Simon Mauras
ABSTRACT
Stable matching in a community consisting of N men and N women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley.
When the input preference profile is generated from a distribution, we study the output distribution of two stable matching procedures: women-proposing-deferred-acceptance and men-proposing-deferred-acceptance. We show that the two procedures are ex-ante equivalent: that is, under certain conditions on the input distribution, their output distributions are identical.
In terms of technical contributions, we generalize (to the non-uniform case) an integral formula, due to Knuth and Pittel, which gives the probability that a fixed matching is stable. Using an inclusion-exclusion principle on the set of rotations, we give a new formula which gives the probability that a fixed matching is the women/men-optimal stable matching. We show that those two probabilities are equal with an integration by substitution.
- Atila Abdulkadiroug lu, Parag A Pathak, and Alvin E Roth. 2005 a. The new york city high school match. American Economic Review, Vol. 95, 2 (2005), 364--367.Google Scholar
Cross Ref
- Atila Abdulkadiroug lu, Parag A Pathak, Alvin E Roth, and Tayfun Sönmez. 2005 b. The Boston public school match. American Economic Review, Vol. 95, 2 (2005), 368--371.Google ScholarCross Ref
- Brian Aldershof, Olivia M Carducci, and David C Lorenc. 1999. Refined inequalities for stable marriage. Constraints, Vol. 4, 3 (1999), 281--292.Google ScholarDigital Library
- Itai Ashlagi, Yash Kanoria, and Jacob D Leshno. 2017. Unbalanced random matching markets: The stark effect of competition. Journal of Political Economy, Vol. 125, 1 (2017), 69--98.Google ScholarCross Ref
- Péter Biró, Katar'ina Cechlárová, and Tamás Fleiner. 2008. The dynamics of stable matchings and half-matchings for the stable marriage and roommates problems. International Journal of Game Theory, Vol. 36, 3--4 (2008), 333--352.Google ScholarCross Ref
- Jose Correa, Rafael Epstein, Juan Escobar, Ignacio Rios, Bastian Bahamondes, Carlos Bonet, Natalie Epstein, Nicolas Aramayo, Martin Castillo, Andres Cristi, et al. 2019. School Choice in Chile. Proceedings of the 2019 ACM Conference on Economics and Computation (2019), 325--343.Google ScholarDigital Library
- Pavlos S Efraimidis and Paul G Spirakis. 2006. Weighted random sampling with a reservoir. Inform. Process. Lett., Vol. 97, 5 (2006), 181--185.Google ScholarCross Ref
- David Gale and Lloyd S Shapley. 1962. College admissions and the stability of marriage. The American Mathematical Monthly, Vol. 69, 1 (1962), 9--15.Google ScholarCross Ref
- Hugo Gimbert, Claire Mathieu, and Simon Mauras. 2019. Two-sided matching markets with correlated random preferences have few stable pairs. arXiv preprint arXiv:1904.03890 (2019).Google Scholar
- Dan Gusfield and Robert W Irving. 1989. The stable marriage problem: structure and algorithms .MIT press.Google ScholarDigital Library
- Avinatan Hassidim, Assaf Romm, and Ran I Shorrer. 2018. Need vs. merit: The large core of college admissions markets. (2018).Google Scholar
- Nicole Immorlica and Mohammad Mahdian. 2015. Incentives in large random two-sided markets. ACM Transactions on Economics and Computation, Vol. 3, 3 (2015), 14.Google ScholarDigital Library
- Bettina Klaus and Flip Klijn. 2006. Procedurally fair and stable matching. Economic Theory, Vol. 27, 2 (2006), 431--447.Google ScholarCross Ref
- Donald E Knuth. 1976. Mariages stables et leurs relations avec d'autres problemes combinatoires: introduction a l'analysis mathematique des algorithmes-. Les Presses de l'Universite de Montreal.Google Scholar
- Donald E Knuth. 1997. Stable marriage and its relation to other combinatorial problems: An introduction to the mathematical analysis of algorithms. Vol. 10. American Mathematical Soc.Google Scholar
- Fuhito Kojima and Parag A Pathak. 2009. Incentives and stability in large two-sided matching markets. American Economic Review, Vol. 99, 3 (2009), 608--27.Google ScholarCross Ref
- SangMok Lee. 2016. Incentive compatibility of large centralized matching markets. The Review of Economic Studies, Vol. 84, 1 (2016), 444--463.Google ScholarCross Ref
- Jinpeng Ma. 1996. On randomized matching mechanisms. Economic Theory, Vol. 8, 2 (1996), 377--381.Google ScholarCross Ref
- Stephan Mertens. 2015. Small random instances of the stable roommates problem. Journal of Statistical Mechanics: Theory and Experiment, Vol. 6 (2015).Google Scholar
- Boris Pittel. 1989. The average number of stable matchings. SIAM Journal on Discrete Mathematics, Vol. 2, 4 (1989), 530--549.Google ScholarDigital Library
- Boris Pittel. 1992. On likely solutions of a stable marriage problem. The Annals of Applied Probability (1992), 358--401.Google Scholar
- Boris Pittel. 2019. On random stable partitions. International Journal of Game Theory, Vol. 48, 2 (2019), 433--480.Google ScholarCross Ref
- Boris G Pittel and Robert W Irving. 1994. An upper bound for the solvability probability of a random stable roommates instance. Random Structures & Algorithms, Vol. 5, 3 (1994), 465--486.Google ScholarCross Ref
- Alvin E Roth and Elliott Peranson. 1999. The redesign of the matching market for American physicians: Some engineering aspects of economic design. American economic review, Vol. 89, 4 (1999), 748--780.Google Scholar
- Alvin E Roth and JH Vande Vate. 1990. Random Paths to Stability in Two-Sided Matching. Econometrica, Vol. 58, 6 (1990), 1475--1480.Google ScholarCross Ref
- Jimmy JM Tan. 1991. A necessary and sufficient condition for the existence of a complete stable matching. Journal of Algorithms, Vol. 12, 1 (1991), 154--178.Google ScholarDigital Library
Index Terms
- Two-sided Random Matching Markets: Ex-ante Equivalence of the Deferred Acceptance Procedures
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