# Monotonicity and Consistency in Matching Markets

## Abstract

Objective: To obtain axiomatic characterizations of the core of one-to-one and one-to-many matching markets. Methods: The axioms recently applied to characterize the core of assignment games were adapted to the models of this paper. Results: The core of one-to-one matching markets is characterized by two different lists of axioms. The first one consists of weak unanimity, population monotonicity, and Maskin monotonicity. The second consists of weak unanimity, population monotonicity, and consistency. If we allow for weak preferences, the core is characterized by weak unanimity, population monotonicity, Maskin monotonicity, and consistency. For one-to-many matchings, the same lists as for the case of strict preferences characterize the core. Conclusions: The cores of the discrete matching markets are characterized by axioms that almost overlap with the axioms characterizing the core of the continuous matching markets. This provides an axiomatic explanation for the observations in the literature that almost parallel properties are obtained for the core of the two models. We observe that Maskin monotonicity is closely related to consistency in matching markets

This is a preview of subscription content, access via your institution.

## References

• Ehlers L (2004) Monotonic and implementable solutions in generalized matching problems. J Econ Theory 114:358–369

• Ergin H I (2002) Efficient resource allocation on the basis of priorities. Econometrica 70:2489–2497

• Eriksson K, Karlander J (2000) Stable matching in a common generalization of the marriage and assignment models. Discrete Math 217:135–156

• Fujishige S, Tamura A (2003) A general two-sided matching market with discrete concave utility functions. RIMS Preprint 1401, Kyoto University, Kyoto

• Gale D, Shapley L (1962) College admission and the stability of marriage. Am Math Monthly 69:9–15

• Kaneko M (1982) The central assignment games and the assignment markets. J Math Econ 10:205–232

• Kara T (1996) Implementation in matching problems. PhD Thesis submitted to University of Rochester

• Kara T, Sönmez T (1996) Nash implementation of matching rules. J Econ Theory 68:425–439

• Kara T, Sönmez T (1997) Implementation of college admission rules. Econ Theory 9:97–218

• Maskin E (1999) Nash equilibrium and welfare optimality. Rev Econ Stud 66:23–38

• Peleg B (1992) Axiomatization of the core. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol 1. Elsevier, Amsterdam, pp 397–412

• Roth A, Rothblum U, Vande Vate J (1993) Stable matchings, optimal assignments, and linear programming. Math Oper Res 18:803–828

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

• Sasaki H (1995) Consistency and monotonicity in assignment problems. Int J Game Theory 24:373–397

• Sasaki H, Toda M (1992) Consistency and characterization of the core of the two-sided matching problems. J Econ Theory 56:218–227

• Shapley L, Shubik M (1972) The assignment game A: the core. Int J Game Theory 1:111–30

• Sonn S.Y (1994) Maskin monotonicity of solutions in many-to-one matching. University of Rochester, Mimeo

• Sotomayor M (2000) Existence of stable outcomes and the lattice property for a unified matching market. Math Soc Sci 32:119–132

• Tadenuma K (1993) Maskin monotonicity in matching problems. University of Rochester, Mimeo

• Tadenuma K, Toda M (1998) Implementable stable solutions to pure matching problem. Math Soc Sci 35:121–132

• Thomson W (1983) The fair division of a fixed supply among a growing population. Math Oper Res 8:319–326

• Thomson W (1995) Population monotonic allocation rules. In: Barnett W. A, Moulin H, Salles M, Schofield NJ (eds) Social choice, welfare and ethics. Cambridge University Press, Cambridge, pp 79–124

• Thomson W (2003) Consistent allocation rules. University of Rochester, Mimeo

• Thomson W (2004) Consistency and its converse: an introduction. University of Rochester, Mimeo

• Toda M (2003) Axiomatization of the core of assignment games. Games Econ Behav (in press)

• Zhou L (1991) A weak monotonicity property of the nucleolus. Int J Game Theory 19:407–411

## Author information

Authors

### Corresponding author

Correspondence to Manabu Toda.

This research is financially supported by Waseda University Grant for Special Research Projects #2000A−887, 21COE-GLOPE, and Grant-in-Aid for Scientific Research #15530125, JSPS. This paper was presented at the 7th. International Meeting of the Society for Social Choice and Welfare held in Osaka, Japan. The comments of the participants are gratefully acknowledged. The author thanks Professors William Thomson, Eiichi Miyagawa and anonymous referees for their valuable comments and suggestions. Any remaining errors are independent

## Rights and permissions

Reprints and Permissions

Toda, M. Monotonicity and Consistency in Matching Markets. Int J Game Theory 34, 13 (2006). https://doi.org/10.1007/s00182-005-0002-5

• Published:

• DOI: https://doi.org/10.1007/s00182-005-0002-5

### Keywords

• Two-sided matchings