We explore whether ambiguous communication can be beneficial to the sender in a persuasion problem, when the receiver (and possibly the sender) is ambiguity averse. Our analysis highlights the necessity of using a collection of experiments that form a splitting of an obedient experiment. Some experiments in the collection must be Pareto-ranked in that both players agree on their payoff ranking. If an optimal Bayesian persuasion experiment can be split in this way, then any not-too-ambiguity-averse sender as well as the receiver benefit. There are no benefits when the receiver has only two actions.

The α-MEU model and the smooth ambiguity model are two popular models in decision making under ambiguity. However, the axiomatic foundations of these two models are not completely understood. We provide axiomatic foundations of these models in a symmetric setting with a product state space S∞. This setting allows marginals over S to be linked behaviorally with (limiting frequency) events. Bets on such events are shown to reveal the i.i.d. measures that are relevant for the decision maker's preferences and appear in the representations. By characterizing both models within a common framework, it becomes possible to better compare and relate them.

This paper compares the efficacy of a centralized and a decentralized rights structure in determining the size of an externality generating project. Consider a central authority and two localities. One locality can operate a variable-size project which produces an externality that affects the other locality. Each locality may have some private information concerning its own net benefit from the project. Under centralization, localities are vertically integrated with a benevolent central authority who effectively possesses all property rights. Under decentralization, localities are separate legal entities (endowed with property rights) who bargain to determine the project size. We examine the performance of these two regimes and show how one or the other may dominate depending on the distributions of private and external benefits from the project. The effect of the size and variation in the externality on this trade-off is of particular interest. JEL: Organizational Behavior; Transaction Costs; Property Rights; Externalities; Asymmetric and Private Information; Structure, Scope, and Performance of Government (D23, D62, D82, H11). ⇤We thank Robin Boadway, Faruk Gul, Lu Hu, Eric Maskin, Dilip Mookherjee, Stefan Reichelstein, Mike Peters, Patrick Rey, Jacques Robert, Lars Stole and Francois Vaillancourt for comments and thank a number of seminar and conference audiences. The first version of this paper was written while the second author was visiting the MEDS Department at Northwestern University. He would like to thank MEDS for its hospitality and support during his visit, and also C.I.R.A.N.O., C.R.S.H. and F.C.A.R. for their financial support. †MEDS, Northwestern University ‡Sciences economiques, Universite de Montreal

Consider a canonical problem in choice under uncertainty: choosing from a convex feasible set consisting of all (Anscombe–Aumann) mixtures of two acts f and g, [Formula: see text]. We propose a preference condition, monotonicity in optimal mixtures, which says that surely improving the act f (in the sense of weak dominance) makes the optimal weight(s) on f weakly higher. We use a stylized model of a sales agent reacting to incentives to illustrate the tight connection between monotonicity in optimal mixtures and a monotone comparative static of interest in applications. We then explore more generally the relation between this condition and preferences exhibiting ambiguity-sensitive behavior as in the classic Ellsberg paradoxes. We find that monotonicity in optimal mixtures and ambiguity aversion (even only local to an event) are incompatible for a large and popular class of ambiguity-sensitive preferences (the c-linearly biseparable class. This implies, for example, that maxmin expected utility preferences are consistent with monotonicity in optimal mixtures if and only if they are subjective expected utility preferences. This incompatibility is not between monotonicity in optimal mixtures and ambiguity aversion per se. For example, we show that smooth ambiguity preferences can satisfy both properties as long as they are not too ambiguity averse. Our most general result, applying to an extremely broad universe of preferences, shows a sense in which monotonicity in optimal mixtures places upper bounds on the intensity of ambiguity-averse behavior. This paper was accepted by Manel Baucells, decision analysis.

We study incomplete information games with ambiguity averse players. Our focus is on equilibrium concepts satisfying sequential optimality—each player’s strategy is optimal at each information set given opponents’ strategies. We show sequential optimality, which does not make any explicit assumption on updating, is equivalent to sequential optimality with respect to beliefs updated using a particular generalization of Bayesian updating. Ambiguity aversion expands the set of equilibria compatible with players sharing common ambiguous beliefs. We connect ambiguity aversion with belief robustness. Examples illustrate new strategic behavior, including strategic use of ambiguity, under ambiguity aversion. (JEL C73, D81, D83)

