Size Versus Truthfulness in the House Allocation Problem

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 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 18 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. By using a characterisation result of Bade, we show that any randomised mechanism that is a symmetrisation of a truthful, non-bossy and Pareto optimal mechanism has an improved lower bound of e e−1 . Since our new mechanism is a symmetrisation of RSDM for strict preferences, it follows that this lower bound is tight. 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.

Fair and Efficient Allocations under Lexicographic Preferences

“…When preferences can include ties, several intricate axiomatic and technical challenges may arise. An interesting future direction is to start from mechanisms proposed by Krysta et al (2014) and Bogomolnaia et al (2005) for achieving strategyproof and PO allocations, and investigate the existence and compatibility of EFX along with other desirable properties.…”

Assignment Maximization

“…The result in item (ii) in Proposition 1 was obtained by Krysta et al (2014). Both items combined set the stage for the rest of the paper.…”

“…One exception is Afacan and Dur (2018), which follows-up to this paper and shows that no strategy-proof and individually rational mechanism systematically matches more students than either of Boston, Gale-Shapley deferred acceptance, and serial dictatorship mechanisms. Krysta et al (2014) consider the problem of producing maximal matchings in a house allocation problem. They show that there is no mechanism that is deterministic, maximal, and strategy-proof, and provide instead a random mechanism that is strategy-proof and yields approximately-maximal outcomes.…”

On the Complexity of Fair House Allocation

“…While house allocation has typically been considered from the economic efficiency and strategyproofness perspectives [Abraham et al, 2004, Krysta et al, 2014, another important concern is fairness: it is desirable that the agents feel fairly treated. For example, the prominent fairness notion of envy-freeness means that agents do not envy one another with respect to their assigned houses.…”