Opting Into Optimal Matchings

TL;DR

This paper investigates the design of optimal matching mechanisms that are individually rational, demonstrating that fixed optimal matchings are likely to be individually rational under certain conditions, with practical mechanisms achieving near-optimality.

Contribution

It introduces a new perspective by analyzing random vertex associations in graphs, showing fixed optimal matchings are likely individually rational, and proposes a simple mechanism that is both practical and nearly optimal.

Findings

Fixed optimal matchings are likely individually rational under certain conditions.

A simple mechanism can be fully individually rational and nearly optimal.

Results have implications for market design and kidney exchange.

Abstract

We revisit the problem of designing optimal, individually rational matching mechanisms (in a general sense, allowing for cycles in directed graphs), where each player --- who is associated with a subset of vertices --- matches as many of his own vertices when he opts into the matching mechanism as when he opts out. We offer a new perspective on this problem by considering an arbitrary graph, but assuming that vertices are associated with players at random. Our main result asserts that, under certain conditions, any fixed optimal matching is likely to be individually rational up to lower-order terms. We also show that a simple and practical mechanism is (fully) individually rational, and likely to be optimal up to lower-order terms. We discuss the implications of our results for market design in general, and kidney exchange in particular.