Shapley-Scarf Housing Markets: Respecting Improvement, Integer Programming, and Kidney Exchange

TL;DR

This paper studies the properties of core allocations in Shapley-Scarf housing markets, showing they respect improvements, and introduces integer programming methods for computing these allocations, with applications to kidney exchange markets.

Contribution

It proves that strong core allocations respect improvements and develops new integer programming formulations for computing core and competitive allocations.

Findings

Strong core allocations respect improvements in preferences.

New integer programming models efficiently compute core and competitive allocations.

Simulations compare game-theoretical solutions with kidney exchange optimization.

Abstract

In a housing market of Shapley and Scarf, each agent is endowed with one indivisible object and has preferences over all objects. An allocation of the objects is in the (strong) core if there exists no (weakly) blocking coalition. In this paper we show that in the case of strict preferences the unique strong core allocation (or competitive allocation) respects improvement: if an agent's object becomes more attractive for some other agents, then the agent's allotment in the unique strong core allocation weakly improves. We obtain a general result in case of ties in the preferences and provide new integer programming formulations for computing (strong) core and competitive allocations. Finally, we conduct computer simulations to compare the game-theoretical solutions with maximum size and maximum weight exchanges for markets that resemble the pools of kidney exchange programmes.