and we present merely a very simple, static example, onereason to think that our approach may be useful here is the particularly clean separationbetween tastes (risk aversion, ambiguity aversion) and beliefs (¹) it provides, which allowsone to be con�dent in doing comparative statics that the intended feature is all that isbeing varied.¥

Our second example will show that ambiguity aversion and risk aversion do not alwaysreinforce each other. In particular, there can be a trade-o¤ between risk and ambiguity.

Example 5 : (Allocating $1 between a risky asset & an ambiguous asset)Here we consider the allocation problem where the assets available are the risky (but

unambiguous) asset underlying l and the ambiguous asset underlying f . If the agentis both ambiguity neutral (® = 0) and risk neutral (a = 0), the optimal allocation isto invest as much as possible in asset l, since its return would �rst-order stochasticallydominate that of f . However, as risk aversion increases, holding ambiguity aversion �xed,the agent will want to diversify into the asset f since it is not perfectly correlated withl, trading-o¤ expected return against risk. On the other hand, as ambiguity aversion

26

increases, holding risk aversion �xed, the ambiguity about the payo¤ from f drives theagent towards l, as f becomes a less e¤ective diversi�er and less valuable. Thus riskaversion and ambiguity aversion work in opposite directions here. Tables 3 and 4 givenumerical illustrations of these e¤ects.

Coe¢cient of ambiguity aversion (®) Optimal amount allocated to asset l0 0.47577

0.02 0.4758072 0.47951820 0.511136100 0.599734200 0.635793

Table 3. Optimal amount out of $1 allocated to the risky asset as ambiguity

aversion increases holding risk aversion at a = 2

Coe¢cient of risk aversion (a) Optimal amount allocated to asset l0 +1 (1)

0.02 21.2026 (1)2 0.4757720 0.344886100 0.335644200 0.334489

Table 4. Optimal amount out of $1 allocated to the risky asset as risk

aversion increases holding ambiguity aversion at ® = 0.

In this case, if such behavior is examined ignoring ambiguity aversion, the agent�slevel of risk aversion will typically be underestimated. This suggests that ambiguity mayplay a role in explaining the underdiversi�cation puzzle � the �nding that the portfoliosof risky assets that individuals hold are not diversi�ed as much as plausible levels of riskaversion say they should be. One example of the underdiversi�cation puzzle is home-bias,where the assets that are not su¢ciently diversi�ed into are those of companies geograph-ically removed from the investor. If one hypothesizes that investors are ambiguity averseand perceive more ambiguity with increased distance then this could generate home-bias.Generation of underdiversi�cation in the context of a model uncertainty framework ap-pears in Uppal and Wang (2002). Epstein and Miao (2001) generates home-bias in aheterogeneous agent dynamic multiple priors setting. ¥

7 Appendix: proofs and related material

7.1 Theorem 1

By Axiom 3, f º g , f2 º2 g2. By Axiom 2, f2 º2 g2 ,RIv (cf (¼i)) d¹ ¸R

Iv (cg (¼i)) d¹: Hence,

f º g ,

ZI

v (cf (¼i)) d¹ ¸

ZI

v (cg (¼i)) d¹: (7)

27

Given Remark 1, v and u must represent the same ordinal preferences on C: Therefore,v (cf (¼i)) = Á (u (cf (¼i))) for some strictly increasing Á. Since v and u are continuous,so is Á. Substituting for v (cf (¼i)) in (7), we get

f º g ,

ZI

Á (u (cf (¼i))) d¹ ¸

ZI

Á (u (cg (¼i))) d¹: (8)

Now, recall,

±cf (¼i) » lif , u (cf (¼i)) =

Z[0;1)

u (lif (r)) dr:

So,

u (cf (¼i))

=X

x2supp(¼if)

u(x)¼if(x) (9)

=X

x2supp(¼if)

u(x)

24 Xf!jf(!;r)=x for some r2[0;1)g

Zfrjf(!;r)=xg

¼i (f! £ [0; 1)g) dr

35 (10)

=X!2­

·Zr2[0;1)

u (f (!; r)) ¼i (f! £ [0; 1)g) dr

¸: (11)

Thus, substituting (11) into (8),

f º g ,

ZI

Á

ÃX!

Z[0;1)

u (f (!; r)) ¼i (f! £ [0; 1)g) dr

!d¹

¸

ZI

Á

ÃX!

Z[0;1)

u (g (!; r)) ¼i (f! £ [0; 1)g) dr

!d¹

This proves the representation claim in the Theorem. To see the uniqueness propertiesof Á, notice,

v (cf (¼i)) = Á (u (cf (¼i))), Á (y) = v¡u¡1 (y)

¢:

Let ~u = ®u+ ¯ and let y 2 U : Then,¡~u¡1

¢(®y + ¯) = fx : ~u (x) = ®y + ¯g

= fx : ®u (x) + ¯ = ®y + ¯g

= fx : u (x) = yg

= u¡1 (y) :

Hence, 8y 2 U , ~Á (®y + ¯) = (v ± ~u¡1) (®y + ¯) = (v ± u¡1) (y) = Á (y). Finally, if thereexists J µ [0; 1] with 0 < ¹ (J) < 1, then v is unique up to positive a¢ne transformationsaccording to Axiom 2, so, �xing u, Á is as well. ¥

We close this section by stating the result mentioned right after Theorem 1.

28

Theorem 6 Let º and º2 be two binary relations on F and F, respectively. The follow-ing statements, (i) and (ii), are equivalent:

i. Axioms 1, 2 and 3 hold.

ii. There exists a continuous, strictly increasing Á : U ! R, a unique �nitely additiveprobability ¹ : 2[0;1] ! [0; 1], continuous and strictly increasing utility functionsv : C ! R and u : C ! R, such that

(a) Á = v ± u¡1

(b) º2 is represented by the preference functional V 2 : F! R given by

V 2(f) =

ZI

v (f (i)) d¹

(c) º is represented by the preference functional V : F ! R given by

V (f) =

ZU

Á (x) d¹f

=

ZI

Á

"X!

Z[0;1)

u (f (!; r)) ¼i (f! £ [0; 1)g) dr

#d¹

´ E¹Á (E¼iu ± f) :

If there exists J µ [0; 1] with 0 < ¹ (J) < 1, then v is unique up to positive a¢netransformations. Moreover, u is unique up to positive a¢ne transformations and if~u = ®u + ¯, ® > 0, then the associated ~Á is such that ~Á(®y + ¯) = Á (y), wherey 2 U .

