Convergence Analysis of the Best Response Algorithm for Time-Varying Games

TL;DR

This paper analyzes the convergence properties of a best response algorithm in strongly monotone, time-varying games, establishing conditions for exponential convergence and tracking accuracy, supported by numerical experiments.

Contribution

It provides a rigorous convergence analysis of the best response scheme for both static and dynamic games, including conditions for exponential convergence and equilibrium tracking.

Findings

Exponential convergence occurs under a strong monotonicity condition.

The best response algorithm may oscillate if the condition is not met.

The algorithm effectively tracks evolving equilibria in time-varying games.

Abstract

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.