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March & April  2015, 2(3&4): 321-330. doi: 10.3934/jdg.2015008

Finding all stable matchings with couples

1. 

Department of Economics, Stanford University, 579 Serra Mall, Stanford, CA 94305, United States

Received  April 2015 Revised  May 2015 Published  November 2015

In two-sided matching markets in which some doctors form couples, a stable matching does not necessarily exist.Wecharacterize the set of stable matchings as the fixed points of a function that is reminiscent of a tâtonnement process. Then we show that this function ismonotone decreasing with respect to a certain partialorder. Based on these results,wepresent an algorithm that finds all the stable matchings wheneverone exists, and otherwise demonstrates that there is no stable matching.
Citation: Fuhito Kojima. Finding all stable matchings with couples. Journal of Dynamics & Games, 2015, 2 (3&4) : 321-330. doi: 10.3934/jdg.2015008
References:
[1]

A. Abdulkadiroglu, Y.-K. Che and Y. Yasuda, Resolving conflicting preferences in school choice: The 'boston' mechanism reconsidered,, American Economic Review, (2009), 399.  doi: 10.2139/ssrn.1465293.  Google Scholar

[2]

A. Abdulkadiroglu, P. A. Pathak and A. E. Roth, Strategy-proofness versus efficiency in matching with indifferences: Redesigning the new york city high school match,, American Economic Review, 99 (2009), 1954.   Google Scholar

[3]

A. Abdulkadiroǧlu and T. Sönmez, School choice: A mechanism design approach,, American Economic Review, 93 (2003), 729.   Google Scholar

[4]

H. Adachi, On a characterization of stable matchings,, Economics Letters, 68 (2000), 43.  doi: 10.1016/S0165-1765(99)00241-4.  Google Scholar

[5]

I. Ashlagi, M. Braverman and A. Hassidim, Stability in large matching markets with complementarities,, Operations Research, 62 (2014), 713.  doi: 10.1287/opre.2014.1276.  Google Scholar

[6]

E. M. Azevedo and J. W. Hatfield, Complementarity and multidimensional heterogeneity in matching markets, 2012,, Mimeo., ().   Google Scholar

[7]

M. Balinski and T. Sönmez, A tale of two mechanisms: student placement,, Journal of Economic Theory, 84 (1999), 73.  doi: 10.1006/jeth.1998.2469.  Google Scholar

[8]

P. Biró, T. Fleiner, R. W. Irving and D. F. Manlove, The college admissions problem with lower and common quotas,, Theoretical Computer Science, 411 (2010), 3136.  doi: 10.1016/j.tcs.2010.05.005.  Google Scholar

[9]

P. Biró, T. Fleiner and R. Irving, Matching couples with scarf's algorithm,, Institute of Economics, ().   Google Scholar

[10]

P. Biró, R. W. Irving and I. Schlotter, Stable matching with couples: an empirical study,, Journal of Experimental Algorithmics (JEA), 16 (2011).  doi: 10.1145/1963190.1963191.  Google Scholar

[11]

P. Biró and F. Klijn, Matching with couples: A multidisciplinary survey,, International Game Theory Review, 15 (2013).  doi: 10.1142/S0219198913400082.  Google Scholar

[12]

P. Biró, D. F. Manlove and I. McBride, The hospitals/residents problem with couples: Complexity and integer programming models,, in Experimental Algorithms, (2014), 10.   Google Scholar

[13]

Y.-K. Che, J. Kim and F. Kojima, Stable Matching in Large Economies,, Technical report, (2013).   Google Scholar

[14]

Y.-K. Che and F. Kojima, Asymptotic equivalence of probabilistic serial and random priority mechanisms,, Econometrica, 78 (2010), 1625.  doi: 10.3982/ECTA8354.  Google Scholar

[15]

B. Dutta and J. Masso, Stability of matchings when individuals have preferences over colleagues,, Journal of Economic Theory, 75 (1997), 464.  doi: 10.1006/jeth.1997.2291.  Google Scholar

[16]

F. Echenique, Finding all equilibria in games with strategic complements,, Journal of Economic Theory, 135 (2007), 514.  doi: 10.1016/j.jet.2006.06.001.  Google Scholar

[17]

