Logic for games of perfect information and epistemic conditions for backward induction and for subgame perfectness

We propose a logical system in which a notion of the structure of a game is formally defined and the meaning of sequential rationality is formulated. We provide a set of decision criteria which, given sufficiently high order of mutual belief of the game structure and of every player following these criteria, entails Backward Induction decisions in generic perfect information games. We say that a player is rational if the player follows these criteria in his/her decisions. The set of mutual beliefs is also necessary, in the sense that any mutual belief of lower order can not entail the Backward Induction decisions. These conditions are determined by the length of the game structure, and they are never involved with common belief. Moreover, we give a set of epistemic conditions for subgame perfect equilibria for any perfect information game, which requires every player follow these decision criteria and there be mutual belief of the the equilibrium strategy and of the game structure. JEL classification: C72, D81