7.2 Lemma 1, Propositions 1 and 2

We begin with a useful lemma (see Theorems 88 and 91 in Hardy, Littlewood, and Polya(1952)).

Lemma 2 Let Á : A µ R ! R be a continuous function de�ned on a convex set A.Then, Á is concave (strictly concave) if and only if there exists ¸ 2 (0; 1) such that, forall x; y 2 A with x 6= y,

Á (¸x+ (1¡ ¸) y) ¸ (>)¸Á (x) + (1¡ ¸)Á (y) : (12)

Next we prove Propositions 1 and 2.

Proof of Proposition 1. (i) implies (ii): By the Jensen inequality, Á¡R

xd¹f¢¸R

Á (x) d¹f . Thus, Á¡e¡¹f¢¢¸RÁ (x) d¹f , which in turn implies ±

u¡1(e(¹f)) º f by

Theorem 1.(ii) implies (i): Suppose I consists of two mutually singular probability measures

¼0 and ¼00, i.e., there is some event E with ¼0 (E) = 1 and ¼00 (E) = 0. Given any

29

x; y 2 U let a = u¡1 (x) and b = u¡1 (y). Hence, a; b 2 C and so f ´ aEb 2 F . Then,u (cf (¼

0)) = u (a) = x and u (cf (¼1)) = u (b) = y. Since, by de�nition, ¹I has fullsupport on I, there is ¸ 2 (0; 1) such that ¹ (0) = ¸ and ¹ (1) = 1¡¸. Thus, ¹I;f (x) = ¸and ¹I;f (y) = 1¡ ¸. By (ii) and the representation,

Á (¸x+ (1¡ ¸) y) ¸ ¸Á (x) + (1¡ ¸)Á (y) : (13)

So, there exists ¸ 2 (0; 1) such that, given any x; y 2 U , Equation (13) holds. By Lemma2, Á is concave. Finally, by Axiom 4, Á is independent of the choice of I above.

Proof of Proposition 2. To prove (1.), apply Proposition 1 and its analoguefor smooth ambiguity love and note that Á both concave and convex is equivalent to Álinear. Now turn to the proof of (2.). Á strictly concave on an open interval J µ Uimplies Á

¡Rxd¹f

¢>RÁ (x) d¹f for all ¹f with non-singleton supp

¡¹f¢µ J by the

strict version of Jensen�s inequality. Thus, Á¡e¡¹f¢¢

>RÁ (x) d¹f , which in turn implies

±u¡1(e(¹f)) Â f for all f 2 F with non-singleton supp

¡¹f¢µ J by Theorem 1. The

reverse direction follows directly from the argument in the proof of Proposition 1 thatsmooth ambiguity aversion implies concavity of Á with the weak inequalities replacedwith strict and attention limited to the restriction of Á to J . Part (3.) follows exactlyas (2.) with concavity replaced by convexity and inequalities reversed.

We close by proving Lemma 1.

Proof of Lemma 1. Suppose Á : U ! R is twice continuously di¤erentiable andis not linear. There exists x0 2 U such that Á00 (x0) 6= 0. For, suppose per contra thatÁ00 (x) = 0 for all x 2 U . Then Á0 (x) = k 2 R for all x 2 U . Hence, Á (x) = kx + c forsome k; c 2 R, a contradiction. We conclude that there is x0 2 U such that Á00 (x0) 6= 0.Since Á00 is continuous, there exists an interval (®; ¯) µ U , with x0 2 [®; ¯], such thatÁ00 (x)Á00 (x0) > 0 for all x 2 (®; ¯), which implies the desired conclusion.

7.3 Proposition 3

Suppose (2) holds. Let E be such that xEy  xBy. By Theorem 1, V (xBy) =Á (u (x)¯ + u (y) (1¡ ¯)), where ¯ = ¼i (B) for all i 2 I. Since Á (u (y)) · V (xEy) ·Á (u (x)), by the continuity of Á there is ¯¤ ¸ ¯ such that

Á (u (x) ¯¤ + u (y) (1¡ ¯¤)) = V (xEy) : (14)

Since ¸ is non-atomic, there is ­ £ B 3 B¤ ¶ B such that ¼i (B¤) = ¯¤ for all i 2 I.

Hence, by (14) and by Theorem 1, xEy » xB¤y. By (2), this implies that yEx » yB¤x.As Á is strictly increasing, Á (u (x) (1¡ ¯¤) + u (y)¯¤) < Á (u (x) (1¡ ¯) + u (y) ¯), andso, by Theorem 1, yB¤x Á yBx. Hence, yEx Á yBx, and we conclude that

xEy  xBy =) yEx Á yBx.

A similar argument proves the converse implication, and so

xEy  xBy () yEx Á yBx.

30

Finally, again a similar argument shows that

xEy Á xBy () yEx  yBx,

as desired. This completes the proof as the �only if� part is trivial.

7.4 Theorem 2

By assumption, º satis�es the conditions in Theorem 1 and so the representation thereapplies. Fix an event E. Suppose that E is ambiguous. This means that there exists aneventB 2 ­£B and x; y 2 X with ±x  ±y such that either [xEy  xBy and yEx º yBx]or [xEy Á xBy and yEx ¹ yBx] or [xEy » xBy and yEx ¿ yBx]. Let ¯ denote ¼i (B)(=¼j (B) for all j 2 I). If ¼i (E) were equal to some �xed ® 2 [0; 1] for ¹-almost-all i, then,by the representation, for all w; z 2 X,

wEz º wBz () ®u (w) + (1¡ ®)u (z) ¸ ¯u (w) + (1¡ ¯) u (z) :

However, this makes it impossible for E to be ambiguous. Therefore ¼i (E) must varywith i. Speci�cally, if ° =

RI¼i (E) d¹, then there exist ¹-non-null sets I 0 µ I and I 00 µ I

such that ¼i (E) < ° for i 2 I 0 and ¼i (E) > ° for i 2 I 00 and the �rst claim in thetheorem is proved.

