Abstract
We study multistage centralized assignment systems to allocate scarce resources based on priorities in the context of school choice. We characterize schools’ capacity-priority profiles under which an additional stage of assignment may improve student welfare when the deferred acceptance algorithm is used at each stage. If the capacity-priority profile is acyclic, then no student prefers any subgame-perfect Nash equilibrium (SPNE) outcome of the 2-stage system to the truthful dominant-strategy equilibrium outcome of the 1-stage system. If the capacity-priority profile is not acyclic, then an SPNE outcome of the 2-stage system may Pareto dominate the truthful dominant-strategy equilibrium outcome of the 1-stage system. If students are restricted to playing truncation strategies, an additional stage unambiguously improves student welfare: no student prefers the truthful dominant-strategy equilibrium outcome of the 1-stage system to any SPNE outcome of the 2-stage system.
Article PDF
Similar content being viewed by others
References
Abdulkadiroğlu, A., Tayfun, S.: School choice: a mechanism design approach. Am. Econ. Rev. 93(3), 729–747 (2003)
Abdulkadiroğlu, A., Pathak, P.A., Roth, A.E.: Strategy-proofness versus efficiency in matching with indifferences: redesigning the NYC high school match. Am. Econ. Rev. 99(5), 1954–78 (2009)
Akbarpour, M., Li, S., Gharan, S.O.: Thickness and information in dynamic matching markets. J. Polit. Econ. 128(3), 783–815 (2020)
Alva, S., Manjunath, V.: Strategy-proof Pareto-improvement. J. Econ. Theory 181, 121–142 (2019)
Andersson, T., Dur, U., Ertemel, S., Kesten, O.: Sequential school choice with public and private schools. Working paper (2018)
Bó, I., Hakimov, R.: The iterative deferred acceptance mechanism. Games Econ. Behav. 135, 411–433 (2022)
Baccara, M., Lee, S.M., Yariv, L.: Optimal dynamic matching. Theor. Econ. 15(3), 1221–1278 (2020)
Bando, K.: On the existence of a strictly strong Nash equilibrium under the student-optimal deferred acceptance algorithm. Games Econ. Behav. 87, 269–287 (2014)
Calsamiglia, C., Haeringer, G., Klijn, F.: Constrained school choice: an experimental study. Am. Econ. Rev. 100(4), 1860–74 (2010)
Chambers, C.P., Yenmez, M.B.: Choice and matching. Am. Econ. J. Microecon. 9, 126–147 (2017)
Chen, Y.: New axioms for deferred acceptance. Soc. Choice Welf. 48(2), 393–408 (2017)
Combe, J.: Reallocation with priorities and minimal envy mechanisms. Econ. Theory (2022). https://doi.org/10.1007/s00199-022-01465-x
Doğan, B., Klaus, B.: Resource allocation via immediate acceptance: characterizations and an affirmative action application. J. Math. Econ. 79, 140–56 (2018)
Doğan, B., Klaus, B., Yenmez, M.B.: Unified versus divided enrollment in school choice: improving student welfare in Chicago. Games Econ. Behav. 118, 366–373 (2019)
Doval, L.: Dynamically stable matching. Theor. Econ. 17(2), 687–724 (2022)
Dubins, L.E., Freedman, D.A.: Machiavelli and the Gale–Shapley algorithm. Am. Math. Mon. 88(7), 485–494 (1981)
Dur, U., Gitmez, A.A., Yılmaz, Ö., Kesten, O.: Sequential versus simultaneous assignment systems and two applications. Econ. Theory 68(2), 251–283 (2018)
Dur, U., Gitmez, A.A., Yılmaz, Ö.: School choice under partial fairness. Theor. Econ. 14(4), 1309–1346 (2019)
Ehlers, L., Morrill, T.: (Il)legal assignments in school choice. Rev. Econ. Stud. 87(4), 1837–1875 (2019)
Ergin, H.I.: Consistency in house allocation problems. J. Math. Econ. 34(1), 77–97 (2000)
Ergin, H.I.: Efficient resource allocation on the basis of priorities. Econometrica 70(6), 2489–2497 (2002)
