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Probabilistic stable rules and Nash equilibrium in two-sided matching problems

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Abstract

We study many-to-many matching with substitutable and cardinally monotonic preferences. We analyze stochastic dominance (sd) Nash equilibria of the game induced by any probabilistic stable matching rule. We show that a unique match is obtained as the outcome of each sd-Nash equilibrium. Furthermore, individual-rationality with respect to the true preferences is a necessary and sufficient condition for an equilibrium outcome. In the many-to-one framework, the outcome of each equilibrium in which firms behave truthfully is stable for the true preferences. In the many-to-many framework, we identify an equilibrium in which firms behave truthfully and yet the equilibrium outcome is not stable for the true preferences. However, each stable match for the true preferences can be achieved as the outcome of such equilibrium.

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Notes

  1. This is an adaptation of the stability definition in Hatfield and Kominers (2012).

  2. Substitutability is an adaptation of the gross substitutability property (Kelso and Crawford 1982) by Roth (1984a) and Roth and Sotomayor (1990) to matching problems without monetary transfers.

  3. The existence of a pairwise-stable match is shown via an algorithm for strict preferences (Roth 1984a) and via a non-constructive proof for preferences that are not necessarily strict (Sotomayor 1999).

  4. See Echenique and Oviedo (2006) and Sotomayor (1999) for different formulations of stability in many-to-many matching problems.

  5. Cardinal monotonicity is called size monotonicity and law of aggregate demand in Alkan and Gale (2003) and Hatfield and Milgrom (2005), respectively.

  6. In the formal description of US hospital-intern market hospitals have preferences over individual students. This suffices to define stability without reference to preferences over groups of students as long as they are assumed to be responsive to preferences over individual students.

  7. Roth (1991) observes that prior to the adoption of a centralized matching procedure the traditional practice among surgeons in Edinburgh was to employ no more than one female student.

  8. In other words, \(P_{v}\) is transitive, antisymmetric (strict) and total.

  9. With a slight abuse of notation we sometimes write x for a singleton \(\{x\}\).

  10. With a slight abuse of notation we sometimes write xy for a set \(\{x,y\}\).

  11. Cardinal monotonicity was introduced by Alkan (2002).

  12. College admissions was first studied by Gale and Shapley (1962).

  13. Martínez et al. (2004) introduced an algorithm to calculate all stable matches when preferences are substitutable.

  14. Unlike in many-to-one matching with substitutable preferences, pairwise-stability is not equivalent to core-stability in many-to-many matching. Indeed, no logical relation exists between the two concepts (Blair 1988).

  15. The theorem is first proved for the class of responsive preferences and later for the strictly larger class of substitutable and separable preferences in many-to-one matching problems (Gale and Sotomayor 1985; Roth 1984b; Martínez et al. 2000). Separability: for each \(S\subseteq S_v\) with \(|S| < c_v\) and each \(v' \notin S; S\cup \{v'\} \mathrel {P_v} S\) if and only if \(v' \mathrel {P_v} \emptyset \) and for each S with \(|S| > c_v\), \(\emptyset \mathrel {P_v} S\).

  16. This name is taken from Thomson (2011). The concept was introduced by d’Aspremont and Peleg (1988). It is referred as ordinal Nash equilibrium in the literature.

References

  • Alkan A (2002) A class of multipartner matching markets with a strong lattice structure. Econ Theory 19(4):737–746

    Article  Google Scholar 

  • Alkan A, Gale D (2003) Stable schedule matching under revealed preference. J Econ Theory 112(2):289–306

    Article  Google Scholar 

  • Blair C (1988) The lattice structure of the set of stable matchings with multiple partners. Math Oper Res 13(4):619–628

    Article  Google Scholar 

  • d’Aspremont C, Peleg B (1988) Ordinal bayesian incentive compatible representations of committees. Soc Choice Welf 5:261–279

    Article  Google Scholar 

  • Echenique F, Oviedo J (2006) A theory of stability in many-to-many matching markets. Theoret Econ 1(2):233–273

    Google Scholar 

  • Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Month 69:9–15

    Article  Google Scholar 

  • Gale D, Sotomayor MAO (1985) Some remarks on the stable matching problem. Discrete Appl Math 11(3):223–232

