This paper proposes an agent-based optimistic policy iteration (OPI) scheme for learning stationary optimal stochastic policies in multi-agent Markov Decision Processes (MDPs), in which agents incur a Kullback-Leibler (KL) divergence cost for their control efforts and an additional cost for the joint state. The proposed scheme consists of a greedy policy improvement step followed by an m-step temporal difference (TD) policy evaluation step. We use the separable structure of the instantaneous cost to show that the policy improvement step follows a Boltzmann distribution that depends on the current value function estimate and the uncontrolled transition probabilities. This allows agents to compute the improved joint policy independently. We show that both the synchronous (entire state space evaluation) and asynchronous (a uniformly sampled set of substates) versions of the OPI scheme with finite policy evaluation rollout converge to the optimal value function and an optimal joint policy asymptotically. Simulation results on a multi-agent MDP with KL control cost variant of the Stag-Hare game validates our scheme's performance in terms of minimizing the cost return.

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