American Institute of Mathematical Sciences

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March & April  2015, 2(3&4): 331-340. doi: 10.3934/jdg.2015009

Why do stable clearinghouses work so well? - Small sets of stable matchings in typical environments, and the limits-on-manipulation theorem of Demange, Gale and Sotomayor

Alvin E. Roth 1,

1. 

Department of Economics, Stanford University, Landau Economics Building, 579 Serra Mall, Room 344, Stanford University, Stanford, CA 94305-6072, United States

Received  April 2015 Revised  August 2015 Published  November 2015

Marilda Sotomayor is one of the pioneers of the theory of stable matching. She has published many important results, including some which are fundamental to subsequent developments. I will concentrate on one fundamental theorem, which today allows us to understand better why stable clearinghouses work so well. Demange, Gale and Sotomayor (1987)[16] proved a theorem which implies that when the set of stable matchings is small, participants in a stable clearinghouse will seldom be able to profit from strategically manipulating their preferences. More recent results show (empirically and theoretically) that the set of stable matchings can be expected to be small in typical applications. Therefore, reporting true preferences will be rewarded in clearinghouses that produce stable matchings in terms of stated preferences, and so there is a reason that such clearinghouses elicit sufficiently good preference data to produce matchings that are stable with respect to true preferences.
Keywords: Stable matching, market design..
Mathematics Subject Classification: 91A, 91B, 91.
Citation: Alvin E. Roth. Why do stable clearinghouses work so well? - Small sets of stable matchings in typical environments, and the limits-on-manipulation theorem of Demange, Gale and Sotomayor. Journal of Dynamics & Games, 2015, 2 (3&4) : 331-340. doi: 10.3934/jdg.2015009
References:
[1]

A. Abdulkadiroglu, P. A. Pathak and A. E. Roth, The New York City high school match,, American Economic Review, 95 (2005), 364. Google Scholar

[2]

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

[3]

A. Abdulkadiroglu, P. A. Pathak, A. E. Roth and T. Sönmez, The Boston public school match,, American Economic Review, 95 (2005), 368. 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]

I. Ashlagi, Y. Kanoria and J. D. Leshno, Unbalanced random matching markets: The stark effect of competition,, Journal of Political Economy, (). Google Scholar

[7]

A. Banerjee, E. Duflo, M. Ghatak and J. Lafortune, Marry for what: Caste and mate selection in modern India,, American Economic Journal: Microeconomics, 5 (2013), 33. Google Scholar

[8]

G. S. Becker, A theory of marriage: Part I,, Journal of Political Economy, 81 (1973), 813. Google Scholar

[9]

G. S. Becker, A theory of marriage: Part II,, Journal of Political Economy, 82 (1974). Google Scholar

[10]

G. S. Becker, A Treatise on the Family,, Harvard University Press, (1981). Google Scholar

[11]

P. A. Coles and R. I. Shorrer, Optimal truncation in matching markets,, Games and Economic Behavior, 87 (2014), 591. doi: 10.1016/j.geb.2014.01.005. Google Scholar

[12]

P. Coles, Y. Gonczarowski and R. I. Shorrer, Strategic behavior in unbalanced matching markets,, Working Paper, (2014). Google Scholar

[13]

V. P. Crawford and E. M. Knoer, Job matching with heterogeneous firms and workers,, Econometrica, 49 (1981), 437. Google Scholar

[14]

G. Demange and D. Gale, The strategy structure of 2-sided matching markets,, Econometrica, 53 (1985), 873. doi: 10.2307/1912658. Google Scholar

[15]

G. Demange, D. Gale and M. Sotomayor, Multi-item auctions,, Journal of Political Economy, 94 (1986), 863. Google Scholar

[16]

G. Demange, D. Gale and M. Sotomayor, A further note on the stable matching problem,, Discrete Applied Mathematics, 16 (1987), 217. doi: 10.1016/0166-218X(87)90059-X. Google Scholar

[17]

L. E. Dubins and D. A. Freedman, Machiavelli and the gale-shapley algorithm,, American Mathematical Monthly, 88 (1981), 485. doi: 10.2307/2321753. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

