Random Ambiguity. Jay Lu. September Abstract

Transcription

1 Random Ambiguity Jay Lu September 2014 Abstract We introduce a model of random ambiguity aversion. A group of agents with heterogeneous levels of ambiguity aversion choose from a set of options. An analyst does not observe this heterogeneity but observes aggregate choice that is probabilistic or random. We characterize a model where the analyst can uniquely identify the distribution of ambiguity aversion from random choice. Moreover, the analyst can also assess when one population is more ambiguity averse than another. The model also applies to a single agent receiving independent shocks to ambiguity aversion. More generally, we analyze the stochastic properties of non-linearity in models of random utility maximization. IamdeeplyindebtedtobothFarukGulandWolfgangPesendorferfortheircontinuousadvice,guidance and encouragement. I would also like to thank Larry Blume, David Easley, Diogo Guillen, Rohit Lamba, Pietro Ortoleva, Stephen Morris and Paulo Natenzon for helpful their suggestions and input.

2 1 Introduction 1.1 Motivation and Overview In many economic situations, ambiguity aversion (i.e. aversion towards Knightian [29] uncertainty) is useful for explaining behavior that cannot be easily addressed with standard models of expected utility maximization. In actual applications, given a population of agents, it is realistic to assume some degree of heterogeneity in the magnitude of ambiguity aversion. 1 If this heterogeneity is unobservable to an analyst (i.e. an outside observer), then the aggregate choice observed by the analyst is probabilistic or random. In this paper, we are interested in situations where the only unobserved heterogeneity is in ambiguity aversion. We show that given the relevant choice data, an analyst can completely identify the distribution of ambiguity aversion in the population. To be concrete, consider a population of agents facing investment problems with exogenous prices. Each investment problem consists of choosing an asset from a set of assets. Suppose that there is heterogeneity in the level of ambiguity aversion in the population. Agents who are more ambiguity averse will choose the safer asset while those who are less ambiguity averse will opt for the more uncertain asset. 2 If this heterogeneity in ambiguity aversion is unknown to an analyst but the proportion of agents choosing each asset in a given investment problem is observable, then the analyst can use this choice data to fully identify the entire distribution of ambiguity aversion in the population. This is especially useful if the analyst is a regulator where the optimal policy may depend heavily on the exact distribution of ambiguity aversion in the population (see Easley and O Hara [12]). Alternatively, one could interpret our model as describing the behavior of a single agent receiving unobservable shocks to his ambiguity aversion. These shocks neither reveal new information nor alter the taste preferences of the agent. They do however affect the magnitude of the agent s ambiguity aversion. 3 If the distribution of these shocks is unknown to the analyst, then from the analyst s perspective, the agent s choice behavior is random. Our 1 Empirical evidence of this heterogeneity has been well-documented (for example, see Abdellaoui et. al. [1] and Ahn et. al. [3]). 2 It is well known for example that ambiguity aversion can explain the surprisingly large fraction of individuals who do not participate in financial markets (see Dow and Werlang [11]). 3 One could interpret the severity of ambiguity aversion as reflective of the amount of uncertainty about the unknown state. For example, in Caballero and Krishnamurthy [8], investors become more ambiguity averse and face an enlarged set of possible priors during surprise defaults or bank runs. They are then compelled to scramble toward safer assets in a phenomenon termed flight to quality. 1

3 results demonstrate that if the analyst has the relevant random choice data, then he can fully identify the distribution of these shocks. We introduce a model of random utility maximization (RUM) to capture both interpretations. 4 In particular, we consider a randomized version of the maxmin expected utility model of Gilboa and Schmeidler [24]. The focus on the maxmin model is due to its tractability and the fact that it is one of the most widely applied models of ambiguity aversion in the literature. Moreover, the magnitude of ambiguity aversion has a natural interpretation as it corresponds exactly to the size of the set of possible priors. In our model, choice is random from the perspective of the analyst only because the heterogeneity in ambiguity aversion is unobservable. We provide an axiomatic foundation for our model and show that when all the relevant choice data is available, the analyst can uniquely identify the distribution of ambiguity aversion from random choice. He can also use the data to assess when one population is more ambiguity averse than another. More generally, we analyze the stochastic properties of non-linearity, in particular quasiconcavity and quasiconvexity, in models of RUM. The more practical questions of performing identification and inference when choice data is only partially available or when there is unobserved heterogeneity beyond ambiguity aversion are left for future research. Formally, fix a finite state space. We model each choice option as a state-contingent payoff called an act. 5 A decision-problem is a finite set of acts. In the investment problem for example, choosing an asset from a set of assets corresponds to choosing an act from a decision-problem. The analyst does not observe the heterogeneity in ambiguity aversion but observes a choice distribution over acts for each decision-problem. We call this mapping from decision-problems to choice distributions a random choice rule (RCR). In Gilboa and Schmeidler [24], maxmin expected utility is characterized by a von Neumann- Morgenstern utility index and a non-empty, closed and convex set of priors. An act is evaluated according to its expected utility using the worst possible prior in the set of priors. We consider a RUM model where the utilities are maxmin expected utilities and the sets of priors satisfy K t := (1 t) K + t K where K K are two non-empty, closed and convex sets of priors and t 2 [0, 1] is a 4 For classic works on RUM, see Block and Marschak [6], Falmagne [17] and McFadden and Richter [32]. 5 Formally, these are known as Anscombe-Aumann [4] acts. 2

