The Robustness of Extensive-Form Rationalizability

As proved by Shimoji and Watson, a strategy of an extensive-form game is rationalizable in the sense of Pearce if and only if it survives the maximal elimination of conditionally dominated strategies. Briefly, this is a process of iteratively eliminating conditionally dominated strategies according to a specific order. We prove, however, that there is nothing special about this order, and that the notion of extensive-form rationalizability (EFR) is actually quite robust from an algorithmic point of view. Indeed, although possibly very different from each other, the sets of strategies surviving two arbitrary elimination orders are always equivalent in a very strong sense. That is, each strategy si of a player i surviving the first order can be identified with a strategy si of i surviving the second order, so that for every strategy profile (s1, . . . , sn) the corresponding profile (s1, . . . , s ′ n) not only generates the same payoff profile, but actually the same terminal node. To prove our results we put forward a new notion of dominance and an elementary characterization of EFR that may be of independent interest. We establish connections between EFR and other existing iterated strategy-elimination procedures, using our notion of dominance and our characterization of EFR.