A random graph model of kidney exchanges: efficiency, individual-rationality and incentives

In kidney exchanges, hospitals share patient lists and receive transplantations. A kidney-paired donation (KPD) mechanism needs to promote full sharing of information about donor-patient pairs, and identify a Pareto efficient outcome that also satisfies participation constraints of hospitals. We introduce a random graph model of the KPD exchange and then fully characterize the structure of the efficient outcome and the expected number of transplantations that can be performed. Random graph theory allows early experimental results to be explained analytically, and enables the study of participation incentives in a methodological way. We derive a square-root law between the welfare gains from sharing patient-donor pairs in a central pool and the individual sizes of hospitals, illustrating the urgent need for the nationwide expansion of such programs. Finally, we establish through theoretical and computational analysis that enforcing simple individual rationality constraints on the outcome can mitigate the negative impact of strategic behavior by hospitals.