4 random parameter. 6 We interpret K as a common set of priors shared by all agents. All heterogeneity in the population is captured by variation in K t.highervaluesoftcorrespond to larger sets of priors and greater levels of ambiguity aversion. The distribution of K t is exactly the distribution of ambiguity aversion in the population. Given a decision-problem, the probability that an act is chosen is equal to the probability that its maxmin expected utility according to K t is higher than that of all other acts in the decision-problem. Call this a random ambiguity representation of the RCR. Our first result, Theorem 1, asserts that under random ambiguity representations, an analyst can completely identify the distribution of ambiguity aversion in the population simply from binary choice. We also introduce a random choice version of comparative ambiguity aversion. One population of agents is more uncertainty averse than another if the probability that constant acts are chosen is higher in the former than in the latter. Proposition 1 shows that greater uncertainty aversion corresponds exactly to first-order stochastic dominance of the distributions of ambiguity aversion. This allows the analyst to assess when one population is more ambiguity averse than another using only the relevant choice data. Since maxmin expected utility are quasiconcave, before providing a full characterization of our representation, we first study the stochastic properties of random non-linear utilities. We focus on two properties of RCRs: extremeness and convexity. Extremeness was introduced by Gul and Pesendorfer [27] in the context of random expected utility and states that interior mixtures will never be chosen with strictly positive probability. Violations of extremeness are exactly the types of behavior characterized by ambiguity aversion. Convexity is novel. It states that the probability that an act is chosen will not be affected by the addition of a new act provided that the original decision-problem contains some mixture of the two acts. Theorem 2 asserts that a random utility is quasiconcave (quasiconvex) if and only if its corresponding RCR is convex (extreme). 7 It follows that any RCR with a random ambiguity representation must be convex. Our work thus extends the axiomatic study of non-linearity to the realm of random choice. We provide a full axiomatic characterization of a random ambiguity representation. The 6 The Minkowski mixture for two sets K and K and t 2 [0, 1] is defined as (1 t) K + t K := (1 t) p + tq (p, q) 2 K K 7 In light of Theorem 2, convexity is aptly named given that under deterministic choice, a utility is quasiconcave if and only if its corresponding preference relation is convex. 3

5 first five axioms (C-linearity, continuity, S-monotonicity, convexity, non-degeneracy) are the random choice analogs of corresponding axioms in the original maxmin expected utility representation. For instance, convexity serves as the random choice version of the uncertainty aversion axiom in the deterministic maxmin model. Monotonicity states that adding new acts cannot increase the probability that an act is chosen in the original decision-problem. It is necessary for any random utility representation. Lastly, we introduce B-independence which states that if the binary choice probability of an act is greater than that of another with respect to two constant acts 8,thentheprobabilityofchoosingthefirstisgreaterthan the second in any decision-problem. It is a weakening of the independence condition that characterizes Luce rules (see Gul, Natenzon and Pesendorfer [26]). Theorem 3 asserts that a RCR has a random ambiguity representation if and only if it satisfies these seven axioms. In general, RUM models have difficulty dealing with indifferences in the random utility. In our model, we address this issue by drawing an analogy with deterministic choice. Under deterministic choice, if two acts are indifferent (i.e. they have the same utility), then the model is silent about which act will be chosen. Similarly, under random choice, if two acts are indifferent (i.e. they have the same random utility), then our model is silent about what the choice probabilities are. This approach has two advantages. First, it allows the analyst to be agnostic about choice data that is beyond the scope of the model and provides some additional freedom to interpret the data. Second, it generalizes the definition of a RCR just enough so that we can include deterministic choice as a special case of random choice. In particular, the standard maxmin model obtains as a degenerate case in our model when K = K is a single prior. 1.2 Related Literature This paper is related to a long literature on random utility. 9 Testable implications of RUM were first studied by Block and Marschak [6], and the model was later characterized by Mc- Fadden and Richter [32] and Falmagne [17]. Gul and Pesendorfer [27] obtain a more intuitive characterization by enriching the choice space with lotteries. More recently, Gul, Natenzon and Pesendorfer [26] characterize a special class of RUM models called attribute rules that 8 An act is constant if it returns the same payoff in every state. 9 RUM is used extensively in discrete choice estimation. Most models in this literature assume specific parametrizations such as the logit, the probit, the nested logit, etc. (see Train [35]). 4

