Empirical Policy Optimization for <i>n</i>-Player Markov Games
Yuanheng Zhu, Weifan Li, Mengchen Zhao, Jianye Hao, Dongbin Zhao
Abstract
In single-agent Markov decision processes, an agent can optimize its policy based on the interaction with the environment. In multiplayer Markov games (MGs), however, the interaction is nonstationary due to the behaviors of other players, so the agent has no fixed optimization objective. The challenge becomes finding equilibrium policies for all players. In this research, we treat the evolution of player policies as a dynamical process and propose a novel learning scheme for Nash equilibrium. The core is to evolve one's policy according to not just its current in-game performance, but an aggregation of its performance over history. We show that for a variety of MGs, players in our learning scheme will provably converge to a point that is an approximation to Nash equilibrium. Combined with neural networks, we develop an empirical policy optimization algorithm, which is implemented in a reinforcement-learning framework and runs in a distributed way, with each player optimizing its policy based on own observations. We use two numerical examples to validate the convergence property on small-scale MGs, and a pong example to show the potential on large games.