Stability in Matching Markets with Sizes

Matching markets such as day care, student exchange, refugee resettlement, and couples problems involve agents of different sizes, that is agents who require different amounts of capacity. I study a matching market between agents and objects where the size of an agent is either one or two. Contrary to canonical models, the set of stable matchings may be empty. I identify a trade-off for existence: it is always possible to either bound the instability to a certain number of units per object or to eliminate waste but the existence of a matching that does both is not guaranteed. I develop two fairness criteria that lie on either side of this trade-off: unit-stability bounds the instability and size-stability eliminates waste. I show that size-stability is more desirable than unit-stability from a welfare point of view.

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