  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 best-response 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).
