Social Choice and Welfare

, Volume 42, Issue 4, pp 793–811 | Cite as

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

  • Paula JaramilloEmail author
  • Çaǧatay Kayı
  • Flip Klijn


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).


Stable Mechanism Potential Partner Stable Matchings Preference List Truncation Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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).


  1. Alkan A (1999) On the properties of stable many-to-many matchings under responsive preferences. In: Alkan A, Aliprantis CD, Yannelis NC (eds) Current trends in economics: theory and applications, vol 8. Studies in economic theory. Springer, BerlinGoogle Scholar
  2. Alkan A (2001) On preferences over subsets and the lattice structure of stable matchings. Rev Econ Des 6(1):99–111Google Scholar
  3. Alkan A (2002) A class of multipartner matching markets with a strong lattice structure. Econ Theory 19(4):737–746CrossRefGoogle Scholar
  4. Ashlagi I, Klijn F (2012) Manipulability in matching markets: conflict and coincidence of interests. Soc Choice Welf 39(1):23–33CrossRefGoogle Scholar
  5. Baïou M, Balinski M (2000) Many-to-many matchings: stable polyandrous polygamy (or polygamous polyandry). Discret Appl Math 101(1):1–12CrossRefGoogle Scholar
  6. Blair C (1988) The lattice structure of the set of stable matchings with multiple partners. Math Oper Res 13(4):619–628CrossRefGoogle Scholar
  7. Coles P, Shorrer R (2012) Optimal truncation in matching markets. Mimeo, Harvard Business School, AllstonGoogle Scholar
  8. Dubins LE, Freedman DA (1981) Machiavelli and the gale-shapley algorithm. Am Math Mon 88(7): 485–494Google Scholar
  9. Echenique F, Oviedo J (2006) A theory of stability in many-to-many matching markets. Theor Econ 1(2):233–273Google Scholar
  10. Ehlers L (2008) Truncation strategies in matching markets. Math Oper Res 33(2):327–335CrossRefGoogle Scholar
  11. Fleiner L (2003) A fixed-point approach to stable matchings and some applications. Math Oper Res 28(1):103–126CrossRefGoogle Scholar
  12. Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69(1):9–15CrossRefGoogle Scholar
  13. Kelso AS, Crawford VP (1982) Job matching, coalition formation, and gross substitutes. Econometrica 50(6):1483–1504CrossRefGoogle Scholar
  14. Klaus B, Walzl M (2009) Stable many-to-many matchings with contracts. J Math Econ 45(7):422–434CrossRefGoogle Scholar
  15. Klijn F, Yazıcı A (2012) A many-to-many ‘Rural Hospital Theorem’. Barcelona Graduate School of Economics, Working Paper 567Google Scholar
  16. Kojima F, Pathak PA (2009) Incentives and stability in large two-sided matching markets. Am Econ Rev 99(3):608–627CrossRefGoogle Scholar
  17. Konishi H, Ünver MU (2006) Credible group-stability in many-to-many matching problems. J Econ Theory 129(1):57–80CrossRefGoogle Scholar
  18. Ma J (2010) The singleton core in the college-admissions problem and its application to the national resident matching program (NRMP). Games Econ Behav 69(1):150–164CrossRefGoogle Scholar
  19. Martínez R, Massó J, Neme A, Oviedo J (2004) An algorithm to compute the set of many-to-many stable matchings. Math Soc Sci 47(2):187–210CrossRefGoogle Scholar
  20. Mongell S, Roth AE (1991) Sorority rush as a two-sided matching mechanism. Am Econ Rev 81(3):441–464Google Scholar
  21. Romm A (2011) Mechanism-free implications of entry and capacity reduction in many-to-one stable matching. Mimeo, Harvard Business School, AllstonGoogle Scholar
  22. Roth AE (1982) The economics of matching: stability and incentives. Math Oper Res 7(4):617–628CrossRefGoogle Scholar
  23. Roth AE (1984a) Stability and polarization of interests in job matching. Econometrica 52(1):47–58CrossRefGoogle Scholar
  24. Roth AE (1984b) The evolution of the labor market for medical interns and residents: a case study in game theory. J Political Econ 92(6):991–1016CrossRefGoogle Scholar
  25. Roth AE (1985a) The college admission problem is not equivalent to the marriage problem. J Econ Theory 36(2):277–288CrossRefGoogle Scholar
  26. Roth AE (1985b) Conflict and coincidence of interest in job matching: some new results and open questions. Math Oper Res 10(3):379–389CrossRefGoogle Scholar
  27. Roth AE (1991) A natural experiment in the organization of entry-level labor markets: regional markets for new physicians and surgeons in the United Kingdom. Am Econ Rev 81(3):415–440Google Scholar
  28. Roth AE, Rothblum UG (1999) Truncation strategies in matching markets—in search of advice for participants. Econometrica 67(1):21–43CrossRefGoogle Scholar
  29. Roth AE, Sotomayor MAO (1989) The college admissions problem revisited. Econometrica 57(3):559–570CrossRefGoogle Scholar
  30. Roth AE, Sotomayor MAO (1990) Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, Econometric Society Monograph Series, New YorkGoogle Scholar
  31. Roth AE, Vande Vate JH (1991) Incentives in two-sided matching with random stable mechanisms. Econ Theory 1(1):31–44CrossRefGoogle Scholar
  32. Sönmez T (1997) Manipulation via capacities in two-sided matching markets. J Econ Theory 77(1):197–204CrossRefGoogle Scholar
  33. Sotomayor MAO (1999a) The lattice structure of the set of stable outcomes of the multiple partners assignment game. Int J Game Theory 28(4):567–583CrossRefGoogle Scholar
  34. Sotomayor MAO (1999b) Three remarks on the many-to-many stable matching problem. Math Soc Sci 38(1):55–70CrossRefGoogle Scholar
  35. Sotomayor MAO (2004) Implementation in the many-to-many matching market. Games Econ Behav 46(1):199–212CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Facultad de EconomíaUniversidad de los AndesBloque WColombia
  2. 2.Facultad de EconomíaUniversidad del RosarioBogotáColombia
  3. 3.Institute for Economic Analysis (CSIC) and Barcelona GSEBarcelonaSpain

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