Stability and Substitutability in Dynamic Matching Markets

We analyze a dynamic matching market where matching between agents is decided for each time period. To analyze this situation, we embed the situation into the framework of many-to-many matching with contracts where the contract includes the time period at which the matching occurs. While a general stability concept is already defined for the matching with contracts framework, in a dynamic matching model, a stable outcome may not exist when contracts exhibit complementarities across time periods. Thus, we define a stability concept called temporal stability that is more suitable to the dynamic nature of the model. We provide sufficient conditions for the existence of a temporally stable outcome, including a corresponding substitutability condition, ordered substitutability, for the dynamic matching model.