H. Günter, A. Hortaçsu and D. Ariely, Matching and sorting in online dating,, American Economic Review, 100 (2010), 130. Google Scholar

[25]

R. Holzman and D. Samet, Matching of like rank and the size of the core in the marriage problem,, Games and Economic Behavior 88 (2014), 88 (2014), 277. doi: 10.1016/j.geb.2014.10.003. Google Scholar

[26]

A. Hylland and R. Zeckhauser, The efficient allocation of individuals to positions,, Journal of Political Economy, 87 (1979), 293. Google Scholar

[27]

N. Immorlica and M. Mahdian, Marriage, honesty, and stability,, SODA 2005 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, (2005), 53. Google Scholar

[28]

J. H. Kagel and A. E. Roth, The dynamics of reorganization in matching markets: A laboratory experiment motivated by a natural experiment,, Quarterly Journal of Economics, 115 (2000), 201. Google Scholar

[29]

M. Kaneko and M. H. Wooders, Cores of partitioning games,, Mathematical Social Sciences, 3 (1982), 313. doi: 10.1016/0165-4896(82)90015-4. Google Scholar

[30]

A. S. Kelso, Jr. and V. P. Crawford, Job matching, coalition formation, and gross substitutes,, Econometrica, 50 (1982), 1483. Google Scholar

[31]

D. E. Knuth, Mariages Stables (French), Stable Mariages,, Les Presses de l'Université de Montreal, (1976). Google Scholar

[32]

D. E. Knuth, R. Motwani and B. Pittel, Stable husbands,, Proceedings of the first annual ACM-SIAM Symposium on Discrete Algorithms, 1 (1990), 1. doi: 10.1002/rsa.3240010102. Google Scholar

[33]

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

[34]

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. Google Scholar

[35]

T. C. Koopmans and M. Beckmann, Assignment problems and the location of economic activities,, Econometrica, 25 (1957), 53. doi: 10.2307/1907742. Google Scholar

[36]

D. G. McVitie, The stable marriage problem and the selection of students for university admission,, M.Sc. Thesis, (1967). Google Scholar

[37]

D. G. McVitie and L. B. Wilson, Stable marriage assignments for unequal sets,, BIT Numerical Mathematics, 10 (1970), 295. Google Scholar

[38]

D. G. McVitie and L. B. Wilson, The application of the stable marriage assignment to university admissions,, Operational Research Quarterly, 21 (1970), 425. Google Scholar

[39]

D. G. McVitie and L. B. Wilson, The stable marriage problem,, Communications of the ACM, 14 (1971), 486. doi: 10.1145/362619.362631. Google Scholar

[40]

A. E. Roth, The economics of matching: Stability and incentives,, Mathematics of Operations Research, 7 (1982), 617. doi: 10.1287/moor.7.4.617. Google Scholar

[41]

A. E. Roth, Incentive compatibility in a market with indivisible goods,, Economics Letters, 9 (1982), 127. doi: 10.1016/0165-1765(82)90003-9. Google Scholar

[42]

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. Google Scholar

[43]

A. E. Roth, Misrepresentation and stability in the marriage problem,, Journal of Economic Theory, 34 (1984), 383. doi: 10.1016/0022-0531(84)90152-2. Google Scholar

[44]

A. E. Roth, The college admissions problem is not equivalent to the marriage problem,, Journal of Economic Theory, 36 (1985), 277. doi: 10.1016/0022-0531(85)90106-1. Google Scholar

[45]

A. E. Roth, New physicians: A natural experiment in market organization,, Science, 250 (1990), 1524. Google Scholar

[46]

A. E. Roth, A natural experiment in the organization of entry level labor markets: Regional markets for new physicians and surgeons in the U.K.,, American Economic Review, 81 (1991), 415. Google Scholar

[47]

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. Google Scholar

[48]

A. E. Roth and M. Sotomayor, Interior points in the core of two-sided matching markets,, Journal of Economic Theory, 45 (1988), 85. doi: 10.1016/0022-0531(88)90255-4. Google Scholar

[49]

A. E. Roth and M.a Sotomayor, The college admissions problem revisited,, Econometrica, 57 (1989), 559. doi: 10.2307/1911052. Google Scholar

[50]

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

[51]

