A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements
证明，在支付函数和信念结构足够丰富时，任何理性化行动都可被扰动为唯一理性化且稳健；多重性仅出现在“刀锋”情形，并据此刻画了精炼的稳健预测仅限于所有理性化策略都成立的预测。
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.