Abstract
This paper studies the one-to-one two-sided marriage model (Gale and Shapley 1962). If agents’ preferences exhibit mutually best (i.e., each agent is most preferred by her/his most preferred matching partner), there is a unique stable matching without rank gaps (i.e., in each matched pair the agents assign one another the same rank). We study in how far this result is robust for matching markets that are “close” to mutually best. Without a restriction on preference profiles, we find that natural “distances” to mutually best neither bound the size of the core nor the rank gaps at stable matchings. However, for matching markets that satisfy horizontal heterogeneity, “local” distances to mutually best provide bounds for the size of the core and the rank gaps at stable matchings.
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Notes
See, e.g., Chiappori and Salanie (2016) for an overview and Frimmel et al. (2013)[p. 908], for a general discussion. More specifically, Hitsch et al. (2010) use data on user attributes and interactions from an online dating site to estimate individuals’ preferences. Stable matchings computed for estimated preferences are similar to the actual matches achieved by the dating site. Moreover, out-of-sample predictions exhibit assortative mating patterns similar to those observed in actual marriages. Interestingly, Hitsch et al. (2010)[p. 161] demonstrate that assortative mating is largely driven by preferences that favor a partner who is similar rather than a vertical differentiation of potential matching partners.
In Eeckhout (2000) [Corollary 4] “horizontal heterogeneity” refers to our condition of mutually best. Our domain of horizontal heterogeneity contains the domain with the same name in Eeckhout (2000) but also contains profiles where all men prefer different women and all women prefer different men but where not all most preferred mates form mutually best pairs.
The condition requires that agents can be relabeled \(m_1,\ldots ,m_n\) and \(w_1,\ldots ,w_n\) such that for each k, man \(m_k\) and woman \(w_k\) are mutually most preferred among agents \(m_k,\ldots ,m_{n}\) and \(w_k,\ldots ,w_{n}\).
Unlike a set, a multiset allows for multiple instances for each of its elements.
Jaramillo et al. (2019) call a matching \(\mu \) rank-fair if \(\Gamma _1(\mu )=0\) or, equivalently, for all \(p\in (0,\infty ]\), \(\Gamma _{p}(\mu )=0\).
In fact, we prove a stronger result: (i) there are problems r and \(r'\) such that for each \(p\in (0,\infty )\), \(\Delta ^{l}_p(r)<\Delta ^{l}_p(r')\) and \(\Delta ^{g}_p(r)>\Delta ^{g}_p(r')\) and (ii) there are problems r and \(r'\) such that \(\Delta ^{l}_\infty (r)<\Delta ^{l}_\infty (r')\) and \(\Delta ^{g}_\infty (r)>\Delta ^{g}_\infty (r')\).
We thank a reviewer for suggesting this alternative approach.
See Manlove (2013) [p.23] for definitions and a more detailed discussion.
At the unique stable matching \(\{(m_1,w_1),(m_2,w_2),(m_3,w_3),(m_4,w_4)\}\) agents \(m_4\) and \(w_4\) are matched to their \(4^{\text {th}}\)–ranked mates.
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The first draft of this paper was written while F. Klijn was LFUI guest professor at Innsbruck University. He gratefully acknowledges the hospitality of Innsbruck University and support from AGAUR-Generalitat de Catalunya (2017-SGR-1359), Ministerio de Ciencia, Innovación y Universidades (ECO2017-88130-P), and the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563 and CEX2019-000915-S). C. Kah and M. Walzl thank the Austrian Science Foundation (FWF) for support through project P-28632-G27 and SFB63. We thank two reviewers, Péter Biró, Lars Ehlers, Pau Milán, Anne van den Nouweland, and seminar and workshop participants at Université de Lausanne, the Dutch Social Choice Colloquium (Maastricht, 2019), 24th Coalition Theory Network Workshop (Aix-en-Provence), and Conference on Economic Design 2019 (Budapest) for useful comments and suggestions. We are particularly grateful to the reviewer who suggested an alternative global distance, as discussed in Sect. 5
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Klijn, F., Walzl, M. & Kah, C. Almost mutually best in matching markets: rank gaps and size of the core. Soc Choice Welf 57, 797–816 (2021). https://doi.org/10.1007/s00355-021-01312-3
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DOI: https://doi.org/10.1007/s00355-021-01312-3