L. S. Shapley and M. Shubik, The assignment game I: The core,, International Journal of Game Theory, 1 (1972), 111. Google Scholar

[52]

M. 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

[53]

M. Sotomayor, The lattice structure of the set of stable outcomes of the multiple partners assignment game,, International Journal of Game Theory, 28 (1999), 567. doi: 10.1007/s001820050126. Google Scholar

[54]

M. Sotomayor, Existence of stable outcomes and the lattice property for a unified matching market,, Mathematical Social Sciences, 39 (2000), 119. doi: 10.1016/S0165-4896(99)00028-1. Google Scholar

[55]

M. Sotomayor, An elementary non-constructive proof of the non-emptiness of the core of the housing market of Shapley and Scarf,, Mathematical Social Sciences, 50 (2005), 298. doi: 10.1016/j.mathsocsci.2005.04.004. Google Scholar

show all references

References:
[1]

A. Abdulkadiroglu, P. A. Pathak and A. E. Roth, The New York City high school match,, American Economic Review, 95 (2005), 364. Google Scholar

[2]

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

[3]

A. Abdulkadiroglu, P. A. Pathak, A. E. Roth and T. Sönmez, The Boston public school match,, American Economic Review, 95 (2005), 368. 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]

I. Ashlagi, Y. Kanoria and J. D. Leshno, Unbalanced random matching markets: The stark effect of competition,, Journal of Political Economy, (). Google Scholar

[7]

A. Banerjee, E. Duflo, M. Ghatak and J. Lafortune, Marry for what: Caste and mate selection in modern India,, American Economic Journal: Microeconomics, 5 (2013), 33. Google Scholar

[8]

G. S. Becker, A theory of marriage: Part I,, Journal of Political Economy, 81 (1973), 813. Google Scholar

[9]

G. S. Becker, A theory of marriage: Part II,, Journal of Political Economy, 82 (1974). Google Scholar

[10]

G. S. Becker, A Treatise on the Family,, Harvard University Press, (1981). Google Scholar

[11]

P. A. Coles and R. I. Shorrer, Optimal truncation in matching markets,, Games and Economic Behavior, 87 (2014), 591. doi: 10.1016/j.geb.2014.01.005. Google Scholar

[12]

P. Coles, Y. Gonczarowski and R. I. Shorrer, Strategic behavior in unbalanced matching markets,, Working Paper, (2014). Google Scholar

[13]

V. P. Crawford and E. M. Knoer, Job matching with heterogeneous firms and workers,, Econometrica, 49 (1981), 437. Google Scholar

[14]

G. Demange and D. Gale, The strategy structure of 2-sided matching markets,, Econometrica, 53 (1985), 873. doi: 10.2307/1912658. Google Scholar

[15]

G. Demange, D. Gale and M. Sotomayor, Multi-item auctions,, Journal of Political Economy, 94 (1986), 863. Google Scholar

[16]

G. Demange, D. Gale and M. Sotomayor, A further note on the stable matching problem,, Discrete Applied Mathematics, 16 (1987), 217. doi: 10.1016/0166-218X(87)90059-X. Google Scholar

[17]

L. E. Dubins and D. A. Freedman, Machiavelli and the gale-shapley algorithm,, American Mathematical Monthly, 88 (1981), 485. doi: 10.2307/2321753. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

H. Günter, A. Hortaçsu and D. Ariely, Matching and sorting in online dating,, American Economic Review, 100 (2010), 130. Google Scholar

[25]

R. Holzman and D. Samet, Matching of like rank and the size of the core in the marriage problem,, Games and Economic Behavior 88 (2014), 88 (2014), 277. doi: 10.1016/j.geb.2014.10.003. Google Scholar

[26]

A. Hylland and R. Zeckhauser, The efficient allocation of individuals to positions,, Journal of Political Economy, 87 (1979), 293. Google Scholar

[27]

N. Immorlica and M. Mahdian, Marriage, honesty, and stability,, SODA 2005 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, (2005), 53. Google Scholar

[28]

J. H. Kagel and A. E. Roth, The dynamics of reorganization in matching markets: A laboratory experiment motivated by a natural experiment,, Quarterly Journal of Economics, 115 (2000), 201. Google Scholar

