In competitive multi-player interactions, simultaneous optimality is a key requirement for establishing strategic equilibria. This property is explicit when the game-theoretic equilibrium is the simultaneously optimal solution of coupled optimization problems. However, no such optimization problems exist for the correlated equilibrium, a strategic equilibrium where the players can correlate their actions. We address the lack of a coupled optimization framework for the correlated equilibrium by introducing an {unnormalized game} -- an extension of normal-form games in which the player strategies are lifted to unnormalized measures over the joint actions. We show that the set of fully mixed generalized Nash equilibria of this unnormalized game is a subset of the correlated equilibrium of the normal-form game. Furthermore, we introduce an entropy regularization to the unnormalized game and prove that the entropy-regularized generalized Nash equilibrium is a sub-optimal correlated equilibrium of the normal form game where the degree of sub-optimality depends on the magnitude of regularization. We prove that the entropy-regularized unnormalized game has a closed-form solution, and empirically verify its computational efficacy at approximating the correlated equilibrium of normal-form games.

翻译：在竞争性多方交互中，同时最优性是建立策略均衡的关键要求。当博弈论均衡是耦合优化问题的同时最优解时，这一性质是显式的。然而，对于关联均衡——一种玩家可以关联其行动的策略均衡，并不存在此类优化问题。我们通过引入非归一化博弈——正规型博弈的扩展形式，其中玩家策略被提升为联合行动上的非归一化测度——解决了关联均衡缺乏耦合优化框架的问题。我们证明，该非归一化博弈的完全混合广义纳什均衡集合是正规型博弈关联均衡的子集。进一步，我们在非归一化博弈中引入熵正则化，并证明熵正则化广义纳什均衡是正规型博弈的次优关联均衡，其次优程度取决于正则化的强度。我们证明熵正则化非归一化博弈具有闭式解，并通过实验验证了其在近似正规型博弈关联均衡时的计算效率。