Bayesian equilibrium

Definition

A strategy vector $\sigma^* = (\sigma_1^*,\sigma_2^*,...,\sigma_n^*)$ is a Bayesian equilibrium if for each player $i \in N$, each type $t_i \in T_i$, and each possible action $a_i \in A_i(t_i)$,

An equivalent way to define Bayesian equilibrium is via an auxiliary game, the agent-form game.

Theorem (Bayesian equilibrium and Nash equilibrium in corresponding agent-form game)

A strategy vector $\sigma^* = (\sigma_i^*)_{i \in N}$ is a Bayesian equilibrium in a game $\Gamma$ with incomplete information if and only if the strategy vector $(\sigma_i^*(t_i))_{i \in N, t_i \in T_i}$ is a Nash equilibrium in the corresponding agent-form game $\hat{\Gamma}$.

Theorem (incomplete information and finite set of actions)

Every game with incomplete information in which the set of types is finite and the set of actions of each type is finite has a Bayesian equilibrium (in behavior strategies).

Theorem (Harsanyi, 1967)

In a game with incomplete information in which the number of types of each player is finite, every Bayesian equilibrium is also a Nash equilibrium, and conversely every Nash equilibrium is also a Bayesian equilibrium.

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 354.