Fictitious Play Outperforms Counterfactual Regret Minimization

In two-player zero-sum games a Nash equilibrium strategy is guaranteed to win (or tie) in expectation against any opposing strategy by the minimax theorem. In games with m ore than two players there can be multiple equilibria with different values to the players, and follow ing one has no performance guarantee; however, it was shown that a Nash equilibrium strategy defeated a variet y of agents submitted for a class project in a 3-player imperfect-information game, Kuhn poker [13]. Thi s demonstrates that Nash equilibrium strategies can be successful in practice despite the fact that they do no t have a performance guarantee. While Nash equilibrium can be computed in polynomial time fo r two-player zero-sum games, it is PPAD-hard to compute for nonzero-sum and games with 3 or mor e agents and widely believed that no efficient algorithms exist [8, 9]. Counterfactual regret mi nimization (CFR) is an iterative self-play procedure that has been proven to converge to Nash equilibrium in two-p layer zero-sum [28].