Next, suppose that º are not smoothly ambiguity neutral, Axioms 4 and 5 hold andE is unambiguous. Proposition 2 implies that Á is strictly concave (or strictly convex)on a non-empty open interval (u1; u2) µ U . Fix k; l 2 U such that u1 < k < l < u2. Let° =

RI¼i (E) d¹: One can think of ° as the decision maker�s �expected� probability of

the event E. According to our representation of preferences the following is true:

V¡u¡1 (l)­£ [0; °) u¡1 (k)

¢= Á(°l + (1¡ °) k);

V¡u¡1 (l)Eu¡1 (k)

¢=

ZI

Á (¼i(E)l + (1¡ ¼i(E)) k) d¹;

V¡u¡1 (k)­£ [0; °) u¡1 (l)

¢= Á(°k + (1¡ °) l);

V¡u¡1 (k)Eu¡1 (l)

¢=

ZI

Á(¼i(E)k + (1¡ ¼i(E)) l)d¹:

Since Á is strictly concave (the strictly convex case follows similarly) on the interval [k; l],Jensen�s inequality (and the de�nition of °) implies that

V¡u¡1 (l)Eu¡1 (k)

¢· V

¡u¡1 (l) ­£ [0; °) u¡1 (k)

¢and

V¡u¡1 (k)Eu¡1 (l)

¢· V

¡u¡1 (k)­£ [0; °) u¡1 (l)

¢with both inequalities strict if it is not the case that ¼i (E) takes on the same valueeverywhere (speci�cally, ¼i (E) = ° for ¹-almost-all i). Suppose that both inequalitiesare indeed strict. This says that

u¡1 (l)Eu¡1 (k) Á u¡1 (l) ­£ [0; °)u¡1 (k)

andu¡1 (k)Eu¡1 (l) Á u¡1 (k) ­£ [0; °) u¡1 (l)

implying that E is ambiguous, a contradiction. Therefore it must be that ¼i (E) = ° for¹-almost-all i and the second claim in the theorem is proved.

31

7.5 Corollary 2

Let º satisfy Axioms 1 through 4. First consider the case where º also satis�es Axiom5(i). In this case, by Corollary 1, preferences are SEU and hence ¤ coincides with §, a¾-algebra and therefore, a ¸-system. Next consider the case where either Axiom 5(ii) orAxiom 5(iii) holds and let E;F 2 § s:t: E \ F = ?: By Theorem 2, E unambiguousimplies there exists a ° 2 [0; 1] such that ¼i(E) = ° for ¹ almost all i. Also, ¼i(E

c) = 1¡°for ¹ almost all i. Similarly, there exists a °0 2 [0; 1] such that ¼i(F ) = °0 and ¼i(F

c) =1¡ °0and for ¹ almost all i. Since ¼i is a probability measure and E \ F = ?, it followsthat ¼i(E [F ) = ° + °0 for ¹ almost all i. Hence, by Theorem 2, E [F is unambiguous.Finally, note that it follows directly from De�nition 5 that S is unambiguous and that ifE is unambiguous then so is Ec.

7.6 Corollary 3

�If part�: Fix an event E. Suppose that ºq is a qualitative probability relation for º ona ¸-system A ¶ E = fE;Ec;­£ Bg : By properties (i), (v), and (vi) in the de�nition ofa qualitative probability relation, E is unambiguous.

�Only if part�: Let ¤ denote the set of all unambiguous events in §: Suppose thatE, and therefore all elements of E , is unambiguous. By Corollary 2, ¤ is a ¸-system.By our hypothesis, E µ ¤: Let º satisfy Axioms 1 through 4. First, suppose º alsosatis�es either Axiom 5(ii) or Axiom 5(iii). By Theorem 2 for every A 2 ¤ there exists a° (A) 2 [0; 1] such that ¼i(A) = ° (A) for ¹ almost all i. So, a probability ° representingthe likelihood relation on ¸-system ¤ ¶ E exists and therefore a qualitative probabilityrelation for º exists on the same ¸-system. Next, suppose º satis�es Axiom 5(i) (inaddition to Axioms 1 through 4). In this case, by Corollary 1, preferences are SEU andhence a probability ° representing the likelihood relation on ¸-system ¤ ¶ E exists andtherefore a qualitative probability relation for º exists on the same ¸-system. ThereforeE unambiguous implies the existence of a qualitative probability relation for º on a¸-system containing E .

7.7 Corollary 4

By the representation (iv) =) (iii). The implications (iii) =) (ii) =) (i) are obvious.By Theorem 2, (i) =) (iv) when º satis�es Axiom 5(ii) or (iii). To see this, observethat if ºq is a qualitative probability relation for º on §, then by Corollary 3 all eventsE 2 § are unambiguous. By Theorem 2, provided º satis�es Axiom 5(ii) or (iii), thisimplies agreement about each event�s probability. As to the case where º satis�es Axiom5(i), recall that º has an SEU representation if º is smoothly ambiguity neutral.

7.8 Theorem 3

The �if� part follows easily from the Jensen inequality. As to the �only if�, set h (x) =¡ÁA ± Á

¡1B

¢(x) for all x 2 U . The function h is clearly strictly increasing. Moreover, since¡

Á¡1A ± ÁA¢(x) = x =

¡Á¡1B ± ÁB

¢(x) for all x 2 U , we have ÁA = h ± ÁB. We want to

show that h is concave if and only if A is more ambiguity averse than B.

32

By De�nition 8,RÁAd¹f ¸ ÁA (u (x)) implies

RÁBd¹f ¸ ÁB (u (x)) for all f 2 F

and x 2 C. Since U is an interval, given any f 2 F there exists xf 2 C such thatRÁAd¹f = ÁA (u (xf)). Hence, f »1 ±xf and so, by (3),

RÁBd¹f ¸ ÁB (u (xf )). In turn

this implies that, for all f 2 F ,

Á¡1B

µZÁBd¹f

¶¸ Á¡1A

µZÁAd¹f

¶

and so

h

µZÁBd¹f

¶¸

ZÁAd¹f =

Z(h ± ÁB) d¹f : (15)

Let ÁB (x) ; ÁB (y) 2 ÁB (U). By proceeding as in the proof of Proposition 1, there is aset I, an act f and a ¸ 2 (0; 1) such that ¹If (x) = ¸ and ¹If (y) = 1 ¡ ¸. Hence, Eq.(15) reduces to

h (¸Á (x) + (1¡ ¸)Á (y)) ¸ ¸h (Á (x)) + (1¡ ¸)h (Á (y)) :

Since ÁB (U) is an interval, by Lemma 2 we conclude that h is concave.

