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On the exhaustiveness of truncation and dropping strategies in many-to-many matching markets

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Abstract

We consider two-sided many-to-many matching markets in which each worker may work for multiple firms and each firm may hire multiple workers. We study individual and group manipulations in centralized markets that employ (pairwise) stable mechanisms and that require participants to submit rank order lists of agents on the other side of the market. We are interested in simple preference manipulations that have been reported and studied in empirical and theoretical work: truncation strategies, which are the lists obtained by removing a tail of least preferred partners from a preference list, and the more general dropping strategies, which are the lists obtained by only removing partners from a preference list (i.e., no reshuffling). We study when truncation/dropping strategies are exhaustive for a group of agents on the same side of the market, i.e., when each match resulting from preference manipulations can be replicated or improved upon by some truncation/dropping strategies. We prove that for each stable mechanism, dropping strategies are exhaustive for each group of agents on the same side of the market (Theorem 1), i.e., independently of the quotas. Then, we show that for each stable mechanism, truncation strategies are exhaustive for each agent with quota 1 (Theorem 2). Finally, we show that this result cannot be extended neither to individual manipulations when the agent’s quota is larger than 1 (even when all other agents’ quotas equal 1—Example 1), nor to group manipulations (even when all quotas equal 1—Example 2).

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Notes

  1. Note that agents only have preferences over potential partners on the other side of the market and not over their colleagues.

  2. An agent has substitutable preferences if the agent continues to want to be partners with an agent even if other agents become unavailable. Note that substitutability excludes complementarities. Substitutability was introduced by Kelso and Crawford (1982) to show the existence of stable matchings in a many-to-one model with money.

  3. In particular, quotas cannot be manipulated (cf. Sönmez 1997).

  4. Responsiveness implies substitutability, and hence the existence of a stable matching.

  5. See, for instance, Roth (1991).

  6. Roth and Vande Vate (1991) studied random stable mechanisms. We rephrase their Theorem 2 to fit it for our framework.

  7. In fact, Kojima and Pathak (2009) also considered strategic manipulation by underreporting quotas. We focus on manipulation via preference lists and aim to establish “exhaustiveness results” (of truncation and dropping strategies) for different classes of quota vectors.

  8. Note that we only focus on (pairwise) stability and do not consider larger blocking coalitions than worker-firm pairs. This is not a conceptual contradiction to our study of joint manipulations, since larger blocking coalitions would involve agents from both sides of the market, while the joint manipulations we study only deal with groups of agents on the same side of the market. It seems more likely that a group of agents on the same side of the market can carry out a group manipulation that is actually binding on its members.

  9. For each many-to-one market, there is a one-to-one correspondence between its stable matchings and those of a related one-to-one market. Hence, many properties of the set of stable matchings in the one-to-one model carry over to the many-to-one model. Yet, with respect to strategic issues, Roth (1985a) showed that the two models are not equivalent.

  10. With some abuse of notation we often write \(x\) for a singleton \(\{x\}\).

  11. See Roth (1985a) and Roth and Sotomayor (1989) for a discussion of this assumption.

  12. A many-to-one matching market is a market where each agent on one given side of the market has quota 1. A one-to-one or marriage market is a market where each agent has quota 1.

  13. Alternatively, by responsiveness condition (r1), a matching \(\mu \) is individually rational if no agent would be better off by breaking a match, i.e., for each \(i\in I\) and each \(j\in \mu (i), \mu (i) \succ _i \mu (i)\backslash j\).

  14. By responsiveness conditions (r1) and (r2), (b2) is equivalent to [  \([|\mu (w)|< q_w\) and \(\mu (w)\cup f \succ _w \mu (w)\,]\) or [there is \(f^{\prime } \in \mu (w)\) such that \(\, (\mu (w)\setminus f^{\prime })\cup f \succ _w \mu (w)\,]\, \)  ]. A similar equivalent statement holds for (b3).

  15. In fact, the set of stable matchings does not depend on the agents’ orderings of the (individual) unacceptable partners either.

