  We study a Markov matching market involving a planner and a set of strategic
agents on the two sides of the market. At each step, the agents are presented
with a dynamical context, where the contexts determine the utilities. The
planner controls the transition of the contexts to maximize the cumulative
social welfare, while the agents aim to find a myopic stable matching at each
step. Such a setting captures a range of applications including ridesharing
platforms. We formalize the problem by proposing a reinforcement learning
framework that integrates optimistic value iteration with maximum weight
matching. The proposed algorithm addresses the coupled challenges of sequential
exploration, matching stability, and function approximation. We prove that the
algorithm achieves sublinear regret.
