Subjective ambiguity, expected utility and Choquet expected utility

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1 Eonomi Theory 20, (2002) Subjetive ambiguity, expeted utility and Choquet expeted utility Jiankang Zhang Department of Eonomis, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, CANADA ( Jiankang Reeived: Marh 23, 2000; revised version: April 24, 2001 Summary. Using the Savage set up, this paper provides a simple axiomatization of the Choquet Expeted Utility model where the apaity is an inner measure. Two attrative features of the model are its speifiity and the transpareny of its axioms. The key axiom states that the deision-maker uses unambiguous ats to approximate ambiguous ones. In addition, the notion of ambiguity is subjetive and derived from preferenes. Keywords and Phrases: Ambiguity, Expeted utility, Choquet expeted utility, Capaity, Inner measure, λ-system. JEL Classifiation Numbers: C69, D81. 1 Introdution 1.1 Motivation Muh empirial evidene, inspired by Ellsberg [3], shows that the Subjetive Expeted Utility (SEU) model, axiomatized by Savage [15], annot aommodate aversion to unertainty or ambiguity. Reently, a number of generalizations of the standard model have been developed that are axiomati and an aommodate the noted aversion as well. Most notable from the perspetive of this paper is the Choquet Expeted Utility (CEU) model, or expeted utility with respet to nonadditive probabilities or apaities, due to Shmeidler [16] and Gilboa [8]. I am deeply indebted to Larry Epstein for his valuable suggestions and important omments. I am grateful to the finanial support from the Canadian SSHRC, to Chew Soo Hong, M. Marinai, U. Segal, and P. Wakker for valuable disussions and suggestions, to the anonymous referee for suggesting substantial improvements.

2 160 J. Zhang Shmeidler uses the Ansombe-Aumann [1] set-up to deliver the CEU model, but this approah presumes the existene of objetive lotteries. Gilboa avoids this drawbak by using the Savage set-up, but his axioms are hard to interpret. Furthermore, it is not lear from these models why and how aversion to ambiguity leads a deision maker (DM) to use a nonadditive probability measure to represent her beliefs about the likelihoods of events and a CEU model to represent her preferenes over ats. Using the Savage set up, this paper provides an axiomati generalization of SEU that is more restritive than the general CEU model, but that an nevertheless aommodate Ellsberg type behavior. In partiular, our model amounts to Choquet expeted utility theory with the added restrition that the apaity is an inner measure, and the inner measure is generated from a probability measure over λ-systems whih will be defined in Setion 2. The greater speifiity of our model is advantageous from the perspetive of proper model development; one wants models that are as lose as possible to SEU and an still explain the evidene at hand. The freedom within the general CEU model to hoose any apaity means that one an explain almost anything by a suitable hoie of apaity. This embarrassment of rihes is evident partiularly in appliations to market data, where disipline similar to that derived within the SEU framework from the rational expetations hypothesis is laking (see [12]). Another important attrative feature of the model is the simpliity of its axioms, whih failitates understanding of the situations where they do or do not have appeal. There are two lassial extreme approahes to modeling deision making under unertainty the Savage probability based model, and the other extreme of omplete ignorane and riteria suh as the maximin. (See [11, Ch. 13] for larifiation of the meaning of omplete ignorane and for an axiomati haraterization.) Our axiomati model provides an intermediate approah by ombining these two extremes and delivering thereby a utility funtion representing preferenes that has two prinipal features. The first is that the DM has suffiient information about the set of payoff-relevant states to form probabilisti beliefs about the likelihoods of events in A, a olletion of events interpreted as unambiguous. That is, there exists an additive probability measure p on A representing beliefs about likelihoods and suh that an expeted utility funtion with p represents preferenes over unambiguous prospets; tehnially, over ats that are A-measurable. The seond prinipal feature of the utility funtion is that there is omplete ignorane of the state spae besides what is modeled by the measure p on A; a rough interpretation is that the information underlying p is the only information available to the DM. These two features together deliver Choquet expeted utility with apaity given by the inner measure generated by A and p. The two lassial extremes appear as speial ases of our model orresponding to alternative speifiations for A. One obtains the Savage model if A is the olletion of all events and the other extreme of pure ignorane if A onsists of only the empty set and the full state spae. An important aspet of the model is the subjetive nature of A; it is derived from preferenes rather than being speified exogenously. For this purpose, the

3 Subjetive ambiguity, expeted utility and Choquet expeted utility 161 paper adopts a definition of ambiguity and subsequent analysis that are related to but distint from those developed in Epstein and Zhang [5]. Clarifiation of the similarities and differenes is provided there. We emphasize that [5] is silent on the nature of preferenes over ambiguous ats, whih is the fous and main ontribution of this paper. 1.2 Two examples The Ellsberg Paradox The Ellsberg Paradox is desribed here. There are 90 balls in an urn, 30 red ones and the rest of 60 either blak or yellow. One ball will be drawn at random from the urn. The following preferenes over ats are typial, f = f = $100 if s R $0 if s B $0 if s Y $100 if s R $0 if s B $100 if s Y $0 if s R $100 if s B $0 if s Y $0 if s R $100 if s B $100 if s Y = g and = g, where R, B and Y denote the events orresponding to the hosen ball being red or blak or yellow and s refers to the ball that is drawn. The DM piks f rather than g, presumably beause the hane of getting $100 in at f is preisely known to be 1/3, but the hane of getting $100 in g is ambiguous sine any number between 0 and 2/3 is possible. Similarly, she prefers g to f, beause the hane of getting $100 in g is preisely known to be 2/3, whereas the hane of getting $100 in f is ambiguous sine any number between 1/3 and 1 is possible. It an be proved easily that the above preferenes an not be represented by an expeted utility funtion E p u( ) with respet to any stritly inreasing vnm utility indexes u and any additive probability measures p. The key point of the Ellsberg Paradox is that there are two kinds of events unambiguous and ambiguous to the DM sometimes. Intuitively, an event is unambiguous if it has preisely known probability. But the SEU model annot distinguish them sine there is only one additive probability measure defined on all the events. For example, the set of unambiguous events in the Ellsberg Paradox is A = {, {R, B, Y }, {R}, {B, Y }}. The following probability measure p on A = {, {R, B, Y }, {R}, {B, Y }} is natural: p( ) =0, p({r, B, Y }) =1, p({r}) =1/3 and p({b, Y }) =2/3. (1.1) It is also intuitive that she uses an expeted utility model E p u( ) to evaluate her unambiguous ats F ua whih are A-measurable (defined preisely below), where u is a vnm index.

