Editing Stable matching problem (section)

{{Short description|Pairing where no unchosen pair prefers each other over their choice}}
In [[mathematics]], [[economics]], and [[computer science]], the '''stable matching problem''' <ref>{{cite web |author=Tesler, G.|url=https://mathweb.ucsd.edu/~gptesler/154/slides/154_galeshapley_20-handout.pdf|website=mathweb.ucsd.edu|title=Ch. 5.9: Gale-Shapley Algorithm|date= 2020|publisher=[[University of California San Diego]]|access-date=26 April 2025|url-status=|archive-url=|archive-date=}}</ref><ref>{{cite web|last1=Kleinberg|first1=Jon |last2=Tardos|first2=Éva |url=https://www.cs.princeton.edu/~wayne/kleinberg-tardos/pdf/01StableMatching.pdf|website=www.cs.princeton.edu|title=Algorithmn Design: 1. Stable Matching|date= 2005|publisher=[[Pearson PLC|Pearson]]-[[Addison Wesley]]: [[Princeton University]]|access-date=26 April 2025|url-status=|archive-url=|archive-date=}}</ref><ref>{{cite web |last1=Goel|first1=Ashish |editor1-last=Ramseyer|editor1-first=Geo  |url=https://web.stanford.edu/~ashishg/cs261/win21/notes/l5_note.pdf|website=web.stanford.edu|title=CS261 Winter 2018- 2019 Lecture 5: Gale-Shapley Algorith|date= 21 January 2019|publisher=[[Stanford University]]|access-date=26 April 2025|url-status=|archive-url=|archive-date=}}</ref> is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element.  A matching is a [[bijection]] from the elements of one set to the elements of the other set.  A matching is ''not'' stable if:

{{Ordered list|list-style-type=numeric
|There is an element ''A'' of the first matched set which prefers some given element ''B'' of the second matched set over the element to which ''A'' is already matched, and
|''B'' also prefers ''A'' over the element to which ''B'' is already matched.
}}

In other words, a matching is stable when there does not exist any pair (''A'', ''B'') which both prefer each other to their current partner under the matching.

The stable marriage problem has been stated as follows:
{{quote|Given ''n'' men and ''n'' women, where each person has ranked all members of the opposite sex in order of preference, marry the men and women together such that there are no two people of opposite sex who would both rather have each other than their current partners. When there are no such pairs of people, the set of marriages is deemed stable.}}
The existence of two classes that need to be paired with each other (heterosexual men and women in this example) distinguishes this problem from the [[stable roommates problem]].