We study the existence and computation of Nash equilibria in concave games where the players' admissible strategies are subject to shared coupling constraints. Under playerwise concavity of constraints, we prove existence of Nash equilibria. Our proof leverages topological fixed point theory and novel structural insights into the contractibility of feasible sets, and relaxes strong assumptions for existence in prior work. Having established existence, we address the question of whether in the presence of coupling constraints, playerwise independent learning dynamics have convergence guarantees. We address this positively for the class of potential games by designing a convergent algorithm. To account for the possibly nonconvex feasible region, we employ a log barrier regularized gradient ascent with adaptive stepsizes. Starting from an initial feasible strategy profile and under exact gradient feedback, the proposed method converges to an $ε$-approximate constrained Nash equilibrium within $\mathcal{O}(ε^{-3})$ iterations.

翻译：本文研究具有共享耦合约束的凹性博弈中纳什均衡的存在性与计算问题。在约束满足玩家凹性的条件下，我们证明了纳什均衡的存在性。该证明运用拓扑不动点理论及对可行集可收缩性的创新结构分析，放宽了既有研究中关于均衡存在的强假设条件。在确立存在性后，我们探讨了在耦合约束存在的情况下，玩家独立学习动态是否具有收敛保证。针对势博弈这一特定类别，我们通过设计收敛算法给出了肯定回答。为处理可能非凸的可行域，我们采用具有自适应步长的对数障碍正则化梯度上升法。从初始可行策略组合出发，在精确梯度反馈条件下，所提方法可在$\mathcal{O}(ε^{-3})$迭代次数内收敛至$ε$近似约束纳什均衡。