Markov $\alpha$-Potential Games

We propose a new framework of Markov $\alpha$-potential games to study Markov games. We show that any Markov game with finite-state and finite-action is a Markov $\alpha$-potential game, and establish the existence of an associated $\alpha$-potential function. Any optimizer of an $\alpha$-potential function is shown to be an $\alpha$-stationary Nash equilibrium. We study two important classes of practically significant Markov games, Markov congestion games and the perturbed Markov team games, via the framework of Markov $\alpha$-potential games, with explicit characterization of an upper bound for $\alpha$ and its relation to game parameters. Additionally, we provide a semi-infinite linear programming based formulation to obtain an upper bound for $\alpha$ for any Markov game. Furthermore, we study two equilibrium approximation algorithms, namely the projected gradient-ascent algorithm and the sequential maximum improvement algorithm, along with their Nash regret analysis, and corroborate the results with numerical experiments.