Random ambiguity

Lu, Jay Article Random ambiguity Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Lu, Jay (2021) : Random ambiguity, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 16, Iss. 2, pp. 539-570, https://doi.org/10.3982/TE3810 This Version is available at: https://hdl.handle.net/10419/253493 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 16 (2021), 539–570 1555-7561/20210539 Random ambiguity Jay Lu Department of Economics, UCLA We introduce a model of random ambiguity aversion. Choice is stochastic due to unobserved shocks to both information and ambiguity aversion. This is modeled as a random set of beliefs in the maxmin expected utility model of Gilboa and Schmeidler (1989). We characterize the model and show that the distribution of ambiguity aversion can be uniquely identified from binary choices. A novel stochastic order on random sets is introduced that characterizes greater uncertainty aversion under stochastic choice. If the set of priors is the Aumann expectation of the random set, then choices satisfy dynamic consistency. This corresponds to an agent who knows the distribution of signals but is uncertain about how to interpret signal realizations. More broadly, the analysis of stochastic properties of random ambiguity attitudes provides a theoretical foundation for the study of other random nonlinear utility models. Keywords. Stochastic choice, ambiguity, random utility, updating. JEL classification. D81, D83. 1. Introduction In many economic situations, ambiguity aversion, i.e., aversion toward Knightian uncertainty (Knight 1921), is useful for explaining behavior that cannot be easily addressed by standard expected utility maximization. In practice, however, it is useful to model choice behavior as stochastic due to unobserved heterogeneity in a population of agents or unobserved shocks to a single agent’s preferences. In this paper, we provide a theory of stochastic choice generated by random ambiguity. To be concrete, consider a population of agents choosing between a safe and an uncertain asset where prices are fixed. Agents have heterogeneous ambiguity attitudes.1 Thosewhoaremoreambiguity-aversechoosethesafeassetwhilethosewhoareless ambiguity averse choose the uncertain one.2An analyst observes only the proportion of Jay Lu: [email protected] This is a significantly revised version of Chapter 2 of my dissertation at Princeton. I am deeply indebted to Faruk Gul and Wolfgang Pesendorfer for their advice, guidance, and encouragement. I would also like to thank Larry Blume, David Easley, Diogo Guillen, Rohit Lamba, Yi-Hsuan Lin, Stephen Morris, Paulo Natenzon, Pietro Ortoleva, and Kota Saito for their helpful comments. Financial support from the NSF under Award SES-1919275 is gratefully acknowledged. 1Empirical evidence of this heterogeneity include Abdellaoui et al. (2011) and Ahn et al. (2014). 2One could also interpret the magnitude of ambiguity aversion as reflective of uncertainty about the unknown state. For example, in Caballero and Krishnamurthy (2008), agents face an enlarged set of possible priors during surprise defaults or bank runs and scramble for safe assets in a flight-to-quality. ©2021 The Author. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE3810 540 Jay Lu Theoretical Economics 16 (2021) agents who choose each asset (i.e., stochastic choice), but not the distribution of ambiguity aversion. Our model shows how the analyst can uniquely identify this distribution using stochastic choice from a rich set of binary options. This is an important exercise if the analyst is a regulator deciding on optimal policy implementation (see Easley and O’Hara 2009). Alternatively, consider a single hiring manager faced with a pool of job applicants. The manager conducts private interviews before deciding whether to hire each worker. Hiring decisions depend on the information acquired from interviews as well as the manager’s perceived ambiguity regarding each worker. An analyst observes only the hiring rate (i.e., stochastic choice) of the manager, but not the distribution of interview outcomes. Our model shows how the analyst can identify this distribution using stochastic choice. We introduce a model of stochastic choice and ambiguity that captures both examples. In the multiple-priors model of Gilboa and Schmeidler (1989), an agent evaluates an option fby minimizing expected utility over a set Kof possible beliefs about the payoff-relevant state. The maxmin expected utility of the option fis given by uK(f) =min π∈Kπ·(u ◦f) In our random ambiguity model, the von Neumann–Morgenstern (vNM) utility uis fixed, but the set of beliefs Kis random. In other words, this is a random utility model where the utilities are maxmin expected utilities that depend on K. The distribution of ambiguity aversion is captured by the distribution of the random set K. Our main results are as follows. First, we show that the distribution of the random set Kcan be fully identified from stochastic choice from binary menus. We then introduce a novel stochastic order on random sets that characterizes greater uncertainty aversion in this context. Next, we show that if the ex ante set of priors is exactly the Aumann expectation of K(i.e., the set of all priors consistent with K), then choice behavior satisfies dynamic consistency. We also provide a full axiomatic characterization of the model. Although we focus on the maxmin model due to its tractability, our analysis of stochastic properties of random ambiguity attitudes provides a theoretical foundation for the general study of random nonlinear utility. Section 2.1 introduces the formal setup. Let Sbe a finite payoff-relevant state space. An act is a state-contingent mapping from Sto payoffs in the form of risky prospects. Following Anscombe and Aumann (1963)andSeo (2009), we distinguish between