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Empirical Analysis of Fictitious Play for Nash Equilibrium Computation in Multiplayer Games

Sam Ganzfried

TL;DR

The paper addresses the challenge of computing Nash equilibria in multiplayer, non-zero-sum games, where CFR has shown strong empirical performance but lacks general convergence guarantees. It compared core FP and CFR across varying numbers of players and actions, including random initializations and GAMUT benchmarks, and found FP often yields closer $\epsilon$-Nash equilibria than CFR in most non-two-player-zero-sum settings. It also demonstrates that random initializations enable FP to converge on classic counterexamples such as Shapley's game and the Doctrines game, indicating FP can reach equilibria from a positive-measure set of initial conditions. These results provide positive evidence for FP as a practical approach in multiplayer contexts and raise open questions about universal convergence guarantees and initialization strategies.

Abstract

While fictitious play is guaranteed to converge to Nash equilibrium in certain game classes, such as two-player zero-sum games, it is not guaranteed to converge in non-zero-sum and multiplayer games. We show that fictitious play in fact leads to improved Nash equilibrium approximation over a variety of game classes and sizes than (counterfactual) regret minimization, which has recently produced superhuman play for multiplayer poker. We also show that when fictitious play is run several times using random initializations it is able to solve several known challenge problems in which the standard version is known to not converge, including Shapley's classic counterexample. These provide some of the first positive results for fictitious play in these settings, despite the fact that worst-case theoretical results are negative.

Empirical Analysis of Fictitious Play for Nash Equilibrium Computation in Multiplayer Games

TL;DR

The paper addresses the challenge of computing Nash equilibria in multiplayer, non-zero-sum games, where CFR has shown strong empirical performance but lacks general convergence guarantees. It compared core FP and CFR across varying numbers of players and actions, including random initializations and GAMUT benchmarks, and found FP often yields closer -Nash equilibria than CFR in most non-two-player-zero-sum settings. It also demonstrates that random initializations enable FP to converge on classic counterexamples such as Shapley's game and the Doctrines game, indicating FP can reach equilibria from a positive-measure set of initial conditions. These results provide positive evidence for FP as a practical approach in multiplayer contexts and raise open questions about universal convergence guarantees and initialization strategies.

Abstract

While fictitious play is guaranteed to converge to Nash equilibrium in certain game classes, such as two-player zero-sum games, it is not guaranteed to converge in non-zero-sum and multiplayer games. We show that fictitious play in fact leads to improved Nash equilibrium approximation over a variety of game classes and sizes than (counterfactual) regret minimization, which has recently produced superhuman play for multiplayer poker. We also show that when fictitious play is run several times using random initializations it is able to solve several known challenge problems in which the standard version is known to not converge, including Shapley's classic counterexample. These provide some of the first positive results for fictitious play in these settings, despite the fact that worst-case theoretical results are negative.

Paper Structure

This paper contains 5 sections, 6 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Fictitious Play and Regret Minimization
  3. Comparison of Fictitious Play and Regret Minimization for Nash Equilibrium Approximation in Multiplayer Games
  4. Random Initialization Solves Notorious Fictitious Play Counterexamples
  5. Conclusion

Figures (2)

  • Figure 1: Heatmap of the strategy profiles out of the first 100 initializations that produce $\epsilon < e-4$ in Shapley's game. The first three columns are player 1's strategy probabilities and next three columns are player 2's probabilities.
  • Figure 2: Heatmap of the strategy profiles out of the first 10,000 initializations that produce $\epsilon < e-4$ in Foster and Young's Doctrines Game. The first eight columns are player 1's strategy probabilities and next eight columns are player 2's probabilities.