Bayesian Game

A game in which players have private information (types) drawn from a common prior.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

A Bayesian game is $\langle N,\{\Theta_i\},p,\{A_i\},\{u_i(a,\theta)\}\rangle$ with type space $\Theta=\prod\Theta_i$ and prior $p\in\Delta(\Theta)$ .

A strategy is $\sigma_i:\Theta_i\to\Delta(A_i)$ ; a Bayes-Nash equilibrium satisfies $\sigma_i^*(\theta_i)\in\arg\max_{a_i}\mathbb{E}_{\theta_{-i}|\theta_i}u_i(a_i,\sigma_{-i}^*(\theta_{-i}),\theta).$

Intuition

Each player best-responds in expectation over the others' possible private information.

Private information is modeled as a "type" that parametrizes payoffs or beliefs.

Worked example

In a first-price auction with i.i.d. private values $v_i\sim U[0,1]$ , the symmetric BNE bid is $b(v)=\frac{n-1}{n}v$ .

Signaling models of labor markets (Spence) and insurance (Rothschild-Stiglitz) are Bayesian.

The math

Harsanyi (1967-68) showed any game of incomplete information has a Bayesian equivalent via the type-space transformation.

Common-prior assumption links beliefs consistently through Bayesian updating.

Where it is used

Online advertising platforms run Bayesian auctions daily: each advertiser has a private value for a click, drawn from a distribution the platform estimates. The generalized second-price auction is analyzed as a Bayesian game of incomplete information.

In insurance markets, adverse selection creates a Bayesian game: buyers know their own risk type but insurers only know the population distribution. Rothschild-Stiglitz screening models analyze equilibrium contract menus under this information asymmetry.

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