    Article  Google Scholar 

  • Hatfield JW, Kominers SD (2012) Contract design and stability in many-to-many matching. Working Paper, Version: June 2015

  • Hatfield JW, Milgrom P (2005) Matching with contracts. Am Econ Rev 95(4):913–935

    Article  Google Scholar 

  • Kelso AS, Crawford VP (1982) Job matching, coalition formation and gross substitutes. Econometrica 50(6):1483–1504

    Article  Google Scholar 

  • Klijn F, Yazıcı A (2014) A many-to-many ‘Rural Hospital Theorem’. J Math Econ 54:63–73

    Article  Google Scholar 

  • Kojima F, Ünver UM (2008) Random paths to pairwise stability in many-to-many matching problems: a study on market equilibriation. Int J Game Theory 36:473–488

    Article  Google Scholar 

  • Ma J (2002) Stable matchings and the small core in Nash equilibrium in the college admissions problem. Rev Econ Des 7:117–134

    Google Scholar 

  • Martínez R, Massó J, Neme A, Oviedo J (2000) Single agents and the set of many-to-one stable matchings. J Econ Theory 91:91–105

    Article  Google Scholar 

  • Martínez R, Massó J, Neme A, Oviedo J (2004) An algorithm to compute the full set of many-to-many stable matchings. Math Soc Sci 47(2):187–210

    Article  Google Scholar 

  • Pais J (2008) Random matching in the college admissions problem. Econ Theory 35:99–116

    Article  Google Scholar 

  • Roth AE (1984a) Stability and polarization of interests in job matching. Econometrica 52:47–58

    Article  Google Scholar 

  • Roth AE (1984b) The evolution of the labor market for medical interns and residents: a case study in game theory. J Polit Econ 92:991–1016

    Article  Google Scholar 

  • Roth AE (1984c) Misrepresentation and stability in the marriage problem. J Econ Theory 34:383–387

    Article  Google Scholar 

  • Roth AE (1985) The college admissions problem is not equivalent to the marriage problem. J Econ Theory 36:277–288

    Article  Google Scholar 

  • Roth AE (1990) New physicians: a natural experiment in market organization. Science 250:1524–1528

    Article  Google Scholar 

  • Roth AE (1991) A natural experiment in the organization of entry level labor markets: regional markets for new physicians and surgeons in the UK. Am Econ Rev 81:415–440

    Google Scholar 

  • Roth AE, Sotomayor MAO (1990) Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Roth AE, Vande Vate JH (1990) Random paths to stability in two-sided matching. Econometrica 58:1475–1480

    Article  Google Scholar 

  • Roth AE, Vande Vate JH (1991) Incentives in two-sided matching with random stable mechanisms. Econ Theory 1:31–44

    Article  Google Scholar 

  • Sotomayor MAO (1999) Three remarks on the many-to-many stable matching problem. Math Soc Sci 38(1):55–70

    Article  Google Scholar 

  • Thomson W (2011) Strategy proof allocation rules. Work in progress, University of Rochester

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  1. Durham University Business School, Mill Hill Lane, DH1 3LB, Durham, UK

    Ayşe Yazıcı

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Correspondence to Ayşe Yazıcı.

Additional information

I am grateful to William Thomson for his very helpful comments and discussions. I would like to thank an associate editor and two referees for their very useful comments, and Paulo Barelli, Onur Kesten, Asen Kochov, Fuhito Kojima, Romans Pancs, Azar Abizada, Battal Doğan, Eun Jeong Heo, İpek Madi, Duygu Nizamoğulları, Ali İhsan Özkes and seminar participants at the University of Rochester, Durham University Business School, Sabancı University, TOBB University of Economics and Technology and the 4th World Congress of Game Theory (GAMES2012) in İstanbul for their helpful comments and discussions. The first draft of this work was written while I was visiting İstanbul Bilgi University. I gratefully acknowledge the hospitality of Department of Economics at İstanbul Bilgi University.

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Yazıcı, A. Probabilistic stable rules and Nash equilibrium in two-sided matching problems. Int J Game Theory 46, 103–124 (2017). https://doi.org/10.1007/s00182-015-0525-3

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