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Competitive equilibrium and singleton cores in generalized matching problems

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Abstract

We study competitive equilibria in generalized matching problems. We show that, if there is a competitive matching, then it is unique and the core is a singleton consisting of the competitive matching. That is, a singleton core is necessary for the existence of competitive equilibria. We also show that a competitive matching exists if and only if the matching produced by the top trading cycles algorithm is feasible, in which case it is the unique competitive matching. Hence, we can use the top trading cycles algorithm to test whether a competitive equilibrium exists and to construct a competitive equilibrium if one exists. Lastly, in the context of bilateral matching problems, we compare the condition for the existence of competitive matchings with existing sufficient conditions for the existence or uniqueness of stable matchings and show that it is weaker than most existing conditions for uniqueness.

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Notes

  1. Sönmez (1999) allows indifference between two distinct allocations, and in this case “essentially” singleton cores are necessary, where all allocations in the core are Pareto indifferent.

  2. Using data for five years 1991–1994 and 1996 for the thoracic surgery market, Roth and Peranson (1999) report that there are two stable matchings in 1992 and 1993 and there is only one stable matching in 1991, 1994, and 1996. Similarly, using data for two school years 2005–2006 and 2006–2007 for Boston Public School student admissions, Pathak and Sönmez (2008) find that there is only one stable matching in either year at grade K2 and there are two stable matchings in either year at grade 6.

  3. Moulin (1995) shows that, if some agents own more than one indivisible good, the set of competitive allocations may be a proper subset of the core.

  4. Since a matching can be considered as a permutation on a finite set, this result follows from the cycle decomposition theorem for permutations (see, for example, Hungerford 1974, Theorem 6.3, Ch. I).

  5. If consumers’ preferences are continuous and strongly monotone, the strict core is equivalent to the weak core defined by strong domination.

  6. Recall that a matching can be represented as the collection of trading cycles at the matching.

  7. Such a restriction on the number of goods purchased would be natural in many-to-many or many-to-one bilateral matching problems. For example, in college admissions problems, a college offers multiple homogeneous seats to be filled by different students.

  8. See Moulin (1995, Sect. 3.2) and Abdulkadiroğlu and Sönmez (2013, Sect. 3.1) for a description of the top trading cycles algorithm and its properties for housing markets.

  9. There is a variant of the top trading cycles algorithm adapted for school choice problems where students have preferences over schools and schools have capacities and priorities over students. Roughly speaking, the top trading cycles algorithm for school choice problems allows students who have the highest priority at some school to trade their priorities with each other (see Abdulkadiroğlu and Sönmez 2003, for a description of the algorithm and its properties). As in housing markets, there is no restriction on feasible trades of priorities, and thus the algorithm always yields a feasible matching.

  10. We define the length of a cycle as the number of distinct elements in the cycle.

  11. A related, though different, result can be found in Kesten (2006), who shows that, in priority-based allocation problems, the top trading cycles algorithm (adapted to the context as for school choice) and the (agent-proposing) deferred acceptance algorithm yield the same matching if and only if the priority structure is acyclic.

  12. Alternative names for top trading pairs in the literature include “pairs of P-reciprocal agents” in Alcalde (1995), “fixed pairs” in Clark (2006), and “top-top matches” in Niederle and Yariv (2009).

  13. Alcalde ’s (1995) definition of \(\alpha \)-reducibility considers only top trading pairs but not singles because he assumes that there are an even number of agents and that every agent is acceptable to all the others. Niederle and Yariv (2009) use the top-top match property instead of \(\alpha \)-reducibility for the same meaning in the context of marriage problems. \(\alpha \)-reducibility corresponds to the top-coalition property in Banerjee et al. (2001) who consider a more general model than bilateral matching problems. Note that, in bilateral matching problems where agents form coalitions of size one or two, there is no difference between the top-coalition property and the weak top-coalition property of Banerjee et al. (2001).

  14. See Gudmundsson (2014) for an excellent review of various conditions for the existence of stable matchings in roommate problems allowing for weak preferences.

  15. See Tan (1991) for the definition of stable partitions, which generalize stable matchings.

  16. As pointed out by Chung (2000, Definition 3), any marriage problem can be expressed as a roommate problem by putting all the other agents on the same side at the bottom of each agent’s preference list.

  17. The kind of weakening from \(\alpha \)-reducibility to iterative \(\alpha \)-reducibility is also suggested in footnote 11 of Banerjee et al. (2001) and in Gudmundsson (2014). Although the idea that iterative \(\alpha \)-reducibility is sufficient for the uniqueness of stable matchings can be found in previous studies, this paper highlights it as a necessary and sufficient condition for the existence of competitive matchings in roommate and marriage problems.

  18. For example, consider the marriage problem with \(M = \{m_1, m_2, m_3\}\), \(W = \{w_1, w_2\}\), and preferences \(P_{m_1}: m_1, w_1, w_2\), \(P_{m_2}: w_1, w_2, m_2\), \(P_{m_3}: w_2, w_1, m_3\), \(P_{w_1}: m_1, m_3, m_2, w_1\), \(P_{w_2}: m_2, m_3, m_1, w_2\). This marriage problem satisfies the SPC, but there are two stable matchings, \(\{ m_1, (m_2,w_1), (m_3,w_2) \}\) and \(\{ m_1, (m_2,w_2), (m_3,w_1) \}\), and thus there is no competitive matching.

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Acknowledgments

I was introduced to the topic through the research of Joe Ostroy whose continued advice and encouragement are greatly appreciated. I am also grateful to three anonymous referees and participants at Conference on Economic Design 2015 for helpful comments and to the Department of Economics at UCLA for its hospitality during my visit.

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  1. School of Economics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, Korea

    Jaeok Park

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Park, J. Competitive equilibrium and singleton cores in generalized matching problems. Int J Game Theory 46, 487–509 (2017). https://doi.org/10.1007/s00182-016-0543-9

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