We propose and characterize a model of preferences over acts such that the decision maker prefers act f to act g if and only if E μ φ( E π u○f) ⩾ E μ φ( E π u○g), where E is the expectation operator, u is a von Neumann-Morgenstern utility function, φis an increasing transformation, and μis a subjective probability over the set Πof probability measures πthat the decision maker thinks are relevant given his subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective beliefs, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes toward pure risk are characterized by the shape of u, as usual, while attitudes toward ambiguity are characterized by the shape of φ. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures Π. One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's paradox). The maxmin expected utility model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as a limiting case of our model with infinite ambiguity aversion. Two illustrative portfolio choice examples are offered. Copyright The Econometric Society 2005.

We characterize, in the Anscombe-Aumann framework, the preferences for which there are a utility functionu on outcomes and an ambiguity indexc on the set of probabilities on the states of the world such that, for all acts f and g, Copyright The Econometric Society 2006.

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The theory of subjective expected utility (SEU) has been recently extended to allow ambiguity to matter for choice. We propose a notion of absolute ambiguity aversion by building on a notion of comparative ambiguity aversion. We characterize it for a preference model which encompasses some of the most popular models in the literature. We next build on these ideas to provide a definition of unambiguous act and event, and show the characterization of the latter. As an illustration, we consider the classical Ellsberg 3-color urn problem and find that the notions developed in the paper provide intuitive answers.

Rustichini, and Tom Sargent, for some very useful comments. Part of this research was done while some of the authors were visiting the Economics Department of Boston University and the Collegio Carlo Alberto, which they thank for their hospitality.

for helpful discussion, Andreas Lange for pointing out an error in a previous version and seminar audiences at the Cowles Foundation workshop "Uncertainty in Economic Theory,"the

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This is a survey of some of the recent decision-theoretic literature involving beliefs that cannot be quanti…ed by a Bayesian prior. We discuss historical, philosophical, and axiomatic foundations of the Bayesian model, as well as of several alternative models recently proposed. The de…nition and comparison of ambiguity aversion and the updating of non-Bayesian beliefs are brie ‡y discussed. Finally, several applications are mentioned to illustrate the way that ambiguity (or "Knightian uncertainty") can change the way we think about economic problems.