Bayesian Game

A finite Bayesian game models incomplete information by giving each player a type and allowing payoffs to depend on the full type profile.

The data consist of players $I$, finite action sets $(A_i)$, finite type sets $(T_i)$, a common prior $p\in\Delta(T)$ on $T=\prod_i T_i$, and payoff functions $$ g_i:A\times T\to\mathbb R, \qquad A=\prod_i A_i. $$ Player $i$ observes only their own type $t_i$ before choosing an action.

The common prior records the ex-ante probability of type profiles. Conditional beliefs about other players' types are derived from the prior whenever the observed type has positive probability.

References

• [MFoGT, Chapter 7, Section 7.4] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Games with incomplete information, also called Bayesian games.