Bayesian Equilibrium

A Bayesian strategy profile $\sigma$ is a Bayesian equilibrium if no player can improve expected payoff by replacing their type-contingent strategy while the other players keep theirs fixed.

In ex-ante form, for every player $i$ and every alternative type-contingent strategy $\tau_i:T_i\to\Delta(A_i)$, $$ \mathbb E_p[g_i(\sigma_i(t_i),\sigma_{-i}(t_{-i}),t)] \ge \mathbb E_p[g_i(\tau_i(t_i),\sigma_{-i}(t_{-i}),t)]. $$

When every type has positive prior probability, this is equivalent to the interim condition that after each observed type $t_i$, the prescribed mixed action is optimal against the conditional distribution of other players' types and actions.

References

• [MFoGT, Chapter 7, Section 7.4] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Bayesian equilibrium as the Nash-style equilibrium concept for Bayesian games.