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Fictitious Play (FP) is a simple and natural dynamic for repeated play inzero-sum games. Proposed by Brown in 1949, FP was shown to converge to a NashEquilibrium by Robinson in 1951, albeit at a slow rate that may depend on thedimension of the problem. In 1959, Karlin conjectured that FP converges at themore natural rate of $O(1/\sqrt{t})$. However, Daskalakis and Pan disproved aversion of this conjecture in 2014, showing that a slow rate can occur,although their result relies on adversarial tie-breaking. In this paper, weshow that Karlin's conjecture is indeed correct for the class of diagonalpayoff matrices, as long as ties are broken lexicographically. Specifically, weshow that FP converges at a $O(1/\sqrt{t})$ rate in the case when the payoffmatrix is diagonal. We also prove this bound is tight by showing a matchinglower bound in the identity payoff case under the lexicographic tie-breakingassumption.

Fast Convergence of Fictitious Play for Diagonal Payoff Matrices