A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements

Abstract

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.

Depth of Reasoning and Higher Order Beliefs

“…A recent literature (e.g., Bergemann and Morris, 2005), motivated by the Wilson (1987) doctrine of successive elimination of the common knowledge assumptions, remains ignorant about the higher order beliefs of the agents and is looking for mechanisms that do well under many different specifications of beliefs. Because in standard solution concepts behavior is discontinuous in beliefs, the designer has to be very conservative in his choice of the mechanism in order to ensure the desired outcome, especially if, following Weinstein and Yildiz (2007), he thinks that the product topology "captures a reasonable restriction on the researcher's ability to observe players' beliefs. "…”

Robust equilibria under non-common priors

“…To establish our results, on the contrary, we construct a dominance solvable incomplete information game such that (1 ) changes of ex ante beliefs are arbitrarily small and (2 ) the profile of ex ante subjective payoffs of the unique rationalizable strategy profile is arbitrarily close to the profile of expected payoffs. Hence, we may say that the same type of statement as that by Weinstein and Yildiz (2007) is obtained by our ex ante approach, provided that the CPA is dropped.…”

“…Our result shows that when we relax the CPA, none of the existing sufficient conditions implies robustness under non-common priors. Weinstein and Yildiz (2007) consider a notion of interim robustness. 7 A Nash equilibrium a * is interim robust in g if for some N ≥ 0 and for any incomplete information game with (or without) common prior where the action sets are same as those of g, there exists a Bayesian Nash equilibrium, say σ, such that in any state of the world at which it is mutually known up to order N that g is the true game, a * is played under σ.…”

“…To prove their main result, Weinstein and Yildiz (2007) show that for any complete information type in the universal type space (see Mertens and Zamir (1985) and Brandenburger and Dekel (1993)) 8 and any rationalizable action profile a * of this game, there exist a dominance solvable incomplete information game and a sequence of types drawn from this game such that (1) these types are arbitrarily close to the complete information type (i.e., this sequence converges to it with respect to the product topology in the universal type space) and (2) each type of the sequence plays a * . 9 Roughly speaking, the former condition requires that changes of interim beliefs be small.…”

“…This implies that the impact of a small probability event is not large enough in the sense that, in some games and for some strict 8 A complete information type is a (degenerate) type in the universal type space where it is common knowledge that payoffs are given by the complete information game. Note that Weinstein and Yildiz (2007) do not necessarily restrict their attention to complete information types.…”