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:在本文中,我们研究了一种基于格奇的推论式多复制性加权动态。 格奇是理论机器学习和算法游戏理论中众所周知的算法。 列奇产生的实验性平均( 利差平均值) 众所周知, 在零和游戏中接近一个微量平衡。 我们将这一结果加以概括, 以显示实验性平均值的加权版本与对称双矩阵游戏等级的均衡相融合, 以降低学习率参数。 我们的动态是第一个( 无论是连续的还是离散的)动态系统, 显示它会向纳什平衡演变, 而不假定报酬结构的单一性或潜在功能的存在。 尽管我们的设置有些限制,但它也是一般的, 因为对称双矩阵游戏的类别可以捕捉到PPAD类的整个计算复杂性( 即使接近平衡 ) 。