Types and Beliefs

Game Theory · Axiom Academy

Understanding incomplete information through player types, belief formation, and strategic decision-making under uncertainty 1. Player Types as Private Information A type represents a player's private information that affects their payoffs. Each player knows their own type but is uncertain about opponents' types. Types can represent various forms of private information: Preferences: Risk aversion, time preferences Payoffs: Valuations, costs, production capabilities Information: Knowledge about market conditions, quality The common prior assumption states that all players share the same probability distribution over type profiles before learning their own types. This is a foundational assumption in Bayesian game theory. The common prior captures shared knowledge about the "state of the world" before players learn their private information. It represents: Public information: What everyone knows about the distribution of types Rationality: Agreement on how uncertainty should be quantified Consistency: Ensures beliefs across players are coherent 3. Belief Formation Over Opponent Types After learning their own type, each player updates their beliefs about opponents' types using conditional probability . These beliefs guide strategic decision-making. The belief formation process follows these steps: Observe own type: Player i learns t i Apply Bayes' rule: Update probabilities conditional on t i Form posterior beliefs: Obtain p i (t -i | t i ) about opponents

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