7.9 Corollary 6

Construct a family of expected utility preferences fºeuI gIµ[0;1] as follows: Fix u and ¹I so

that they match those for ºI and take Áeu to be the identity. Suppose Á is concave. ThenÁ is an increasing, concave transformation of Áeu. By Theorem 3, º is more ambiguityaverse than ºeu. In the other direction, suppose º is more ambiguity averse than someºeu. Then by Theorem 3, Á is an increasing, concave transformation of Áeu. Since Áeu

must be linear (as Axiom 4 implies Á is the same for each I µ [0; 1]), this implies Á isconcave.

7.10 Proposition 4

W.l.o.g., assume that U = [0; 1]. Let k 2 (0; 1) and set Uk = [0; 1¡ k]. Let Ck µ C besuch that u (Ck) = Uk and consider

Fk = ff 2 F : f (s) 2 Ck for each s 2 Sg :

De�ne ºkI on F

k as follows: f ºkI g if and only if

ZÁk

ÃX!

Z[0;1)

u (f (!; r)) ¼i (f! £ [0; 1)g) dr

!d¹I

¸

ZÁk

ÃX!

Z[0;1)

u (g (!; r))¼i (f! £ [0; 1)g) dr

!d¹I ;

33

where Ák (x) = Á (x+ k) for each x 2 Uk. We have:

f º kI±x ()

ZÁk

ÃX!

Z[0;1)

u (f (!; r)) ¼i (f! £ [0; 1)g) dr

!d¹I ¸ Ák (u (x))

()

ZÁ

ÃX!

Z[0;1)

(u (f (!; r)) + k) ¼i (f! £ [0; 1)g) dr

!d¹I ¸ Á (u (x) + k)

()

ZÁ

ÃX!

Z[0;1)

u (f 0 (!; r))¼i (f! £ [0; 1)g) dr

!d¹I ¸ Á (u (x) + k)

() f 0 ºI ±u¡1(u(x)+k) () f ºI ±x;

where the last equivalence follows from De�nition 9. Hence, ºk is as ambiguity averse asº, when restricted to Fk. By Theorem 3, there exist a (k) > 0 and b (k) 2 R such that,for all x 2 [0; 1¡ k],

Á (x+ k) = Ák (x) = a (k)Á (x) + b (k) . (16)

Since k was arbitrary, we conclude that the functional equation (16) holds for all k 2 (0; 1)and all x 2 (0; 1) such that x+k · 1. This is a variation of Cauchy�s functional equation(see p. 150 of Aczel (1966)), and its only strictly increasing solutions are (up to positivea¢ne transformations) Á (x) = x or Á (x) = ¡ 1

®e¡®x, ® 6= 0 .

7.11 Proposition 5

The proposition is a consequence of the following lemma, which we prove for the sake ofcompleteness.

Lemma 3 Let ¢(X) be the set of all �nitely supported probabilities de�ned on a conse-quence space X. Let ¸n be the Arrow-Pratt coe¢cient of un. Let % be a vN-M orderingon ¢(X). There exists a sequence f%ng

1n=1 of vN-M orderings on ¢(X), with %1=%,

such that

i all %n agree on X,

ii limn ¸n (x) = +1 and ¸n (x) ¸ ¸n¡1 (x) for all n ¸ 1 and all x 2 X,

Given any p and q in ¢(X), if it holds, eventually, that p %n q, then

minx2supp(p)

u (x) ¸ minx2supp(q)

u (x) ; (17)

whilemin

x2supp(p)u (x) > min

x2supp(q)u (x) (18)

implies that, eventually, p Ân q.

34

Proof of Lemma 3. Let %n be the vN-M ordering whose vN-M utility index isgiven by u1 (x) = u (x), and un (x) = ¡u (x)¡n for all n > 1. Since un is obtained fromu through the increasing transformation ¡x¡n, point (i) is satis�ed. On the other hand,¸n (x) = n¸ (x), and so point (ii) as well is satis�ed.

We begin by showing that

limn

µ¡

Z¡u (x)¡n dp

¶¡ 1n

= minx2supp(p)

u (x) : (19)

To prove it, �rst observe that, given any bounded X -measurable function à : X ! R, wehave (see, e.g., Lemma 12.1 in Aliprantis and Border (1999)):

limn

µZ(Ã (x))n dp

¶ 1n

= maxx2supp(p)

à (x) . (20)

As,

limn

µ¡

Z¡u (x)¡n dp

¶¡ 1n

=

µmin

x2supp(p)u (x)

¶limn

ÃZ µu (x)

minx2supp(p) u (x)

¶¡ndp

!¡ 1n

;

and

limn

ÃZ µu (x)

minx2supp(p) u (x)

¶¡ndp

!¡ 1n

= limn

1³R ³minx2supp(p) u(x)

u(x)

´ndp´ 1n

= 1;

we conclude that Eq. (19) holds.

On the other hand, since ¡x¡1n is an increasing transformation, it is easy to check

that, given any p and q in ¢(X), we haveR¡u (x)¡n dp ¸

R¡u (x)¡n dq if and only ifµ

¡

Z¡u (x)¡n dp

¶¡ 1n

¸

µ¡

Z¡u (x)¡n dq

¶¡ 1n

: (21)

Hence, p %n q implies Eq. (21), and so if eventually it holds p %n q, we have

limn

µ¡

Z¡u (x)¡n dp

¶¡ 1n

¸ limn

µ¡

Z¡u (x)¡n dq

¶¡ 1n

:

By Eq. (19), we �nally conclude that Eq. (17).To complete the proof, assume that Eq. (18) holds. By Eq. (19), there exists n0 large

enough so that, for all n ¸ n0,¯̄̄¯̄µ¡ Z ¡u (x)¡n dp

¶¡ 1n

¡ minx2supp(p)

u (x)

¯̄̄¯̄ < minx2supp(p) u (x)¡minx2supp(q) u (x)

2;

¯̄̄¯̄µ¡ Z ¡u (x)¡n dq

¶¡ 1n

¡ minx2supp(q)

u (x)

¯̄̄¯̄ < minx2supp(p) u (x)¡minx2supp(q) u (x)

2:

Hence, for all n ¸ n0 it holdsµ¡

Z¡u (x)¡n dp

¶¡ 1n

>

µ¡

Z¡u (x)¡n dq

¶¡ 1n

;

which in turn implies p Ân q.

