## Abstract

This paper shows that the college admissions model with responsive preferences is not always satisfactory for representing real college admissions markets. Simple examples are used to illustrate real situations, in which the knowledge of the preferences of the institutions over all possible assignments of candidates is necessary for the analysis of relevant problems for the markets under consideration.

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## Notes

A detailed description of the college admissions model is presented in Roth and Sotomayor (1990).

Indeed, this result is more general. It is not necessary to require that the matching rule yields the optimal stable matching for a given side for any preference profile Q. The conclusion also holds if this requirement is just for the profile of true preferences P.

Kojima and Pathak (2009) refer to procedures by which final ordered lists of preferences are obtained using an acceptable rule “dropping strategies”. These authors proved the following:

*Consider an arbitrary stable mechanism. Fix preferences of institutions other than i. Suppose the mechanism produces de matching x under some report P′(i) (not necessarily i’s true preference P(i)). Then there exists a dropping strategy Q(i) that produces a matching that i weakly prefers to x according to its true preferences.*Of course, if P′(i) = P(i), then Q(i) = x(i) is a dropping strategy which produces the same matching x.This result allowed the establishment of other conclusions for the college admissions model with responsive preferences, as for example, the lattice property of the set of stable matchings, the polarization of interests between students and colleges (Roth and Sotomayor 1990), as well as the general manipulability theorem and the general impossibility theorem (Sotomayor 2012).

The substitutes condition is defined in Hatfield and Milgrom (2005). Notice that the discrete many-to-many matching model with responsive preferences can be regarded as the particular case of a matching model with contracts, in which one contract, at most, can be associated to a doctor-hospital pair and the doctors are allowed to sign more than one contract. Therefore, the first example of this section also illustrates that the approach considered in Hatfield and Milgrom (2005) may be deficient for the identification of stable matchings in the many-to-many case.

In the college admissions model, the choice set of an institution with a quota of q, from any set of candidates, is either the q most preferred acceptable candidates in the set, or all the acceptable candidates in the set, whichever is the smaller number. Then, if for example, an institution i has a quota of two and P(i) = c

_{1}, c_{2}, c_{3},c_{4}is its list of preferences over individual candidates, the responsiveness of the preferences of i implies that {c_{1},c_{3}} >_{i}{c_{2},c_{3}}. However these preferences cannot be derived from i’s choice function. Facing {c_{1}, c_{2}, c_{3}}, this function only informs that {c_{1}, c_{2}} >_{i}S, for every subset S of {c_{1}, c_{2}, c_{3}}. Therefore, the problems that arise when responsiveness is used as a primitive do not disappear if, instead, we use the choice function framework.Hatfield and Kojima (2010) presents a more appropriate mathematical model for representing a matching market with contracts. In this model, the preference relations of the agents are primitives and the choice sets are defined as being consistent with these preferences. The special case, where a single contract can be associated with a doctor-hospital pair, is the discrete many-to-one matching model with substitutable and strict preferences presented in Roth and Sotomayor (1990), which is the discrete version of the matching model of Kelso and Crawford (1982). The extension of this model to the many-to-many case is due to Roth (1984). Sotomayor (1999) generalizes Roth’s model by assuming not-necessarily strict preferences.

This example also illustrates that, in spite of the fact that the reformulation proposed in Aygün and Sönmez (2013) guarantees the existence of stable allocations in the model of Hatfield and Milgrom, it may be deficient, even for the identification of stable matchings, when the doctors are allowed to sign more than one contract.

## References

Aygün, Sönmez (2013) Matching with contracts: comment. Am Econ Rev 103(5):2050–2051

Dubins LE, Freedman DA (1981) Machiavelli and the Gale–Shapley algorithm. Am Math Mon 88:485–494

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

Hatfield JW, Kojima F (2010) Substitutes and stability for matching with contracts. J Econ Theory 145:1704–1723

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

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

Kojima F, Pathak P (2009) Incentives and stability in large two-sided matching markets. Am Econ Rev 99(3):608–627

Roth A (1982) The economics of matching: stability and incentives. Math Oper Res 7:617–628

Roth A (1984) Stability and polarization of interest in job matching. Econometrica 53:7–57

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

Roth A, Sotomayor M (1989) The college admissions problem revisited. Econometrica 57:559–570

Roth A, Sotomayor M (1990) Two-sided matching. A study in game-theoretic modeling and analysis. Econometric Society Monographs, n. 18. Cambridge, MA; Cambridge University Press

Sotomayor M (1996) Admission mechanisms of students to colleges. A game-theoretic modeling and analysis. Braz Rev Econom 16(1):25–63 (First published in 1995, Annals of Brazilian Meeting of Econometrics, Florianópolis, Brazil)

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

Sotomayor M (2012) A further note on the college admissions game. Int J Game Theory 41:179–193

Sotomayor M (2016) Modeling cooperative decision situations: the deviation function form and the equilibrium concept. Int J Game Theory 45(3):743–768

## Acknowledgements

This paper was partially supported by CNPq-Brazil, Grant no. 304973/2015-7. The author is grateful to David Perez-Castrillo and Bertan Turhan for helpful remarks on the previous version of this work. Thanks are also due to anonymous referees and the associate editor for helpful suggestions. Special thanks are due to Michael Borns for his careful reading and English corrections. Earlier versions circulated under the title: “The college admissions model is deficient for the purpose of serving as vehicle for the manipulability analysis of a stable matching rule” and “The college admissions model with responsive preferences revisited”.

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Sotomayor, M. Some remarks on the modeling of discrete matching markets.
*Int J Game Theory* **51**, 155–167 (2022). https://doi.org/10.1007/s00182-021-00788-8

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DOI: https://doi.org/10.1007/s00182-021-00788-8

### Keywords

- Cooperative equilibrium
- Stability
- Matching
- Pairwise-stability
- Choice-function

### JEL Classification

- C78
- D78