Best-response dynamics, playing sequences, and convergence to equilibrium in random games

Abstract

We show that the playing sequence-the order in which players update their actions-is a crucial determinant of whether the best-response dynamic converges to a Nash equilibrium. Specifically, we analyze the probability that the best-response dynamic converges to a pure Nash equilibrium in random n-player m-action games under three distinct playing sequences: clockwork sequences (players take turns according to a fixed cyclic order), random sequences, and simultaneous updating by all players. We analytically characterize the convergence properties of the clockwork sequence best-response dynamic. Our key asymptotic result is that this dynamic almost never converges to a pure Nash equilibrium when n and m are large. By contrast, the random sequence bestresponse dynamic converges almost always to a pure Nash equilibrium when one exists and n and m are large. The clockwork best-response dynamic deserves particular attention: we show through simulation that, compared to random or simultaneous updating, its convergence properties are closest to those exhibited by three popular learning rules that have been calibrated to human game-playing in experiments (reinforcement learning, fictitious play, and replicator dynamics).

Exact time-dependent dynamics of discrete binary choice models

“…Other exciting results in decision theory can be obtained with random interactions. Agents interacting through random games are already known to produce very rich dynamics, in particular because multiple Nash equilibria emerge as the games become more complex [54] and because minute details such as the order in which players update their actions has an impact on the existence of an equilibrium [55].…”

Cost Minimization for Equilibrium Transition

“…They find that the best response dynamics consistently converges to a pure Nash equilibrium in such games. Heinrich et al (2022) analyze the performance of the best-response dynamic across all normal-form games using a random games approach. They show that the best-response dynamic converges to a pure Nash equilibrium in a vanishingly small fraction of all large games when players take turns according to a fixed cyclic order.…”