randomization that is ex ante (i.e., before the state is realized) and ex post (i.e., after the state is realized). Formally, each choice option is a lottery, an ex ante mixture over acts. The main primitive is a stochastic choice over menus of lotteries that specify the choice probability for each lottery pin the menu A. In our model, the probability of choosing a lottery p∈Ais precisely the probability that pattains the highest utility in the menu A. Since ambiguity is with respect to the state, ex ante mixtures are evaluated using standard expected utility. Thus, this is a random utility model where utilities are expected utilities with respect to ex ante randomization and maxmin with respect to ex post randomization. Theoretical Economics 16 (2021) Random ambiguity 541 Distinguishing between ex ante and ex post randomization in this stochastic choice model provides several advantages. First, the use of ex ante randomization facilitates the axiomatic characterization (see Section 3) as characterizing random utility in general without additional structure is a known difficulty.3Second, the use of lotteries over acts allows for sharper identification results (Section 2.2) and cleaner comparative statics (Section 2.3). Section 2.2 considers identification. Theorem 1 shows that the distribution of the random set Kcan be identified from simple binary menus where one option is constant (i.e., yields the same payoff regardless of the state). This is achieved by leveraging the richness of ex ante randomization. In the absence of ex ante randomization, identification can still be achieved, but requires stochastic choice data beyond binary menus (see Theorem 7 in the Appendix); this is possible despite the well known difficulties in identifying models of random nonexpected utility (see Lin 2020). Section 2.3 considers comparative statics. Call one stochastic choice more uncertainty-averse than another if constant acts are chosen more frequently in the first than in the second. Theorem 2 shows that greater uncertainty aversion is exactly characterized by a novel stochastic order called stochastic c-dominance. A random set of beliefs is greater than another in this order if it puts greater weight on all increasing, convex, and closed families of beliefs. This notion is weaker than the standard stochastic order on random sets that corresponds to first-order stochastic dominance with respect to set inclusion. In Section 2.4, we extend our model to address updating and dynamic consistency. Consider a random set of beliefs K. If a random belief is always in the random set, then we call it a posterior selection of K. Suppose the agent’s initial set of priors is the Aumann expectation of K, i.e., the set of priors of all posterior selections of K. This could correspond to an agent who knows the distribution of signals but is uncertain about how to interpret signal realizations. As a result, he has in mind a family of possible signal structures. Theorem 3 shows that if the agent’s ex ante preferences are represented by a maxmin expected utility where the set of ex ante priors is the Aumann expectation, then the agent satisfies dynamic consistency. This indicates that stochastic models of ambiguity may suggest new approaches toward updating with multiple priors. Section 3 provides the axiomatic characterization. The first axiom (monotonicity, ex ante independence, ex ante extremeness, continuity) consists of conditions that characterize random expected utility from Gul and Pesendorfer (2006). A novel axiom, ex post hedging, characterizes stochastic aversion toward uncertainty. It states that adding an act will not affect the choice of another act as long as a hedge option (i.e., mixture of the two acts) is available in the menu. Certainty reversal of order states that when mixing with constant acts, ex ante and ex post randomization are the same. This is because constant acts involve no uncertainty, so the agent is indifferent to the timing of randomization. Finally, the last three axioms (certainty determinism, dominance, and nondegeneracy) are conditions from the information representation of Lu (2016). Theorem 4 states that these axioms exactly characterize the random ambiguity model. 3For instance, compare the characterization of Falmagne (1978)withthatofGul and Pesendorfer (2006) in the richer lottery setup. The use of ex ante randomization allows us to adopt the latter approach. 542 Jay Lu Theoretical Economics 16 (2021) Finally, in Section 4, we study the general stochastic properties of nonlinear random utility. In the random ambiguity model, Bernoulli utilities are maxmin expected utility and exhibit ambiguity aversion. As a result, ex post randomization is desirable and utilities are quasiconcave (with respect to ex post mixtures). We show that for random utilities, quasiconcavity is exactly characterized by the ex post hedging axiom. Alternatively, if the agent is ambiguity-loving, then ex post randomization is not desirable and utilities are quasiconvex. For random utility, quasiconvexity is exactly characterized by the ex post version of the extremeness axiom of Gul and Pesendorfer (2006). Theorem 5 thus provides a characterization of quasiconcavity and quasiconvexity for random utilities. We then apply the result to characterize betweenness for random utility (Corollary 1) and ambiguity neutrality in our random ambiguity model (Corollary 2), where choice is stochastic only due to information. More generally, these results provide a theoretical foundation for studying other stochastic ambiguity models beyond random maxmin expected utility. All proofs are provided in the Appendix unless stated otherwise. 