A conflict between sequential rationality and consistency principles

It is shown that no solution concept that selects sequentially rational (perfect, proper, persistent, or members of some stable set of) equilibria satisfies the following consistency property. Suppose that in every solution of the game G, player i's action is a, and denote by G^a the game in which player i is restricted to choose a. Then some player j ≠ i has an action c that is used with positive probability in both some solution of G and some solution of G^a. This result illustrates a conflict between a mild consistency condition and sequential rationality.