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Social Choice and Welfare

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01-02-2014 | Original Paper

Pareto-optimal assignments by hierarchical exchange

Author: Sophie Bade

Published in: Social Choice and Welfare | Issue 2/2014

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Abstract

Papai’s 2000 hierarchical exchange mechanisms for house allocation problems determine matchings as the outcome of multiple rounds of trading cycles. Any hierarchical exchange mechanism can be defined through a structure of ownership, which determines the ownership of houses after any round of trading cycles. Given a permutation of agents, a “permuted” hierarchical exchange mechanism can be constructed by consistently permuting agents over the entire structure of ownership. The paper shows that for any Pareto-efficient matching and any hierarchical exchange mechanism, there is a permutation of agents in the ownership structure such that the induced permuted hierarchical exchange mechanism leads to this matching.

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The expression \(hR_i h^{\prime }\) means that agent i weakly prefers house h to house h’. Since preferences are assumed to be strict \(hR_i h^{\prime }\) and \(h^{\prime }R_i h\) both obtain only if \(h=h^{\prime }\).

Note that the assumption that preferences are strict implies that at least some agent must strictly prefer \(\sigma ^{\prime }\) to \(\sigma \), since these two matchings differ and since all agents weakly prefer \(\sigma ^{\prime }\) to \(\sigma \).

In Pycia and Unver (2011) this requirement is called “consistency” in their discussion of Papai’s 2000 hierarchical exchange mechanisms on page 12. The same requirement appears as (R4) “persistence of ownership” on page 18 in the general definition of trading cycles mechanisms.

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Title: Pareto-optimal assignments by hierarchical exchange
Author: Sophie Bade
Publication date: 01-02-2014
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2014
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-013-0730-6

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