  16. In particular, quotas cannot be manipulated (cf. Sönmez 1997).

  17. We do not suppress the notation \(q\) since the quotas play a role in the definition of stability. Moreover, our results are also conditional on the values of the quotas.

  18. The listed potential partners are interpreted as the acceptable potential partners. The other potential partners are unacceptable and, since we focus on stable mechanisms, their relative ordering is irrelevant.

  19. However, some stable mechanisms are strategy-proof for one side of the market if each agent on that side of the market has quota 1 (Roth 1982, Theorem 5).

  20. The proof of Theorem 1 shows that any group of agents on the same side of the market only needs to consider dropping strategies in which the number of acceptable firms they report is at most their quota. However, they cannot only focus on dropping strategies in which the number of acceptable firms is equal to their quota (Example 3). We would like to thank the associate editor for pointing out this fact. We refer to Sect. 4 for further details.

  21. Proposition 6 in Alkan (2002) is an extension of part of a result that is known as the Rural Hospital Theorem (Roth 1984b) which states that each agent is matched to the same number of partners in every stable matching.

  22. Truncation strategies have been extensively studied in the matching literature (Roth and Vande Vate 1991; Ehlers 2008; Romm 2011; Ashlagi and Klijn 2012; Coles and Shorrer 2012, among others). Moreover, they have been used in practice, for instance, in the sorority rush (Mongell and Roth 1991).

  23. The preferences \(P\) are adapted from Roth (1985a, p. 283, Table I) and Roth and Sotomayor (1990, p. 146).

  24. For each pair of truncation strategies of \(w_1\) and \(w_2\), we can construct a pair of dropping strategies that consist of the acceptable firms that they are matched to at \(\varphi (Q_{w_1},Q_{w_2}, P_{-\{w_1,w_2\}})\) in the true relative order (as described in the proof of Theorem 1). Then, these dropping strategies yield the same matches for \(w_1\) and \(w_2\).

  25. A setwise stable matching is an individually rational matching that cannot be blocked by a coalition that forms new matches only among its members, but may preserve some of its matches outside of the coalition. See Roth (1984a), Sotomayor (1999b), and Echenique and Oviedo (2006) for a discussion on setwise stability.

  26. Note that by introducing additional workers and firms, the result here can be extended in a straightforward way to many-to-one matching markets.

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Acknowledgments

We would like to thank Lars Ehlers, Jordi Massó, William Thomson, Hanzhe Zhang, an associate editor, and two anonymous referees for helpful comments on an earlier draft of the paper. We thank the seminar participants at Universidad de los Andes, Universidad del Rosario, GAMES 2012, and First Caribbean Game Theory Conference for valuable discussions. Ç. Kayı gratefully acknowledges the hospitality of Institute for Economic Analysis (CSIC) and financial support from Colciencias/CSIC (Convocatoria No: 506/2010), El Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación, Francisco José de Caldas. The first draft of this paper was written while F. Klijn was visiting Universidad del Rosario. He gratefully acknowledges the hospitality of Universidad del Rosario and financial support from CSIC/Colciencias through grant 2010C00013 and the Spanish Ministry of Economy and competitiveness through Plan Nacional I+D+i (ECO2011–29847) and the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2011-0075).

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Authors and Affiliations

  1. Facultad de Economía, Universidad de los Andes, Calle 19A # 11 – 37, Bloque W, Bogotá, Colombia

    Paula Jaramillo

  2. Facultad de Economía, Universidad del Rosario, Bogotá, Colombia

    Çaǧatay Kayı

  3. Institute for Economic Analysis (CSIC) and Barcelona GSE, Barcelona, Spain

    Flip Klijn

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  1. Paula Jaramillo
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  2. Çaǧatay Kayı
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Correspondence to Paula Jaramillo.

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Jaramillo, P., Kayı, Ç. & Klijn, F. On the exhaustiveness of truncation and dropping strategies in many-to-many matching markets. Soc Choice Welf 42, 793–811 (2014). https://doi.org/10.1007/s00355-013-0746-y

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