Design and analysis of multi-hospital kidney exchange mechanisms using random graphs

Abstract

Kidney exchanges enable transplants when a pair of a patient and an incompatible donor is matched with other similar pairs. In multi-hospital kidney exchanges pairs are pooled from multiple hospitals, and each hospital is able to decide which pairs to report and which to hide and match locally. Modeling the problem as a maximum matching on a random graph, we first establish that the expected benefit from pooling scales as the square-root of the number of pairs in each hospital. We design the xCM mechanism, which achieves efficiency and incentivizes hospitals of moderate-to-large size to fully report their pairs. Reciprocal pairs are crucial in the design, with the probabilistic uniform rule used to ensure incentive alignment. By grouping certain pair types into so-called virtual-reciprocal pairs, xCM extends to handle 3-cycles. We validate the performance of xCM in simulation, demonstrating its efficiency and incentive advantages over the Bonus mechanism [3].Keywords: kidney exchange, random graphs, mechanism design, maximum $ This is a significantly revised version of an extended abstract that appeared in ACM EC'11. In addition to a modified mechanism and an extension to 3-cycles, we also provide complete proofs and extensive simulation results. We wish to thank Itai Ashlagi, Ian Kash, Felix Fischer, Les Valiant and the reviewers from EC'11 for comments on an earlier version, as well as the reviewers and the Associate Editor at GEB.

Opting Into Optimal Matchings

“…The two papers that are most closely related to ours are the ones by Ashlagi and Roth [7] and Toulis and Parkes [29]. Ashlagi and Roth show that under some technical assumptions, and under a random graph model of kidney exchange, large random graphs admit an individually rational matching that is optimal up to a certain constant fraction of the number of vertices, with high probability.…”

“…Ashlagi and Roth show that under some technical assumptions, and under a random graph model of kidney exchange, large random graphs admit an individually rational matching that is optimal up to a certain constant fraction of the number of vertices, with high probability. Toulis and Parkes [29] independently study a very similar random graph model (it does make different assumptions about the size of hospitals), and obtain a similar result regarding individual rationality.…”

“…These important results have inspired our own work, but -in addition to a number of significant technical advantages 2 -we believe our high-level approach is significantly more compelling. In a nutshell, the random graph model studied by Ashlagi and Roth [7] and Toulis and Parkes [29] draws blood types for each donor and patient from a distribution that gives each of the four blood types (O, A, B, and AB) constant probability. For each pair of blood type compatible vertices (e.g., an O donor is blood type compatible with an A patient, but a B donor is not), a directed edge exists with constant probability.…”

“…More generally, players can choose to reveal a subset of their vertices, and internally match the rest. Several papers seek to design mechanisms that incentivize players to reveal all their vertices, either as a dominant strategy [5,23] or in equilibrium [7,29]. These known results are quite limited; obtaining stronger results is a central open problem.…”

Can Everyone Benefit from Social Integration?

“…Matching mechanisms that satisfy pro-integration criteria would be particularly useful in practice because they would promote the operation of a large centralized clearinghouse instead of several small ones. For example, they would promote that hospitals voluntarily enroll all patient-donor pairs in multi-hospital kidney exchange programs, instead of conducting kidney exchanges locally (Ashlagi and Roth, 2014;Toulis and Parkes, 2015). They would also encourage charter and district-run schools to collaborate in the assignment of students to educational institutions (Manjunath and Turhan, 2016;Dogan and Yenmez, 2017;Ekmekci and Yenmez, 2018).…”