Fictitious Play in Markov Games with Single Controller

TL;DR

This paper investigates the convergence of fictitious play dynamics to Nash equilibria in Markov games, specifically focusing on a new variant for n-player identical-interest games with a single controller, bridging a gap in game-theoretic analysis.

Contribution

The paper introduces a new fictitious play variant for Markov games and proves its convergence in n-player identical-interest MGs with a single controller, extending existing results.

Findings

Fictitious play converges to NE in certain MGs with a single controller.

Establishes FPP for two-player zero-sum MGs and n-player identical-interest MGs with a single controller.

Bridges the gap between fully competitive and fully cooperative game models.

Abstract

Certain but important classes of strategic-form games, including zero-sum and identical-interest games, have the fictitious-play-property (FPP), i.e., beliefs formed in fictitious play dynamics always converge to a Nash equilibrium (NE) in the repeated play of these games. Such convergence results are seen as a (behavioral) justification for the game-theoretical equilibrium analysis. Markov games (MGs), also known as stochastic games, generalize the repeated play of strategic-form games to dynamic multi-state settings with Markovian state transitions. In particular, MGs are standard models for multi-agent reinforcement learning -- a reviving research area in learning and games, and their game-theoretical equilibrium analyses have also been conducted extensively. However, whether certain classes of MGs have the FPP or not (i.e., whether there is a behavioral justification for equilibrium analysis or not) remains largely elusive. In this paper, we study a new variant of fictitious play dynamics for MGs and show its convergence to an NE in n-player identical-interest MGs in which a single player controls the state transitions. Such games are of interest in communications, control, and economics applications. Our result together with the recent results in [Sayin et al. 2020] establishes the FPP of two-player zero-sum MGs and n-player identical-interest MGs with a single controller (standing at two different ends of the MG spectrum from fully competitive to fully cooperative).