Rationalizability and Common Knowledge of
Men tracht un Got lacht
(Mortals scheme and God laughs)
To determine what a rational player will do in a game, eliminate strategies that violate the cannons of rationality. Whatever is left we call rationalizable. We show that rationalizability in normal form games is equivalent to the iterated elimination of strongly dominated strategies, and the epistemological justification of rationalizability depends on the common knowledge of rationality (Tan and Werlang 1988).
If there is only one rationalizable strategy profile, it must be a Nash equilibrium, and it must be the choice of rational players, provided there is common knowledge of rationality.
There is no plausible set of epistemic conditions that imply the common knowledge of rationality. This perhaps explains the many non-obvious, indeed perplexing, arguments surrounding the iterated elimination of strongly dominated strategies, some of which are presented and analyzed below.
The Nash equilibrium criterion (§2.4) does not refer to the knowledge or beliefs of players. If players are Bayesian rational (§1.5), however, they then have beliefs concerning the behavior of the other players, and they maximize their expected utility by choosing best responses given these beliefs. Thus, to investigate the implications of Bayesian rationality, we must incorporate beliefs into the description of the game.
An epistemic game G consists of a normal form game with players i = 1,…, n and a finite pure-strategy set Si for each player i, sois the set of pure-strategy profiles for G, with payoffs :S→R. In addition, G includes a set of possible states Ω of the game, a knowledge partition Pi