1.1 Related literature This paper is related to a long literature on stochastic choice and random utility. Early seminal works include Block and Marschak (1960), McFadden and Richter (1990), and Falmagne (1978). More recent works have leveraged enriched choice domains to obtain more structured characterizations. These include Gul and Pesendorfer (2006), Ahn and Sarver (2013), Fudenberg and Strzalecki (2015), Lu (2016), Lu and Saito (2018), Duraj (2018), Frick et al. (2019), and Lin (2019). In particular, this paper contributes to our understanding of random nonlinear utility. Violations of linearity in a stochastic context have been well documented (see Becker et al. 1963 and Kahneman and Tversky 1979). In Section 4, we study the stochastic properties of random nonlinear utility and provide characterizations of random quasiconcave and quasiconvex utility. Recently, Lin (2020) shows that contrary to random linear utility, models of random nonlinear utility may lack unique identification. Recent papers that have also studied ambiguity and stochastic choice include Fudenberg et al. (2015), Saito (2015), and Agranov and Ortoleva (2017). In all these models, the agent deliberately randomizes over choice options due to a preference for ex ante randomization. In contrast, the agent in our model has a preference for ex post randomization and any observed stochastic choice is due to random ambiguity. For stochastic binary choices, Ryan (2018) characterizes a Fechner model of ambiguity aversion. Epstein and Kopylov (2007) study a model of menu choice where a sophisticated agent anticipates future shocks to ambiguity aversion. While their model is ex ante where choices are due to the anticipation of cold feet shocks, our model is ex post where choices are the direct result of such shocks. Seo (2009) considers a deterministic choice model that also distinguishes between ex ante and ex post randomization. He also relaxes the reversal of order axiom and assumes the agent is impartial to ex ante randomization. The paper is also related to a large literature on updating under ambiguity. Approaches include Gilboa and Schmeidler (1993), Sarin and Wakker (1998), Epstein and Theoretical Economics 16 (2021) Random ambiguity 543 Schneider (2003), Hanany and Klibanoff (2007), and Siniscalchi (2011). When updating with multiple priors, a tension arises between accommodating rich ambiguity attitudes and satisfying dynamic consistency and consequentialism. We show that if the original set of priors is the Aumann expectation of the random set of posteriors, then the agent satisfies dynamic consistency and consequentialism. Thus, stochastic choice models with random sets of beliefs may suggest new restrictions on updating under ambiguity. Finally, this paper is related to a large literature on applications of heterogeneous ambiguity aversion. Dow and da Costa Werlang (1992)andEpstein and Wang (1994) study the effects of ambiguity aversion on market nonparticipation and asset prices. Bose et al. (2006) investigate varying ambiguity aversion in an auction setting. Caballero and Krishnamurthy (2008) study investors who receive random shocks to ambiguity aversion due to surprise defaults or bank runs and scramble to safer assets in a phenomenon called flight-to-quality. Easley and O’Hara (2009) study the role of regulation in a heterogeneous population with different levels of ambiguity aversion. Epstein and Schneider (2010) provide a review of various limited participation problems involving heterogeneous ambiguity aversion. 2. A model of random ambiguity 2.1 Setup and model Let Sbe a finite state space and let Xbe a finite set of prizes. Let S and X be their respective probability simplexes. We interpret S as the set of all beliefs about Sand interpret X as the set of payoffs that consists of risky prospects. As in Anscombe and Aumann (1963), an act corresponds to a state-contingent payoff f:S→X.Anactis constant if f(s)isthesameforalls∈S.LetHdenote the set of all acts and let Hc⊂H denote the set of constant acts. Following Anscombe and Aumann (1963)andSeo (2009), we consider lotteries (i.e., Borel probability measures) over acts. Let H denote the set of all lotteries. Given an act f∈H,letδf∈H denote the degenerate lottery that yields ffor sure. We distinguish between ex ante and ex post randomization. For instance, given two lotteries δfδg∈ H and a∈[01],themixture p=aδf⊕(1−a)δg∈H corresponds to ex ante mixing of payoffs where pyields the act fwith probability aand yields gwith probability 1−a. Since randomization occurs before the state is realized, it is ex ante. Alternatively, given two acts f g ∈Hand a∈[01],themixture h=af +(1−a)g ∈H corresponds to ex post mixing of payoffs where h(s) =af (s) +(1−a)g(s) for all states s∈S. Here, randomization is ex post as it occurs after the state is realized. 544 Jay Lu Theoretical Economics 16 (2021) Call a finite set of lotteries a menu and let Adenote the set of all menus. We endow H with the topology of weak convergence and endow Awith the corresponding Hausdorff metric. Note that Ais a mixture space under Minkowski mixing for ex ante randomization.4Stochastic choice corresponds to a choice distribution across all lotteries in a menu. Let (H) denote the set of all choice distributions over lotteries. Definition 1. A stochastic choice is a mapping ρ:A→(H) such that ρA(A) =1for all A∈A. For any menu A∈Aand lottery p∈A,ρA(p) specifies the probability that the lottery pis chosen in the menu A. For binary menus A={pq}, we employ the more succinct notation ρ(pq) =ρA(p). As with any model of random utility, there is a need to address ties. Following Lu (2016), we model ties by relaxing the restriction that all choice probabilities have to be fully specified. For example, if two lotteries pand qare tied, then the stochastic choice does not specify choice probabilities for either por qspecifically. Formally, we model this as nonmeasurability with respect to the choice probability and let ρdenote the corresponding outer measure