The Logic of Backward Induction

Abstract

In [1], it was shown that in games of perfect information, common knowledge of rationality entails that the backward induction outcome is reached, and is consistent (for short, entails BI). That work has been criticized because it assumes that a player chooses rationally at all his nodes, even those that he knows he will not reach, indeed even those that he himself precluded by one of his own previous moves. Here we derive an epistemic characterization of BI that is reminiscent of [1], but avoids these issues. Specifically, say that a player strongly believes a proposition at a node of the game tree if he believes the proposition unless it is logically inconsistent with that node having been reached. We show that with a definition of rationality that is weaker than (i.e., implied by) that of [1], common strong belief of rationality entails BI; as with knowledge, the word “common” signifies truth, strong belief, strong belief of strong belief, and so on an infinitum. Our treatment is syntactic. A semantic notion of common strong belief that is conceptually closely related to ours has been advanced by Battigalli and Sinischalchi [6]. The relationship of our result to theirs is discussed.