4 162 J. Zhang After evaluating the unambiguous ats, a natural and simple way to evaluate the ambiguous ats she an use is to approximate them by the unambiguous ones from below. 1 That is, the utility of at h is defined by U (h) = sup{e p u(h ): h h F ua }, where h h means h(s) h (s) for all s {R, B, Y }. Aordingly, U (f ) > U (g) and U (f ) < U (g ). That is, our model resolves the Ellsberg Paradox The four olor example 2 The set of unambiguous events in Setion is an algebra. However, this is not true in general, beause the set of unambiguous events annot be expeted to be losed with respet to intersetions. The following example will show this. Suppose there are 100 balls in an urn and that a ball s olor may be blak (B), red (R), grey (G) or white (W). The sum of blak and red ball is 50 and the sum of blak and grey ball is also 50. Similarly, one ball will be drawn at random from the urn. We onsider the following preferenes: f 1 = g 1 = h 1 = $1 if s B $100 if s R $0 if s G $0 if s W $1 if s B $100 if s R $100 if s G $0 if s W $1 if s B $100 if s R $100 if s G $100 if s W f 2 = g 2 = h 2 = $100 if s B $0 if s R $0 if s G $0 if s W $100 if s B $0 if s R $100 if s G $0 if s W $100 if s B $0 if s R $100 if s G $100 if s W,, and. The DM piks f 1 rather than f 2, mainly beause the hane of getting $100 in f 1 is the same as in f 2, but also with additional hane to get $1 in at f 1. The only differene between pairs {g 1,g 2 } and {f 1, f 2 } is the payoff at state G. Though both events {B} and {G} are ambiguous, the ombination of them, however, leads to {B, G} being unambiguous sine it has preisely known probability of one half. Thus, piking at g 2 leads her to get $100 with probability of one half, while piking at g 1 leads her only to get $1 with probability of one half and no idea with how muh probability to get $100. Thus, she prefers g 2 to g 1. Finally, the same reasons as with the pair of ats {f 1, f 2 }, she will prefer h 1 to h 2. An important and interesting phenomenon has happened: Changing the ommon outome on event {G} in the pair of ats {f 1, f 2 } leads the DM to hange 1 Here below reflets her attitude towards ambiguity. In other words, she is ambiguity averse. 2 This example is based on a suggestion by U. Segal.

5 Subjetive ambiguity, expeted utility and Choquet expeted utility 163 the ranking, while hanging the ommon outome on event {G, W } in the pair of ats {f 1, f 2 } leaves the ranking of ats unhanged. What is the key differene between events {G} and {G, W }? Intuitively, event {G} is ambiguous, sine the probability of event {G} is unknown to the DM. While event {G, W } is unambiguous, sine the probability of event {G, W } is known to her. This example suggests that unambiguous and ambiguous events an be derived from preferenes. Roughly, an event A is unambiguous if replaing a ommon outome for all states in A by any other ones does not hange the ranking of the pair of ats being ompared (for formal expression, see Setion 3). Intuitively, the set of unambiguous events is A = {φ, {B, R, G, W }, {B, G}, {R, W }, {B, R}, {G, W }}. However, A is not an algebra, sine it is not intersetion-losed. For example, both {B, G} and {B, R} are unambiguous, but {B} = {B, G} {B, R} is ambiguous. Though A is not an algebra, she still an assign probabilities over it. The following are natural to her: p(s )=1, p(φ) =0,p({B, G}) =p({r, W }) =p({b, R}) =p({g, W }) =1/2. (1.2) If she onforms to our model, then she has an expeted utility funtion E p u( ) to represent her preferenes over F ua and for other ats, she uses the utility funtion U (h) = sup{e p u(h ):h h F ua }. As a result, U (f 1 ) > U (f 2 ), U (g 1 ) < U (g 2 ), and U (h 1 ) > U (h 2 ). 1.3 Related literature The papers in the literature that are most losely related to ours are Sarin and Wakker [14], Jaffray and Wakker [10] and Mukerji [13] 3. Sarin and Wakker also employ unambiguous events, but they do not expliitly derive them and simply assume that there exist the two kinds of events. In addition, Sarin and Wakker s key axiom differs substantially from ours; this is by neessity sine the general CEU model is axiomatized in Sarin and Wakker [14], while we axiomatize the speial ase orresponding to an inner measure. Jaffray and Wakker exploit and adapt the well-known Dempster-Shafer representation of belief funtions to show that CEU with a apaity that is a belief funtion aptures preisely the information struture desribed earlier a lass of unambiguous events and omplete ignorane otherwise. In omparison to the lassi Savage model and the model we present, the formal model in Jaffray & Wakker [10] employs two added primitives taken from the Dempster-Shafer representation. These are an auxiliary spae of states and a (multi-valued) mapping that links it with S and delivers a definition for unambiguous events in S. As a result, verifiation of the axioms requires that the modeler observe or hypothesize these auxiliary 3 The Jaffray-Wakker and Mukerji are similar and we disuss only the former here.

