A Counter-example to Karlin's Strong Conjecture for Fictitious Play

Abstract

Fictitious play is a natural dynamic for equilibrium play in zero-sum games, proposed by Brown [6], and shown to converge by Robinson [33]. Samuel Karlin conjectured in 1959 that fictitious play converges at rate O(t[superscript -1/2]) with respect to the number of steps t. We disprove this conjecture by showing that, when the payoff matrix of the row player is the n × n identity matrix, fictitious play may converge (for some tie-breaking) at rate as slow as Ω(t[superscript -1/n]).