Doxastic Conditional Logic, Rationality, and Backward Induction

Abstract

Rationality in a game depends on a player’s preferences and two broad kinds of belief: belief about other players and belief about counterfactuals. Logical analyses relating to beliefs about other players are well represented in epistemic game theory. Less explored, however, is the logic of counterfactuals in a gametheoretic setting. If a rational player chooses some move m over another move m′, it is because they believe that the eventual outcome were they to playm′ would or might be worse than the eventual outcome were they to playm. This kind of counterfactual reasoning is not well described by the tools of standard modal logic. This thesis introduces a new logic, DCLEG, to analyze rational behavior in the class of finite perfect-information extensive-form (FPIE) games. DCLEG combines premise semantics for counterfactuals with multi-agent doxastic and temporal logic to better capture players’ belief structures in FPIE games. We apply DCLEG to the analysis of rationality and backward induction. We find a sufficient belief condition to guarantee that players reach a backward induction solution. We then turn to Gustafsson and Rabinowicz’s “new riddle of backward induction”, and their proposed solution: conditioned dominance. We introduce a similar concept, conditioned rationality, which can be expressed in DCLEG and find a sufficient belief condition to guarantee that a rational player obeys conditioned rationality.