Rationality, Nash Equilibrium and Backward Induction in Perfect Information Games
We say that a player is certain of an event A if she assigns probability 1 to A. There is common certainty (CC) of A if the event A occurred, each player is certain of A, each player is certain that every other player is certain of A, and so forth. It is shown that in a generic perfect-information game the set of outcomes that are consistent with common certainty of rationality (CCR) at the beginning of the game coincides with the set of outcomes that survive one deletion of weakly dominated strategies and then iterative deletion of strongly dominated strategies. Thus, the backward induction outcome is not the only outcome that is consistent with CCR. In particular, cooperation in Rosenthal's (1981) centipede game, and fighting in Selten's (1978) chainstore game are consistent with CCR at the beginning of the game. Next, it is shown that, if in addition to CCR, there is CC that each player assigns a positive probability to the true strategies and beliefs of the other players, and if there is CC of the support of the beliefs of each player, then the outcome of the game is a Nash equilibrium outcome.