We can often make sharper predictions about the possible outcomes of a game if we are willing to make stronger assumptions. Up until now we have assumed that the players are rational but we haven’t even assumed that each knows that the others are rational. Beyond that we could further assume that each player knows that the other players know that the others are all rational. We could continue adding additional layers of such assumptions ad nauseam. We summarize the entire infinite hierarchy of such assumptions by saying that the rationality of the players is common knowledge.
Rationality constrains players to choose best responses to their beliefs but does not restrict those beliefs. Common knowledge of rationality imposes a consistency requirement upon players’ beliefs about others’ actions.
By assuming that the players’ rationality is common knowledge, we can justify an iterative process of outcome rejection—the iterated elimination of strictly dominated strategies—which can often sharpen our predictions. Outcomes which do not survive this process of elimination cannot plausibly be played when the rationality of the players is common knowledge.
A similar, and weakly stronger, process—the iterated elimination of strategies which are never best responses—leads to the solution concept of rationalizability. The surviving outcomes of this process constitute the set of rationalizable outcomes. Each such outcome is a plausible result—and these are the only plausible results—when the players’ rationality is common knowledge. In two-player games the set of rationalizable outcomes is exactly the set of outcomes which survive the iterated elimination of strictly dominated strategies. In three-or-more-player games, the set of rationalizable outcomes can be strictly smaller than the set of outcomes which survives the iterated elimination of strictly dominated strategies.
In a rationalizable outcome players’ beliefs about the same question can differ, and hence some are incorrect; and a player can find—after the others’ choices are revealed—that she would have preferred to have made a different choice.