We study the existence and computation of Nash equilibria in continuous static games where the players' admissible strategies are subject to shared coupling constraints, i.e., constraints that depend on their \emph{joint} strategies. Specifically, we focus on a class of games characterized by playerwise concave utilities and playerwise concave constraints. Prior results on the existence of Nash equilibria are not applicable to this class, as they rely on strong assumptions such as joint convexity of the feasible set. By leveraging topological fixed point theory and novel structural insights into the contractibility of feasible sets under playerwise concave constraints, we give an existence proof for Nash equilibria under weaker conditions. Having established existence, we then focus on the computation of Nash equilibria via independent gradient methods under the additional assumption that the utilities admit a potential function. 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 $\epsilon$-approximate constrained Nash equilibrium within $\mathcal{O}(\epsilon^{-3})$ iterations.

翻译：本文研究连续静态博弈中纳什均衡的存在性与计算问题，其中玩家的可行策略受共享耦合约束限制，即依赖于其联合策略的约束条件。具体而言，我们聚焦于具有玩家凹效用函数与玩家凹约束的一类博弈。现有关于纳什均衡存在性的结论均不适用于此类博弈，因其依赖于可行集的联合凸性等强假设。通过运用拓扑不动点理论，并结合对玩家凹约束下可行集可收缩性的新颖结构分析，我们在更弱的条件下给出了纳什均衡的存在性证明。在确立存在性后，我们进一步在效用函数存在势函数的附加假设下，研究通过独立梯度法计算纳什均衡的方法。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。从初始可行策略剖面出发，在精确梯度反馈下，所提方法可在$\mathcal{O}(\epsilon^{-3})$次迭代内收敛至$\epsilon$近似约束纳什均衡。