Bayesian Game

In classical game theory, every player knows the structure of the game completely: who the players are, what actions are available, and what payoffs result from each combination of choices. A Bayesian game — also called a game of incomplete information — relaxes this assumption. Each player possesses private information, called a type, that affects her payoffs and possibly her available actions. The other players do not observe this type directly; instead, they hold beliefs about it, encoded as a probability distribution called the common prior.

This framework, introduced by John C. Harsanyi in a landmark trilogy of papers (1967–1968), transformed game theory from a tool for analyzing stylized parlor games into a language capable of modeling auctions, bargaining under asymmetric information, signaling in labor markets, mechanism design, and virtually every strategic interaction in which agents hold private knowledge.

Formal Definition

A Bayesian game is defined by a tuple (N, A, T, p, u) consisting of a set of players, action spaces, type spaces, a common prior over type profiles, and utility functions.

Each player observes her own type tᵢ but not the types of others. She then chooses an action aᵢ from her action set. A strategy for player i is a function σᵢ : Tᵢ → Aᵢ mapping types to actions. The solution concept is the Bayes-Nash equilibrium: a profile of strategies in which each player's strategy maximizes her expected utility given her type and her beliefs about others' types.

Harsanyi's Contribution

Before Harsanyi, games of incomplete information were considered intractable. If a player does not know her opponent's payoffs, she must reason about what her opponent knows — and about what her opponent believes she knows, generating an infinite regress of higher-order beliefs. Harsanyi's ingenious solution was to introduce a fictitious move by "Nature" that selects each player's type from a commonly known prior distribution. This converts the infinite regress into a single, well-defined Bayesian inference problem.

Canonical Applications

Auctions

In a sealed-bid auction, each bidder's valuation of the item is her private type. Under the independent private values model, each valuation is drawn independently from a known distribution. The revenue equivalence theorem, proved by Vickrey (1961) and generalized by Myerson (1981), shows that all standard auction formats — first-price, second-price, English, Dutch — yield the same expected revenue when bidders play Bayes-Nash equilibrium strategies. Myerson's optimal auction design is a direct application of Bayesian game theory: the seller designs a mechanism to maximize expected revenue given the prior distribution over bidder types.

Signaling Games

In Spence's job-market signaling model, a worker's ability is her private type. She chooses an education level (a costly signal), and an employer observes the signal and forms Bayesian beliefs about her ability. In a separating equilibrium, different types choose different signals, and the employer can infer the type perfectly. In a pooling equilibrium, all types choose the same signal, and the employer learns nothing. The analysis of which equilibria survive refinements — like the intuitive criterion of Cho and Kreps (1987) — is a central topic in Bayesian game theory.

Mechanism Design

Mechanism design is the "inverse" of game theory: instead of analyzing equilibria of a given game, the designer chooses the game's rules to achieve a desired outcome. The revelation principle states that for any Bayes-Nash equilibrium of any mechanism, there exists an equivalent direct mechanism in which each player truthfully reports her type. This dramatically simplifies the design problem by restricting attention to truth-telling equilibria.

"I have tried to show that games with incomplete information can be reduced, in a natural way, to games with complete but imperfect information, by use of Bayesian probability." — John C. Harsanyi, Nobel Prize Lecture (1994)

Connection to Bayesian Inference

The "Bayesian" in Bayesian games refers specifically to each player's use of Bayes' theorem to update beliefs about opponents' types after observing signals or actions. In dynamic Bayesian games — where players move sequentially and can observe previous actions — the notion of perfect Bayesian equilibrium (PBE) formalizes the requirement that beliefs be updated via Bayes' rule at every information set reached with positive probability. Off-equilibrium beliefs, where Bayes' rule does not apply (because the information set has zero probability under the equilibrium strategy), must be specified separately, leading to a rich theory of equilibrium refinements.

Modern Extensions

Contemporary research extends Harsanyi's framework in several directions. Robust mechanism design (Bergemann and Morris, 2005) asks what mechanisms perform well across a range of possible priors, rather than for a single known prior. Information design (Bayesian persuasion) studies how an informed sender can strategically disclose information to influence a receiver's Bayesian beliefs and actions. Global games (Carlsson and van Damme, 1993) show how small amounts of private information can select unique equilibria in coordination games that have multiple equilibria under complete information.