This paper presents an axiomatic model of decision making under uncertainty which incorporates objective but imprecise information. Information is assumed to take the form of a probability-possibility set, that is, a set P of probability measures on the state space. The decision maker is told that the true probability law lies in P and is assumed to rank pairs of the form (P, f ) where f is an act mapping states into outcomes. The representation result delivers maxmin expected utility at each probability-possibility set. The model explains how "beliefs" vary with information:there is a mapping that gives for each probability-possibility set the revealed set of probability distributions. This allows both expected utility when the set is reduced to a singleton and extreme pessimism when the decision maker takes the worst case scenario in the entire probability-possibility set. We define a notion of comparative imprecision aversion and show it is characterized by inclusion of the sets of revealed probability distributions, irrespective of the utility functions that capture risk attitude. We also identify an explicit attitude toward imprecision that underlies usual hedging axioms. Finally, we characterize, under extra axioms, a more precise functional form, in which the set of revealed probability distributions is obtained by (i) solving for the "mean value" of the probability-possibility set, and (ii) shrinking the probability-possibility set toward the mean value to a degree determined by preferences.

This paper explores risk-sharing and equilibrium in a general equilibrium set-up wherein agents are non-additive expected utility maximizers. We show that when agents have the same convex capacity, the set of Pareto-optima is independent of it and identical to the set of optima of an economy in which agents are expected utility maximizers and have same probability. Hence, optimal allocations are comonotone. This enables us to study the equilibrium set. When agents have different capacities, matters are much more complex (as in the vNM case). We give a general characterization and show how it simplifies when Pareto-optima are comonotone. We use this result to characterize Pareto-optima when agents have capacities that are the convex transform of some probability distribution. comonotonicity of Paretooptima is also shown to be true in the two-state case if the intersection of the core of agents' capacities is non-empty; Pareto-optima may then be fully characterized in the two-agent, two-state case. This comonotonicity result does not generalize to more than two states as we show with a counter-example. Finally, if there is no-aggregate risk, we show that non-empty core intersection is enough to guarantee that optimal allocations are full-insurance allocation. This result does not require convexity of preferences.

We review recent advances in the field of decision making under uncertainty or ambiguity. We start with a presentation of the general approach to a decision problem under uncertainty, as well as the 'standard' Bayesian treatment and issues with this treatment. We present more general approaches (Choquet expected utility, maximin expected utility, smooth ambiguity and so forth) that have been developed in the literature under the name of models of ambiguity sensitive preferences. We draw a distinction between fully subjective models and models incorporating explicitly some information. We review definitions and characterizations of ambiguity aversion in these models. We mention the challenges posed by some of the models presented. We end with a review of part of the experimental literature and applications of these models to economic settings.