without loss of generality.5With this definition, ρ(pq) =ρ(qp) =1whenever pand qare tied. In the classic Raiffa (1961) critique, an agent who is ambiguity-averse may strictly prefer randomization for hedging purposes. Differentiating between ex ante and ex post randomization is one way the literature has resolved this issue (see Saito 2015). Since ambiguity is with respect to the state, ex ante randomization that occurs before the state is realized does not help. Stochastic choice adds a third layer of randomization, as the agent can now deliberately randomize between acts (see Cerreia-Vioglio et al. 2019). However, since deliberate randomization also occurs before the state is realized, an agent who is impartial to ex ante randomization would likely also be impartial to deliberate randomization. Adopting this approach allows us to interpret any observed stochastic choice as the result of random ambiguity rather than deliberate randomization. We now describe the random ambiguity model. Let Kbe the set of all nonempty compact convex subsets of S endowed with the Hausdorff metric. Note that Kis also a mixture space under Minkowski mixing. In the multiple-priors model of Gilboa and Schmeidler (1989), maxmin expected utility is characterized by a von Neumann– Morgenstern (vNM) utility u:X →Rand a set of beliefs K∈K. Formally, the expected utility of an act f∈Hgiven a belief π∈S is π·(u ◦f)= s π(s)uf(s) 4The Minkowski mixture of two menus AB ∈Aand a∈[01]is given by aA ⊕(1−a)B ={ap ⊕(1−a)q : p∈Aq ∈B}. 5Stochastic choice naturally subsumes a σ-algebra Bon H. Given any menu A∈A,thecorresponding choice distribution ρAis a measure on the σ-algebra generated by B∪{A}.Weletρdenote the outer measure with respect to this σ-algebra without loss. For details, see Lu (2016). Theoretical Economics 16 (2021) Random ambiguity 545 Maxmin expected utility evaluates each act according to the worst possible belief in the set K∈K. In other words, the maxmin expected utility of an act fis given by uK(f) := min π∈Kπ·(u ◦f) We extend the utility to lotteries by defining the expected utility UK(p) := HuK(f)dp, where the Bernoulli utility uK(·)is maxmin. Note that this implies that the agent is indifferent to ex ante randomization as in Seo (2009). In our model, choice is stochastic due to unobserved shocks to both information and ambiguity attitudes. We model this by allowing the set of beliefs Kto be random. Formally, let μbe a Borel probability measure on Kwith the σ-algebra induced by the Hausdorff metric. A random element taking values in Kwith distribution μis called a random set. Given a vNM utility u, a distribution μis regular if the utilities of two options are either always or never equal, i.e., UK(p) =UK(q) with probability 0or 1. This relaxes the standard restriction in standard random utility models that assumes that utilities are never equal. Going forward, let (μu) consist of a regular distribution μand a nonconstant u. Define a random ambiguity representation as follows. Definition 2 (Random ambiguity). Stochastic choice ρis represented by (μu) if ρA(p) =μK∈K:UK(p) ≥UK(q) for all q∈A This is a random utility model where the random Bernoulli utilities are maxmin expected utilities that depend on the random set of beliefs K. The probability of choosing p∈Ais precisely the probability that pattains the highest utility in the menu A. In the group interpretation of the model, there is a population of maxmin agents, and stochastic choice reflects unobserved heterogeneity in both information and ambiguity in the population. This heterogeneity is captured by μ,whichisthepopulation distribution of sets of priors.6For example, suppose fcorresponds to a safe asset (e.g., a constant act), while gcorresponds to an asset with uncertain payoffs. In this case, ρ(δfδg)reflects the proportion of agents in the population who are sufficiently ambiguity-averse (i.e., large enough sets of priors) that they choose the safe asset over the uncertain asset. In the individual interpretation of the model, there is a single maxmin agent, and stochastic choice reflects the agent’s private information and ambiguity attitudes. This is captured by μ, which is a distribution of signals where each signal realization corresponds to a set of priors for the agent. Choice frequencies can be calculated from repeated choices over a series of independent decisions or over time. For example, suppose the agent is a hiring manager, and the acts fand gcorrespond, respectively, to 6Note that in the maxmin model, the size of the set of priors can be interpreted either as the magnitude of ambiguity aversion or the amount of perceived subjective uncertainty about the state. Ghirardato et al. (2004) show that the two interpretations can be distinguished with additional restrictions on choice data. A similar exercise can be performed here. 546 Jay Lu Theoretical Economics 16 (2021) hiring and not hiring a worker from a pool of job applicants. In this case, ρ(δfδg)reflects the hiring rate of the manager, which depends on ambiguity attitudes as well as private information gleaned from interviews. Alternatively, suppose fcorresponds to constant consumption every period, while gcorresponds to uncertain consumption every period. In this case, ρ(δfδg)reflects the time-averaged long-run frequency that the agent is sufficiently ambiguity-averse and chooses constant consumption (see Lu and Saito 2020). The following examples are two special cases of the random ambiguity model. In the first example, information is fixed but ambiguity attitudes are random, while in the second, ambiguity attitudes are fixed but information is random. Example 1. Suppose the random set of beliefs satisfies K=(1−){π}+J where ∈[01]is a random parameter, π∈S is a fixed belief, and J∈Kis a fixed set of beliefs such that π∈J. This is a simple one-dimensional