Bayesian Games

Game Theory · Axiom Academy

Unit 5: Games with Incomplete Information The key insight: we can model incomplete information by introducing an imaginary player called Nature who moves first, before any real player acts. Incomplete Information: Players are uncertain about the structure of the game itself (e.g., payoffs, available actions, or other players' characteristics). Imperfect Information: Players observe Nature's move imperfectly - each player learns their own type but not others' types. This transformation allows us to use the tools we've already developed for extensive-form games with imperfect information to analyze situations where players have different private information. 2. Type Spaces and Prior Distributions Each player i has a type space Θ i representing all possible private characteristics they might have. Nature assigns each player a type θ i ∈ Θ i according to a common prior probability distribution. Type: A complete specification of a player's private information (payoff function, beliefs, knowledge, etc.) Type Space Θ i : The set of all possible types for player i Prior Distribution p: A probability distribution over the type profile space Θ = Θ 1 × Θ 2 × ... × Θ n Each player knows their own type but must form beliefs about other players' types based on the common prior distribution. A Bayesian game unfolds in three stages: Nature moves: Assigns a type to each player according to prior distribution p Players observe: Each player privately learns their own type θ i

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