  While fictitious play is guaranteed to converge to Nash equilibrium in
certain game classes, such as two-player zero-sum games, it is not guaranteed
to converge in non-zero-sum and multiplayer games. We show that fictitious play
in fact leads to improved Nash equilibrium approximation over a variety of game
classes and sizes than (counterfactual) regret minimization, which has recently
produced superhuman play for multiplayer poker. We also show that when
fictitious play is run several times using random initializations it is able to
solve several known challenge problems in which the standard version is known
to not converge, including Shapley's classic counterexample. These provide some
of the first positive results for fictitious play in these settings, despite
the fact that worst-case theoretical results are negative.