F. Echenique and J. Oviedo, Core many-to-one matchings by fixed point methods,, Journal of Economic Theory, 115 (2004), 358.  doi: 10.1016/S0022-0531(04)00042-1.  Google Scholar

[18]

F. Echenique and J. Oviedo, A theory of stability in many-to-many matching,, Theoretical Economics, 1 (2006), 233.  doi: 10.2139/ssrn.691443.  Google Scholar

[19]

F. Echenique and M. B. Yenmez, A solution to matching with preferences over colleagues,, Games and Economic Behavior, 59 (2007), 46.  doi: 10.1016/j.geb.2006.07.003.  Google Scholar

[20]

A. Erdil and H. Ergin, What's the matter with tie-breaking? improving efficiency in school choice,, American Economic Review, 98 (2008), 669.  doi: 10.1257/aer.98.3.669.  Google Scholar

[21]

T. Fleiner, A fixed-point approach to stable matchings and some applications,, Mathematics of Operations Research, 28 (2003), 103.  doi: 10.1287/moor.28.1.103.14256.  Google Scholar

[22]

D. Fragiadakis and P. Troyan, Market design under distributional constraints: Diversity in school choice and other applications, 2014,, Mimeo., ().   Google Scholar

[23]

D. Fragiadakis, A. Iwasaki, P. Troyan, S. Ueda and M. Yokoo, Strategyproof matching with minimum quotas,, mimeo., ().   Google Scholar

[24]

D. Gale and L. S. Shapley, College admissions and the stability of marriage,, American Mathematical Monthly, 69 (1962), 9.  doi: 10.2307/2312726.  Google Scholar

[25]

D. Gale and M. A. O. Sotomayor, Ms. machiavelli and the stable matching problem,, American Mathematical Monthly, 92 (1985), 261.  doi: 10.2307/2323645.  Google Scholar

[26]

D. Gale and M. A. O. Sotomayor, Some remarks on the stable matching problem,, Discrete Applied Mathematics, 11 (1985), 223.  doi: 10.1016/0166-218X(85)90074-5.  Google Scholar

[27]

M. Goto, N. Hashimoto, A. Iwasaki, Y. Kawasaki, S. Ueda, Y. Yasuda and M. Yokoo, Strategy-proof matching with regional minimum quotas,, in AAMAS2014, (2014).   Google Scholar

[28]

M. Goto, A. Iwasaki, Y. Kawasaki, Y. Yasuda and M. Yokoo, Improving fairness and efficiency in matching markets with regional caps: Priority-list based deferred acceptance mechanism,, Mimeo (the latest version is available at , ().   Google Scholar

[29]

J. Hatfield and P. Milgrom, Matching with contracts,, American Economic Review, 95 (2005), 913.  doi: 10.1257/0002828054825466.  Google Scholar

[30]

J. W. Hatfield and F. Kojima, Matching with contracts: Comment,, American Economic Review, 98 (2008), 1189.  doi: 10.1257/aer.98.3.1189.  Google Scholar

[31]

J. W. Hatfield and S. D. Kominers, Contract design and stability in matching markets,, Harvard University and Stanford University working paper., ().   Google Scholar

[32]

N. Immorlica and M. Mahdian, Marriage, honesty, and stability,, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, (2005), 53.   Google Scholar

[33]

Y. Kamada and F. Kojima, Stability and strategy-proofness for matching with constraints: A problem in the japanese medical match and its solution,, American Economic Review P&P, 102 (2012), 366.  doi: 10.1257/aer.102.3.366.  Google Scholar

[34]

Y. Kamada and F. Kojima, General theory of matching under distributional constraints, 2014,, Mimeo., ().   Google Scholar

[35]

Y. Kamada and F. Kojima, Stability concepts in matching with distributional constraints, 2014,, Mimeo., ().   Google Scholar

[36]

Y. Kamada and F. Kojima, Efficient matching under distributional constraints: Theory and applications,, American Economic Review, 105 (2015), 67.  doi: 10.1257/aer.20101552.  Google Scholar

[37]

O. Kesten, School choice with consent,, The Quarterly Journal of Economics, 125 (2010), 1297.  doi: 10.1162/qjec.2010.125.3.1297.  Google Scholar

[38]

B. Klaus and F. Klijn, Stable matchings and preferences of couples,, Journal of Economic Theory, 121 (2005), 75.  doi: 10.1016/j.jet.2004.04.006.  Google Scholar

