Arbitrarily Slow Polynomial Convergence of Fictitious Play

2026-10-06

AI Summary

This study addresses the long-standing open problem regarding the lower bound on the convergence rate of fictitious play in zero-sum games and the unresolved Karlin conjecture. By recursively constructing payoff matrices and analyzing unique best responses, this work establishes instances exhibiting arbitrarily slow convergence for any integer k, rigorously proving that the duality gap decays at a rate of Θ(t^{-1/k}). These results refute Karlin's conjecture and generalize the previously known slowest convergence rates to arbitrary polynomial orders. Ultimately, this research establishes a theoretical lower bound demonstrating that fictitious play can converge at an arbitrarily slow polynomial rate, providing critical theoretical foundations for understanding the fundamental performance limits of this algorithm.

Abstract

We show that fictitious play can converge at arbitrarily slow polynomial rates in two-player zero-sum games. For every integer $k \ge 2$, we construct a payoff matrix with $(k+1)^2 - 5$ actions per player for which the duality gap of the empirical strategies decays as $Θ(t^{-1/k})$ after $t$ steps. The family starts from the standard rock-paper-scissors matrix, with each higher-order game constructed recursively from the preceding one. After a prescribed common initial action, every subsequent best response under fictitious play is unique. For $k \ge 3$, these games give counterexamples to Karlin's conjectured $O(t^{-1/2})$ convergence rate, and they extend the recent $Θ(t^{-1/3})$ construction of Wang (2025) to arbitrarily slow polynomial rates.

Problem

• Fictitious Play
• Zero-sum Games
• Convergence Rate
• Karlin's Conjecture
• Duality Gap

Innovation

• Fictitious Play
• Polynomial Convergence
• Two-player Zero-sum Games
• Karlin's Conjecture
• Duality Gap