parametrization of random ambiguity aversion where εmeasures the dispersion of beliefs from the fixed belief π.It is a stochastic version of the classic -contamination model that has been widely used in robust statistics.7Note that given this parametrization of K, it is straightforward to rewrite the maxmin expected utility as uK=u{π}−(u{π}−uJ) The random utility can be thus decomposed into a fixed subjective expected utility u{π} minus a stochastic cost of ambiguity aversion (u{π}−uJ). If we interpret as the magnitude of ambiguity aversion, then choice is stochastic due to random ambiguity aversion, while information is fixed. ♦ Example 2. Suppose Kis a singleton almost surely or, in other words, K={π},where πis a random belief in S. In this case, the maxmin expected utility of p, UK(p) =H π·(u ◦f)dp=π·(u ◦fp) is just the subjective expected utility of fp:= Ep[f], the average act under p.Asaresult, ρA(p) =μπ∈S :π·(u ◦fp)≥π·(u ◦fq)for all q∈A In this example, choice is stochastic due to information, while the agent is ambiguityneutral. Note that this corresponds to the information representation of Lu (2016). Corollary 2 in Section 4 provides a characterization of this special case. ♦ 7It was even suggested by Ellsberg (1961) as a simple functional form to address his namesake paradox. See Gajdos et al. (2008) and Kopylov (2009) for axiomatic characterizations of deterministic models of - contamination. Theoretical Economics 16 (2021) Random ambiguity 553 3. Characterization We now provide an axiomatic characterization of the random ambiguity model. First, we leverage ex ante randomization to obtain a random (ex ante) expected utility representation. We then impose conditions to ensure that the random (ex post) Bernoulli utilities are maxmin expected utilities. The first axiom (Axiom 1) consists of four conditions from Gul and Pesendorfer (2006) with respect to ex ante randomization. Monotonicity ensures that the probability an option is chosen is decreasing in the size of the menu and is necessary for any random utility model. Ex ante independence is the standard independence axiom applied to ex ante randomization. A lottery p∈Ais (ex ante) extreme in Aif it is not in the interior of the convex hull (with respect to ex ante randomization) of A.Letext⊕(A) denote the set of extreme lotteries of A.Ex ante extremeness asserts that ex ante randomization is not helpful, so only extreme lotteries are chosen, barring ties. Continuity says that the stochastic choice mapping restricted to the domain A◦⊂Aof menus without ties is continuous. Axiom 1.1 (Monotonicity). For all p∈A⊂Band AB ∈A,ρA(p) ≥ρB(p) . Axiom 1.2 (Ex ante independence). For all p∈A∈A,q∈H,anda>0,ρA(p) = ρaA⊕(1−a)q(ap ⊕(1−a)q). Axiom 1.3 (Ex ante extremeness). For all A∈A,ρA(ext⊕(A)) =1. Axiom 1.4 (Continuity). The mapping ρ:A◦→(H) is continuous. The next axiom is the stochastic analog to uncertainty aversion and preference for hedging. To illustrate, consider two acts fand g, and the ex post mixture h=af +(1−a)g for a∈(01).Notethathis the hedge option. For deterministic preferences, uncertainty aversion means that his ranked higher than the worst of fand g. This implies four possible preference rankings: fhgghf hfghgf Among the four rankings, fis preferred to both gand hif and only if fis preferred to just h. For stochastic preferences, this means that the probability of choosing fover both gand his the same as the probability of just choosing fover h.Inotherwords,one can ignore the other option gwhen the hedge option his available. Axiom 2 (ex post hedging) generalizes this concept. Removing an option does not affect the choice of another option as long as the hedge option (i.e., mixture of the two options) is available. In Section 4, we show that this axiom characterizes random quasiconcave utility and is exactly the stochastic analog of convex preferences in the deterministic setting. We use the notation δFto denote the menu of degenerate acts δF:= {δf:f∈F} for any finite set of acts F⊂H. 554 Jay Lu Theoretical Economics 16 (2021) Axiom 2 (Ex post hedging). For any f∈Fand g∈H,ifaf +(1−a)g ∈F,thenρδF(δf)= ρδF∪{g}(δf). An interesting implication of ex post hedging combined with monotonicity is that an option is chosen less frequently when other options are “closer” to it via randomization. To illustrate, consider h=af +(1−a)g as before, so the act fis closer to hthan it is to g. If we let F={f h g}, then by ex post hedging and monotonicity, ρ(δfδh)=ρδF(δf)≤ρ(δfδg) Thus, the act fis chosen less frequently when other acts are closer. This is the stochastic analog of uncertainty aversion under deterministic choice, i.e., fhimplies fg. Note, however, that this is weaker than ex post hedging and is insufficient to ensure the random utility is quasiconcave. The next axiom relaxes reversal of order and states that the agent does not distinguish between ex ante and ex post randomization when mixing with constant acts. To illustrate, recall that under maxmin expected utility, deterministic preferences satisfy the independence axiom when mixed with constant acts (the certainty-independence axiom of Gilboa and Schmeidler 1989). In other words, if δfis indifferent to δg,then δaf +(1−a)h is also indifferent to δag+(1−a)h for any constant h. Suppose gis also constant and the agent is indifferent to timing whenever there is no uncertainty. Since constant acts convey no uncertainty, δag+(1−a)h is indifferent to aδg⊕(1−a)δh. Finally, since the agent satisfies the independence axiom for ex ante randomization, aδf⊕(1−a)δh∼aδg⊕(1−a)δh∼δaf +(1−a)h Axiom 3 (certainty reversal of order) extends this reasoning for stochastic choice. It states that ex post and ex ante randomization with a constant act are interchangeable in binary comparisons. Axiom 3 (Certainty reversal of order). If h∈His constant, then for any f∈Hand p∈ H, ρaδf⊕(1−a)δhp=ρ(δaf +(1−a)hp) The last three axioms are conditions from the information representation of Lu (2016). Axiom 4 (certainty determinism) states that choice is deterministic over menus consisting of constant acts. It ensures that all stochasticity is due to ambiguity attitudes or information. We say the menu δFis constant if Fconsists only of constant acts. Axiom 4 (Certainty determinism). For all constant δF,ρδF(δf)∈{01}. Axiom 5 (dominance) states that if an act is the best regardless of which state occurs, then it must be chosen for sure. It is the stochastic analog of the standard state monotonicity axiom necessary for maxmin expected utility. Given f∈Hand s∈S,letfs∈Hc Theoretical Economics 16 (2021) Random ambiguity 555 denote the constant act such that fs(s)=f(s)for all s∈S. Given finite F⊂H,let Fs:= {fs:f∈F} denote the set of constant acts in state s∈S. Note that the menu δFsis constant. Axiom 5 (Dominance). If ρδFs(δfs)=1for all s∈S,thenρδF(δf)=1for any f∈F. Finally, nondegeneracy rules out the trivial case of universal indifference. Axiom 6 (Nondegeneracy). There exists Fand some f∈Fsuch that ρδF(δf)<1. We are now ready to present the main representation result. Theorem 4. Stochastic choice ρsatisfies Axioms 1–6if and only if it is represented by (μ u). The primary difficulty of the proof lies in the construction of a random utility representation. Incorporating ex ante randomization in our setup allows us to leverage the random expected utility characterization of Gul and Pesendorfer (2006). Intuitively, linearity lowers the dimensionality of the space of possible utilities, allowing for a simpler axiomatization. In the absence of ex ante randomization, a clean and intuitive characterization of random nonlinear utility would be difficult without imposing additional parametric restrictions.11 The Gul and Pesendorfer (2006) characterization involves lotteries over a finitedimensional set, while we have lotteries over the set of all acts, an infinite-dimensional set. Lu and Saito (2020) provide an infinite-dimensional extension of random expected utility, but the extension applies when utilities are Lipschitz continuous with common bound. The main insight of the proof is that our axioms are sufficient for Lipschitz continuity (see Step 3 in Appendix A.4). Once a random utility representation is established, it is straightforward to show that our axioms imply that the random utility must be maxmin expected utility almost surely. It is worth pointing out that this approach can be used to obtain characterizations of other ambiguity representations beyond maxmin expected utility (e.g., a random Choquet expected utility). 4. Stochastic ambiguity attitudes In this section, we study the general stochastic properties of nonlinearity in models of random utility. For simplicity, we focus exclusively on ex post randomization in this section. In the random ambiguity model, Bernoulli utilities are maxmin expected utilities and exhibit ambiguity aversion. As a result, ex post randomization is desirable and utilities are quasiconcave. We show that for random utilities, quasiconcavity is exactly characterized by ex post hedging. Alternatively, if the agent is ambiguity-loving, then ex post 11Of course, one can always directly impose the Block–Marschak inequalities. 556 Jay Lu Theoretical Economics 16 (2021) randomization is not desirable and utilities are quasiconvex. We show that for random utilities, quasiconvexity is exactly characterized by the ex post version of the extremeness axiom (Axiom 1.3). Our results thus provide a theoretical foundation for studying other stochastic ambiguity models beyond random maxmin expected utility. We say a stochastic choice is ex post hedging if it satisfies Axiom 2 above. Definition 8. Stochastic choice ρis ex post hedging if af +(1−a)g ∈Fimplies ρδF(δf)=ρδF∪{g}(δf)for any f∈Fand g∈H. We can weaken ex post hedging to ex post mixture-loving, which applies only to menus that contain two acts and their mixture. Note that when there is a onedimensional ordering, this is exactly the strong centrality axiom of Apesteguia et al. (2017) that characterizes single-peakedness. Definition 9. Stochastic choice ρis ex post mixture-loving if h=af +(1−a)g implies ρδ{fh}(δf)=ρδ{fhg}(δf)for any f g ∈H. Alternatively, ex post randomization would not be attractive for an ambiguity-loving agent. As a result, ex post mixtures of acts are never chosen in any menu. An act f∈Fis (ex post) extreme in a finite F⊂Hif it is not in the interior of the convex hull of F.Let ext(F) denote the set of extreme acts of F. Ex post extremeness asserts that only extreme acts are chosen, barring ties. Definition 10. Stochastic choice ρis ex post extreme if ρδF(δext(F))=1for all finite F⊂H. We can weaken ex post extremeness to the following model, which applies only to menus that contain two acts and their mixture. Definition 11. Stochastic choice ρis ex post mixture-averse if ρδ{faf +(1−a)gg}({δfδg})= 1for all f g ∈H. We now consider a general random utility model that may be ambiguity-averse or ambiguity-loving. Let Vbe the set of continuous utilities v:H→Rand let νbe a probability measure on V. We say that ρis represented by νif ρδF(δf)=νv∈V:v(f) ≥v(g) for all g∈F For instance, in the random ambiguity model, νis a distribution on maxmin expected utilities. We can now define quasiconcave and quasiconvex as stochastic properties of the random utility. Definition 12. The utility distribution νhas the following properties: (i) It is quasiconcave if v(f) ≥v(g) implies v(af +(1−a)g) ≥v(g) almost surely. (ii) It is quasiconvex if v(f) ≥v(g) implies v(f) ≥v(af +(1−a)g) almost surely. Theoretical Economics 16 (2021) Random ambiguity 557 The main result of this section shows that random quasiconcave utilities are characterized by ex post hedging (or mixture loving), while random quasiconvex