Approximate State Abstraction for Markov Games

TL;DR

This paper develops a state abstraction method for two-player zero-sum Markov games to simplify computation of equilibria, providing bounds on the approximation error and demonstrating effectiveness in a Markov Soccer example.

Contribution

It introduces a novel state abstraction framework for Markov games, with bounds on the duality gap to evaluate approximation quality, and applies it to a soccer simulation.

Findings

State abstraction reduces computational complexity in Markov games.

Bounds on the duality gap quantify the approximation error.

Applied method successfully to Markov Soccer, computing approximate equilibria.

Abstract

This paper introduces state abstraction for two-player zero-sum Markov games (TZMGs), where the payoffs for the two players are determined by the state representing the environment and their respective actions, with state transitions following Markov decision processes. For example, in games like soccer, the value of actions changes according to the state of play, and thus such games should be described as Markov games. In TZMGs, as the number of states increases, computing equilibria becomes more difficult. Therefore, we consider state abstraction, which reduces the number of states by treating multiple different states as a single state. There is a substantial body of research on finding optimal policies for Markov decision processes using state abstraction. However, in the multi-player setting, the game with state abstraction may yield different equilibrium solutions from those of the ground game. To evaluate the equilibrium solutions of the game with state abstraction, we derived bounds on the duality gap, which represents the distance from the equilibrium solutions of the ground game. Finally, we demonstrate our state abstraction with Markov Soccer, compute equilibrium policies, and examine the results.