Ergin, H., Sönmez, T.: Games of school choice under the Boston mechanism. J. Public Econ. 90, 215–237 (2006)
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962)
Haeringer, G., Klijn, F.: Constrained school choice. J. Econ. Theory 144(5), 1921–1947 (2009)
Haeringer, G., Klijn, F., Iehlé, V.: Gradual college admission. J. Econ. Theory 198, 105378 (2021)
Jackson, M.O.: Mechanism theory. In: Derigs, U. (ed.) Optimization and Operations Research. Encyclopedia of Life Support Systems, vol. 3. EOLSS, Oxford (2003)
Kelso, A.S., Crawford, V.P.: Job matching, coalition formation, and gross substitutes. Econometrica 50, 1483–1504 (1982)
Kesten, O.: On two competing mechanisms for priority-based allocation problems. J. Econ. Theory 127(1), 155–171 (2006)
Kesten, O.: School choice with consent. Q. J. Econ. 125(3), 1297–1348 (2010)
Kesten, O., Kurino, M.: Strategy-proof improvements upon deferred acceptance: a maximal domain for possibility. Games Econ. Behav. 117, 120–143 (2019)
Klaus, B., Meo, C.: The core for housing markets with limited externalities. Econ. Theory (2023). https://doi.org/10.1007/s00199-022-01478-6
Kojima, F., Ünver, M.U.: The “Boston’’ school-choice mechanism: an axiomatic approach. Econ. Theory 55(3), 515–544 (2014)
Kurino, M.: House allocation with overlapping generations. Am. Econ. J. Microecon. 6(1), 258–89 (2014)
Manjunath, V., Turhan, B.: Two school systems, one district: what to do when a unified admissions process is impossible. Games Econ. Behav. 95, 25–40 (2016)
Morrill, T.: Making just school assignments. Games Econ. Behav. 92, 18–27 (2015)
Roth, A.E., Sotomayor, M.: Matching, Two-sided: A Study in Game-Theoretic Modelling and Analysis. Econometric Society Monographs, vol. 18. Cambridge University Press, Cambridge (1990)
Roth, A.E., Sotomayor, M., Rothblum, U.G.: Truncation strategies in matching markets-in search of advice for participants. Econometrica 67(1), 21–43 (1999)
Sönmez, T., Ünver, U.: House allocation with existing tenants: a characterization. Games Econ. Behav. 69(2), 425–445 (2010)
Sotomayor, M.: The stability of the equilibrium outcomes in the admission games induced by stable matching rules. Int. J. Game Theory 36, 621–640 (2008)
Thomson, W.: The consistency principle. In: Ichiishi, T., Neyman, A., Tauman, Y. (eds.) Game Theory and Applications, pp. 187–215. Academic Press, New York (1990)
Troyan, P., Delacrétaz, D., Kloosterman, A.: Essentially stable matchings. Games Econ. Behav. 120, 370–390 (2020)
Ünver, M.U.: Dynamic kidney exchange. Rev. Econ. Stud. 77(1), 372–414 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Lars Ehlers, Guillaume Haeringer, Bettina Klaus, Scott Duke Kominers, Alexey Kushnir, Vikram Manjunath, William Thomson, anonymous referees, and participants at several seminars and conferences for helpful comments. Battal Doğan gratefully acknowledges financial support from the Swiss National Science Foundation (SNSF) and the British Academy/Leverhulme Trust (SRG1819\190133), and the hospitality of the Center of Mathematical Sciences and Applications (CMSA, Harvard University) where part of this paper was written. This paper was circulated as a part of our previous 2017 working paper entitled “How to improve student assignment in Chicago: unified enrollment in school choice,” which is obsolete now.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Doğan, B., Yenmez, M.B. When does an additional stage improve welfare in centralized assignment?. Econ Theory 76, 1145–1173 (2023). https://doi.org/10.1007/s00199-023-01488-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-023-01488-y