Rationality and the definition of consistent pairs

A consistent pair specifies a set of rational strategies for both players such that a strategy is rational if and only if it is a best reply to a Bayesian belief that gives positive probability to every rational strategy of the opponent and probability zero otherwise. Although the idea underlying consistent pairs is quite intuitive, the original definition suffers from non-existence problems. In this article, we propose an alternative formalization of consistent pairs. According to our definition, a strategy is rational if and only if it is a best reply to some lexicographic probability system that satisfies certain consistency conditions. These conditions imply in particular that a player's probability system gives infinitely more weight to rational strategies than to other strategies. We show that modified consistent pairs exist for every game.This article is based on Chapter 3 of my Ph.D. thesis finished at the University of Bonn in fulfillment of the requirements of the European Doctoral Programme. For helpful comments and discussions, I would like to thank Eddie Dekel, Larry Blume, Tilman Börgers, Martin Dufwenberg, Frank Schuhmacher, Ariel Rubinstein, Avner Shaked, and seminar participants at Tel Aviv and Iowa City. Financial assistance by the German Academic Exchange Service is gratefully acknowledged.