6 164 J. Zhang primitives; suh verifiation is not possible through observation exlusively of hoies between alternative ats over S. This is in ontrast to our model, where all axioms are expressed in terms of preferenes over Savage style ats over the (payoff-relevant) state spae S, without reliane on auxiliary state spaes or ad ho assumptions regarding A. Another differene from our paper is that the two papers deliver different lasses of preferenes. Jaffray and Wakker admit any belief funtion while our axioms deliver inner measures whih are not neessary belief funtions. The use of inner measures as a way of modeling unertainty is due to Fagin and Halpern [6], but their notion of inner measure is more restritive than ours. Ours is generated from a probability measure on a λ-system, while Fagin and Halpern begin with a probability measure on a σ-algebra (or algebra). Fagin and Halpern take the perspetive of the artifiial intelligene literature where beliefs are primitive rather than derived from preferenes. Our ontribution is to show how their analysis may be adapted to a deision theoreti framework with a more general notion of inner measures. This paper proeeds as follows. λ-systems, inner measures and inner integrals are defined next. The axioms and the representation results are desribed in Setion 3. Some examples are provided in Setion 4. All proofs are olleted in appendies. 2 λ-systems, inner measures and inner integrals Let S be a state spae with assoiated σ-algebra given by the power set. The usual way of representing a deision maker s beliefs about the likelihoods of events is by means of an (additive) probability measure on 2 S. This struture is too restritive as desribed in the examples in Setion 1.2. In this paper, we generalize this standard approah in two ways: (i) we relax the additivity of beliefs on all events to additivity on a subset A of 2 S ; (ii) we require that A be a λ-system but not neessarily an algebra. 2.1 λ-systems Definition 2.1. A nonempty lass of subsets A of S is a λ system if λ.1. S A; λ.2. A A = A A; and λ.3. A n A, n =1, 2,... and A i A j =, i /= j = n A n A. This definition and terminology appear in [2, p. 36]. Think of A as the olletion of DM s unambiguous events and that she an assign probability to eah event in A. λ.1 is a usual assumption, sine the DM knows that event S will happen without ambiguity. Assumption λ.2 is natural. If she an assign probability to event A, it is natural for her to assign the differene between the probabilities of S and A to A. The intuition for λ.3

7 Subjetive ambiguity, expeted utility and Choquet expeted utility 165 is that if she an assign probabilities to two disjoint events A and B, then it is natural for her to assign the sum of probabilities of events A and B to the union A B. On the other hand, even if she an assign probabilities to both events A and B, she still annot assign probability to the intersetion A B when A B is ambiguous ; reall the example in the Setion The key differene between a σ algebra and a λ system is that the latter is not required to be intersetion-losed. It seems evident that a λ system is more natural for modeling unambiguous events. 2.2 Probabilities and integrals Though A is not a σ algebra, a probability measure an still be defined on it. Say that a funtion p : A [0, 1] is a ountably additive probability measure over A if: 1. p( ) =0, p(s ) = 1; and 2. p( i=1 A i )=Σ i=1 p(a i ), A i A j = φ, i /= j, i, j =1, 2,... Denote by (S, A, p) aλ system probability spae. Say that a probability p is onvex-ranged if for all A A and 0 < r < 1, there exists B A, B A, suh that p(b) =rp(a). Given a λ-system probability spae (S, A, p), say that a finite-ranged funtion f : S R 1 is A measurable, if for any x R 1, {s S : f (s) x} A. Denote by F = {f : f has finite range} and F ua = {f F : f is A measurable}. The integral of f F ua with respet to p is defined as follows: fdp = Σ i x i p({s : f (s) =x i }). 2.3 Inner and outer measures and integrals Think of a λ-system A as onsisting of all unambiguous events and of the probability measure p as representing the DM s beliefs about the likelihoods of events in A. Suppose that the information underlying p is all that is available to her and that she is extremely pessimisti. Then one way to assess likelihoods for ambiguous events, that is, events not in A, is by means of the inner measure p orresponding to (S, A, p), defined by: For all events B, p (B) sup {p(a) : A A, A B}. (2.1) Note that p = p is additive on A, but generally non-additive outside A. Similarly, if she is extremely optimisti, then she might assess likelihoods for ambiguous events by means of the outer measure p orresponding to (S, A, p), defined by: For all events B,

8 166 J. Zhang p (B) inf {p(a) : A A, A B}. (2.2) The inner, outer measures p (B), p (B) provide lower, upper bounds on the likelihood of event B respetively. The use of a lower, rather than upper bound, reflets aversion to ambiguity. Similarly, the use of an upper, rather than lower bound, reflets an affinity for ambiguity. Evidently, the upper and lower bounds agree on A. Corresponding to inner and outer measures, we introdue inner and outer integrals as follows: fdp = sup{ gdp : f g F A }, f F and fdp = inf{ gdp : f g F A }, f F, where f g means that f (s) g(s), s S. Their intuitive meanings are similar to those for inner and outer measures. 2.4 Capaities and Choquet integrals Inner and outer measures are speial ases of apaities. Say that ν :2 S [0, 1] is a apaity if: 1. A B = ν(a) ν(b); and 2. ν( ) =0,ν(S )=1. Call ν the onjugate of ν, where ν(a) =1 ν(a ). A apaity is superadditive if: For all disjoint events A and B in 2 S,ν(A B) ν(a)+ν(b). A apaity is subadditive if: For all disjoint events A and B in 2 S, ν(a B) ν(a)+ν(b). A apaity is onvex if: For all events A and B in 2 S, ν(a B)+ν(A B) ν(a)+ν(b). It is not hard to verify the following: Lemma 2.2. Let p and p be the inner and outer measures defined in (2.1) and (2.2). Then (a): p and p are onjugates and (b): p is superadditive. However, the outer measure p is not subadditive in general. The inner measure p is not onvex in general. For example, onsider the p generated from A and p in example But p is onvex if A is an algebra. See Proposition 3.1 in [6] for an elementary proof for the ase of σ-algebra. Convexity has been interpreted by Shmeidler [16] as refleting ambiguity aversion of the DM. But the preferenes given in example are obviously ambiguity averse and the apaity p orresponding to them is not onvex. For more detailed explanation, see Epstein [4] and Epstein and Zhang [5]. Appliations of apaities to deision theory employ the notion of Choquet integration. For any apaity ν and integrand f : S R 1, the Choquet integral is defined by 0 fdν ν({s : f (s) t}) dt + [ν({s : f (s) t}) 1] dt. (2.3) 0

