On Tolerance Analysis of Games with Belief Revision

Abstract

Aumann’s Rationality Theorem claims that in perfect information games, common knowledge of rationality yields backward induction (BI). Stalnaker argued that in the belief revision setting, BI does not follow from Aumann’s assumptions. However, as shown by Artemov, if common knowledge of rationality is understood in the robust sense, i.e., if players do not forfeit their knowledge of rationality even hypothetically, then BI follows. A more realistic model would bound the number of hypothetical non-rational moves by player i that can be tolerated without revising the belief in i’s rationality on future moves. We show that in the presence of common knowledge of rationality, if n hypothetical non-rational moves by any player are tolerated, then each game of length less than 2n + 3 yields BI, and that this bound on the length of model is tight for each n. In particular, if one error per player is tolerated, i.e., n = 1, then games of length up to 4 are BI games, whereas there is a game of length 5 with a non-BI solution.