Since at least de Finetti (Annales de l’Institut Henri Poincare 7:1–68, 1937), preference symmetry assumptions have played an important role in models of decision making under uncertainty. In the current paper, we explore (1) the relationship between the symmetry assumption of Klibanoff et al. (KMS) (Econometrica 82:1945–1978, 2014) and alternative symmetry assumptions in the literature, and (2) assuming symmetry, the relationship between the set of relevant measures, shown by KMS (2014) to reflect only perceived ambiguity, and the set of measures (which we will refer to as the Bewley set) developed by Ghirardato et al. (J Econ Theory 118:133–173, 2004), Nehring (Ambiguity in the context of probabilistic beliefs, working paper, 2001, Bernoulli without Bayes: a theory of utility-sophisticated preference, working paper, 2007) and Ghirardato and Siniscalchi (A more robust definition of multiple priors, working paper, 2007, Econometrica 80:2827–2847, 2012). This Bewley set is the main alternative offered in the literature as possibly representing perceived ambiguity. Regarding symmetry assumptions, we show that, under relatively mild conditions, a variety of preference symmetry conditions from the literature [including that in KMS (2014)] are equivalent. In KMS (2014), we showed that, under symmetry, the Bewley set and the set of relevant measures are not always the same. Here, we establish a preference condition, No Half Measures, that is necessary and sufficient for the two to be the same under symmetry. This condition is rather stringent. Only when it is satisfied may the Bewley set be interpreted as reflecting only perceived ambiguity and not also taste aspects such as ambiguity aversion.

Consider a canonical problem in choice under uncertainty: choosing from a convex feasible set consisting of all (Anscombe-Aumann) mixtures of two acts f and g, {αf + (1− α)g : α ∈ [0, 1]}. We propose a preference condition, Monotonicity in Mixtures, which says that clearly improving the act f (in the sense of weak dominance) makes putting more weight on f more desirable. We show that this property has strong implications for preferences exhibiting behavior as in the classic Ellsberg (1961) paradoxes. For example, we show that maxmin expected utility (MEU) preferences (Gilboa and Schmeidler 1989) satisfy Monotonicity in Mixtures if and only if they are expected utility preferences. Thus, for MEU, Monotonicity in Mixtures and Ellsberg behavior are incompatible. We extend this stark finding in several directions. Moreover, we demonstrate that the incompatibility is not between Monotonicity in Mixtures and Ellsberg behavior (or even global ambiguity aversion) per se. For example, in addition to deriving general implications of Monotonicity in Mixtures, we show that smooth ambiguity preferences (Klibanoff, Marinacci and Mukerji 2005) can satisfy both properties as long as they are not too ambiguity averse. ∗We thank Gavriel Hirsch for excellent research assistance. We thank Nemanja Antic, Sarah Auster, RoseAnne Dana, Itzhak Gilboa, Mark Machina, Sujoy Mukerji, Chris Shannon, Marciano Siniscalchi, Jean-Marc Tallon and several seminar audiences for helpful comments. †Cowles Foundation and School of Management, Yale University, New Haven CT USA. e-mail: soheil.ghili@yale.edu ‡Kellogg School of Management, Northwestern University, Evanston IL USA. e-mail: peterk@kellogg.northwestern.edu

This is the web appendix of the paper "Experiments on Compound Risk in Relation to Simple Risk and Ambiguity", forthcoming in Management Science.

We conduct experiments measuring individual behavior under compound risk, simple risk, and ambiguity. We focus on 1 treatment of compound risks relative to simple risks and 2 the relationship between compound risk attitudes and ambiguity attitudes. We find that compound risks are valued differently than corresponding reduced simple risks. These differences measure compound risk attitudes. These attitudes display more aversion as the reduced probability of the winning event increases. Like Halevy [Halevy Y 2007 Ellsberg revisited: An experimental study. Econometrica 75:503-536], we find an association between compound risk reduction and ambiguity neutrality. However, in contrast to the almost perfect identification in Halevy's data, we find a substantially weaker relation in both directions. First, a majority of our ambiguity-neutral subjects fail to reduce compound risk. Second, almost a quarter of our subjects who reduce compound risk are nonneutral to ambiguity. All of the latter come from the more quantitatively sophisticated part of our subject pool. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2014.1953 . This paper was accepted by Peter Wakker, decision analysis.

We axiomatize preferences that can be represented by a monotonic aggregation of subjective expected utilities generated by a utility function and some set of i.i.d. probability measures over a product state space, S1. For such preferences, we define relevant measures, show that they are treated as if they were the only marginals possibly governing the state space and connect them with the measures appearing in the aforementioned representation. These results allow us to interpret relevant measures as reflecting part of perceived ambiguity, meaning subjective uncertainty about probabilities over states. Under mild conditions, we show that increases or decreases in ambiguity aversion cannot affect the relevant measures. This property, necessary for the conclusion that these measures reflect only perceived ambiguity, distinguishes the set of relevant measures from the leading alternative in the literature. We apply our findings to a number of well-known models of ambiguity-sensitive preferences. For each model, we identify the set of relevant measures and the implications of comparative ambiguity aversion.