9 Subjetive ambiguity, expeted utility and Choquet expeted utility 167 When f is a finite-ranged funtion, we an suppose that f assumes the values x 1 < < x n on the events E 1,...,E n respetively. In this ase, fdν = Σ n i=1 x i [ν( n j =i E j ) ν( n j =i+1 E j )], where ν( n n+1 E j ) 0. Sine any inner measure is a apaity, Choquet integration is defined also for inner and outer measures. In these ases, Choquet integrals are related to inner and outer integrals, as shown next. Theorem 2.3. Let p, p be the inner, outer measures defined in (2.1) and (2.2). Then: 1. For any funtion f F, fdp fdp fdp fdp; and 2. If p (A B) =p(a)+p (B) for all A A and B A, then fdp = fdp and fdp = fdp. (2.4) Under the ondition in part 2, (2.4) provide a representation and novel intuition for Choquet integrals that are based on inner and outer measures. This representation is used to prove our main result, Theorem The model and main results 3.1 Unambiguous events Savage s set-up is adopted throughout. Denote by S the set of states of the world, by 2 S the set of all events and by X the set of outomes. Prospets are modeled via simple ats, mappings from S to X having finite range. The set of ats is F = {...,f,...}. The DM has a preferene relation on the set of ats. The following definition is entral: Definition 3.1. An event A is unambiguous if: (i) For all ats f, f and outomes x, y X, ( ) ( ) f if s A f if s A = x if s A x if s A ( ) ( ) (3.1) f if s A f if s A ; y if s A y if s A and (ii) The ondition obtained if A is everywhere replaed by A in (i) is also satisfied. Otherwise, A is alled ambiguous. Define A = {A 2 S : A is unambiguous} F ua = {f F : f is A measurable}. Both sets A and F ua are nonempty beause and S are unambiguous.

10 168 J. Zhang It merits emphasis that ambiguity is subjetive; it is endogenously derived from the DM s preferene order. This distinguishes the present analysis from that of Fishburn [7], who takes ambiguity as a primitive, and from [14], where the issue of the derivation of the lass of unambiguous events is not expliitly addressed and an exogenously speified lass of unambiguous events is used. To understand definition 3.1, reall Savage s key axiom. Sure-Thing Priniple (STP): For all events A and all ats f, f,gand g, ( ) ( ) f if s A f if s A = g if s A g if s A ( ) ( ) (3.2) f if s A f if s A g if s A g. if s A An impliation of STP is that for all events A and all ats f, f, and outomes x, y, ( ) ( ) f if s A f if s A = x if s A x if s A ( ) ( ) (3.3) f if s A f if s A. y if s A y if s A Therefore, if two ats imply different subats (f ( ) versus f ( )) over an event A, but the same outome over event A, the ranking of these ats will not depend on this ommon outome. This axiom implies that preferenes are separable aross mutually exlusive events, whih is the key property of expeted utility preferenes, either over objetive probability distributions or over ats. However, this priniple is ontradited by the typial hoies in the Ellsberg Paradox. We interpret suh hoies as evidene that the separability required by the STP between outome in A and those in A S \A is desriptively (and perhaps even normatively) problemati when the event A is ambiguous. In suh ases, hanging a ommon outome in for some states in A an ast an entirely new light on the pair of ats being ompared. Though the STP is no longer appealing in similar situations, it is still of use in building an alternative formal model. If one views ambiguity as the only soure of violation of the Sure-Thing Priniple, then one is led to use STP to give formal meaning to ambiguity. This leads to our definition 3.1. It is immediate that preferenes satisfy STP if and only if all events are unambiguous. Another alternative to define unambiguous events is to fully use STP. Denote by A = {A : A and A satisfies (3.2) for all f, f,g,g}. (3.4) But A is intersetion-losed, making it unsuitable as the lass of unambiguous events. In addition, A is too small in general. For instane, A = {, S } in example

11 Subjetive ambiguity, expeted utility and Choquet expeted utility Preferenes over unambiguous ats A natural and interesting question is that without assuming Savage s STP, an we still find reasonable axioms suh that the DM s beliefs are represented by an additive probability measure over A and restrited over F ua is represented by a SEU? The answer is yes. For example, Axioms 1-6 in Epstein and Zhang [5] will do, but we omit their statement for the sake of brevity. To state our main result Theorem 3.3 onisely and learly, we define Savage-representable preferene order over F ua as follows: Definition 3.2. A preferene order is Savage-representable on F ua if there exist a unique ountably additive onvex-ranged probability measure p on A and a (nononstant) utility index u : X R 1 suh that f g u(f )dp u(g)dp, f,g F ua. (3.5) As a result, is Savage-representable on F ua if it satisfies Axioms 1-6 in [5]. While Savage-representable preferene order desribes the nature of preferenes over unambiguous ats, our primary onern is the nature of the ordering of ambiguous ats. We turn to this aspet of preferenes next. 3.3 Ambiguous ats and Choquet expeted utilities The remaining task is to relate the ordering of ambiguous ats to the ordering of unambiguous ones. The next axioms restrit both the set of outomes X and. The following notation is adopted: f g means that f (s) g(s) s S. (3.6) Axiom 1 (Weak Monotoniity). For any two ats f, g with f F ua or g F ua,f g if f g. Axiom 2 (Certainty Equivalent). For any f F, there exists x X suh that f x. Axioms 1 and 2 are self-explanatory and ommon. The final axiom is the entral one. Axiom 3 (Approximation from Below). For any f F,x X, if f x, then there exists an at g F ua suh that f g and g x. The interpretation of Axiom 3 is that in order to evaluate an arbitrary at f, the DM approximates f from below by an unambiguous at g. Given Weak Monotoniity, any suh g provides a lower bound for the utility derived from f. Roughly speaking, Approximation from Below requires further that f be ranked only as highly as the most preferred of suh approximating unambiguous ats g. In other words, the DM faing the ambiguous at f asks what is the worst that