6 can approximate any RUM model. Fudenberg and Strzalecki [20] study the relationship between stochastic choice and preferences for larger or smaller option sets in a dynamic setting with RUM. Our main contribution to this literature is providing an axiomatic characterization of a non-linear RUM model. Moreover, the stochastic properties of non-linearity that we study extend easily to other choice domains. For example, when choice options are lotteries, Kahneman and Tversky [28] record violations of the classic independence axiom. In much earlier work, Becker, DeGroot and Marschak [5] record violations of extremeness. Our Theorem 2 provides a precise connection between observable choice behavior and violations of linearity in the form of quasiconcavity and quasiconvexity. Some recent papers that also look at the relationship between ambiguity aversion and stochastic choice are Agranov and Ortoleva [2], Fudenberg, Iijima and Strzalecki [19] and Saito [33]. In all these models, a sophisticated agent deliberately randomizes over choice options due to ambiguity aversion. In contrast, the source of random choice in our model is the analyst s unobservability of the heterogeneity of ambiguity aversion. Also, the population interpretation of random choice in our model has no counterpart in their models. In addition to the maxmin model of Gilboa and Schmeidler [24], our model is also related to some other non-stochastic models of ambiguity aversion. Our parametrization of the sets of priors includes the "-contamination model as a special case. This has been widely used in robust statistics and was even suggested by Ellsberg [13] as a simple functional form to address his namesake paradox. Both Kopylov [30] and Gajdos et. al. [21] provide axiomatic characterizations of "-contamination. The former uses observable deferred choice while the latter uses objective sets of priors. In contrast, we provide an axiomatic characterization via random choice. Epstein and Kopylov [14] study an agent who exhibits a preference for commitment in anticipation of future ambiguity aversion which they interpret as cold feet. In contrast, the agent in our model is either naively unaware of or powerless against these shocks to his ambiguity aversion. Finally, this paper is related to a large literature on applications of heterogeneity in ambiguity aversion. In Caballero and Krishnamurthy [8], investors receive random shocks to ambiguity aversion due to surprise defaults or bank runs. As a result, they scramble toward safer assets in a phenomenon called flight to quality. Easley and O Hara [12] study the role of regulation in a heterogeneous population with different levels of ambiguity aversion. Bose, Ozdenoren and Pape [7] investigate varying ambiguity aversion in an auction setting. 5

7 Dow and Werlang [11] and Epstein and Wang [16] study the effects of ambiguity aversion on market non-participation and asset prices. Epstein and Schneider [15] review other various limited participation problems that are addressed using heterogeneity in ambiguity aversion. 2 Random Choice Rules Let S be a finite objective state space and X be a finite set of prizes. Let S and X be their respective probability simplexes. Interpret S as the set of all beliefs about S and X as the set of lotteries over prizes. Each choice option corresponds to a state-contingent payoff called an act. Following the setup of Anscombe and Aumann [4], an act is formally amappingf : S! X. Let H be the set of all acts. Call a finite set of acts a decisionproblem. metric. 10 Let D be the set of all decision-problems, which we endow with the Hausdorff For notational convenience, let f denote the singleton set {f} whenever there is no risk of confusion. The primitive (i.e. the choice data) is a random choice rule (RCR) that specifies a choice distribution over acts for every decision-problem. Thus, the RCR specifies the frequency distribution of choices in a population if every agent in the population is faced with the same decision-problem. In the single-agent interpretation of our model, the RCR specifies the frequency distribution of choices by the agent if he chooses from the same decisionproblem repeatedly. In the classic model of rational choice, if two acts are indifferent (i.e. they have the same utility), then the model is silent about which act will be chosen. We introduce an analogous innovation to address indifferences under RUM. If two acts are indifferent (i.e. they have the same random utility), then we declare that the RCR is unable to specify choice probabilities for each act in the decision-problem. For instance, it could be that one act is chosen over another with probability a half, but any other probability would also be perfectly consistent with the model. Similar to how the classic model is silent about which act will be chosen in the case of indifference, our model is silent about what the choice probabilities are. In both 10 For two sets F and G, thehausdorffmetricisgivenby d h (F, G) := max sup f2f inf f g2g g, sup inf f g g2f f2g! 6

8 cases, indifference is interpreted as choice behavior that is beyond the scope of the model. This provides the analyst with additional freedom to interpret data. Formally, indifference is modelled as non-measurability with respect to some -algebra H on H. For example, if H is the Borel algebra, then this corresponds to the benchmark case where every act is measurable and there are no indifferences. In general, H can be coarser than the Borel algebra. Given any decision-problem, the decision-problem itself must be measurable. This is because we know that some act in it will be chosen for sure. For F 2D, let H F be the -algebra generated by H[{F }. 11 Let be the set of all probability measures on any measurable space of H. Definition. A random choice rule (RCR) is a (, H) where : D! and (F ) is a measure on (H, H F ) with support F 2D. Let F denote the measure (F ) for every F 2D. A RCR thus assigns a probability measure on (H, H F ) for each decision-problem F 2Dsuch that F (F )=1. 12 Interpret F (G) as the probability that some act in G will be chosen in the decision-problem F 2K.Forease of exposition, we denote RCRs by with the implicit understanding that it is associated with some H. ToaddressthefactthatG F may not be H F -measurable, we use the outer measure 13 F (G) := inf F (G 0 ) G G 0 2H F As both and coincide on measurable sets, let denote without loss of generality. A RCR is deterministic iff all choice probabilities are either zero or one. What follows is an example of a deterministic RCR. Its purpose is to highlight (1) the use of non-measurability to model indifferences and (2) the modeling of classic deterministic choice as a special case of random choice. Example 1. Let S = {s 1,s 2 } and X = {x, y}. Without loss of generality, let f =(a, b) 2 [0, 1] 2 denote the act f 2 H where f (s 1 )=a x +(1 f (s 2 )=b x +(1 a) y b) y 11 This definition imposes a form of common measurability across all decision-problems. It can be relaxed if we strengthen our monotonicity axiom. 12 The definition of H F ensures that F (F ) is well-defined. 13 Lemma A1 in the Appendix ensures that this is well-defined. 7

