Bayesian Nash Equilibrium

Game Theory · Axiom Academy

LESSON Bayesian Nash Equilibrium Understanding equilibria in games with incomplete information 1. Definition of Bayesian Nash Equilibrium Unlike regular Nash equilibrium where players choose actions directly, in Bayesian games players choose strategies that map their private information (type) to actions . Types (θ): Private information that affects payoffs (e.g., cost, valuation, card strength) Strategies (s): Functions from types to actions: s i : Θ i → A i Beliefs (p): Probability distributions over opponents' types Payoffs (u): Depend on actions and types of all players 2. Strategies as Functions from Types to Actions The crucial insight: In Bayesian games, a strategy is not just an action—it's a complete contingency plan that specifies what action to take for every possible type you might have. Example - First-Price Sealed-Bid Auction: Type: Your private valuation v ∈ [0, 100] Strategy: A function b(v) that says how much to bid for each possible valuation Example strategy: b(v) = 2v/3 (bid two-thirds of your value) 3. Expected Payoffs Given Beliefs Since you don't know your opponents' types, you must calculate expected payoffs using your beliefs about their type distribution. The expectation is taken over all possible type profiles of your opponents, weighted by your beliefs about how likely each profile is. Components of Expected Payoff: Your type θ i : Known to you (affects your payoffs) Your action a i : What you're considering doing

This is the written version of the interactive lesson above. See the full Game Theory course.