Proper Equilibrium

A mixed profile $\sigma\in\Sigma$ is a proper equilibrium of a finite normal-form game if there are completely mixed profiles $\sigma^n\in\operatorname{int}\Sigma$ and numbers $\epsilon_n\to 0$ such that:

• each $\sigma^n$ is $\epsilon_n$-proper;
• $\sigma^n\to\sigma$.

Proper equilibrium refines perfect equilibrium by ranking mistakes: worse deviations must become infinitely less likely than better deviations.

References

• [MFoGT, Def. 6.5.6] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Myerson proper equilibrium as a limit of epsilon-proper completely mixed profiles.