9 Let H be the -algebra generated by sets of the form B [0, 1] where B is a Borel set on [0, 1]. Consider the RCR (, H) where F (f) =1if f 1 g 1 for all g 2 F. Acts are ranked based on how likely they will yield prize x if state s 1 occurs. This could describe an agent who prefer x to y and believes that s 1 will realize for sure. Let F = {f,g} where f 1 = g 1 and note that neither f nor g is H F -measurable; the RCR is unable to specify choice probabilities for f or g. This is because both acts yield x with the same probability in state s 1. The two acts are thus indifferent. Observe that corresponds exactly to classic deterministic choice where f is preferred to g iff f 1 g 1. We now introduce continuity for RCRs. Given a RCR, let D 0 Dbe the set of decisionproblems where every act in the decision-problem is measurable with respect to the RCR. In other words, F 2D 0 iff f 2H F for all f 2 F. Let 0 be the set of all Borel measures on H endowed with the topology of weak convergence. Since all acts in F 2D 0 are H F - measurable, F 2 0 for all F 2D 0 without loss of generality. 14 Call continuous iff it is continuous on the restricted domain D 0. Definition (Continuity). is continuous iff : D 0! 0 is continuous. If H is the Borel algebra, then D 0 = D. Inthiscase,ourcontinuityaxiomcondensesto standard continuity. In general though, the RCR is not continuous over all decision-problems. In fact, the RCR is discontinuous precisely at decision-problems that contain indifferences. In other words, choice data that is beyond the scope of the model exhibits discontinuities with respect to the RCR. However, every decision-problem is arbitrarily (Hausdorff) close to some decision-problem in D 0,socontinuityispreservedoveralmostalldecision-problems. 3 The Random Ambiguity Representation 3.1 Representation and Uniqueness We now describe the random ambiguity representation. In Gilboa and Schmeidler [24], maxmin expected utility is characterized by a von Neumann-Morgenstern utility index and anon-empty,closedandconvexsetofpriors.anactisevaluatedaccordingtoitsexpected utility using the worst possible prior in the set of priors. In our model, heterogeneity in 14 We can easily complete F so that it is Borel measurable. 8

10 ambiguity aversion is captured by heterogeneity in the size of the sets of priors. In particular, we consider a single-dimensional affine parametrization of the sets of priors. Since this heterogeneity in the sets of priors is not observable by the analyst, choice is random and can be modelled by the analyst as a RCR. Call this a random ambiguity representation of the RCR. Let u : X! R be a von Neumann-Morgenstern utility index and K be the set of all compact, convex and non-empty subsets of S. Given a prior q in a set of priors K 2K, the expected utility of an act f 2 H is q (u f). 15 Maxmin expected utility evaluates each act according to the worst possible prior in K. In other words, given K 2K,themaxmin expected utility of an act f is U K (f) :=min q2k q (u f) Fix two sets of priors K, K Ksuch that K K. Consider the following Minkowski mixture of set of priors K t := (1 t) K + t K where t 2 T := [0, 1] is a random parameter. Let K T Kbe the set of all such K t and µ be aprobabilitymeasureonk where µ (K T )=1. In the maxmin model, the size of the set of priors K 2Kcan either be interpreted as the magnitude of the agent s ambiguity aversion or the amount of the agent s perceived uncertainty about S. Absent any additional restrictions on choice data or objective data about uncertainty, the model is unable to distinguish between greater ambiguity aversion or greater subjective uncertainty. As a result, without loss of generality, we interpret µ as the distribution of ambiguity aversion in the population. 16 In the single-agent interpretation of our model, µ corresponds to the distribution of unobserved shocks to ambiguity aversion. Given a fixed u, theambiguityaversiondistributionµ is regular iff the maxmin expected utilities of two acts are either always are never equal. This relaxes the standard restriction in traditional RUM where utilities are never equal. Definition. µ is regular iff U Kt (f) =U Kt (g) with µ-measure zero or one. Let (µ, u) consist of a regular µ and a non-constant u. Define a random ambiguity 15 For any act f 2 H, letu f 2 R S denote its utility vector where (u f)(s) =u (f (s)) for all s 2 S. 16 Ghirardato, Maccheroni and Marinacci [23] do differentiate between ambiguity aversion and amount of uncertainty with additional restrictions on choice data. A similar exercise could be carried out in our random choice framework. 9

11 representation as follows. Definition (Random Ambiguity Representation). is represented by (µ, u) iff for f 2 F 2D F (f) =µ {K t 2K T U Kt (f) U Kt (g) 8g 2 F } This is a RUM model where the random utilities are maxmin expected utilities and they depend on the size of the set of priors K t. If a RCR is represented by (µ, u), thenthe probability of choosing f 2 F is precisely the probability that f attains the highest maxmin expected utility in F. Choice is random only due to unobserved heterogeneity in ambiguity aversion. Note that choice over acts that give the same payoff in every state is deterministic. Since the ambiguity aversion distribution µ is unobservable, the analyst can only infer about µ by studying the RCR. Our model provides a simple one-dimensional parametrization of ambiguity aversion such that heterogeneity is complete captured by two fixed sets of priors and a scalar distribution. In addition, it lends itself to several natural interpretations. For example, we can interpret K t as the set of priors that are close to the baseline set of priors K according to some divergence measurement. 17 In this case, t serves as a measurement of the dispersion of beliefs from the set of common priors K. By the additivity of support functions 18,wecan rewrite the maxmin expected utility as U Kt =(1 t) U K + tu K In the single-agent interpretation of our model, this captures the behavior of an agent who has two sets of priors in mind but chooses randomly depending on how attractive either premise is. In the special case where K is a singleton, the random utility decomposes to expected utility plus a random cost of ambiguity aversion distributed according to µ. We now consider the uniqueness properties of the random ambiguity representation. Call an act constant iff f (s) is the same for all s 2 S. Thus, a constant act gives the same payoff in every state. For example, a safe asset such as a risk-less bond would correspond to a constant act. Theorem 1 asserts that the analyst can completely identify the distribution of ambiguity aversion from binary choice data. In fact, binary choices where one of the acts 17 In particular, K t has an affine parametrization if the divergence is linear; that is, if r is equidistant from p and q, thenitisalsoequidistantfromap+(1 a) r and aq+(1 a) r for a 2 [0, 1]. The "-Gini-contamination for example satisfies this property (see Grant and Kajii [25]). 18 For elementary properties of support functions, see Theorem of Schneider [34] 10

