A pessimist’s approach to one-sided matching

Inspired by real-world applications such as the assignment of pupils to schools or the allocation of social housing, the one-sided matching problem studies how a set of agents can be assigned to a set of objects when the agents have preferences over the objects, but not vice versa. For fairness reasons, most mechanisms use randomness, and therefore result in a probabilistic assignment. We study the problem of decomposing these probabilistic assignments into a weighted sum of ex-post (Pareto-)efficient matchings, while maximizing the worst-case number of assigned agents. This decomposition preserves all the assignments’ desirable properties, most notably strategy-proofness. Next to discussing the complexity of the problem, we obtain tight lower and upper bounds on the optimal worst-case number of assigned agents. Moreover, we propose two alternative column generation frameworks for the introduced problem, which prove to be capable of finding decompositions with the theoretically best possible worst-case number of assigned agents, both for randomly generated data, and for real-world school choice data from the Belgian cities Antwerp and Ghent. Lastly, the proposed column generation frameworks are inherently flexible, and can therefore also be applied to settings where other ex-post criteria are desirable, or to find decompositions that satisfy other worst-case measures.