12 170 J. Zhang an happen if I hoose f? and answers this question by relying exlusively on the ats that she understands well, namely on unambiguous ats. This algorithm seems inappropriate for situations where the distintion between slightly ambiguous and highly ambiguous events (and ats) is important. Our DM thinks in terms of blak or white rather than in terms of shades of grey. Note that Axiom 3 does not require unambiguous ats to be rih. The DM ranks f x beause she an find g F ua suh that f g and g x. Now we an state our main result the representation for preferenes on F. Theorem 3.3. Let be a preferene relation on F and A the orresponding set of unambiguous events. Then is Savage-representable on F ua and satisfies axioms 1-3 iff A is a λ-system and there exist a unique onvex-ranged, ountably additive probability measure p on A, and a nononstant utility index u : X R 1 having onvex range suh that f g u(f )dp u(g)dp, f,g F, (3.7) where p is the inner measure generated from the λ-probability spae (S, A, p) and integration is in the sense of Choquet. Theorem 3.3 tells us that it is omplete ignorane outside A and approximation from below that lead the DM to use a apaity p to represent her beliefs and a Choquet Expeted Utility to represent her preferenes. Aordingly, omplete ignorane outside A and approximation from below apture the heuristi meaning of ambiguity aversion. The interpretation of the utility representation, partiularly the subjetive nature of A, warrants emphasis. It annot be argued that exlusive reliane (via p ) on events in A to ompute likelihoods of other events reflets an extreme or unreasonable degree of ignorane or ambiguity aversion. 4 At a formal level, that is beause A is a omponent of the utility representation that is inseparable from the use of inner measure. Less formally, the above argument is unsupportable beause A is subjetive; whether or not there is a large degree of ignorane or ambiguity aversion implied depends on the size of A. Indeed, a primary role of A is to model the degree of ambiguity aversion of the DM. Finally, there is an obvious mirror image of this analysis in whih optimism, unertainty affinity and Approximation From Above replae pessimism, unertainty aversion and Approximation From Below. That is, replae axiom 3 by the following axiom: Axiom 4 (Approximation from Above). For any f F,x X, if f x, then there exist an at g F ua suh that f g and g x. This delivers the CEU model of preferenes with apaity given by the outer measure p. 4 There is a parallel with the question of how to interpret the set of priors and the minimization over that set that appears in the multiple-priors model of preferene in Gilboa & Shmeidler [9].

13 Subjetive ambiguity, expeted utility and Choquet expeted utility 171 We lose this setion with a brief disussion of the extreme ase of a Savage preferene, whih judges all events as unambiguous. Intuitively, suh a preferene should satisfy both approximation from below and above. The following Theorem proves this is true: Theorem 3.4. Let be a preferene relation on F and A the orresponding set of unambiguous events. Then is Savage-representable on F ua and satisfies axioms 1 4 iff A =2 S, F = F ua and there exist a unique onvex-ranged, ountably additive probability measure p on 2 S, and a nononstant utility index u : X R 1 having onvex range suh that for all ats f,g F, f g u(f )dp u(g)dp. 4 Examples 4.1 Choquet expeted utility model Now suppose that the DM s preferene order is represented by CEU with a monotone, superadditive apaity ν. That is, f g u(f )dν u(g)dν, f,g F, (4.1) where u : X R 1 is a nononstant utility index with a onvex range. Sine the attitude towards ambiguity is inluded in the apaity ν, the preise relation between ν and A is of interest. Define Σ = {A 2 S : ν(a B) =ν(a)+ν(b), B A } Σ = {A 2 S : ν(a)+ν(a )=1}. Theorem 4.1. Let on F be defined in 4.1. Then: 1. If Σ is symmetri in the sense that A Σ A Σ, then Σ = A and Σ is losed with respet to disjoint unions. 2. If A Σ, then A Σ.Σ is an algebra if ν is onvex, but not more generally. 3. Σ Σ and they are not equal in general. 4. If ν is onvex, then Σ = Σ. Colletion Σ represents an extreme possible approah to defining unambiguous events diretly in terms of the apaity ν. But Σ is not losed with respet to disjoint unions. For example: Example 4.2. Let S = {ω 1,ω 2,ω 3 } and ν be defined on (S, 2 S ) as follows: ν(s )=1,ν( ) =0=ν({ω 1 }) ν({ω 2 })=ν({ω 3 })=1/3 ν({ω 1,ω 2 })=ν({ω 1,ω 3 })=ν({ω 2,ω 3 })=2/3.