12 is constant is enough for full identification of µ. This is a particularly appealing feature that is unique to our model and does not hold in general for any RUM model with maxmin expected utilities. Theorem 1. Let and be represented by (µ, u) and (, v) respectively. Then the following are equivalent (1) f[g (f) = f[g (f) for all f and constant g (2) = (3) (µ, u) =(, v + ) for >0 Proof. See Appendix. We follow with two examples of random ambiguity representations. Example 2. Let S = {s 1,s 2 }, X = {x, y} and u (a x +(1 a) y ) = a 2 [0, 1]. Let K = 1 3, 2 3, K = k, 2 3 k k 2 0, 1 2 and µ be uniform. Each act corresponds to a point in the unit square. For u f =(a, b) 2 [0, 1] 2, U Kt (f) = min q 1 2[ 1 3, 1 3 +t 1 2] (aq 1 +(1 q 1 ) b) = Let be represented by (µ, u). In the region a 8 < : a+2b 3 if a b a+2b 3 + t a b 3 if a<b b, is deterministic with indifference lines 1 b = apple a. For a<b, the indifference lines are b = apple 1+ta with t distributed uniformly 2 2 t on T. This RCR describes an agent whose set of priors on s 1 range uniformly from 1 to 3 1, 5.Notethatthepriorons1 is bounded below by 1 and the only factor driving random choice is the stochastic nature of the prior s upper bound. Example 3. Let S = {s 1,s 2 }, X = {x, y} and u (a x +(1 a) y ) = a 2 [0, 1]. Let K = K = {(1, 0)} so agents are expected utility maximizers and believe that s 1 will surely occur. Let (µ, u) represent and F = {f,g}. Ifu (f (s 1 )) u (g (s 1 )), then F (f) =µ {K t 2K T U Kt (f) U Kt (g)} =1 If u (f (s 1 )) = u (g (s 1 )), thenu Kt (f) =U Kt (g) µ-a.s. so F (f) = F (g) =1 11

13 and neither f nor g is H F -measurable. This is exactly the RCR described in Example 1 above. Example 3 above is exactly the standard expected utility model where agents believe that s 1 will realize for sure. It serves to demonstrate how our random choice model includes deterministic choice as a special case. We end this section with a technical remark about regularity. As mentioned above, indifferences in traditional RUM must occur with probability zero. Since all choice probabilities are specified, these models run into difficulty when there are indifferences in the random utility. 19 Our definition of regularity circumvents this by allowing for just enough flexibility so that we can model indifferences using non-measurability. 20 In Example 3, if U Kt (f) =U Kt (g) µ-a.s., then neither f nor g is H f[g -measurable. Acts that have the same utility µ-a.s. correspond exactly to non-measurable singletons. Note that the deterministic maxmin expected utility representation obtains as a special case if K = K µ-a.s.. Our definition still imposes certain restrictions on µ. Forexample,multiplemasspointsarenot allowed if µ is regular. 3.2 Assessing Ambiguity Aversion In this section, we show that the analyst can use choice data to assess when one population is more ambiguity averse than another. We introduce a notion of comparative uncertainty aversion in the context of stochastic choice. One population is more uncertainty averse than another if constant acts are chosen more frequently in the former than the latter. Definition. is more uncertainty averse than iff F (f) F (f) for all constant f If exhibits more uncertainty aversion than, thentheprobabilityofchoosingaconstant act (which yields a certain fixed payoff) is higher under than under. This is the observable characterization of more uncertainty averse and corresponds exactly to first-order stochastic 19 Note that if we assumed that acts are mappings f : S! [0, 1], thenwecouldobtainaconsistentmodel by assuming that indifferences never occur. Nevertheless, this would not allow us to include deterministic choice as a special case. 20 More precisely, our definition of regularity permits strictly positive measures on sets that have less than full dimension. Regularity in Gul and Pesendorfer [27] on the other hand, requires full-dimensionality (see their Lemma 2). See Block and Marschak [6] for the case of finite alternatives. 12

