Correlated optimin

TL;DR

This paper introduces the concept of correlated optimin, extending the optimin notion to correlated distributions, and demonstrates its existence and properties in finite and zero-sum games.

Contribution

It defines correlated optimin, proves its existence in finite games, and shows how it relates to correlated equilibrium and maximin values.

Findings

Correlated optimin exists in every finite game.

In two-player zero-sum games, correlated optimin equals the correlated equilibrium and the maximin value.

Correlated optimin can strictly Pareto dominate correlated equilibrium payoffs.

Abstract

We extend the optimin notion of Ismail (2025) from mixed strategy profiles to correlated distributions. A correlated distribution is evaluated by the worst expected payoff each player can receive when opponents may either obey their private recommendations or make unilateral recommendation-contingent deviations that are strictly profitable under the posterior induced by the distribution. Correlated optimins are Pareto optimal with respect to this vector of guaranteed payoffs. We show that a correlated optimin exists in every finite game. In addition, for every correlated equilibrium, there exists a correlated optimin such that every player's guaranteed payoff is weakly higher than his or her correlated equilibrium payoff. In two-player zero-sum games, correlated optimin coincides with correlated equilibrium and yields the maximin value. Outside zero-sum games, correlated optimin may strictly improve upon all correlated equilibria. We illustrate this with a simple 2x2 game with a unique correlated and coarse correlated equilibrium, in which there exists a correlated optimin that strictly Pareto dominates the equilibrium payoff.