35

7.12 Proposition 6

Suppose ¼i(E £ [0; 1)) = 0, for ¹ almost all i: By Theorem 1,

f =

µx if s 2 E £ [0; 1)

y if s 2 Ec £ [0; 1)

¶»

µz if s 2 E £ [0; 1)

y if s 2 Ec £ [0; 1)

¶= g;

i¤ ZI

Á

"X!2E

Z[0;1)

u (x) ¼i (f! £ [0; 1)g) dr +X!2Ec

Z[0;1)

u (y) ¼i (f! £ [0; 1)g) dr

#d¹

=

ZI

Á

"X!2E

Z[0;1)

u (z)¼i (f! £ [0; 1)g) dr +X!2Ec

Z[0;1)

u (y)¼i (f! £ [0; 1)g) dr

#d¹

,

ZI

Á

"X!2Ec

Z[0;1)

u (y) ¼i (f! £ [0; 1)g) dr

#d¹

=

ZI

Á

"X!2Ec

Z[0;1)

u (y) ¼i (f! £ [0; 1)g) dr

#d¹:

Therefore E is null. To prove the other direction, suppose E is null. To generate acontradiction assume ¼i(E £ [0; 1)) > 0 for i 2 H where ¹(H) = c > 0: Fix x and z suchthat ±x  ±z. Set y = 0. By Theorem 1,

f =

µx if s 2 E £ [0; 1)

y if s 2 Ec £ [0; 1)

¶»

µz if s 2 E £ [0; 1)

y if s 2 Ec £ [0; 1)

¶= g;

i¤ ZI

Á

"X!2E

Z[0;1)

u (x) ¼i (f! £ [0; 1)g) dr +X!2Ec

Z[0;1)

u (0) ¼i (f! £ [0; 1)g) dr

#d¹

=

ZI

Á

"X!2E

Z[0;1)

u (z)¼i (f! £ [0; 1)g) dr +X!2Ec

Z[0;1)

u (0)¼i (f! £ [0; 1)g) dr

#d¹

,

ZI

Á

"X!2E

Zfr2[0;1)j(!;r)2Eg

u (x)¼i (f! £ [0; 1)g) dr

#d¹

=

ZI

Á

"X!2E

Zfr2[0;1)j(!;r)2Eg

u (z)¼i (f! £ [0; 1)g) dr

#d¹

, u(x) = u(z);

a contradiction.

7.13 Theorem 4

(a) Fix any event E that is unambiguous according to De�nition 5. Suppose,0BB@

x¤ if s 2 Ax if s 2 B

h (s) if s 2 C ´ Ecn (A [ B)z if s 2 E

1CCA º

0BB@

x if s 2 Ax¤ if s 2 Bh (s) if s 2 C ´ Ecn (A [ B)z if s 2 E

1CCA .

36

Then according to our representation,ZI

¡e¡®[u(z)¼i(E)+u(x¤)¼i(A)+u(x)¼i(B)+

RCu(h(s))d¼i]d¹ (22)

¸

ZI

¡e¡®[u(z)¼i(E)+u(x)¼i(A)+u(x¤)¼i(B)+

RCu(h(s))d¼i]d¹:

Since E is unambiguous according to De�nition 5 and Á is strictly concave, Theorem 2implies ¼i (E) is almost everywhere constant in i and hence the u (z) term can be takenoutside the integral on both sides and canceled. This can clearly be done, in exactly thesame way, if z is replaced by some z0 in the acts above. Therefore for any z0 2 C,0

BB@x¤ if s 2 Ax if s 2 B

h (s) if s 2 C ´ Ecn (A [ B)z0 if s 2 E

1CCA º

0BB@

x if s 2 Ax¤ if s 2 Bh (s) if s 2 C ´ Ecn (A [ B)z0 if s 2 E

1CCA .

Since, under De�nition 5, E unambiguous implies that Ec is unambiguous, a similarpreference implication can be derived replacing E by Ec. Thus E is unambiguousaccording to the de�nition in Epstein and Zhang (2001).

(b)8 Fix any E µ ­ such that E £ [0; 1) is ambiguous according to de�nition 5.The strategy for showing that this event is ambiguous according to Epstein and Zhang(2001) after a suitable perturbation of the measures in ¦ is as follows: We eventuallyselect a particular pair of acts for which we will show that there is the type of conditionallikelihood reversal required by Epstein and Zhang (2001). Using our representation, weare able to �nd equations (23, 24) that are necessary for no reversal to occur. It turnsout that there are essentially only two ways these two equations can hold simultaneouslygiven that ¼i (E £ [0; 1)) is not constant. One way is for the relative probabilities of theevents in the complement of E £ [0; 1) to remain unchanged, as happened in Example2. A second, related way is that although the relative probabilities of the events in thecomplement do change, they do so in a way which happens to exactly �cancel out� onlevel sets of ¼i (E £ [0; 1)). We construct the perturbation of ¦ precisely to prevent thesetwo circumstances from happening (and also to maintain the ambiguity/unambiguity ofall events of the form ! £ [0; 1) as required by part b(ii) of the theorem). We then areable to conclude that the needed reversal in preference does occur and E£ [0; 1) is indeedambiguous according to Epstein and Zhang (2001).

We begin by constructing the needed perturbation to prevent both the �cancellingout� on level sets problem (by making sure all such sets are singletons) and the constancyof relative probabilities problem (by direct perturbation). Since E £ [0; 1) is ambiguousand unambiguous events are closed under disjoint unions, there must exist an !1 2 Esuch that !1 £ [0; 1) is ambiguous. Fix such an !1. Choose !2, !3 2 ­ as follows. ByTheorem 2, there exist i; j 2 I such that ¼i (E £ [0; 1)) 6= ¼j (E £ [0; 1)). Given such iand j, since probabilities sum to 1, there must be an !0 2 ­nE such that !0 £ [0; 1) isambiguous. Set !2 equal to some such !0. Choose !3 to be any element of ­n (E [ f!2g)

8We are grateful to Rakesh Vohra for his help in constructing this proof. (Full responsibility lies withthe authors for any mistakes.)

