Size versus truthfulness in the House Allocation problem

TL;DR

This paper investigates truthful mechanisms for the House Allocation problem, proposing a randomized mechanism with a good approximation ratio for Pareto optimal matchings, and establishing bounds on the performance of such mechanisms.

Contribution

It introduces a natural extension of the Random Serial Dictatorship Mechanism for preferences with ties and provides tight bounds on approximation ratios for truthful mechanisms.

Findings

Proposed a universally truthful randomized mechanism with approximation ratio e/(e-1).

Established a lower bound of 18/13 for universally truthful mechanisms with strict preferences.

Connected the problem to the secretary problem, showing optimality of the proposed strategy.

Abstract

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomised mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomised mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of $\frac{e}{e-1}$. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of $\frac{18}{13}$ on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy with an additional technical assumption, we show by utilising a result of Bade that an improved lower bound of $\frac{e}{e-1}$ holds. This lower bound is tight since RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomised strategy of the administrator who interviews the applicants.