Abstract

Population games are games with a finite set of available strategies and an infinite number of players, in which the reward for choosing a given strategy is a function of the distribution of players over strategies. The paper shows that, in a certain class of maxmin optimization problems, it is possible to associate a population game to a given maxmin problem in such a way that solutions to the optimization problem are found from Nash equilibria of the associated game. Iterative solution methods for maxmin optimization problems can then be derived from systems of differential equations whose trajectories are known to converge to Nash equilibria. In particular, we use a discrete-time version of the celebrated replicator equation of evolutionary game theory, also known in machine learning as the exponential multiplicative weights algorithm. The resulting algorithm can be viewed as a generalization of the Iteratively Reweighted Least Squares (IRLS) method, which is well known in numerical analysis as a useful technique for solving Chebyshev function approximation problems on a finite grid. Examples are provided to show the use of the generalized IRLS method in collective investment and in decision making under model uncertainty.

中文翻译：

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通过群体博弈解决 Maxmin 优化问题

摘要

群体博弈是具有有限的可用策略集和无限数量的玩家的游戏，其中选择给定策略的奖励是玩家在策略上的分布的函数。该论文表明，在一类 maxmin 优化问题中，可以将总体博弈与给定的 maxmin 问题关联起来，从而从关联博弈的纳什均衡中找到优化问题的解。然后，可以从其轨迹已知收敛于纳什均衡的微分方程组导出最大最小优化问题的迭代求解方法。特别是，我们使用了著名的进化博弈论复制方程的离散时间版本，在机器学习中也称为指数乘法权重算法。由此产生的算法可以被视为迭代重加权最小二乘法 (IRLS) 方法的推广，该方法在数值分析中众所周知，是一种在有限网格上求解切比雪夫函数逼近问题的有用技术。举例说明了广义 IRLS 方法在集体投资和模型不确定性下的决策中的应用。