37

such that !3 £ [0; 1) is non-null. Without loss of generality we may assume that such a!3 exists.

9

Given ¦, construct ¦(²n) as follows:Step 1 : Recall that I µ [0; 1]. Divide I into sets of the form

I· ´ fi 2 I j ¼i (E £ [0; 1)) = ·g :

Let J 0 denote the collection of sets I· including exactly each such set that is not asingleton, has · 6= 0 and · 6= 1, and has ¼i (!2 £ [0; 1)) > 0 for all i 2 I·. Let J= fi j i 2 I· for some I· 2 J 0g. Let H 0 denote the collection of sets I· including exactlythose that are not in J 0, and have · 6= 0. Let H = fi j i 2 I· for some I· 2 H 0g. Choose²n > 0 no larger than the minimum of {half the minimum distance between the · values as-sociated with the I· sets, half ofmini2J (1¡ ¼i (E £ [0; 1))), half ofmini2J ¼i (!2 £ [0; 1)),half of mini2H (1¡ ¼i (!2 £ [0; 1))), half of mini2H ¼i (E £ [0; 1)), half of

maxi;j2I

j¼i (!2 £ [0; 1))¡ ¼j (!2 £ [0; 1))j ;

and half of

mini;j2I;!2E s.t. ¼i(!£[0;1))6=¼j(!£[0;1))

j¼i (! £ [0; 1))¡ ¼j (! £ [0; 1))jg:

If there are ! 2 E such that ! £ [0; 1) is unambiguous, then transfer a total mass of ²nto !1 taken evenly from these unambiguous states for each i.

For each i 2 J , let

¼1i (²n) (!1 £ [0; 1)) = ¼i (!1 £ [0; 1)) + i²n

and¼1i (²n) (!2 £ [0; 1)) = ¼i (!2 £ [0; 1))¡ i²n:

Note that the way we have chosen ²n and the fact that I µ [0; 1] ensures that E £ [0; 1),!1 £ [0; 1) and !2 £ [0; 1) remain ambiguous according to de�nition 5 and that no new�bunchings� of ¼i (E £ [0; 1)) have been created.

Step 2 : For each i 2 H, let (if there was mass transferred to !1 from the unambiguousstates in E in the previous step)

¼1i (²n) (!1 £ [0; 1)) = ¼i (!1 £ [0; 1))¡ i²n;

otherwise

subtract the i²n from the ambiguous states in E in any way that doesn�t push any below 0

and¼1i (²n) (!2 £ [0; 1)) = ¼i (!2 £ [0; 1)) + i²n:

9If no such !3 exists, since there were assumed to be at least three ! 2 ­ such that ! £ [0; 1) isnon-null, it must be that at least two of these lie in E. If E £ [0; 1) above is replaced by (­nE)£ [0; 1),then (­nE)£[0; 1) will be ambiguous according to de�nition 5 and !2; !3 2 E with the desired propertieswill exist. The proof may then be carried through for (­nE)£ [0; 1). This will then imply the resultfor E £ [0; 1) since both de�nitions of ambiguous event are closed under complementation.

38

Note that the way we have chosen ²n and the fact that I µ [0; 1] ensures that E £ [0; 1),!1 £ [0; 1) and !2 £ [0; 1) and any other ambiguous ! 2 E remain ambiguous accordingto de�nition 5 and that no new �bunchings� of ¼i (E £ [0; 1)) have been created.

Step 3: For all points and for all i 2 I at which ¼1i (²n) has yet to be de�ned, let

¼1i (²n) (:) = ¼i (:) :

The end result of Steps 1 through 3 is a set of probabilities ¦1 (²n) ´ f¼1i (²n)gi2I thathas the same set of ambiguous events of the form ! £ [0; 1) as ¦ but, except (possibly)where ¼1i (²n) (E £ [0; 1)) = 0, ¼1i (²n) (E £ [0; 1)) takes on distinct values for each i 2 Iand ¼1i (²n) (!2 £ [0; 1)) > 0 for all i 2 I.

Step 4 : This step should only be undertaken if the ratio¼1i (²n)(!3£[0;1))

¼1i (²n)(!2£[0;1))is constant

across all i 2 I such that ¼1i (²n) (E £ [0; 1)) > 0. Notice that since !2 £ [0; 1) is am-biguous, this can only occur if !3 £ [0; 1) is ambiguous as well. Select an m 2 I suchthat

¼1m (²n) (E £ [0; 1)) > 0:

Set¼2m (²n) (!3 £ [0; 1)) = ¼1m (²n) (!3 £ [0; 1)) +

m

2²n

and¼2m (²n) (!2 £ [0; 1)) = ¼1m (²n) (!2 £ [0; 1))¡

m

2²n:

Step 5 : For all points and for all i 2 I at which ¼2i (²n) has yet to be de�ned, let

¼2i (²n) (:) = ¼1i (:) :

Then de�ne ¦(²n) = ¦2 (²n). Notice, because of Step 4, that the ratio ¼i(²n)(!3£[0;1))¼i(²n)(!2£[0;1))

cannot be constant across all i 2 I such that ¼i (²n) (E £ [0; 1)) > 0: This fact will beused in the proof below.

Above, ²n was allowed to be real number strictly between zero and an upper boundthat was the minimum of several terms. To form a sequence, select any sequence ofnumbers f²ng

1n=1 in this range such that the limn!1 ²n = 0: Observe, that given our

construction, limn!1¦(²n) = ¦ (speci�cally, limn!1 ¼i (²n) (E) = ¼i (E) for all i 2 Iand all E 2 §): Now we proceed to show that, given any such ¦(²n) ;the event E £ [0; 1)is ambiguous according to the de�nition in Epstein and Zhang (2001). Consider thepair of acts depicted below (where E; !2 and !3 are as in the construction of ¦(²n) andC ´ (E £ [0; 1))c n ((!3 £ [0; t)) [ (!2 £ [t; 1)))).0

BB@1 if s 2 !3 £ [0; t)0 if s 2 !2 £ [t; 1)0 if s 2 C0 if s 2 E £ [0; 1)

1CCA ,

0BB@

0 if s 2 !3 £ [0; t)1 if s 2 !2 £ [t; 1)0 if s 2 C0 if s 2 E £ [0; 1)

1CCA .

