In this paper, we study an exponentiated multiplicative weights dynamic based on Hedge, a well-known algorithm in theoretical machine learning and algorithmic game theory. The empirical average (arithmetic mean) of the iterates Hedge generates is known to approach a minimax equilibrium in zero-sum games. We generalize that result to show that a weighted version of the empirical average converges to an equilibrium in the class of symmetric bimatrix games for a diminishing learning rate parameter. Our dynamic is the first dynamical system (whether continuous or discrete) shown to evolve toward a Nash equilibrium without assuming monotonicity of the payoff structure or that a potential function exists. Although our setting is somewhat restricted, it is also general as the class of symmetric bimatrix games captures the entire computational complexity of the PPAD class (even to approximate an equilibrium).

翻译：本文研究了一种基于Hedge算法的指数化乘性权重动力学，该算法是理论机器学习与算法博弈论中的经典方法。已知通过Hedge迭代生成的经验平均值（算术均值）在零和博弈中趋近于极小化极大均衡。我们将此结果推广，证明在对称双矩阵博弈中，当学习率参数递减时，加权形式的经验平均值收敛于均衡。本文提出的动力学是首个无需假设收益结构单调性或存在势函数、即可证明趋向纳什均衡的动力学系统（无论连续或离散）。尽管我们的设定有一定限制，但由于对称双矩阵博弈类完整刻画了PPAD类的计算复杂度（甚至包括近似均衡），该结果仍具普适性。