Tie-breaking Agnostic Lower Bound for Fictitious Play

2025-07-14

AI Summary

This paper refutes the weak form of Samuel Karlin’s (1959) conjecture that fictitious play (FP) converges at rate $O(t^{-1/2})$ in two-player zero-sum games. Method: Without assuming adversarial tie-breaking, we construct an explicit $10 imes 10$ zero-sum matrix game and rigorously analyze the FP dynamic—combining precise trajectory analysis of initial strategy evolution with a carefully engineered payoff structure—to establish a lower bound on the convergence rate of time-averaged strategies. Contribution/Results: We prove that the average strategy sequence converges no faster than $Omega(t^{-1/3})$, thereby falsifying Karlin’s weak conjecture under standard FP dynamics without tie-breaking artifacts. This is the first such counterexample in the canonical FP setting and yields the tightest known convergence lower bound for zero-sum games, revealing an intrinsic slow-convergence property of FP in this class.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Adversarial Search

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 $Ω(t^{-1/3})$, and no ties occur except for the first step.

Problem

• Disproves Karlin's conjecture on FP convergence rate
• Analyzes tie-breaking impact in fictitious play dynamics
• Demonstrates slower convergence in 10x10 zero-sum games

Innovation

• Disproves Karlin's conjecture on FP convergence
• Uses 10-by-10 zero-sum matrix game
• Achieves Ω(t^{-1/3}) convergence rate