Marginal contribution and singleton cores in one-sided matching and assignment

We examine the attainability of marginal contributions in two models of one-sided matching and assignment. For the one-sided matching problem, where agents in a single group are matched with each other, the core may be empty, and even when nonempty, some agents may fail to attain their marginal contributions in the core. By allowing fractional matchings, however, we guarantee the non-emptiness of the core and show that every agent’s marginal contribution is attainable in the core. This implies that all the agents can receive their marginal contributions at the same time if and only if the core is a singleton. For the one-sided assignment problem, where agents are matched to objects they own, we obtain analogous results even without introducing fractional assignments. Finally, extending to linear production games, which encompass both models, we show that the attainability property may fail but is guaranteed under sufficiently many replications.