Rationalizable strategies in random games

We study point-rationalizable and rationalizable strategies in random games. In a random n x n symmetric game, an explicit formula is derived for the distribution of the number of point-rationalizable strategies, which is of the order root n in probability as n -> infinity. The number of rationalizable strategies depends on the payoff distribution, and is bounded by the number of point-rationalizable strategies (lower bound), and the number of strategies that are not strictly dominated by a pure strategy (upper bound). Both bounds are tight in the sense that there exists a payoff distribution such that the number of rationalizable strategies reaches the bound with a probability close to one. We also show that given a payoff distribution with a finite third moment, as n -> infinity, all strategies are rationalizable with probability one. Our results qualitatively extend to two-player asymmetric games, but not to games with more than two players. (C) 2019 Elsevier Inc. All rights reserved.