The copyright to this Article is held by the Econometric Society. It may be downloaded, printed and reproduced only for educational or research purposes, including use in course packs. No downloading or copying may be done for any commercial purpose without the explicit permission of the Econometric Society. For such commercial purposes contact the Office of the Econometric Society (contact information may be found at the website http://www.econometricsociety.org or in the back cover of Econometrica). This statement must be included on all copies of this Article that are made available electronically or in any other format. 1 We axiomatize preferences that can be represented by a monotonic aggregation of subjective expected utilities generated by a utility function and some set of i.i.d. probability measures over a product state space, S ∞. For such preferences, we define relevant measures, show that they are treated as if they were the only marginals possibly governing the state space, and connect them with the measures appearing in the afore-mentioned representation. These results allow us to interpret relevant measures as reflecting part of perceived ambiguity, meaning subjective uncertainty about probabilities over states. Under mild conditions, we show that increases or decreases in ambiguity aversion cannot affect the relevant measures. This property, necessary for the conclusion that these measures reflect only perceived ambiguity, distinguishes the set of relevant measures from the leading alternative in the literature. We apply our findings to a number of well-known models of ambiguity-sensitive preferences. For each model, we identify the set of relevant measures and the implications of comparative ambiguity aversion.

We axiomatize preferences that can be represented by a monotonic aggregation of subjective expected utilities generated by a utility function and some set of i.i.d. probability measures over a product state space, S∞. For such preferences, we define relevant measures, show that they are treated as if they were the only marginals possibly governing the state space and connect them with the measures appearing in the aforementioned representation. These results allow us to interpret relevant measures as reflecting part of perceived ambiguity, meaning subjective uncertainty about probabilities over states. Under mild conditions, we show that increases or decreases in ambiguity aversion cannot affect the relevant measures. This property, necessary for the conclusion that these measures reflect only perceived ambiguity, distinguishes the set of relevant measures from the leading alternative in the literature. We apply our findings to a number of well-known models of ambiguity-sensitive preferences. For each model, we identify the set of relevant measures and the implications of comparative ambiguity aversion.

We offer a theory of polarization as an optimal response to ambiguity. Suppose individual A's beliefs first-order stochastically dominate individual B's. They observe a common signal. They exhibit polarization if A's posterior dominates her prior and B's prior dominates her posterior. Given agreement on conditional signal likelihoods, we show that polarization is impossible under Bayesian updating or after observing extreme signals. However, we also show that polarization can arise after intermediate signals as ambiguity averse individuals implement their optimal prediction strategies. We explore when this polarization will occur and the logic underlying it. (JEL D81, D82, D83)

We find that Epstein's (2010) Ellsberg-style thought experiments pose, contrary to his claims, no paradox or difficulty for the smooth ambiguity model of decision making under uncertainty developed by Klibanoff, Marinacci, and Mukerji (2005). Not only are the thought experiments naturally handled by the smooth ambiguity model, but our reanalysis shows that they highlight some of its strengths compared to models such as the maxmin expected utility model (Gilboa and Schmeidler (1989)). In particular, these examples pose no challenge to the model's foundations—interpretation of the model as affording a separation of ambiguity and ambiguity attitude or the potential for calibrating ambiguity attitude in the model.

We show equivalence among various symmetry axioms in the literature.

We define a behavioral concept of relevance in the context of decision making under uncertainty. We argue that this concept provides a sensible answer to the question "What probabilistic environments do an individuals' preferences reveal as mattering to her decisions?" under a symmetry assumption. This question has important implications for economic modeling. It is often the case that a modeler desires to restrict the probabilistic environments a decision maker considers. Without a concept of relevant beliefs, it is impossible to check from preferences whether a model is reflecting what the modeler intended. This checking is essential to isolating the effect of changing information while holding tastes fixed. We show that a single concept of relevance delivers this for a wide range of models, including models that allow for ambiguity attitude. We also use symmetry and relevance to provide insight into the foundations of the α-MEU and smooth ambiguity models of decision-making under uncertainty.

We examine a variety of preference-based definitions of ambiguous events in the context of the smooth ambiguity model. We first consider the definition proposed in Klibanoff et al. (Econometrica 73(6):1849–1892, 2005 ) based on the classic Ellsberg two-urn paradox (Ellsberg Q J Econ 75:643–669, 1961 ) and show that it satisfies several desirable properties. We then compare this definition with those of Nehring (Math Soc Sci 38(2):197–213, 1999 ), Epstein and Zhang (Econometrica 69:265–306, 2001 ), Zhang (Econ Theory 20:159–181, 2002 ), and Ghirardato and Marinacci (J Econ Theory 102:251–289, 2002 ). Within the smooth ambiguity model, we show that Ghirardato and Marinacci (J Econ Theory 102:251–289, 2002 ) would identify the same set of ambiguous and unambiguous events as our definition while Epstein and Zhang (Econometrica 69:265–306, 2001 ) and Zhang (Econ Theory 20:159–181, 2002 ) would yield a different classification. Moreover, we discuss and formally identify two key sources of the differences compared to Epstein and Zhang (Econometrica 69:265–306, 2001 ) and Zhang (Econ Theory 20:159–181, 2002 ). The more interesting source is that these two definitions can confound non-constant ambiguity attitude and the ambiguity of an event.