14 dominance of the distribution of ambiguity aversion. Define µ on larger sets of priors than. when µ puts more weight Definition. µ iff µ {K 2K K L} apple {K 2K K L} for all L 2K. The following proposition asserts that first-order stochastic dominance of ambiguity aversion distributions is exactly characterized by more uncertainty aversion. It allows the analyst to assess which population is more ambiguity averse simply by looking at choice probabilities of constant acts. Proposition 1. Let and be represented by (µ, u) and (, v) respectively. Then is more uncertainty averse than iff µ and u = v + for >0. Proof. See Appendix. Note that our comparative notion of more uncertainty averse is incomplete and there are distributions of ambiguity aversion that are incomparable according to our definition. It is however a random choice generalization of corresponding results under deterministic choice Stochastic Properties of Non-Linearity In the classic maxmin model, violations of linearity is associated with behavior reflective of ambiguity aversion. 22 Hence, before we proceed to the characterization of random ambiguity representations, we first study the stochastic properties of non-linearity in RUM models in general. We focus on two properties of RCRs, convexity and extremeness, which will correspond to the RUM notions of quasiconcavity and quasiconvexity respectively. First, consider the benchmark random expected utility model of Gul and Pesendorfer [27]. Two necessary conditions are linearity and extremeness. We present these two conditions in our setting. Given a decision-problem F 2Kand an act g, letaf +(1 a) g denote the decision-problem obtained by mixing F with g for a 2 [0, 1]. Linearity asserts that choice probabilities remain the same when decision-problems are mixed with other acts. 21 See Theorem 17(ii) of Ghirardato and Marinacci [22]. 22 In general though, an agent can have a non-linear utility and still exhibit probabilistically sophisticated behavior (see Machina and Schmeidler [31]). 13

15 Definition. is linear iff F (f) = af +(1 a)g (af +(1 a) g) for f 2 F and a 2 (0, 1). An act f 2 F is extreme in F iff it is not in the interior of the convex hull of F.LetextF denote the set of extreme acts of F 2D. 23 chosen. Definition. is extreme iff F (extf )=1. Extremeness asserts that only extreme acts are Linearity is the stochastic equivalent of the standard independence axiom. In particular, it is the version of the standard independence axiom that is actually tested in experimental settings (see Kahneman and Tversky [28] for example). If we impose linearity in a RUM model, then linearity of the induced RCR follows as a natural consequence. On the other hand, extremeness is a condition that is unique to random choice. It requires that extreme acts of a decision-problem are always chosen, or vice-versa, interior acts are never chosen. Since linear utilities admit indifference sets that are hyperplanes, the resulting RCR must be extreme (indifferences aside). Given that utilities in a random ambiguity representation are non-linear, one would expect the induced RCR to permit violations of both linearity and extremeness. The following example demonstrates. Example 4. Return to the setup in Example 2 above. Let u f =(0, 1), u g =(1, 0) and h = 1 2 f g.letting be the Lebesgue measure on T,wehave f[g (f) = {t 2 T 2 t 1} =1 3 f[h (f) = t 2 T 2 t = Since 1 2 {f,g} f = {f,h}, we have f[g (f) > f[h (f) violating linearity. F := {f,g,h}, then F (h) = t 2 T 3 2 max {1, 2 t} = 1 2 > 0 If we let violating extremeness. Note that since h has less uncertainty than either f or g, ambiguity aversion implies that it should be chosen with some probability. As Example 4 illustrates, violations of extremeness is precisely the behavior reflective of ambiguity aversion. Interior mixtures are chosen with the probability that ambiguity aversion is strong enough so that they are more attractive than other more extreme acts. 23 Formally, f 2 extf 2Diff f 2 F and f 6= ag +(1 a) h for some {g, h} F and a 2 (0, 1). 14

16 We introduce a novel condition called convexity to allow for violations of both linearity and extremeness. It asserts that the probability that an act is chosen will not be affected by the addition of a new act provided some mixture of the two acts is in the original decisionproblem. Definition. is convex iff F (f) = F [g (f) for af +(1 a) g 2 F and a 2 (0, 1). Let f and g be two acts and h = af +(1 a) g be any mixture of the two for some a 2 (0, 1). Convexity requires that for any decision-problem that already includes h, adding g will not affect the probability that f is chosen. The following example illustrates. Example 5. Return to Example 4 above. Note that 3 F (f) = t 2 T 2 t max 2, 1 = 1 2 = f[h (f) satisfying convexity. We now provide justification for our choice of convexity. Let V be the set of all measurable v : H! R and let µ be a probability measure on V. Foreaseofexposition,weassumethat µ has no indifferences. In other words, for all {f,g} H, v (f) =v (g) with µ-measure zero. The following is a definition of RUM in this setting. Definition. is represented by µ iff for all f 2 F 2D, F (f) =µ {v 2V v (f) v (g) 8g 2 F } Note that although maxmin expected utility is non-linear, it is quasiconcave. Recall that under deterministic choice, a utility is quasiconcave iff the preference relation it represents is convex. 24 In the realm of random choice, quasiconcavity under RUM corresponds exactly to convexity of the RCR. Define quasiconcavity and quasiconvexity under RUM as follows. Definition. µ is quasiconcave iff for all {f,g} H and a 2 (0, 1), µ-a.s. It is quasiconvex iff µ-a.s. v (af +(1 a) g) min {v (f),v(g)} v (af +(1 a) g) apple max {v (f),v(g)} 24 Apreferencerelation is convex iff f g implies af +(1 a) g g for all a 2 (0, 1). 15

