Stability and substitutability in multi-period matching markets

We analyze a multi-period 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 multi-period matching model, a stable outcome may not exist when contracts exhibit complementarities across time periods. Thus, we define a weaker stability concept called temporal stability by taking into account 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 multi-period matching model.