Although it has been known since the 1970s that a globally optimal strategy profile in a common-payoff game is a Nash equilibrium, global optimality is a strict requirement that limits the result's applicability. In this work, we show that any locally optimal symmetric strategy profile is also a (global) Nash equilibrium. Furthermore, we show that this result is robust to perturbations to the common payoff and to the local optimum. Applied to machine learning, our result provides a global guarantee for any gradient method that finds a local optimum in symmetric strategy space. While this result indicates stability to unilateral deviation, we nevertheless identify broad classes of games where mixed local optima are unstable under joint, asymmetric deviations. We analyze the prevalence of instability by running learning algorithms in a suite of symmetric games, and we conclude by discussing the applicability of our results to multi-agent RL, cooperative inverse RL, and decentralized POMDPs.

对于对称策略空间中的本地最优对称策略，该研究证明任何局部最优对称策略都是（全局）纳什均衡，这个结果适用于机器学习，并为找到对称策略空间中的局部最优的梯度方法提供全局性保证，最后，总结了研究结果在多智能体RL，合作逆RL和分散式 POMDPs中的应用。

对称团队学习中，局部最优解是全局 Nash 均衡