Dynamic matching games: Stationary equilibria under varying commitments

This paper examines equilibria in dynamic two-sided matching games, extending Gale and Shapley's foundational model to a non-cooperative, decentralized, and dynamic framework. We focus on markets where agents have utility functions and commitments vary. Specifically, we analyze a dynamic matching game in which firms make offers to workers in each period, considering three types of commitment: (ⅰ) no commitment from either side, (ⅱ) firm commitment, and (ⅲ) worker commitment.Our main contributions are threefold: (ⅰ) we show that stable matchings can be supported as stationary equilibria under different commitment scenarios, depending on the strategies adopted by firms and workers; (ⅱ) we characterize the conditions under which agents are willing to switch partners, highlighting the role of discount factors in shaping equilibrium outcomes; and (ⅲ) we provide a unified framework that connects dynamic incentives with classical stability, bridging the gap between cooperative and non-cooperative approaches to matching.