Dynamic and many-to-many matching

This paper expands on the notion of the core developed in “A Theory of Stability in Dynamic Matching Markets” when arrivals are stochastic. This is done by pointing out a connection between a dynamic matching market with stochastic arrivals and a many­to-­many matching market. The connection between the two models is useful because it simplifies defining blocking coalitions, and analyzing different stability concepts using well-­known tools in a novel setting. Moreover, it clarifies why properties that hold when arrivals are deterministic (for instance, the lattice property) fail to hold when arrivals are stochastic. The core in the dynamic environment is defined as the Blair core in the corresponding many-­to-­many matching environment. When preferences in the corresponding many-to-many environment satisfy substitutability, pairwise stable allocations are shown to exist by using the algorithm in Echenique and Oviedo. Moreover, these allocations are weakly setwise stable, and hence in the Blair core. When preferences don’t satisfy substitutability, the Blair core may be empty.

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