17 Theorem 2. Let be represented by µ. Thenµ is quasiconcave (quasiconvex) iff is convex (extreme). Proof. See Appendix. Theorem 2 asserts that convexity and extremeness are the stochastic properties that characterize quasiconcavity and quasiconvexity respectively under RUM. Note that since random expected utility is both quasiconcave and quasiconvex, its RCR must be both convex and extreme. In fact, any random utility that satisfies betweenness 25 must also be both convex and extreme. Our result extend easily to other choice domains (including lotteries) and is a first step toward generalizing the axiomatic study of non-linearity to models of random choice. 5 Characterization We now provide an axiomatic characterization of random ambiguity representations. Given an act f and state s 2 S, letf s 2 H denote the constant act that yields the fixed payoff f (s) in every state. For F 2D,letF s := S f2f f s be the decision-problem consisting of constant acts f s for all f 2 F.Weassumef 2 F 2Dthroughout. The first five axioms are the stochastic counterparts to the original axioms of maxmin expected utility. Axiom 1 (C-linearity). F (f) = af +(1 a)ag (af +(1 a) g) for constant g and a 2 (0, 1). Axiom 2 (Continuity). is continuous. Axiom 3 (S-monotonicity). Fs (f s )=1for all s 2 S implies F (f) =1. Axiom 4 (Convexity). is convex. Axiom 5 (Non-degeneracy). f[g (f) =0for some constant f and g. Certainty-linearity (or C-linearity) imposes linearity only for mixtures with constant acts. It is a weakening of linearity that directly corresponds to C-independence, the weakening of the standard independence axiom under deterministic choice. Continuity is standard albeit adjusted for issues with indifferences discussed above. State-monotonicity (or 25 That is min {v (f),v(g)} applev (af +(1 a) g) apple max {v (f),v(g)} µ-a.s.. See Dekel [10] for an axiomatic characterization of betweenness in the lottery framework. 16

18 S-monotonicity) statesthatifallagentsagreethatanactisthebestineverystate,then they must all choose it for sure. It is the random choice analog of the monotonicity axiom in the original maxmin model. In fact, it is both necessary and sufficient for any RUM model that satisfies state-by-state monotonicity. Convexity is exactly the stochastic version of the uncertainty aversion axiom in the deterministic model and plays the same role by enforcing quasiconcavity. Finally, non-degeneracy rules out the trivial case of universal indifference. The last two axioms are particular to random choice. Monotonicity states that the probability that an act is chosen weakly decreases if the decision-problem is enlarged. This is necessary (but not sufficient) for any RCR with a RUM representation. Axiom 6 (Monotonicity). F G implies F (f) G (f). One interesting implication of monotonicity in conjunction with convexity is that an act is chosen less frequently when other acts are closer to it. To illustrate this, let f 2 F and h = af +(1 a) g for some a 2 (0, 1). Thus, h is closer to f than g is to f. Bymonotonicity and convexity, F [g (f) F [g[h (f) = F [h (f) Thus, an act become less prominent in choice if other acts are closer toward it. This is an observable characteristic of quasiconcavity under RUM that is completely unique to random choice. Before we present the final axiom, we first consider a special class of RUM models known as Luce rules. One feature of Luce rules is that they satisfy an appealing form of stochastic independence: iftheprobabilityofchoosinganactoverasetofactsisgreaterthanthatof choosing another act over that same set, then the same must hold with respect to any set of acts. In our setup, we formally define it as follows. Definition. is independent iff f[g (f) g[g (g) implies f[f (f) g[f (g) for all F 2D. Gul, Natenzon and Pesendorfer [26] show that under a richness condition, independence completely characterizes the Luce rule. RCRs with a random ambiguity representation however, do not satisfy independence as the following example illustrates. Example 6. Return to the setup of Example 2 above. Let u f = 1, 1, u h = 3, 3,

19 g = 1 2 f h and u ĥ = 5 7, 5 7 f[h (f) =. Now, by C-linearity t 2 T t 9 4 = 1 2 = g[h (g) However, f[ ĥ (f) = t 2 T t 15 7 = < = t 2 T t 15 7 = g[ ĥ (g) violating independence. Nevertheless, RCRs with random ambiguity representations satisfy a weakened form of independence which we call binary-independence (or B-independence). Under B-independence, if the binary choice probability of an act is greater than that of another with respect to two constant acts, then the probability of choosing the first is greater than that of choosing the second over any set of acts. Axiom 7. (B-independence). If there are constants h 1 and h 2 such that then f[f (f) g[f (g) for all F 2D. 1= f[h1 (f) > g[h1 (g) f[h2 (f) > g[h2 (g) =0 B-independence follows from the single-dimensional parametrization of sets of priors in random ambiguity representations. 26 The conditions that f[h1 (f) =1and g[h2 (g) =0 are important; they ensure that the comparisons between f and g are stark enough for f[f (f) g[f (g) to hold for all F 2 D. The following illustrates an example of B- independence. Example 7. Return to Example 6 above. Let u ĝ = 1 4, 3 4, u h 1 = 1 2, 1 2 and h 2 = h. Now, f[h1 (f) = t 2 T t 3 2 =1> 1 2 = t 2 T t 3 2 = ĝ[h1 (ĝ) and f[h2 (f) = 1 2 > 0= t 2 T t 9 4 = ĝ[h2 (ĝ) 26 The affine parametrization of K t restricts the utilities our representation to be random affine functions on T. 18

20 By B-independence, f[f (f) all t 2 T, ĝ[f (ĝ) for any F 2D. This follows from the fact that for U Kt (f) = 10 2t 4 > 7 2t 4 = U Kt (ĝ) We now present the representation theorem. Taken together, Axioms 1-7 are necessary and sufficient for a random ambiguity representation. Note that the uniqueness properties of this representation are specified in Theorem 1 above. Theorem 3. satisfies Axioms 1-7 iff it is has a random ambiguity representation. Proof. See Appendix. We provide a brief outline of the sufficiency argument for Theorem 3. First, note that the lower contour sets in maxmin expected utility are translated convex cones. If we consider the two-dimensional linear subspace containing some act and two distinct constant acts, then the random utilities are linear for all decision-problems in this subspace. The crux of the argument rests on showing that convexity and B-independence are sufficient for establishing linearity and extremeness in this subspace. This allows us to construct a random expected utility representation of the RCR restricted this subspace. B-independence then allows us to map all decision-problems into this subspace and admit a random ambiguity representation. Necessity of the axioms follow easily from the representation. 19

