This paper proposes a new notion of Markov $\alpha$-potential games to study Markov games. Two important classes of practically significant Markov games, Markov congestion games and the perturbed Markov team games, are analyzed in this framework of Markov $\alpha$-potential games, with explicit characterization of the upper bound for $\alpha$ and its relation to game parameters. Moreover, any maximizer of the $\alpha$-potential function is shown to be an $\alpha$-stationary Nash equilibrium of the game. Furthermore, two algorithms for the Nash regret analysis, namely the projected gradient-ascent algorithm and the sequential maximum improvement algorithm, are presented and corroborated by numerical experiments.

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