utilities are characterized by ex post extremeness (or mixture aversion). For simplicity, assume ν excludes ties, i.e., v(f) =v(g) occurs with zero probability.12 Theorem 5. Let ρbe represented by νthat excludes ties. (i) Distribution νis quasiconcave if and only if ρis ex post mixture-loving if and only if ρis ex post hedging. (ii) Distribution νis quasiconvex if and only if ρis ex post mixture-averse if and only if ρis ex post extreme. Theorem 5 states that ex post hedging (or mixture loving) and ex post extremeness (or mixture aversion) are exactly the stochastic properties that characterize quasiconcavity and quasiconvexity, respectively, for random utilities. Hedging and extremeness are dual properties: while the former focuses on hedge options, the latter focuses on extreme options. Recall that for deterministic preferences, a utility is quasiconcave if and only if the preference relation it represents is convex.13 Thus, ex post hedging (or mixture loving) is exactly the stochastic choice analog of convex preferences. When stochastic choice is both ex post mixture-loving and -averse, we say it is ex post mixture-neutral. A random utility νsatisfies betweenness if it is both quasiconcave and quasiconvex, i.e., the random utilities satisfy betweenness almost surely.14 The following characterization of random betweenness is immediate from Theorem 5. Corollary 1. Let ρbe represented by νthat exclude ties. Then νsatisfies betweenness if and only if ρis ex post mixture-neutral. The most well known case of a random utility that satisfies betweenness is random expected utility. While an ex post version of the independence axiom (Axiom 1.2)is necessary for random expected utility, it is not sufficient. In fact, Lin (2020)provides an example of a random nonexpected utility that satisfies ex post independence. Under random utility and ex post independence, it is straightforward to show that ex post hedging and ex post extremeness are equivalent. This suggests an alternate axiomatization of random expected utility where we substitute the hedging axiom for the extremeness axiom of Gul and Pesendorfer (2006). Finally, we return to our random ambiguity model. It is easy to see that ex post extremeness is sufficient to ensure that all random utilities are subjective expected utilities. This provides a simple characterization of the special case where the agent is ambiguityneutral almost surely and choice is stochastic only due to information (see Example 2). 12If we allow for ties to occur with probability 1as in regular utility distributions, then ex post hedging is equivalent to a slightly stronger statement of quasiconcavity. See Theorem 6 in the Appendix for details. 13That is, fgimplies af +(1−a)g gfor all a∈[01]. 14Betweenness means that min{v(f)v(g)}≤v(af +(1−a)g) ≤max{v(f)v(g)}for all a∈[01].Dekel (1986) provides an axiomatic characterization of betweenness preferences. 558 Jay Lu Theoretical Economics 16 (2021) Corollary 2. Suppose ρis represented by (μu) and excludes ties. Then ρis ex post extremeifandonlyifKis a singleton almost surely. The proof follows from Theorem 5 and the fact that a maxmin expected utility that satisfies betweenness is linear. The results in this section provide a foundation for the study of more general models of stochastic ambiguity. Although we focused only on ex post randomization in this section, our results on the stochastic properties of random nonlinear utilities extend to any mixture space more generally. Appendix A.1 Proof of Theorem 1 Let ρand τbe represented by (μu) and (ν v), respectively. That (ii) implies (i) is straightforward. We show that (i) implies (ii). Suppose (i) is true and note that for constant f g ∈Hc,u(f) ≥u(g) if and only if ρ(δfδg)=1if and only if τ(δfδg)=1if and only if v(f) ≥v(g).Thus,u=αv +βfor α>0. Without loss of generality, assume u=v and 1=u(¯ x) ≥u(x) ≥u(x)=0for ¯ xx ∈Xand all x∈X. Let W:= [01]n,wheren=|S|. Define the mapping ψ:H→Wsuch that ψ(h) = 1−u◦h. Note that UK(p) =H uK(h)dp =H min π∈Kπ·(u ◦h)dp =W1−max π∈Kπ·wdr =1−σK·r where r=p◦ϕ−1∈W and σKis the support function of K∈K.Letf=(1−α)¯ x+αx, which is constant. Thus, ρ(δfp)=μK∈K:u(f ) ≥UK(p) =μ{K∈K:1−α≥1−σK·r} =μ{K∈K:σK·r≥α} Since ρ(δfp)=τ(δfp),σK·rhas the same distribution under μand νfor all r∈W . Let denote the set of support functions σKand let C() denote the set of continuous functions on .Let⊂C() denote the set of functions φ(σK)= i aieλiσK·ri for ai∈R,λi≥0,andri∈W . Since each σK·rihas the same distribution under μand ν, K φ(σK)dμ=K i aieλiσK·ridμ =K i aieλiσK·ridν =K φ(u)dν Theoretical Economics 16 (2021) Random ambiguity 559 We show that is dense in C(). First note that is a vector space that includes constants, since e0σK·r=1∈.Considera1eλ1σK·r1a2eλ2σK·r2∈.Ifλ1+λ2>0, then a1eλ1σK·r1a2eλ2σK·r2=a1a2e(λ1+λ2)σK·(λ1 λ1+λ2r1+(1−λ1 λ1+λ2)r2)∈ Alternatively, if λ1+λ2=0, then λ1=λ2=0and a1eλ1σK·r1a2eλ2σK·r2=a1a2∈ This means that is closed under multiplication. Next we show that separates points in . Suppose σKσJ∈such that σK= σJ. Thus, there is some w∈Wsuch that σK(w) > σJ(w) without loss of generality. If we let r=δw, then σK·r=σK(w) > σJ(w) =σJ·r,soeσK·r>e σJ·r. This establishes that  separates points in . Since Kis compact (Theorem 3.85 of Aliprantis and Border 2006;henceforthAB),the homeomorphism between and Kimplies that is also compact. Since is a subalgebra, contains the constant function, and separates points in ,is uniformly dense in C() by the Stone–Weierstrass theorem (Theorem 9.13 of AB). This means that for any φ∈C(), we can find φi∈such that φi→φuniformly. By dominated convergence, K φ(σK)dμ=lim iK φi(σK)dμ=lim iK φi(σK)dν=K φ(σK)dν By Theorem 15.1 of AB, support functions have the same distribution under μand ν.In other words, for any J∈K, μ{K∈K:σK≤σJ}=ν{K∈K:σK≤σJ} μ{K∈K:K⊂J}=ν{K∈K:K⊂J} Since containment functionals completely