[39]

B. Klaus, F. Klijn and J. Masso, Some things couples always wanted to know about stable matchings (but were afraid to ask),, Review of Economic Design, 11 (2007), 175.  doi: 10.1007/s10058-006-0017-9.  Google Scholar

[40]

F. Kojima and P. A. Pathak, Incentives and stability in large two-sided matching markets,, American Economic Review, 99 (2009), 608.  doi: 10.1257/aer.99.3.608.  Google Scholar

[41]

F. Kojima, P. A. Pathak and A. E. Roth, Matching with couples: Stability and incentives in large markets,, Quarterly Journal of Economics, 128 (2013), 1585.  doi: 10.1093/qje/qjt019.  Google Scholar

[42]

F. Kojima, A. Tamura and M. Yokoo, Designing matching mechanisms under constraints: An approach from discrete convex analysis, 2015,, Mimeo., ().   Google Scholar

[43]

H. Konishi and U. Unver, Credible group stability in multi-partner matching problems,, Journal of Economic Theory, 129 (2006), 57.  doi: 10.1016/j.jet.2005.02.001.  Google Scholar

[44]

E. J. McDermid and D. F. Manlove, Keeping partners together: algorithmic results for the hospitals/residents problem with couples,, Journal of Combinatorial Optimization, 19 (2010), 279.  doi: 10.1007/s10878-009-9257-2.  Google Scholar

[45]

D. G. McVitie and L. Wilson, Stable marriage assignments for unequal sets,, BIT, 10 (1970), 295.  doi: 10.1007/BF01934199.  Google Scholar

[46]

T. Nguyen and R. Vohra, Near feasible stable matchings with complementarities,, PIER Working Paper, (2014).  doi: 10.2139/ssrn.2500824.  Google Scholar

[47]

M. Ostrovsky, Stability in supply chain networks,, American Economic Review, (): 897.   Google Scholar

[48]

P. A. Pathak and T. Sönmez, Leveling the playing field: Sincere and sophisticated players in the boston mechanism,, The American Economic Review, 98 (2008), 1636.  doi: 10.1257/aer.98.4.1636.  Google Scholar

[49]

P. A. Pathak and T. Sönmez, School admissions reform in chicago and england: Comparing mechanisms by their vulnerability to manipulation,, American Economic Review, 103 (2013), 80.  doi: 10.1257/aer.103.1.80.  Google Scholar

[50]

M. Pycia, Stability and preference alignment in matching and coalition formation,, Econometrica, 80 (2012), 323.  doi: 10.3982/ECTA7143.  Google Scholar

[51]

E. Ronn, Np-complete stable matching problems,, Journal of Algorithms, 11 (1990), 285.  doi: 10.1016/0196-6774(90)90007-2.  Google Scholar

[52]

A. E. Roth, The evolution of the labor market for medical interns and residents: A case study in game theory,, Journal of Political Economy, 92 (1984), 991.  doi: 10.1086/261272.  Google Scholar

[53]

A. E. Roth, On the allocation of residents to rural hospitals: A general property of two-sided matching markets,, Econometrica, 54 (1986), 425.  doi: 10.2307/1913160.  Google Scholar

[54]

A. E. Roth, A natural experiment in the organization of entry-level labor markets: regional markets for new physicians and surgeons in the united kingdom,, The American economic review, (): 415.   Google Scholar

[55]

A. E. Roth and E. Peranson, The redesign of the matching market for american physicians: Some engineering aspects of economic design,, American Economic Review, 89 (1999), 748.  doi: 10.1257/aer.89.4.748.  Google Scholar

[56]

A. E. Roth and M. A. Sotomayor, Two-sided Matching: A Study in Game-Theoretic Modeling and Analysis,, Econometric Society monographs, (1990).  doi: 10.1017/CCOL052139015X.  Google Scholar

[57]

T. Sönmez and M. U. Ünver, Course bidding at business schools,, International Economic Review, 51 (2010), 99.  doi: 10.1111/j.1468-2354.2009.00572.x.  Google Scholar

[58]

M. A. O. Sotomayor, A non-constructive elementary proof of the existence of stable marriages,, Games and Economic Behavior, 13 (1996), 135.  doi: 10.1006/game.1996.0029.  Google Scholar

[59]

