Fast Convergence of Fictitious Play for Diagonal Payoff Matrices

TL;DR

This paper proves that Fictitious Play converges at a rate of O(1/√t) for diagonal payoff matrices with lexicographic tie-breaking, confirming Karlin's conjecture in this specific setting.

Contribution

It establishes the convergence rate of Fictitious Play for diagonal payoff matrices with lexicographic tie-breaking, confirming a long-standing conjecture.

Findings

Fictitious Play converges at O(1/√t) for diagonal payoff matrices.

The convergence rate is tight, with a matching lower bound shown.

The result depends on the assumption of lexicographic tie-breaking.

Abstract

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