characterize the distribution of convex compact sets (Theorem 7.8 of Molchanov 2005), μ=νas desired. A.2 Proof of Theorem 2 Suppose ρis more uncertainty-averse than τ.Notethatifu= αv +βfor α>0, then wecanfindtwoconstantactsf g ∈Hsuch that τ(δfδg)=1>0=ρ(δfδg),which contradicts the fact that ρis more uncertainty-averse than τ. Hence, we can assume u=vwithout loss of generality, and 1=u( ¯ x) ≥u(x) ≥u(x)=0for ¯ xx ∈Xand all x∈X. Let C(W ) denote the set of all continuous functions on the compact set W:= [01]n, where n=|S|.Notethatwecandefinethedualpair ϕ·r:= W ϕ(w) dr where ϕ∈C(W ) and ris a signed Borel measure on W(see Corollary 14.15 of AB). Fix some increasing, closed, and convex J, and define CJ:= ϕ∈C(W ):ϕ≥σJfor some J∈J First, we prove two claims about CJ. 560 Jay Lu Theoretical Economics 16 (2021) Claim 1. AsetK∈Jif and only if σK∈CJ. Proof.IfK∈J,thenσK∈CJtrivially. Suppose σK∈CJso σK≥σJfor some J∈J. Thus, K⊃Jand since Jis increasing, K∈J. Claim 2. The set CJis closed and convex. Proof. We first show that CJis closed. Consider ϕk∈CJsuch that ϕk→ϕ.Thus, ϕk≥σJkfor Jk∈J. Since J⊂Kis closed and Kis compact, Jis also compact. Thus, we can find a convergent subsequence Ji→J∈Jand σJi→σJ.Thus,ϕ≥σJ,soϕ∈CJ. We now show that CJis convex. Suppose ϕ1ϕ2∈CJ,soϕi≥σJifor J1J2∈J. Thus, aϕ1+(1−a)ϕ2≥aσJ1+(1−a)σJ2=σaJ1+(1−a)J2 Since Jis convex, aJ1+(1−a)J2∈J,soaϕ1+(1−a)ϕ2∈CJ. With these claims, we now continue the proof of Theorem 2.Claim 2 implies we can express CJas the intersection of a countable collection of closed half-spaces that support it (Corollary 7.49 of AB). In other words, CJ= iϕ∈C(W ):ϕ·ri≥ai(3) Moreover, for every i, there exists some ϕ∈CJsuch that ϕ·ri=ai. Consider some closed E⊂Wand note that by Urysohn’s lemma, we can find a sequence of ϕk∈C(W ) such that ϕk→ϕ+1Eand ϕk≥ϕ. Since ϕk≥ϕ∈CJ,ϕk∈CJby the definition of CJ. Thus, ϕk·ri≥aifrom (3). Since Wis compact, it follows from dominated convergence that ai≤lim kϕk·ri=(ϕ +1E)·ri=ϕ·ri+ri(E) =ai+ri(E) Thus, ri(E) ≥0for all closed E. Since riis a regular Borel measure, we can assume riis a probability measure without loss of generality. Finally, note that by the definition of CJ, there must exist some J∈Jsuch that ϕ≥σJ. Since σJ≥0, ai=ϕ·ri≥σJ·ri≥0 Define the mapping ψ:H→Wsuch that ψ(h) =1−u◦hand let pi=1 2(ri◦ψ) +1 2δgi where gi=ai¯ x+(1−ai)x.Thus, UK(pi)=1 2H uK(h)dpi+1 2ai=1 2W min π∈Kπ·(1−w)dri+1 2ai =1 2W1−max π∈Kπ·wdri+1 2ai=1 2(1−σK·ri)+1 2ai Theoretical Economics 16 (2021) Random ambiguity 561 Let f=1 2¯ x+1 2x,souK(f ) ≥UK(pi)if and only if 1 2≥1 2(1−σK·ri)+1 2ai ai≤σK·ri Let Ak={δfp1pk},sobyClaim 1 and (3), μ(J)=μ{K∈K:σK∈CJ}=μ i {K∈K:σK·ri≥ai} =lim kμ i≤k {K∈K:σK·ri≥ai}=lim kρAk(δf) Since ρAk(δf)≥τAk(δf),wehaveμ(J)≥ν(J)as desired. Now suppose μ≥cν.Letf∈Hbe constant, where u(f) =1−aand A= {f p1pk}. Again, define the mapping ψ:H→Wsuch that ψ(h) =1−u◦h.Note that u(f) ≥UK(pi)if and only if 1−a≥H uK(h)dpi=H min π∈Kπ·(u ◦h)dpi=1−σK·ri where ri=pi◦ψ−1. Define J:=  i≤k {K∈K:σK·ri≥a} so ρA(δf)=μ(J).NotethatJis closed, convex, and increasing, so ρA(δf)=μ(J)≥ν(J)=τA(δf) as desired. A.3 Proofs for Section 4 A.3.1 Extremeness In this section, we demonstrate the relationship between ex post extremeness and mixture aversion. Since this result is useful for later analysis, for this section, we consider an arbitrary compact metric space Z.Letρbe a stochastic choice on Z. For finite A⊂Z,letext(A) denote the extreme points of A. We define extremeness and mixture aversion for ρexactly as in Section 4.Wesayρis monotone if A⊂B implies ρA(p) ≥ρB(p). Lemma 1. Suppose ρis monotone. Then ρis mixture-averse if and only if it is extreme. Proof.Notethatifρis extreme, then ρis mixture-averse trivially. Suppose ρis monotone and mixture-averse. We show that ρis extreme by contradiction. Suppose ρA(ext(A)) < 1, so there must be some p/∈ext(A) such that ρA(p) > 0. Without loss 562 Jay Lu Theoretical Economics 16 (2021) of generality, assume pis not tied with any point in A(see Lemmas A.2 and A.3 from Lu 2016). By monotonicity, we can also assume A=ext(A) ∪{p}without loss. By the Krein–Milman theorem, we can write p= i≤k αiqi for i≤kαi=1,αi∈(01),andqi∈ext(A) ⊂A. Define r1=q1and λi:= αi  j≤i αj  Recursively define ri:= λiqi+(1−λi)ri−1 and note that p=rk.LetB:= A∪{r1rk−1}, so by monotonicity, ρBext(A)≤ρAext(A)<1 Thus, there must be some i∈{2k}such that riis not tied with anything in ext(A) and ρB(ri)>0. By monotonicity again, ρ{ri−1qiri}(ri)≥ρB(ri)>0 Since ri=λiqi+(1−λi)ri−1and ρis mixture-averse, it means that rimust be tied with qior ri−1. Since qi∈ext(A), this means that riis tied with ri−1,soρB(ri−1)=ρB(ri)>0. By induction, we conclude that riis tied with r1=q1, which contradicts the fact that riis not tied with anything in ext(A).Thus,ρmust be extreme. A.3.2 Proof of Theorem 5 In this section, we prove a more general version of Theorem 5 by allowing νto be regular,thatis,v(f) =v(g) with probability 0or 1.Asaresult,weconsider a slightly stronger version of quasiconcavity called quasiconcavity∗.Ifνexcludes ties, then quasiconcavity∗and quasiconcavity are equivalent. Definition 13. Distribution νis quasiconcave∗if almost surely (a.s.) v(f) ≥v(g) implies v(af +(1−a)g) ≥v(g), and with strictness if v(f) > v(g) and a∈(01). Theorem 6. Let ρbe represented by a regular ν. (i) Distribution νis quasiconcave∗if and only if ρis ex post mixture-loving if and only if ρis ex post hedging. (ii) Distribution νis quasiconvex if and only if ρis ex post mixture-averse if and only if ρis ex post extreme. Theoretical Economics 16 (2021) Random ambiguity 569 Ghirardato, Paolo, Fabio Maccheroni, and Massimo Marinacci (2004), “Differentiating ambiguity and ambiguity attitude.” Journal of Economic Theory, 118, 133–173. 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Manuscript received 6 June, 2019; final version accepted 25 June, 2020; available online 28 July, 2020.