Multi-agent learning is a challenging problem in machine learning that has applications in different domains such as distributed control, robotics, and economics. We develop a prescriptive model of multi-agent behavior using Markov games. Since in many multi-agent systems, agents do not necessary select their optimum strategies against other agents (e.g., multi-pedestrian interaction), we focus on models in which the agents play "exploration but near optimum strategies". We model such policies using the Boltzmann-Gibbs distribution. This leads to a set of coupled Bellman equations that describes the behavior of the agents. We introduce a set of conditions under which the set of equations admit a unique solution and propose two algorithms that provably provide the solution in finite and infinite time horizon scenarios. We also study a practical setting in which the interactions can be described using the occupancy measures and propose a simplified Markov game with less complexity. Furthermore, we establish the connection between the Markov games with exploration strategies and the principle of maximum causal entropy for multi-agent systems. Finally, we evaluate the performance of our algorithms via several well-known games from the literature and some games that are designed based on real world applications.
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