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We analyze the performance of the best-response dynamic across allnormal-form games using a random games approach. The playing sequence -- theorder in which players update their actions -- is essentially irrelevant indetermining whether the dynamic converges to a Nash equilibrium in certainclasses of games (e.g. in potential games) but, when evaluated across allpossible games, convergence to equilibrium depends on the playing sequence inan extreme way. Our main asymptotic result shows that the best-response dynamicconverges to a pure Nash equilibrium in a vanishingly small fraction of all(large) games when players take turns according to a fixed cyclic order. Bycontrast, when the playing sequence is random, the dynamic converges to a pureNash equilibrium if one exists in almost all (large) games.

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