Given our representation of preferences, the decision maker is indi¤erent between thesetwo acts i¤

kXi=1

¡e¡®t¼i(²n)(!3£[0;1))¹ (i) =kX

i=1

¡e¡®(1¡t)¼i(²n)(!2£[0;1))¹ (i) : (23)

39

Let t¤ be the unique t that solves equation (23). Such a t¤exists since the left-hand sideis continuously increasing in t and is equal to ¡1 at t = 0 while the right-hand side iscontinuously decreasing in t and is equal to ¡1 at t = 1. If E £ [0; 1) were unambiguousaccording to Epstein and Zhang (2001), it would be the case that0

BB@1 if s 2 !3 £ [0; t¤)0 if s 2 !2 £ [t¤; 1)0 if s 2 C

u¡1(c) if s 2 E £ [0; 1)

1CCA »

0BB@

0 if s 2 !3 £ [0; t¤)1 if s 2 !2 £ [t¤; 1)0 if s 2 C

u¡1(c) if s 2 E £ [0; 1)

1CCA

for all c 2 U : In terms of our representation, this means

kXi=1

¡e¡®[t¤¼i(²n)(!3£[0;1))+c¼i(²n)(E£[0;1))]¹ (i)

=kX

i=1

¡e¡®[(1¡t¤)¼i(²n)(!2£[0;1))+c¼i(²n)(E£[0;1))]¹ (i) for all c 2 U : (24)

Taking the derivative of both sides of equation (24) with respect to c and evaluating atc = 0,

kXi=1

¡e¡®t¤¼i(²n)(!3£[0;1))¹ (i)¼i (²n) (E £ [0; 1))

=kX

i=1

¡e¡®(1¡t¤)¼i(²n)(!2£[0;1))¹ (i)¼i (²n) (E £ [0; 1)) : (25)

In fact, since equation (24) is an identity in c, we can di¤erentiate both sides as manytimes as we wish while maintaining equality. Calculating this out, we see

kXi=1

¡e¡®t¤¼i(²n)(!3£[0;1))¹ (i) [¼i (²n) (E £ [0; 1))]m

=kX

i=1

¡e¡®(1¡t¤)¼i(²n)(!2£[0;1))¹ (i) [¼i (²n) (E £ [0; 1))]m (26)

for all m = 0; 1; 2; ::: . Since by construction of f¼i (²n)gki=1, either ¼i (²n) (E £ [0; 1)) = 0;

in which case those terms in the system of equations (26) get no weight, or, for theremaining i�s we can strictly order the ¼i (²n)�s by the weight they give to E£ [0; 1) : Leti¤ be the index of the ¼i (²n) that gives the largest such weight. Divide both sides of theeach equation in the system of equations above by [¼i¤ (²n) (E £ [0; 1))]m. We can thenrewrite the system as, for all m = 0; 1; 2; : : : ;

¡e¡®t¤¼i¤(²n)(!3£[0;1))¹ (i¤)

+Xi 6=i¤

¡e¡®t¤¼i(²n)(!3£[0;1))¹ (i) [¼i (²n) (E £ [0; 1))]m

[¼i¤ (²n) (E £ [0; 1))]m

= ¡e¡®(1¡t¤)¼i¤(²n)(!2£[0;1))¹ (i¤)

+Xi 6=i¤

¡e¡®(1¡t¤)¼i(²n)(!2£[0;1))¹ (i) [¼i (²n) (E £ [0; 1))]m

[¼i¤ (²n) (E £ [0; 1))]m: (27)

40

A necessary condition for the above system of equations to hold is that

¡e¡®t¤¼i¤ (²n)(!3£[0;1))¹ (i¤) = ¡e¡®(1¡t

¤)¼i¤(²n)(!2£[0;1))¹ (i¤) : (28)

To see this, observe that for any ± > 0 there exists an M such that for all m > M ,

± >Xi 6=i¤

¡e¡®t¤¼i(²n)(!3£[0;1))¹ (i) [¼i (²n) (E £ [0; 1))]m

¼ni¤ (²n) (E £ [0; 1))

¡Xi 6=i¤

¡e¡®(1¡t¤)¼i(²n)(!2£[0;1))¹ (i) [¼ni (²n) (E £ [0; 1))]m

¼ni¤ (²n) (E £ [0; 1))> ¡±:

This is true because 0 · ¼i (²n) (E £ [0; 1)) < ¼i¤ (²n) (!1 £ [0; 1)) · 1 and all the otherterms are bounded.

Given equation (28), we can cancel the i¤ terms from both sides of the equations inthe system (26). This gives a new system of equations. For this new system �nd the isuch that ¼i (²n) (E £ [0; 1)) gives the largest weight and repeat the above steps to showthat

¡e¡®t¤¼i(²n)(!3£[0;1))¹ (i) = ¡e¡®(1¡t

¤)¼i(²n)(!2£[0;1))¹ (i) ;

for that i: Canceling and repeating k ¡ 2 more times (or until the largest remaining¼i (²n) (E £ [0; 1)) = 0), we �nd

¡e¡®t¤¼i(²n)(!3£[0;1)) = ¡e¡®(1¡t

¤)¼i(²n)(!2£[0;1));

for all i 2 I such that ¼i (²n) (!1 £ [0; 1)) > 0: This is only possible if

¼i (!3 £ [0; 1))

¼i (!2 £ [0; 1))=

1¡ t¤

t¤

for all i 2 I such that ¼i (²n) (!1 £ [0; 1)) > 0: As noticed above in the construction of¦(²n), this cannot be true. Therefore we have a contradiction and it cannot be thatE £ [0; 1) unambiguous according to the de�nition in Epstein and Zhang (2001).

7.14 Theorem 5

We �rst report a result from the working paper version of Ghirardato and Marinacci(2002), where it is stated as Theorem 4.

Theorem 7 Let º be an ambiguity averse monotonic preference relation. For all >1,>2 belonging to the set of benchmark preferences corresponding to º, we have

Hgm>1

= Hgm>2´ Hgm and ¤gm

>1= ¤gm

>2´ ¤gm:

We can now prove Theorem 5.

