Mutual knowledge of rationality and correct beliefs in $n$-person games: An impossibility theorem

Abstract

There are two well-known sufficient conditions for Nash equilibrium: common knowledge of rationality, and common prior, which exogenously assumes a profile of beliefs that are correct. However, it is not known how players arrive at a common prior \textit{before} playing the original game. In this note, I assume, in addition to (objective and subjective) rationality, that players' beliefs \textit{will be} correct once the game is played, but a common prior is not assumed. I study whether and under what conditions players endogenously arrive at a common prior. The main finding is an impossibility theorem, which states that mutual knowledge of rationality and mutual knowledge of correct beliefs are not in general logically consistent in $n$-person games. However, the two assumptions are consistent in two-player zero-sum games