发表机构
University of Passau（帕绍大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究原子可拆分拥堵博弈的单调性与稳定性，通过曲率不等式刻画相关变分不等式算子单调的成本类，揭示其与学习动力学联系，表明特定条件下该曲率条件刻画稳定性，还研究了单纯形上博弈的局部指数稳定性。
AI 中文摘要
我们研究原子可拆分拥堵博弈的通用单调性和弗兰克 - 沃尔夫稳定性性质。具体而言，我们刻画了对于每个博弈，其相关变分不等式算子单调的最大资源成本类。此刻画由涉及允许成本函数的一阶和二阶导数以及参与者数量的曲率不等式给出。我们的框架对通用单调性、严格单调性和强单调性给出了精确刻画；严格和强变体需要相应更严格的曲率条件。接着，我们揭示了与原子可拆分拥堵博弈中学习动力学的惊人联系。我们表明，只要成本类在正仿射变换下封闭，相同的曲率条件也刻画了欧几里得正则化弗兰克 - 沃尔夫动力学在任意凸策略空间上的通用局部和全局稳定性。最后，我们研究单纯形上的博弈，表明正则化弗兰克 - 沃尔夫动力学的内部均衡即使没有曲率条件也是局部指数稳定的。
英文摘要
We study atomic splittable congestion games with $n\ge2$ players and nondecreasing $C^2$ resource costs $f$ with convex $x\mapsto xf(x)$. We characterize the largest resource cost class for which the Nash equilibrium is unique. This characterization combines a curvature inequality involving the first two derivatives and the number of players with strict increase of the scalar marginal costs $x\mapsto f(x)+xf'(x)$. The proof rests on first characterizing the largest resource cost classes for which the associated variational inequality operator is monotone, strictly monotone, or strongly monotone. We also draw a perhaps surprising connection to learning dynamics: the same curvature inequality characterizes universal local and global stability of Euclidean-regularized Frank--Wolfe dynamics on arbitrary compact convex strategy spaces. Both the uniqueness and stability characterizations assume closure under positive affine transformations. Both characterizations remain exact for network games; the necessity constructions use acyclic directed graphs whose arc costs can be chosen strictly positive. Finally, we show that on parallel-link networks, every interior equilibrium of the regularized Frank--Wolfe dynamics is locally exponentially stable even without the curvature condition.
Comments38 pages, added characterization on uniqueness and network representations