Proof of Theorem 5. Let >, the benchmark, be the special case of º where

Á is linear. That is, > is represented by V>(f) =P

!

hR[0;1)

u (f (!; r)) driº (!), where

º (!) =RI¼i (f! £ [0; 1)g) d¹ for all ! 2 ­:

41

Suppose Á is linear (i.e., Axiom 5(i) holds). Then Hgm = F . On the other hand, byCorollaries 3 and 4, Á linear means that all events are unambiguous. Hence, H = F aswell. For the rest of the proof we suppose Á is strictly concave or strictly convex oversome open interval J µ U (i.e., Axiom 5(ii) or (iii) holds)

First we prove that H µ Hgm: By Theorem 2, ¼i = ¼j ´ ¹¼ on ¤. Hence, we have, forf 2 H,

V (f) = Á

ÃX!

·Z[0;1)

u (f (!; r)) dr

¸¹¼ (!)

!: (29)

Since Á is strictly increasing, the way º ranks acts belonging to H is equivalently givenby the functional, X

!

·Z[0;1)

u (f (!; r)) dr

¸¹¼ (!) :

Notice, V>(f) as well reduces to the functional (29). Hence, º and > agree on H. SinceH contains all the constant acts this proves that part (A) of De�nition 12 holds for H.On the other hand, it is immediate to see that part (B) holds as well. Thus, since H>

is the largest subset satisfying (A) and (B), it follows that H µ Hgm> . Furthermore, by

Theorem 7, Hgm> = Hgm:

As to the converse inclusion, we consider the case of Á strictly concave on J (thestrictly convex case being similar). We take an act f =2 H and go on to show thatf =2 Hgm

> either. It is enough to consider acts f such that supp¡¹f¢µ J . To see why

this is the case, assume f is such that supp¡¹f¢* J . Let fx1; :::; xng be the range of the

�nite-valued act f . Since u is strictly increasing on the interval C, it is di¤erentiable on C,except on at most a countable subsetM of C. The function u is therefore locally Lipschitzon C ¡M . Since J is an open interval and u is strictly increasing and continuous, u¡1 (J)is an open interval. Hence, u¡1 (J) \ (C ¡M) 6= ;, and so there exists c 2 u¡1 (J) atwhich u is locally Lipschitz. Let (c¡ "; c+ ") be a neighborhood of c over which u islocally Lipschitz. Since J is an open interval, by taking " small enough, we can assumethat [u (c)¡ "; u (c) + "] µ u¡1 (J). It is easy to check that there exist ® 6= 0 and ¯ 2 Rsuch that j®xi + ¯ ¡ cj · " for each i = 1; :::n. Hence, by the local Lipschitz property,ju (®xi + ¯)¡ u (c)j · ", and so¯̄̄

¯Z

u (®f + ¯) d¼ ¡ u (c)

¯̄̄¯ · "

for all probabilities ¼ 2 ¢. Hence,

supp¡¹®f+¯

¢µ

·inf¼2¢

Zu (®f + ¯) d¼; sup

¼2¢

Zu (®f + ¯) d¼

¸µ [u (c)¡ "; u (c) + "] µ J .

Set g = ®f + ¯. As g (s) = g (s0) if and only if f (s) = f (s0) for all s; s0 2 S, we haveff¡1(x) : x 2 Cg = fg¡1(x) : x 2 Cg. Hence, by (B) of De�nition 12, f 2 Hgm

> if and onlyif g 2 Hgm

> and by De�nition 6, f 2 H if and only if g 2 H. All this shows that in what

follows we can assume that supp¡¹f¢µ J .

42

As in De�nition 4, take the constant act ±u¡1(e(¹f)): As f =2 H, either ±

u¡1(e(¹f)) Â

f or ±u¡1(e(¹f)) » f . Suppose, �rst, the preference is strict. Since for the benchmark

preference >, ±u¡1(e(¹f)) and f are indi¤erent, part (A) of De�nition 12 is violated for f .

Next, suppose ±u¡1(e(¹f)) » f . By the strict concavity of Á (on J ¶ supp

¡¹f¢) this

implies that ¹f has a singleton support. For, if ¹f did not have a singleton support thenby strict concavity of Á,

V (f) =

ZÁ (x) d¹f < Á

µZxd¹f

¶= Á

¡e¡¹f¢¢

= V³±u¡1(e(¹f))

´; (30)

and so ±u¡1(e(¹f)) Â f . Hence it remains to consider the case f =2 H with ¹f having

singleton support. We show, next, that in this case the condition (B) of De�nition 12 isviolated for f .

Since f =2 H there exists an x 2 C such that the event f¡1 (x) is ambiguous. Call thatevent A (x). De�ne g : S ¡! C as follows,

g (s) = f (s) if s =2 A (x) ,

g (s) = x0 if s 2 A (x) ,

with x0 =2 ff (s) : s 2 Sg (recall that all our acts are �nite valued). Since A (x) is am-biguous there exist ¹-non-null sets I 0 µ I and I 00 µ I such that for any i 2 I 0 and j 2 I 00,¼i (A (x)) 6= ¼j (A (x)). Since, ¹f has singleton support, for ¹-almost-all i; j, E¼i (u ± f) =E¼j (u ± f) : Hence, for ¹-almost-all i 2 I 0 and j 2 I 00, E¼i (u ± f) = E¼j (u ± f). For allsuch i; j pairs,

E¼iu (g)¡E¼ju (g) = E¼iu (f)¡ ¼i (A (x)) [u (x)¡ u (x0)]

¡E¼ju (f) + ¼j (A (x)) [u (x)¡ u (x0)]

= [u (x)¡ u (x0)] [¼j (A (x))¡ ¼i (A (x))]

6= 0:

Therefore, ¹g does not have a singleton support since it is not true that E¼i (u ± f) =E¼j (u ± f) for ¹-almost-all i; j. By strict concavity of Á (see Eq. 30) we then concludethat ±

u¡1(e(¹g)) Â g. This violates part (B) of the De�nition 12.

Hence, summing up, we may conclude f =2 Hgm> , and so, again by Theorem 7, f =2 Hgm:

This completes the proof that H = Hgm:Since all acts are �nite-valued, the ¸-system ¤ can be viewed as the set of all pre-

image sets of the acts in H. As ¤gm is de�ned as the collection of all pre-image sets ofacts in Hgm, it follows that ¤ = ¤gm.

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