14 172 J. Zhang In this example, ν is superadditive but not onvex. And Σ = {, S, {ω 1,ω 2 }, {ω 1,ω 3 }, {ω 2 }, {ω 3 }} is not losed with respet to disjoint unions sine the union {ω 2,ω 3 } is not in Σ. 4.2 A ounterexample This example illustrates violation of our model. It is a variation of the example in subsetion Let B + R = 80 and B + G = 10. One ball is to be drawn at random. The following preferenes are intuitive: f 1 = $0 if s B $100 if s R $0 if s G $0 if s W $100 if s B $0 if s R $100 if s G $100 if s W = f 2. The DM piks f 1 rather than f 2 mainly beause she knows from B + R =80 and B + G = 10 that R = G + 70 and W = B +10. Though the events {R} and R = {B, G, W } are both ambiguous to the DM, however, the probability of {R} is objetively at least 0.7, and therefore greater than two times that of its omplement {B, G, W }. It is intuitive that all unambiguous events in this example are: A = {φ, S, {B, G}, {R, W }, {B, R}, {G, W }}, where S = {B, R, G, W }. And the DM s beliefs over A an be represented by the following probability measure p(s ) = 1, p(φ) =0andp({B, G}) =0.1, p({r, W }) =0.9 p({b, R}) = 0.8, p({g, W }) =0.2 However, u(f 1 )dp < u(f 2 )dp, where p is the inner measure indued from p and the integral is in the sense of Choquet. What is wrong with the inner measure model? The problem is that the deision-maker has more information than just the probabilities of events in A, e.g., she knows that R = G + 70 and W = B + 10, even though {R, G} and {W, B} are ambiguous. One suggestion to solve the problem is as follows: The DM knows the true law is an additive probability measure p on 2 S and both p and p agree on A, but she does not know what p looks like outside A. Thus, any one probability measure in

15 Subjetive ambiguity, expeted utility and Choquet expeted utility 173 Π(p) ={p : p is an extension of (S, A, p) to2 S } is possible. Due to ambiguity aversion, she may use min p Π(p) u(f )dp to evaluate ats f. As a result, min p Π(p) u(f 1 )dp > min p Π(p) u(f 2 )dp. Extension of our model to aommodate suh partial information is the subjet of urrent researh. A. Appendix A Theorem 2.3 is proved here. To prove it, the following Lemma is needed: Lemma A.3. Let (S, A, p) be a λ-system probability spae. If p (A B) = p(a)+p (B), A A and B A, then: (i) For any A 2 S, there exists a sequene {A n } in A satisfying A n A n+1 A, n =1, 2,..., suh that p (A) = lim p(a n)=p( n A n ). n (ii) For any hain = D 0 D 1 D 2 D k 1 D k = S, there exists a hain = B 0 B 1 B 2 B k 1 B k = SinAwith B i D i for i =0, 1, 2,...,k suh that p(b i )=p (D i ) for i =0, 1, 2,...,k. Proof. Proof of (i). If p (A) = 0, the onlusion is obviously true. Next, assume p (A) > 0. Take 0 <ɛ<p (A). By the definition of p, there exists A 1 A with A 1 A suh that p(a 1 ) > p (A) ɛ>0. If p(a 1 )=p (A), then let A n = A 1, n =1, 2,..., and the onlusion is proved. Next, let p(a 1 ) < p (A). Then p (A\A 1 ) > 0 follows from p (A) =p (A 1 (A\A 1 )) = p(a 1 )+p (A\A 1 ). Similarly, there exists A A with A A\A 1 suh that p(a ) > p (A\A 1 ) ɛ/2. Denote A 2 = A 1 A A. As a result, p(a 2 ) = p(a 1 A )=p(a 1 )+p(a ) = p (A)+(p(A ) p (A\A 1 )) > p (A) ɛ/2. By indution, there exists an inreasing sequene {A n } n=1 in A suh that p(a n ) > p (A) ɛ/n, n =1, 2,... (A.1) Define B = A n. Therefore,

16 174 J. Zhang p (A) ɛ/n < p(a n ) p(b) p (A), for all n. That is, p (A) = lim p(a n)=p(b). n Proof of (ii). We only need to prove that for any two events D 1 D 2, there exist two events B 1, B 2 in A with B 1 B 2 and B i D i, i =1, 2 suh that p(b i )=p (D i ), i =1, 2. By part (i), there exists event B 1 in A with B 1 D 1 suh that p(b 1 )=p (D 1 ). Aordingly, p (D 2 )=p (B 1 (D 2 \B 1 )=p(b 1 )+p (D 2 \B 1 ). Again by part (i), there exists event C 1 in A with C 1 D 2 \B 1 suh that p(c 1 )=p (D 2 \B 1 ). Let B 2 = B 1 C 1, then B 2 A, B 2 D 2, B 1 B 2 and p(b 2 )=p (D 2 ). Proof of Theorem 2.3. (1): Sine p(a) p (A), A A, gdp = gdp fdp, f g F A. And this implies Similarly, we an prove fdp = sup{ gdp : f g F F } fdp fdp. fdp. Finally, fdp fdp diretly follows from p (A) p (A), A 2 S. (2): We prove only fdp = fdp here, the other is similar. By part (1), we need only prove that sup{ gdp : f g F F } fdp, f F. Let and x 1 < x 2 < < x n. x 1 if s E 1 x f (s) = 2 if s E 2 x n if s E n.

17 Subjetive ambiguity, expeted utility and Choquet expeted utility 175 By the definition of Choquet integral, fdp = Σ n i=1x i [p ( n j =i E j ) p ( n j =i+1e j )], where = n j =n+1 E j n j =n E j n j =2 E j n j =1 E j = S is a hain. By Lemma A.3 (ii), there exists a hain = B 0 B 1 B n = S in A with B i n j =n+1 i E j suh that p ( n j =n+1 i E j )=p(b i ), i =0, 1, 2,...,n. Therefore, fdp = Σi=1 n x i [p ( n j =i E j ) p ( n j =i+1 E j )] = = Σi=1 n x i [p(b n+1 i ) p(b n i )] Σi=1 n x i p(b n+1 i \B n i )= fdp, where f (s) = x 1 x 2 if s B n \B n 1 if s B n 1 \B n 2 x n 1 if s B 2 \B 1 x n if s B n. Obviously, f F A and f f. Therefore, fdp = fdp sup{ gdp : g F A,g f }. B. Appendix B To prove Theorem 3.3, we need to prove the following two Lemmas first: Lemma B.1. If an at f has the following form f = { x If s A y If s A, then sup{ u(h)dp : h F ua, h f } = u(f )dp, (B.1) where x y and the integral of left side of (B.1) is in the sense of Choquet. Proof. For any integer n > 1, by the definition of inner measures, there is A n A with A n A suh that p(a n ) > p (A) 1/n. Let { x if s An g n (s) = y if s A n. Obviously, g n F ua and g n f. Therefore,

