Choice & Inference

Whether rationality and common belief in rationality among the players of centipede games jointly entail the backward inductive outcome has long been argued without consensus. Arguments that these conditions are not sufficient often turn on the claim that there is no justification for supposing that players with initial common belief in rationality would retain such beliefs upon witnessing an unexpected move by an opponent. I will describe one of the more compelling such arguments, due to Stalnaker, who argues that we should use our best theory of belief revision, AGM, to determine which belief changes during a game are rational given the players’ prior beliefs. Stalnaker claims that the AGM axioms do not justify any special assumptions about how players revise from initial common belief in rationality, so that — critically — rational players may decide that their opponents are irrational if those players make non-backward inductive moves; departures from backward induction are therefore permissible. I disagree, arguing that while the content of the common belief (i.e. rationality) does not justify any additional assumptions about belief revision, the structure of common belief does, in accordance with the principles of AGM; I will employ Grove’s revealing sphere-based modelling of AGM to demonstrate this. I then prove a general theorem: Given my proposed constraint on rational belief revision, for all finite, n-player, extensive form, perfect information games with a unique backward induction solution, if there is initial common belief in rationality, then the backward inductive outcome is guaranteed. Further, if the longest branch of the game tree has n+1 decision nodes, initial n’th level mutual belief in rationality suffices for the result.