Two-Sided Random Matching Markets: Ex-Ante Equivalence of the Deferred Acceptance Procedures
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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2020
- DOI:
- 10.48550/arXiv.2005.08584
- arXiv:
- arXiv:2005.08584
- Bibcode:
- 2020arXiv200508584M
- Keywords:
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- Computer Science - Computer Science and Game Theory;
- Computer Science - Discrete Mathematics;
- Computer Science - Data Structures and Algorithms;
- Economics - Theoretical Economics
- E-Print:
- Accepted for publication in the 21st ACM Conference on Economics and Computation (EC'20)