M. A. O. Sotomayor, Three remarks on the many-to-many stable matching problem,, Mathematical social sciences, 38 (1999), 55.  doi: 10.1016/S0165-4896(98)00048-1.  Google Scholar

[60]

M. A. O. Sotomayor, Implementation in the many-to-many matching market,, Games and Economic Behavior, 46 (2004), 199.  doi: 10.1016/S0899-8256(03)00047-2.  Google Scholar

show all references

References:
[1]

A. Abdulkadiroglu, Y.-K. Che and Y. Yasuda, Resolving conflicting preferences in school choice: The 'boston' mechanism reconsidered,, American Economic Review, (2009), 399.  doi: 10.2139/ssrn.1465293.  Google Scholar

[2]

A. Abdulkadiroglu, P. A. Pathak and A. E. Roth, Strategy-proofness versus efficiency in matching with indifferences: Redesigning the new york city high school match,, American Economic Review, 99 (2009), 1954.   Google Scholar

[3]

A. Abdulkadiroǧlu and T. Sönmez, School choice: A mechanism design approach,, American Economic Review, 93 (2003), 729.   Google Scholar

[4]

H. Adachi, On a characterization of stable matchings,, Economics Letters, 68 (2000), 43.  doi: 10.1016/S0165-1765(99)00241-4.  Google Scholar

[5]

I. Ashlagi, M. Braverman and A. Hassidim, Stability in large matching markets with complementarities,, Operations Research, 62 (2014), 713.  doi: 10.1287/opre.2014.1276.  Google Scholar

[6]

E. M. Azevedo and J. W. Hatfield, Complementarity and multidimensional heterogeneity in matching markets, 2012,, Mimeo., ().   Google Scholar

[7]

M. Balinski and T. Sönmez, A tale of two mechanisms: student placement,, Journal of Economic Theory, 84 (1999), 73.  doi: 10.1006/jeth.1998.2469.  Google Scholar

[8]

P. Biró, T. Fleiner, R. W. Irving and D. F. Manlove, The college admissions problem with lower and common quotas,, Theoretical Computer Science, 411 (2010), 3136.  doi: 10.1016/j.tcs.2010.05.005.  Google Scholar

[9]

P. Biró, T. Fleiner and R. Irving, Matching couples with scarf's algorithm,, Institute of Economics, ().   Google Scholar

[10]

P. Biró, R. W. Irving and I. Schlotter, Stable matching with couples: an empirical study,, Journal of Experimental Algorithmics (JEA), 16 (2011).  doi: 10.1145/1963190.1963191.  Google Scholar

[11]

P. Biró and F. Klijn, Matching with couples: A multidisciplinary survey,, International Game Theory Review, 15 (2013).  doi: 10.1142/S0219198913400082.  Google Scholar

[12]

P. Biró, D. F. Manlove and I. McBride, The hospitals/residents problem with couples: Complexity and integer programming models,, in Experimental Algorithms, (2014), 10.   Google Scholar

[13]

Y.-K. Che, J. Kim and F. Kojima, Stable Matching in Large Economies,, Technical report, (2013).   Google Scholar

[14]

Y.-K. Che and F. Kojima, Asymptotic equivalence of probabilistic serial and random priority mechanisms,, Econometrica, 78 (2010), 1625.  doi: 10.3982/ECTA8354.  Google Scholar

[15]

B. Dutta and J. Masso, Stability of matchings when individuals have preferences over colleagues,, Journal of Economic Theory, 75 (1997), 464.  doi: 10.1006/jeth.1997.2291.  Google Scholar

[16]

F. Echenique, Finding all equilibria in games with strategic complements,, Journal of Economic Theory, 135 (2007), 514.  doi: 10.1016/j.jet.2006.06.001.  Google Scholar

[17]

F. Echenique and J. Oviedo, Core many-to-one matchings by fixed point methods,, Journal of Economic Theory, 115 (2004), 358.  doi: 10.1016/S0022-0531(04)00042-1.  Google Scholar

[18]

F. Echenique and J. Oviedo, A theory of stability in many-to-many matching,, Theoretical Economics, 1 (2006), 233.  doi: 10.2139/ssrn.691443.  Google Scholar

[19]

