Mix and Match

TL;DR

This paper introduces the Mix-and-Match mechanism for graph matching problems with self-interested agents, ensuring strategyproofness and achieving near-optimal efficiency, with applications like kidney exchange.

Contribution

The paper presents a novel randomized mechanism that is strategyproof and guarantees a 2-approximation for maximum matching in strategic settings.

Findings

The Mix-and-Match mechanism is strategyproof.

It achieves a 2-approximation of maximum matching.

Lower bounds show the mechanism's near optimality.

Abstract

Consider a matching problem on a graph where disjoint sets of vertices are privately owned by self-interested agents. An edge between a pair of vertices indicates compatibility and allows the vertices to match. We seek a mechanism to maximize the number of matches despite self-interest, with agents that each want to maximize the number of their own vertices that match. Each agent can choose to hide some of its vertices, and then privately match the hidden vertices with any of its own vertices that go unmatched by the mechanism. A prominent application of this model is to kidney exchange, where agents correspond to hospitals and vertices to donor-patient pairs. Here hospitals may game an exchange by holding back pairs and harm social welfare. In this paper we seek to design mechanisms that are strategyproof, in the sense that agents cannot benefit from hiding vertices, and approximately maximize efficiency, i.e., produce a matching that is close in cardinality to the maximum cardinality matching. Our main result is the design and analysis of the eponymous Mix-and-Match mechanism; we show that this randomized mechanism is strategyproof and provides a 2-approximation. Lower bounds establish that the mechanism is near optimal.