Tie-breaking Agnostic Lower Bound for Fictitious Play

TL;DR

This paper demonstrates that fictitious play in certain zero-sum games converges at a rate slower than previously conjectured, even without tie-breaking, challenging earlier assumptions about its efficiency.

Contribution

It provides a counterexample showing that fictitious play can converge at a rate of (t^{-1/3}) in a specific 10-by-10 zero-sum game, disproving a longstanding conjecture.

Findings

Fictitious play can have a convergence rate of (t^{-1/3}) in some zero-sum games.

The constructed game involves no tie-breaking after the initial step.

Disproves Karlin's conjecture on the convergence rate of fictitious play.

Abstract

Fictitious play (FP) is a natural learning dynamic in two-player zero-sum games. Samuel Karlin conjectured in 1959 that FP converges at a rate of $O(t^{-1/2})$ to Nash equilibrium, where $t$ is the number of steps played. However, Daskalakis and Pan disproved the stronger form of this conjecture in 2014, where \emph{adversarial} tie-breaking is allowed. This paper disproves Karlin's conjecture in its weaker form. In particular, there exists a 10-by-10 zero-sum matrix game, in which FP converges at a rate of $\Omega(t^{-1/3})$, and no ties occur except for the first step.