F. Echenique and M. B. Yenmez, A solution to matching with preferences over colleagues,, Games and Economic Behavior, 59 (2007), 46.  doi: 10.1016/j.geb.2006.07.003.  Google Scholar

[20]

A. Erdil and H. Ergin, What's the matter with tie-breaking? improving efficiency in school choice,, American Economic Review, 98 (2008), 669.  doi: 10.1257/aer.98.3.669.  Google Scholar

[21]

T. Fleiner, A fixed-point approach to stable matchings and some applications,, Mathematics of Operations Research, 28 (2003), 103.  doi: 10.1287/moor.28.1.103.14256.  Google Scholar

[22]

D. Fragiadakis and P. Troyan, Market design under distributional constraints: Diversity in school choice and other applications, 2014,, Mimeo., ().   Google Scholar

[23]

D. Fragiadakis, A. Iwasaki, P. Troyan, S. Ueda and M. Yokoo, Strategyproof matching with minimum quotas,, mimeo., ().   Google Scholar

[24]

D. Gale and L. S. Shapley, College admissions and the stability of marriage,, American Mathematical Monthly, 69 (1962), 9.  doi: 10.2307/2312726.  Google Scholar

[25]

D. Gale and M. A. O. Sotomayor, Ms. machiavelli and the stable matching problem,, American Mathematical Monthly, 92 (1985), 261.  doi: 10.2307/2323645.  Google Scholar

[26]

D. Gale and M. A. O. Sotomayor, Some remarks on the stable matching problem,, Discrete Applied Mathematics, 11 (1985), 223.  doi: 10.1016/0166-218X(85)90074-5.  Google Scholar

[27]

M. Goto, N. Hashimoto, A. Iwasaki, Y. Kawasaki, S. Ueda, Y. Yasuda and M. Yokoo, Strategy-proof matching with regional minimum quotas,, in AAMAS2014, (2014).   Google Scholar

[28]

M. Goto, A. Iwasaki, Y. Kawasaki, Y. Yasuda and M. Yokoo, Improving fairness and efficiency in matching markets with regional caps: Priority-list based deferred acceptance mechanism,, Mimeo (the latest version is available at , ().   Google Scholar

[29]

J. Hatfield and P. Milgrom, Matching with contracts,, American Economic Review, 95 (2005), 913.  doi: 10.1257/0002828054825466.  Google Scholar

[30]

J. W. Hatfield and F. Kojima, Matching with contracts: Comment,, American Economic Review, 98 (2008), 1189.  doi: 10.1257/aer.98.3.1189.  Google Scholar

[31]

J. W. Hatfield and S. D. Kominers, Contract design and stability in matching markets,, Harvard University and Stanford University working paper., ().   Google Scholar

[32]

N. Immorlica and M. Mahdian, Marriage, honesty, and stability,, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, (2005), 53.   Google Scholar

[33]

Y. Kamada and F. Kojima, Stability and strategy-proofness for matching with constraints: A problem in the japanese medical match and its solution,, American Economic Review P&P, 102 (2012), 366.  doi: 10.1257/aer.102.3.366.  Google Scholar

[34]

Y. Kamada and F. Kojima, General theory of matching under distributional constraints, 2014,, Mimeo., ().   Google Scholar

[35]

Y. Kamada and F. Kojima, Stability concepts in matching with distributional constraints, 2014,, Mimeo., ().   Google Scholar

[36]

Y. Kamada and F. Kojima, Efficient matching under distributional constraints: Theory and applications,, American Economic Review, 105 (2015), 67.  doi: 10.1257/aer.20101552.  Google Scholar

[37]

O. Kesten, School choice with consent,, The Quarterly Journal of Economics, 125 (2010), 1297.  doi: 10.1162/qjec.2010.125.3.1297.  Google Scholar

[38]

B. Klaus and F. Klijn, Stable matchings and preferences of couples,, Journal of Economic Theory, 121 (2005), 75.  doi: 10.1016/j.jet.2004.04.006.  Google Scholar

[39]

B. Klaus, F. Klijn and J. Masso, Some things couples always wanted to know about stable matchings (but were afraid to ask),, Review of Economic Design, 11 (2007), 175.  doi: 10.1007/s10058-006-0017-9.  Google Scholar

[40]

F. Kojima and P. A. Pathak, Incentives and stability in large two-sided matching markets,, American Economic Review, 99 (2009), 608.  doi: 10.1257/aer.99.3.608.  Google Scholar