18 176 J. Zhang u(gn )dp = u(x)p(a n )+u(y)p(a n)=(u(x) u(y))p(a n )+u(y) > (u(x) u(y))p (A)+u(y) (u(x) u(y))/n. Aordingly, sup{ u(h)dp : h F ua, h f } lim n [(u(x) u(y))p (A)+u(y) (u(x) u(y))/n] =(u(x) u(y))p (A)+u(y) = u(f )dp. The onlusion follows from (B.2) and part (1) of Theorem 2.3. (B.2) Lemma B.2. For any A A, p (A B) =p(a)+p (B), B A. Proof. For any two disjoint events A and B, p (A B) p (A)+p (B) follows from the superadditivity of p. Thus, we only need to prove that for any event A A, p (A B) p(a)+p (B), B A. (B.3) Suppose (B.3) is not true, then there exists an unambiguous event A A and event B A suh that p (A B) > p(a)+p (B). Sine u is nononstant and onvex ranged, there exist outomes x 1, y 1, y 2, and x 2 suh that u(x 1 ) > u(y 1 ) > u(y 2 ) > u(x 2 ). (B.4) Let x 1 if s B y 1 if s B f 1 (s)= x 2 if s A \B, g 1 (s)= y 2 if s A \B and x 1 if s A x 1 if s A x 1 if s B y 1 if s B f 2 (s)= x 2 if s A \B,g 2 (s) = y 2 if s A \B. y 1 if s A y 1 if s A (B.5) Therefore, by (B.9) and Lemma B.1, f 1 g 1 sup{ u(h)dp : h F ua, h f 1 } > sup{ u(h)dp : h F ua, h g 1 } u(f 1 )dp > sup{ u(h)dp : h F ua, h g 1 } and f 2 g 2 sup{ u(h)dp : h F ua, h f 2 } < sup{ u(h)dp : h F ua, h g 2 }. sup{ u(h)dp : h F ua, h f 2 } < u(g 2 )dp.

19 Subjetive ambiguity, expeted utility and Choquet expeted utility 177 If outomes {x 1, x 2, y 1, y 2 } satisfy u(f 1 )dp > u(g 1 )dp and u(f 2 )dp < u(g2 )dp, then sup{ u(h)dp : h F ua, h f 1 } = u(f 1 )dp > u(g 1 )dp sup{ u(h)dp : h F ua, h g 1 } and sup{ u(h)dp : h F ua, h f 2 } u(f 2 )dp < u(g 2 )dp = sup{ u(h)dp : h F ua, h g 2 }. By diret omputations, u(f1 )dp > u(g 1 )dp [u(x 2 ) u(y 2 )][1 p (A B)]+[u(x 1 ) u(y 1 )][p (A B) p (A)] > 0 (B.6) and u(f2 )dp < u(g 2 )dp (B.7) [u(x 2 ) u(y 2 )][1 p (A B)]+[u(x 1 ) u(y 1 )]p (B) < 0. Sine p (A B) > p (A) +p (B) and the range of u is onvex, there exist outomes x 1, x 2, y 1, y 2 satisfying (B.4), (B.6) and (B.7). This implies that f 1 g 1, but f 2 g 2. This ontradits that A is unambiguous. Proof of Theorem 3.3. (= ): It is enough to prove (3.7) assuming (i) the representation (3.5) where A satisfies λ.1 and λ.2 (not neessarily λ.3) and (ii) Axioms 1-3 for. Sine onstant ats are in F ua, (3.5) implies x y u(x) u(y). Sine p is onvex-ranged, it follows from (3.5) and Certainty Equivalent for F ua measurable ats that u has onvex range. Lemma B.3. (i) For any at f F,f x implies that u(x) = sup { u(g)dp : f g F ua }. (ii) The ordering on F is represented by U where, for any f F, U (f ) = sup{ u(g)dp : f g F ua }. (B.8) Proof. (i) By Weak Monotoniity, f g implies x f g. By (3.5), u(g)dp u(x), g F ua, this means that sup{ u(g)dp : f g F ua } u(x). For the other diretion, it is true if x is the worst outome in X. Next, suppose that there exists x x. Sine u has onvex range, for any suffiiently small ɛ>0, there exists x suh that u(x) ɛ<u(x ) < u(x). Now suppose that

