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.

翻译：我们扩展了Ismail（2025）提出的最优最小概念，从混合策略组合推广至相关分布。在给定相关分布后，若对手可能选择服从其私人推荐，或在该分布诱导的后验下采取单方面与推荐相联的严格有利偏离策略，则每个玩家所能获得的最坏期望收益被用于评估该相关分布。关联最优最小是针对这一保障收益向量的帕累托最优解。我们证明每个有限博弈均存在关联最优最小。此外，对每个关联均衡，存在一个关联最优最小使得每位玩家的保障收益弱高于其关联均衡收益。在两人零和博弈中，关联最优最小与关联均衡一致，并产生最大最小值。在非零和博弈中，关联最优最小可能严格优于所有关联均衡。我们通过一个2×2简单博弈进行说明，该博弈具有唯一的关联均衡和粗糙关联均衡，其中存在严格帕累托优于均衡收益的关联最优最小。