Two-Sided Random Matching Markets: Ex-Ante Equivalence of the Deferred Acceptance Procedures

• Published in: arXiv.org
• Main Author: Mauras, Simon
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 18.05.2020
• Subjects: Combinatorial analysis; Computer Science - Computer Science and Game Theory; Computer Science - Data Structures and Algorithms; Computer Science - Discrete Mathematics; Equivalence; Matching
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2005.08584

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.