20 178 J. Zhang u(x) ɛ>sup{ u(g)dp : f g F ua } for some ɛ>0. Then x g for all g F ua. This ontradits f x and Approximation from Below. (ii) follows from (i), Certainty Equivalent and transitivity of. It remains to show that the utility funtion defined in (B.8) is the required Choquet integral with respet to the inner measure p, that is, U (f ) u(f )dp. (B.9) And this diretly follows from Lemma B.2 and part (ii) of Theorem 2.3. To prove Theorem 3.4, we need to prove some Lemmas first. Lemma B.4. Let (S, A, p) be a λ-system probability spae. If p (A B) = p(a)+p (B) for any A A and B A, then {A : p (A B) =p (A)+p (B), B A } = {A : p (A)+p (A )=1}. Proof. Let Σ 2 = {A : p (A B) =p (A)+p (B), B A } and Σ 3 = {A : p (A)+p (A )=1}. Obviously, Σ 2 Σ 3. Next, we prove that Σ 3 Σ 2. First, we prove that Σ 2 is losed with respet to disjoint unions. Take A 1, A 2 Σ 2 are disjoint. For any event B (A 1 A 2 ), p ((A 1 A 2 ) B)=p (A 1 (A 2 B)) = p (A 1 )+p (A 2 B) = p (A 1 )+p (A 2 )+p (B) =p (A 1 A 2 )+p (B). Next, we prove if A Σ 2, p (A) =0andA 1 A, then A 1 Σ 2. For any B A 1, p (A 1 B) p (A B) =p (A (B\A)) = p (A)+p (B\A) p (A 1 )+p (B). p (A 1 B) p (A 1 )+p (B) diretly follows from superadditivity of p. This ompletes the onlusion. Now let A Σ 3. From Lemma A.3 (i), there exist two events A 1 and A 2 in A satisfying A 1 A and A 2 A suh that p (A) =p(a 1 ) and p (A )=p(a 2 ). As a result, (A 1 A 2 ) A Σ 2 and p((a 1 A 2 ) )=0. Sine A\A 1 (A 1 A 2 ) and (A 1 A 2 ) Σ 2, therefore, A\A 1 Σ 2. Sine Σ 2 is losed with respet to disjoint unions, A = A 1 (A\A 1 ) Σ 2. Lemma B.5. If p (A) =p (A) for event A, then p (A)+p (A )=1.

21 Subjetive ambiguity, expeted utility and Choquet expeted utility 179 Proof. Suppose, p (A) =p (A) for event A. Then, we have p (A )=1 p (A) = 1 p (A) =p (A ). If p (A) +p (A ) /= 1, then p (A) +p (A ) < 1bythe superadditivity of p. Aordingly, p (A)+p (A ) = [1 p (A )]+[1 p (A)] = 2 [p (A)+p (A )] > 1 > p (A)+p (A ). But this ontradits the assumption. Proof of Theorem 3.4. Obviously from Theorem 3.3, we have for any ats f,g F f g u(f )dp u(g)dp u(f )dp u(g)dp. (B.10) Claim 1 : For any at f, u(f )dp = u(f )dp. (B.11) Proof of Claim 1. If B.11 is not true, then there exists an at f suh that u(f )dp < u(f )dp. Now pik outome x suh that u(f )dp < u(x) < u(f )dp. But this ontradits B.10. Claim 2 : For any event A, p (A) =p (A). Proof of Claim 2. Suppose this is not true, then there exists an event A suh that p (A) > p (A). Let ( ) x if s A f = x if s A, where x x. Then u(f )dp = u(x)[1 p (A)] + u(x )p (A) =u(x)+[u(x ) u(x)]p (A) < u(x)+[u(x ) u(x)]p (A) = u(f )dp. But this ontradits Claim 1. Then A =2 S diretly follows from Lemmas B.4, B.5, B.2 and part 1 of Theorem 4.1. C. Appendix C Proof of Theorem 4.1 is provided here: Proof of (1). We only prove A Σ here. The other parts an be proved by routine verifiation. Suppose it is not true. Without loss of generality, then there exists an unambiguous event A A and B A suh that ν(a B) >ν(a)+ν(b).

22 180 J. Zhang Sine u is nononstant and onvex ranged, there exist outomes satisfying Let u(x 1 ) > u(y 1 ) > u(y 2 ) > u(x 2 ). x 1 if s B y 1 if s B f 1 (s)= x 2 if s A \B, g 1 (s)= y 2 if s A \B and x 1 if s A x 1 if s A x 1 if s B y 1 if s B f 2 (s)= x 2 if s A \B,g 2 (s) = y 2 if s A \B. y 1 if s A y 1 if s A By diret omputations, u(f1 )dν> u(g 1 )dν (C.1) [u(x 2 ) u(y 2 )][1 ν(a B)]+[u(x 1 ) u(y 1 )][ν(a B) ν(a)] > 0 and u(f2 )dν< u(g 2 )dν (C.2) [u(x 2 ) u(y 2 )][1 ν(a B)]+[u(x 1 ) u(y 1 )]ν(b) < 0. (C.3) Sine ν(a B) >ν(a)+ν(b) and the range of u is onvex, there exist x 1, x 2, y 1, y 2 satisfying (C.1), (C.2) and (C.3). Thus, f 1 g 1 and f 2 g 2. This ontradits that A is unambiguous. Proof of 2. Obviously A Σ = A Σ and Ω, φ are in Σ. To prove that Σ is an algebra when ν is onvex, it suffies to show that if A 1, A 2 are in Σ, then A 1 A 2 and A 1 A 2 are also in Σ. That ν is onvex implies that ν is onave. Therefore, ν(a 1 )+ν(a 2 ) ν(a 1 A 2 )+ν(a 1 A 2 ) ν(a 1 A 2 )+ν(a 1 A 2 ) ν(a 1 )+ν(a 2 )=ν(a 1 )+ν(a 2 ). Beause ν(a 1 A 2 ) ν(a 1 A 2 ) and ν(a 1 A 2 ) ν(a 1 A 2 ), then ν(a 1 A 2 )=ν(a 1 A 2 ) and ν(a 1 A 2 )=ν(a 1 A 2 ). Proof of 3. The set inlusions follow from the definitions of Σ and Σ. Example 4.2 shows that the sets differ in general. Proof of 4. We only need to prove that Σ Σ. Sine ν is onvex, ν(a) = inf{p(a) :p is in the ore of ν}. Let ν(a) =1 ν(a ). Then we prove p(a) =q(a), for any p, q in the ore of ν. Suppose this is not true. That is, there are p, q in the ore of ν suh that p(a) > q(a) and p(a ) < q(a ). Thus, ν(a) q(a), ν(a ) p(a ) and ν(a)+ν(a ) q(a)+p(a ) < q(a)+q(a ) = 1, a ontradition. If p(a) =q(a) for any p, q in the ore of ν, then A is in Σ.

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