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This paper takes the Anscombe-Aumann framework with horse and roulette lotteries, and applies the Savage axioms to the horse lotteries and the von Neumann-Morgenstern axioms to the roulette lotteries. The resulting repre-sentation of preferences yields a subjective probability measure over states and two utility functions, one governing risk attitudes and one governing ambigu-ity attitudes. The model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity.

We consider mechanism design problems with Knightian uncertainty formalized using incomplete preferences, as in Bewley (1986). Without completeness, decision making depends on a set of beliefs, and an action is preferred to another if and only if it has larger expected utility for all beliefs in this set. We consider two natural notions of incentive compatibility in this setting: maximal incentive compatibility requires that no strategy has larger expected utility than reporting truthfully for all beliefs, while optimal incentive compatibility requires that reporting truthfully has larger expected utility than all other strategies for all beliefs. In a model with a continuum of types, we show that optimal incentive compatibility is equivalent to ex-post incentive compatibility under fairly general conditions on beliefs. In a model with a discrete type space, we characterize full extraction of rents generated from private information. We show that full extraction is generically possible with maximal incentive compatible mechanisms, but requires sufficient disagreement across types, which neither holds nor fails generically, with optimal incentive compatible mechanisms.

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We review recent advances in the field of decision making under uncertainty or ambiguity.

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This paper provides a method for identifying components of preference re-flecting perceived ambiguity separate from increases and decreases in ambiguity aversion. Important to this method is the identification of a unique set of re-vealed probability assignments (called relevant measures) from preferences over acts. We show this method works for a large class of preferences that treat the state space as if it had a symmetric “i.i.d. with unknown parameters ” structure. Within this class, we characterize the relevant measures, show where they ap-pear in a canonical representation and provide conditions under which they are separate from increases and decreases in ambiguity aversion. We apply our find-ings to a number of well-known representations of ambiguity-sensitive preferences. For each model, by identifying the set of relevant measures and the implications of comparative ambiguity aversion, we find components of these representations that reflect perceived ambiguity and others that reflect comparative ambiguity aversion.

A preference is invariant with respect to a transformation τ if its ranking of acts is unaffected by a reshuffling of the states under τ. We show that any invariant preference must be parametric: there is a unique sufficient set of parameters such that the preference ranks acts according to their expected utility given the parameters. This property holds for all non-trivial preferences, provided only that they are reflexive, transitive, monotone, continuous and mixture linear. ∗The present paper subsumes DeCastro and Al-Najjar (2009), available on the authors’ webpages. Many of the ideas presented here, including subjective sufficient statistic theo-rem, parametric representations, and parameter-based acts appeared there, as did the major technical results on ergodic theory. †We thank Paolo Ghirardato, Ben Polak and Marciano Siniscalchi for extensive discus-sions and thoughtful comments when we presented to them the main ideas of this project during the summer and fall of 2009. We also thank Simone Galperti for his research

We de…ne 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 individual’s 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 re-‡ecting what the modeler intended. This checking is essential to isolating the e¤ect of changing information while holding tastes …xed. 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-MEU and smooth ambiguity models of decision-making under uncertainty.