Perfect Bayesian Equilibrium

Game Theory · Axiom Academy

LESSON Perfect Bayesian Equilibrium Unit 5 - Games with Incomplete Information 1. PBE Definition: Strategies and Beliefs A Perfect Bayesian Equilibrium consists of two components working together: These two components must satisfy both sequential rationality and belief consistency conditions. 2. Sequential Rationality Given Beliefs At every information set, each player's strategy must be optimal given: Their beliefs about which node they are at within that information set The strategies of other players from that point forward This extends subgame perfection to situations with incomplete information. Unlike standard backward induction, we must account for beliefs about unobserved past actions or types. 3. Belief Consistency via Bayes' Rule Beliefs must be derived from strategies using Bayes' rule whenever possible. Given a prior probability distribution over types and the strategies played, we update beliefs about which node we're at: This ensures beliefs are internally consistent with the equilibrium strategies being played. 4. On-path vs Off-path Beliefs A critical distinction in PBE is between information sets that are reached in equilibrium and those that are not: Off-path beliefs are crucial for deterrence and credible threats. They determine whether deviations are profitable, thus supporting the equilibrium. Perfect Bayesian Equilibrium refines Bayesian Nash Equilibrium for dynamic games by adding sequential rationality requirements:

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