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Size versus truthfulness in the House Allocation problem

Size versus truthfulness in the House Allocation problem
Size versus truthfulness in the House Allocation problem
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. In the case that the mechanism must additionally be non-bossy with an additional technical assumption, we show by utilizing a result of Bade that an improved lower bound of 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.
House allocation problem, Assignment problem, Pareto optimal matching, Randomised mechanisms, Truthfulness
0178-4617
3422-3463
Krysta, Piotr
5f86bae7-3079-4f96-afc9-9346e2eba9a1
Manlove, David
3a51f8db-34ee-43c2-ad3e-5340780e402a
Rastegari, Baharak
6ba9e93c-53ba-4090-8f77-c1cb1568d7d1
Zhang, Jinshan
3ccb88b5-ca97-44e4-b8b1-a2c40a997a3d
Krysta, Piotr
5f86bae7-3079-4f96-afc9-9346e2eba9a1
Manlove, David
3a51f8db-34ee-43c2-ad3e-5340780e402a
Rastegari, Baharak
6ba9e93c-53ba-4090-8f77-c1cb1568d7d1
Zhang, Jinshan
3ccb88b5-ca97-44e4-b8b1-a2c40a997a3d

Krysta, Piotr, Manlove, David, Rastegari, Baharak and Zhang, Jinshan (2019) Size versus truthfulness in the House Allocation problem. Algorithmica, 81 (9), 3422-3463. (doi:10.1007/s00453-019-00584-7).

Record type: Article

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. In the case that the mechanism must additionally be non-bossy with an additional technical assumption, we show by utilizing a result of Bade that an improved lower bound of 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.

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More information

Accepted/In Press date: 5 May 2019
e-pub ahead of print date: 18 May 2019
Published date: September 2019
Keywords: House allocation problem, Assignment problem, Pareto optimal matching, Randomised mechanisms, Truthfulness

Identifiers

Local EPrints ID: 430917
URI: http://eprints.soton.ac.uk/id/eprint/430917
ISSN: 0178-4617
PURE UUID: c36b420e-ddd2-405a-8988-7e67ba5f6d33
ORCID for Baharak Rastegari: ORCID iD orcid.org/0000-0002-0985-573X

Catalogue record

Date deposited: 17 May 2019 16:30
Last modified: 16 Nov 2021 05:04

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Contributors

Author: Piotr Krysta
Author: David Manlove
Author: Baharak Rastegari ORCID iD
Author: Jinshan Zhang

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