From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point $b$ corresponding to a Nash equilibrium of such map $f$ is usually repelling, it is globally Cesaro attracting on the diagonal, that is, \[ \lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}f^k(x)=b \] for every $x$ in the minimal invariant interval. This solves a known open question whether there exists a nontrivial smooth map other than $x\mapsto axe^{-x}$ with centers of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.

中文翻译：

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来自博弈论的混沌映射族

从两个代理都应用乘法权重更新算法的双代理、双策略拥塞博弈中，我们获得了单位正方形到自身的双参数映射族。有趣的动力学出现在不变对角线上，在该对角线上，双峰区间图的双参数族表现出周期性轨道和混沌。虽然对应于这种映射 $f$ 的纳什均衡的不动点 $b$ 通常是排斥的，但它在对角线上是全局 Cesaro 吸引的，即 \[ \lim_{n\to\infty}\frac1n\sum_{ k=0}^{n-1}f^k(x)=b \] 对于最小不变区间中的每个 $x$。这解决了一个已知的悬而未决的问题，是否存在除 $x\mapsto axe^{-x}$ 之外的所有周期轨道的质心重合的非平凡光滑映射。我们还研究了动力学对两个参数的依赖性。