[29]

M. Kaneko and M. H. Wooders, Cores of partitioning games,, Mathematical Social Sciences, 3 (1982), 313. doi: 10.1016/0165-4896(82)90015-4. Google Scholar

[30]

A. S. Kelso, Jr. and V. P. Crawford, Job matching, coalition formation, and gross substitutes,, Econometrica, 50 (1982), 1483. Google Scholar

[31]

D. E. Knuth, Mariages Stables (French), Stable Mariages,, Les Presses de l'Université de Montreal, (1976). Google Scholar

[32]

D. E. Knuth, R. Motwani and B. Pittel, Stable husbands,, Proceedings of the first annual ACM-SIAM Symposium on Discrete Algorithms, 1 (1990), 1. doi: 10.1002/rsa.3240010102. Google Scholar

[33]

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

[34]

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. Google Scholar

[35]

T. C. Koopmans and M. Beckmann, Assignment problems and the location of economic activities,, Econometrica, 25 (1957), 53. doi: 10.2307/1907742. Google Scholar

[36]

D. G. McVitie, The stable marriage problem and the selection of students for university admission,, M.Sc. Thesis, (1967). Google Scholar

[37]

D. G. McVitie and L. B. Wilson, Stable marriage assignments for unequal sets,, BIT Numerical Mathematics, 10 (1970), 295. Google Scholar

[38]

D. G. McVitie and L. B. Wilson, The application of the stable marriage assignment to university admissions,, Operational Research Quarterly, 21 (1970), 425. Google Scholar

[39]

D. G. McVitie and L. B. Wilson, The stable marriage problem,, Communications of the ACM, 14 (1971), 486. doi: 10.1145/362619.362631. Google Scholar

[40]

A. E. Roth, The economics of matching: Stability and incentives,, Mathematics of Operations Research, 7 (1982), 617. doi: 10.1287/moor.7.4.617. Google Scholar

[41]

A. E. Roth, Incentive compatibility in a market with indivisible goods,, Economics Letters, 9 (1982), 127. doi: 10.1016/0165-1765(82)90003-9. Google Scholar

[42]

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. Google Scholar

[43]

A. E. Roth, Misrepresentation and stability in the marriage problem,, Journal of Economic Theory, 34 (1984), 383. doi: 10.1016/0022-0531(84)90152-2. Google Scholar

[44]

A. E. Roth, The college admissions problem is not equivalent to the marriage problem,, Journal of Economic Theory, 36 (1985), 277. doi: 10.1016/0022-0531(85)90106-1. Google Scholar

[45]

A. E. Roth, New physicians: A natural experiment in market organization,, Science, 250 (1990), 1524. Google Scholar

[46]

A. E. Roth, A natural experiment in the organization of entry level labor markets: Regional markets for new physicians and surgeons in the U.K.,, American Economic Review, 81 (1991), 415. Google Scholar

[47]

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. Google Scholar

[48]

A. E. Roth and M. Sotomayor, Interior points in the core of two-sided matching markets,, Journal of Economic Theory, 45 (1988), 85. doi: 10.1016/0022-0531(88)90255-4. Google Scholar

[49]

A. E. Roth and M.a Sotomayor, The college admissions problem revisited,, Econometrica, 57 (1989), 559. doi: 10.2307/1911052. Google Scholar

[50]

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

[51]

L. S. Shapley and M. Shubik, The assignment game I: The core,, International Journal of Game Theory, 1 (1972), 111. Google Scholar

[52]

M. 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

[53]

M. Sotomayor, The lattice structure of the set of stable outcomes of the multiple partners assignment game,, International Journal of Game Theory, 28 (1999), 567. doi: 10.1007/s001820050126. Google Scholar

[54]

M. Sotomayor, Existence of stable outcomes and the lattice property for a unified matching market,, Mathematical Social Sciences, 39 (2000), 119. doi: 10.1016/S0165-4896(99)00028-1. Google Scholar

[55]

M. Sotomayor, An elementary non-constructive proof of the non-emptiness of the core of the housing market of Shapley and Scarf,, Mathematical Social Sciences, 50 (2005), 298. doi: 10.1016/j.mathsocsci.2005.04.004. Google Scholar

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