On the Convergence of Fictitious Play: A Decomposition Approach

TL;DR

This paper extends the understanding of fictitious play convergence to mixed game types using decomposition techniques, unifying cooperative and competitive scenarios, and analyzing non-convergent cases like the Shapley game.

Contribution

It introduces new convergence conditions for fictitious play in combined game settings and unifies cooperative and competitive games through a linear relationship.

Findings

Derived new convergence conditions for FP in mixed games.

Unified cooperation and competition via a linear relationship.

Analyzed the Shapley game and provided conditions for FP convergence.

Abstract

Fictitious play (FP) is one of the most fundamental game-theoretical learning frameworks for computing Nash equilibrium in $n$-player games, which builds the foundation for modern multi-agent learning algorithms. Although FP has provable convergence guarantees on zero-sum games and potential games, many real-world problems are often a mixture of both and the convergence property of FP has not been fully studied yet. In this paper, we extend the convergence results of FP to the combinations of such games and beyond. Specifically, we derive new conditions for FP to converge by leveraging game decomposition techniques. We further develop a linear relationship unifying cooperation and competition in the sense that these two classes of games are mutually transferable. Finally, we analyze a non-convergent example of FP, the Shapley game, and develop sufficient conditions for FP to converge.