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24 Appendix A Given a collection of sets G and F 2D,let G\F := {G \ F G 2G} Note that if G is an algebra, then G\F is the trace algebra of G on F 2D.ForG F 2D, let G F := \ G G 0 2H F G 0 denote the smallest H F -measurable set containing G. Lemma (A1). Let G F 2D. (1) H F \ F = H\F (2) G F = Ĝ \ F 2H F for some Ĝ 2H (3) F F 0 2Dimplies G F = G F 0 \ F Proof. Let G F 2D. (1) Recall that H F := (H[{F}) so H H F implies H\F H F \ F.Let G := {G H G \ F 2H\F } We first show that G is an algebra. Let G 2Gso G \ F 2H\F.Now G c \ F =(G c [ F c ) \ F =(G \ F ) c \ F = F \ (G \ F ) 2H\F as H\F is the trace algebra on F. Thus, G c 2G.ForG i G, G i \ F 2H\F so! [ G i \ F = [ (G i \ F ) 2H\F i i Thus, G is an algebra Note that H Gand F 2Gso H[{F } G. Thus, H F = (H[{F }) G.Hence, so H F \ F = H\F. H F \ F G\F = {G 0 \ F G 0 = G \ F 2H\F } H\F 23

25 (2) Since H F \ F H F,wehave G F := G 0 G G 0 2H F \ \ G G 0 2H F \F Now suppose g 2 T G G 0 2H F \F G0. Let G 0 be such that G G 0 2H F. Now, G G 0 \ F 2H F \ F so by the definition of g, wehaveg 2 G 0 \ F.Sincethisistruefor all such G 0,wehaveg 2 G F.Hence, \ \ G F = G 0 = G G 0 2H F \F G G 0 2H\F where the second equality follows from (1). Since F is finite, we can find Ĝi 2Hwhere G Ĝi \ F for i 2{1,...,k}. Hence, G F = \ Ĝi \ F = Ĝ \ F i G 0 G 0 where Ĝ := T i Ĝi 2H.NotethatG F 2H F follows trivially. o (3) By (2), let G F = Ĝ \ F and G F 0 = Ĝ0 \ F 0 for nĝ, Ĝ 0 H.SinceF F 0, G G F 0 \ F = Ĝ0 \ F 2H F so G F G F 0 \F by the definition of G F. Now, by the definition of G F 0, G F 0 Ĝ\F 0 2 H F 0 so G F 0 \ F Ĝ \ F 0 \ F = Ĝ \ F = G F Hence, G F = G F 0 \ F. Let be a RCR. By Lemma A1, we can now define F (G) := inf F (G 0 )= F (G F ) G G 0 2H F for G F 2D. Going forward, we simply let denote without loss of generality. For {G, F } D,weusethecondensednotation GaF := ag +(1 a) F 24

26 We also employ the notation (F, G) := F [G (F ) Definition. f and g are tied iff (f,g) = (g, f) =1 Lemma (A2). For {f,g} F 2D,thefollowingareequivalent. (1) f and g are tied (2) g 2 f F (3) f F = g F Proof. We prove that (1) implies (2) implies (3) implies (1). Let {f,g} F 2D. First, suppose f and g are tied so (f,g) = (g, f) =1.Iff f[g = f, theng =(f [ g) \f F 2H f[g so g f[g = g. Thus, (f,g)+ (g, f) =2> 1 a contradiction. Thus, f f[g = f [ g. Now, since f [ g F,byLemmaA1,f [ g = f f[g = f F \ (f [ g) so g 2 f F. Thus, (1) implies (2). Now, suppose g 2 f F so g 2 g F \f F.ByLemmaA1,g F \f F 2H F so g F g F \f F which implies g F f F. If f 62 g F,thenf 2 f F \g F 2H F. Thus, f F f F \g F implying g F = Ø a contradiction. Thus, f 2 g F,sof 2 g F \ f F which implies f F g F \ f F and f F g F. Hence, f F = g F so (2) implies (3). Finally, assume f F = g F so f [ g f F by definition. By Lemma A1 again, f f[g = f F \ (f [ g) =f [ g so (f,g) = f[g (f [ g) =1. By symmetric reasoning, (g, f) =1so f and g are tied. Thus, (1) iff (2) iff (3). Lemma (A3). Let be monotonic. (1) For f 2 F 2D, F (f) = F [g (f) if g is tied with some g 0 2 F (2) Let F := S i f i, G := S i g i and assume f i and g i are tied for all i 2{1,...,n}. Then F (f i )= G (g i ) for all i 2{1,...,n}. Proof. We prove the lemma in order (1) By Lemma A2, we can find unique h i 2 F such that h 1 F,...hk F forms a partition on F.Withoutlossofgenerality,assumeg is tied with some g 0 2 h 1 F.ByLemmaA2 again, h 1 F [g = h1 F [ g and hi F [g = hi F for i>1. Bymonotonicity,foralli F h i F = F h i F [g h i = F [g h i F [g 25