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

TL;DR

This paper disproves Karlin's conjecture by showing that fictitious play can converge arbitrarily slowly in certain zero-sum games, specifically with the identity payoff matrix, challenging previous assumptions about its convergence rate.

Contribution

The paper provides a counter-example demonstrating that fictitious play's convergence rate can be much slower than previously conjectured, specifically as slow as (t^{-1/n}) in certain cases.

Findings

Fictitious play may converge as slowly as (t^{-1/n}) in specific zero-sum games.

Karlin's conjecture of an O(1/( )) convergence rate is false.

The identity payoff matrix is used to construct the counter-example.

Abstract

Fictitious play is a natural dynamic for equilibrium play in zero-sum games, proposed by [Brown 1949], and shown to converge by [Robinson 1951]. Samuel Karlin conjectured in 1959 that fictitious play converges at rate $O(1/\sqrt{t})$ with the number of steps $t$. We disprove this conjecture showing that, when the payoff matrix of the row player is the $n \times n$ identity matrix, fictitious play may converge with rate as slow as $\Omega(t^{-1/n})$.