博弈均衡中单纯形乘积域上的定点丛方法
The fixed-point bundle method over product-of-simplex domains arising from game equilibria
AI总结：
本文提出单纯形乘积域上变分不等式的定点丛框架，通过纤维丛同构建立与单纯形域框架的等价性，使预测-校正路径跟踪算法继承全局收敛保证，并在5600个随机博弈实例中全部收敛。
AI中文摘要：
本文将从单纯形域上的有限维变分不等式（VI）的定点丛框架推广到单纯形乘积域，该框架可直接应用于求解纳什均衡。单纯形乘积域上的变分不等式的定点丛揭示了复合纤维丛结构。关键创新在于在单纯形域上构造一个等价的变分不等式，并通过纤维丛同构建立两个定点丛框架之间的等价性。利用这一几何等价性，针对单纯形乘积域上变分不等式的预测-校正路径跟踪算法被证明继承了单纯形域框架的收敛保证，即全局收敛且具有线性间隙缩减至解附近。在5600个随机生成的实例上进行的数值实验，实例维度从2个参与者128个动作到128个参与者2个动作不等，展示了算法的稳健性能。该算法在所有测试实例中均收敛。
英文摘要：
This paper extends the fixed-point bundle framework for finite-dimensional variational inequalities (VIs) from the simplex domain to the product-of-simplex domain, which is directly applicable to solving Nash equilibria. The fixed-point bundle for VIs on the product-of-simplex domain reveals a composite fiber bundle structure. The key innovation is to construct an equivalent VI on the simplex domain and establish the equivalence between the two fixed-point bundle frameworks via a fiber bundle isomorphism. Exploiting this geometric equivalence, the predictor-corrector path-following algorithm for the VI on the product-of-simplex domain is shown to inherit the convergence guarantee of the simplex-domain framework, namely, global convergence with linear gap reduction near solutions. Numerical experiments on 5600 randomly generated instances with dimensions ranging from 2-player 128-action to 128-player 2-action demonstrate robust performance. The algorithm converges in every tested instance.
