INTRODUCTION

It is by now rather well understood that the notion of common knowledge (first introduced by Lewis 1969 and later formalized by Aumann 1976) plays a central role in the theory of games. (An event E is common knowledge between two individuals, if each knows E, each knows the other knows E, etc. …) Indeed, most justifications of Nash's (1951) equilibrium concept usually include (perhaps only implicitly) the assumption that it is common knowledge among the players that both the Nash equilibrium in question will be played by all and that all players are expected utility maximizers. (We shall henceforth call expected utility maximizers “rational.”) We hope to illustrate in an informal manner that there is in fact a large class of extensive-form games, in which each of which it is not possible for rationality to be common knowledge throughout the game.

The consequences of this for many well-known extensive-form refinements of Nash equilibrium are quite serious. Consider, for example, Selten's (1965) notion of subgame perfect Nash equilibrium. The requirements on a solution here are not only that the strategies form a Nash equilibrium of the game as a whole, but also that the strategies induce on every proper subgame a Nash equilibrium. If, however, there are proper subgames beginning at (singleton) information sets at which it is not possible for rationality to be common knowledge, then Nash behavior in that subgame can no longer be justified on common-knowledge grounds.