Rationalizability is a central solution concept of game theory. Economic models often have many rationalizable outcomes, motivating economists to use refinements of rationalizability, including equilibrium refinements. In this paper we try to achieve a general understanding of when this multiplicity occurs and how one should deal with it. Assuming that the set of possible payoff functions and belief structures is sufficiently rich, we establish a revealing structure of the correspondence of beliefs to sets of rationalizable outcomes. We show that, for any rationalizable action a of any type, we can perturb the beliefs of the type in such a way that a is uniquely rationalizable for the new type. This unique outcome will be robust to further small changes. When multiplicity occurs, then we are in a "knife-edge" case, where the unique rationalizable outcome changes, sandwiched between open sets of types where each of the rationalizable actions is uniquely rationalizable. As an immediate application of this result, we characterize, for any refinement of rationalizability, the predictions that are robust to small misspecifications of interim beliefs. These are only those predictions that are true for all rationalizable strategies, that is, the predictions that could have been made without the refinement. Copyright The Econometric Society 2007.

We provide a critical assessment of the ambiguity aversion literature, which we characterize in terms of the view that Ellsberg choices are rational responses to ambiguity, to be explained by relaxing Savage's sure thing principle and adding an ambiguity-aversion postulate. First, admitting Ellsberg choices as rational leads to behavior, such as sensitivity to irrelevant sunk cost, or aversion to information, which most economists would consider absurd or irrational. Second, we argue that the mathematical objects referred to as 'beliefs' in the ambiguity aversion literature have little to do with how an economist or game theorist understands and uses the concept. This is because of the lack of a useful notion of updating. Third, the anomaly of the Ellsberg choices can be explained simply and without tampering with the foundations of choice theory. These choices can arise when decision makers form heuristics that serve them well in real-life situations where odds are manipulable, and misapply them to experimental settings. * We are grateful to Drew Fudenberg, Edi Karni, Bart Lipman, Marciano Siniscalchi, Costis Skiadas, and Rakesh Vohra for detailed comments that substantially improved the paper. We also thank

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Abstract. In some games, the impact of higher-order uncertainty is

We show that a simple "reputation-style" test can always identify which of two experts is informed about the true distribution. The test presumes no prior knowledge of the true distribution, achieves any desired degree of precision in some fixed finite time, and does not use "counterfactual" predictions. Our test relies on a simple reputation argument due to Fudenberg and Levine (1992).We then use our setup to shed some light on the apparent paradox that a strategically motivated expert can ignorantly pass any test. We point out that this paradox is a consequence of the fact that, in the single-expert setting, any mixed strategy for Nature is reducible to a pure strategy, thus eliminating any meaningful sense in which Nature can randomize. Comparative testing reverses the impossibility result because the presence of an informed expert eliminates the reducibility of Nature's mixed strategies.