We formulate a Markov potential game with final-time reach-avoid objectives by integrating potential game theory with stochastic reach-avoid control. Our focus is on multi-player trajectory planning where players maximize the same multi-player reach-avoid objective: the probability of all participants reaching their designated target states by a specified time, while avoiding collisions with one another. Existing approaches require centralized computation of actions via a global policy, which may have prohibitively expensive communication costs. Instead, we focus on approximations of the global policy via local state feedback policies. First, we adapt the recursive single player reach-avoid value iteration to the multi-player framework with local policies, and show that the same recursion holds on the joint state space. To find each player's optimal local policy, the multi-player reach-avoid value function is projected from the joint state to the local state using the other players' occupancy measures. Then, we propose an iterative best response scheme for the multi-player value iteration to converge to a pure Nash equilibrium. We demonstrate the utility of our approach in finding collision-free policies for multi-player motion planning in simulation.

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