This paper studies a class of strongly monotone games involving non-cooperative agents that optimize their own time-varying cost functions. We assume that the agents can observe other agents' historical actions and choose actions that best respond to other agents' previous actions; we call this a best response scheme. We start by analyzing the convergence rate of this best response scheme for standard time-invariant games. Specifically, we provide a sufficient condition on the strong monotonicity parameter of the time-invariant games under which the proposed best response algorithm achieves exponential convergence to the static Nash equilibrium. We further illustrate that this best response algorithm may oscillate when the proposed sufficient condition fails to hold, which indicates that this condition is tight. Next, we analyze this best response algorithm for time-varying games where the cost functions of each agent change over time. Under similar conditions as for time-invariant games, we show that the proposed best response algorithm stays asymptotically close to the evolving equilibrium. We do so by analyzing both the equilibrium tracking error and the dynamic regret. Numerical experiments on economic market problems are presented to validate our analysis.

翻译：本文研究一类强单调博弈，其中非合作代理优化各自时变成本函数。假设代理能观察其他代理的历史行为并选择对他人先前行为的最优响应动作，我们称之为最优响应方案。首先分析该方案在标准时不变博弈中的收敛速率：具体而言，给出时不变博弈强单调参数的一个充分条件，在该条件下所提最优响应算法能够指数收敛至静态纳什均衡。进一步证明，当该充分条件不满足时算法可能出现振荡，表明该条件具有紧致性。随后分析时变博弈中的最优响应算法——此类博弈中每个代理的成本函数随时间变化。在类似时不变博弈的条件下，证明所提最优响应算法能渐近跟踪演化均衡，通过考察均衡跟踪误差与动态遗憾指标完成分析。最后在经济市场问题上的数值实验验证了理论分析结果。