[41]

F. Kojima, P. A. Pathak and A. E. Roth, Matching with couples: Stability and incentives in large markets,, Quarterly Journal of Economics, 128 (2013), 1585.  doi: 10.1093/qje/qjt019.  Google Scholar

[42]

F. Kojima, A. Tamura and M. Yokoo, Designing matching mechanisms under constraints: An approach from discrete convex analysis, 2015,, Mimeo., ().   Google Scholar

[43]

H. Konishi and U. Unver, Credible group stability in multi-partner matching problems,, Journal of Economic Theory, 129 (2006), 57.  doi: 10.1016/j.jet.2005.02.001.  Google Scholar

[44]

E. J. McDermid and D. F. Manlove, Keeping partners together: algorithmic results for the hospitals/residents problem with couples,, Journal of Combinatorial Optimization, 19 (2010), 279.  doi: 10.1007/s10878-009-9257-2.  Google Scholar

[45]

D. G. McVitie and L. Wilson, Stable marriage assignments for unequal sets,, BIT, 10 (1970), 295.  doi: 10.1007/BF01934199.  Google Scholar

[46]

T. Nguyen and R. Vohra, Near feasible stable matchings with complementarities,, PIER Working Paper, (2014).  doi: 10.2139/ssrn.2500824.  Google Scholar

[47]

M. Ostrovsky, Stability in supply chain networks,, American Economic Review, (): 897.   Google Scholar

[48]

P. A. Pathak and T. Sönmez, Leveling the playing field: Sincere and sophisticated players in the boston mechanism,, The American Economic Review, 98 (2008), 1636.  doi: 10.1257/aer.98.4.1636.  Google Scholar

[49]

P. A. Pathak and T. Sönmez, School admissions reform in chicago and england: Comparing mechanisms by their vulnerability to manipulation,, American Economic Review, 103 (2013), 80.  doi: 10.1257/aer.103.1.80.  Google Scholar

[50]

M. Pycia, Stability and preference alignment in matching and coalition formation,, Econometrica, 80 (2012), 323.  doi: 10.3982/ECTA7143.  Google Scholar

[51]

E. Ronn, Np-complete stable matching problems,, Journal of Algorithms, 11 (1990), 285.  doi: 10.1016/0196-6774(90)90007-2.  Google Scholar

[52]

A. E. Roth, The evolution of the labor market for medical interns and residents: A case study in game theory,, Journal of Political Economy, 92 (1984), 991.  doi: 10.1086/261272.  Google Scholar

[53]

A. E. Roth, On the allocation of residents to rural hospitals: A general property of two-sided matching markets,, Econometrica, 54 (1986), 425.  doi: 10.2307/1913160.  Google Scholar

[54]

A. E. Roth, A natural experiment in the organization of entry-level labor markets: regional markets for new physicians and surgeons in the united kingdom,, The American economic review, (): 415.   Google Scholar

[55]

A. E. Roth and E. Peranson, The redesign of the matching market for american physicians: Some engineering aspects of economic design,, American Economic Review, 89 (1999), 748.  doi: 10.1257/aer.89.4.748.  Google Scholar

[56]

A. E. Roth and M. A. Sotomayor, Two-sided Matching: A Study in Game-Theoretic Modeling and Analysis,, Econometric Society monographs, (1990).  doi: 10.1017/CCOL052139015X.  Google Scholar

[57]

T. Sönmez and M. U. Ünver, Course bidding at business schools,, International Economic Review, 51 (2010), 99.  doi: 10.1111/j.1468-2354.2009.00572.x.  Google Scholar

[58]

M. A. O. Sotomayor, A non-constructive elementary proof of the existence of stable marriages,, Games and Economic Behavior, 13 (1996), 135.  doi: 10.1006/game.1996.0029.  Google Scholar

[59]

M. A. O. Sotomayor, Three remarks on the many-to-many stable matching problem,, Mathematical social sciences, 38 (1999), 55.  doi: 10.1016/S0165-4896(98)00048-1.  Google Scholar

[60]

M. A. O. Sotomayor, Implementation in the many-to-many matching market,, Games and Economic Behavior, 46 (2004), 199.  doi: 10.1016/S0899-8256(03)00047-2.  Google Scholar

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