Rationality, Nash Equilibrium and Backward Induction in Perfect Information Games

We say that a player is certain of an event A if he gives A probability 1. 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] centepede game, and fighting in Selten's [1978] chain—store game are consistent with CCR at the beginning of the game. Intuitively, the problem with the backward induction argument, an argument which seems to follow from CCR, is that it assumes that players reason according to a certain logic even at vertices which are inconsistent with that logic (i.e. vertices that would not have been reached had the players followed the backward induction logic). Next, it is shown that if, in addition to CCR, there is CC that each player gives 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. The paper concludes with an extension of the model which allows for the possibility of mistakes in the implementation of the strategies (as opposed to mistakes in the reasoning of the players). It is shown that if, in addition to CCR, there is CC that there is a small probability of a mistake at every vertex, then the players choose the backward induction strategies.