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{{Short description|Procedure for finding a stable matching}}

In [[mathematics]], [[economics]], and [[computer science]], the '''Gale–Shapley algorithm''' (also known as the '''deferred acceptance algorithm''') is an [[algorithm]] for finding a solution to the [[stable matching problem]], named for [[David Gale]] and [[Lloyd Shapley]].

{{Good article}}

It takes [[polynomial time]], and the time is [[linear time|linear]] in the size of the input to the algorithm. Depending on how it is used, it can find either the solution that is optimal for the participants on one side of the matching, or for the participants on the other side. It is a [[truthful mechanism]] from the point of view of the participants for whom it provides the optimal solution.

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In [[mathematics]], [[economics]], and [[computer science]], the '''Gale–Shapley algorithm''' (also known as the '''deferred acceptance algorithm''',{{r|roth}} '''propose-and-reject algorithm''',{{r|carter-price}} or '''Boston Pool algorithm'''{{r|roth}}) is an [[algorithm]] for finding a solution to the [[stable matching problem]]. It is named for [[David Gale]] and [[Lloyd Shapley]], who published it in 1962, although it had been used for the [[National Resident Matching Program]] since the early 1950s. Shapley and [[Alvin E. Roth]] (who pointed out its prior application) won the 2012 [[Nobel Memorial Prize in Economic Sciences|Nobel Prize in Economics]] for work including this algorithm.

The stable matching problem seeks to pair up equal numbers of participants of two types, using preferences from each participant. The pairing must be stable: no pair of matched participants should mutually prefer each other to their assigned match. In each round of the Gale–Shapley algorithm, unmatched participants of one type propose a match to the next participant on their preference list. Each proposal is accepted if its recipient prefers it to their current match. The resulting procedure is a [[truthful mechanism]] from the point of view of the proposing participants, who receive their most-preferred pairing consistent with stability. In contrast, the recipients of proposals receive their least-preferred pairing. The algorithm can be implemented to run in time quadratic in the number of participants, and [[linear time|linear]] in the size of the input to the algorithm.

The stable matching problem, and the Gale–Shapley algorithm solving it, have widespread real-world applications, including matching American medical students to residencies and French university applicants to schools. For more, see {{slink|Stable marriage problem|Applications}}.

==Background==

{{main|stableStable matching problem}}

The stable matching problem, in its most basic form, takes as input equal numbers of two types of participants ({{mvar|n}} menjob and {{mvar|n}} women, or {{mvar|n}} medical studentsapplicants and {{mvar|n}} internshipsemployers, for example), and an ordering for each participant giving their preference for whom to be matched to among the participants of the other type. A matching pairs each participant of one type with a participant of the other type. A matching is ''not'' stable if:

{{Ordered list|list-style-type=numeric

Line 11 ⟶ 19:

}}

In other words, a matching is stable when there doesis not exist anyno matchpair (''A'', ''B'') whichwhere both participants prefer each other to their currentmatched partnerpartners. underIf such a pair exists, the matching is not stable, in the sense that the members of this pair would prefer to leave the system and be matched to each other, possibly leaving other participants unmatched. A stable matching always exists, and the algorithmic problem solved by the Gale–Shapley algorithm is to find one.{{r|gusfield-irving}}

A stable matching always exists, and the algorithmic problem solved by the Gale–Shapley algorithm is to find one.

The stable matching problem has also been called the ''stable marriage problem'', using a metaphor of marriage between men and women, and many sources describe the Gale–Shapley algorithm in terms of [[Marriage proposal|marriage proposals]]. However, this metaphor has been criticized as both sexist and unrealistic: the steps of the algorithm do not accurately reflect typical or even stereotypical human behavior.{{r|wagner|giagkousi}}

==Solution==

[[File:Gale-Shapley.gif|thumb|rightupright=1.35|Animation showing an example of the Gale–Shapley algorithm]]

In 1962, [[David Gale]] and [[Lloyd Shapley]] proved that, for any equal number of menparticipants andof each womentype, it is always possible to solvefind thea SMPmatching andin makewhich all marriagespairs are stable. They presented an [[algorithm]] to do so.<ref>{{cite journal r|first=D. gale-shapley|last=Gale |first2=L. S. |last2=Shapley |title=College Admissions and the Stability of Marriage |journal=[[American Mathematical Monthly]] |volume=69 |issue= 1|pages=9–14 |year=1962 |jstor=2312726 |doi=10.2307/2312726|url=http://www.dtic.mil/get-tr-doc/pdf?AD=AD0251958 mairson}}</ref><ref>[[Harry Mairson]]: "The Stable Marriage Problem", ''The Brandeis Review'' 12, 1992 ([http://www1.cs.columbia.edu/~evs/intro/stable/writeup.html online]).</ref> In 1984, [[Alvin E. Roth]] observed that essentially the same algorithm had already been in practical use since the early 1950s, inas the "Boston Pool algorithm" used by the [[National Resident Matching Program]].<ref>{{cite journalr|last=Bergstrom|first=Theodore C.|date=June 1992|issue=2|journal=[[Journal of Economic Literature]]|jstor=2727713|pages=896–898|title=Review of ''Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis'' by A. E. Roth and M. A. O. Sotomayorroth|volume=30bergstrom}}</ref>

The Gale–Shapley algorithm involves a number of "rounds" (or "[[iteration]]s"). In terms of job applicants and employers, it can be expressed as follows:{{r|jeffe}}

* In each round, one or more employers with open job positions each make a job offer to the applicant they prefer, among the ones they have not yet already made an offer to.

* Each applicant who has received an offer evaluates it against their current position (if they have one). If the applicant is not yet employed, or if they receive an offer from an employer they like better than their current employer, they accept the best new offer and become matched to the new employer (possibly leaving a previous employer with an open position). Otherwise, they reject the new offer.

* This process is repeated until all employers have either filled their positions or exhausted their lists of applicants.

===Implementation details and time analysis===

The '''Gale–Shapley algorithm''' involves a number of "rounds" (or "[[iteration]]s"):

To implement the algorithm efficiently, each employer needs to be able to find its next applicant quickly, and each applicant needs to be able to compare employers quickly. One way to do this is to number each applicant and each employer from 1 to <math>n</math>, where <math>n</math> is the number of employers and applicants, and to store the following [[data structure]]s:{{r|kt}}

* In the first round, first ''a'') each unengaged man proposes to the woman he prefers most, and then ''b'') each woman replies "maybe" to her suitor she most prefers and "no" to all other suitors. She is then provisionally "engaged" to the suitor she most prefers so far, and that suitor is likewise provisionally engaged to her.

*A [[Set (abstract data type)|set]] of employers with unfilled positions

* In each subsequent round, first ''a'') each unengaged man proposes to the most-preferred woman to whom he has not yet proposed (regardless of whether the woman is already engaged), and then ''b'') each woman replies "maybe" if she is currently not engaged or if she prefers this man over her current provisional partner (in this case, she rejects her current provisional partner who becomes unengaged). The provisional nature of engagements preserves the right of an already-engaged woman to "trade up" (and, in the process, to "jilt" her until-then partner).

*A one-dimensional [[Array (data structure)|array]] indexed by employers, specifying the preference index of the next applicant to whom the employer would send an offer, initially 1 for each employer

* This process is repeated until everyone is engaged.

*A one-dimensional array indexed by applicants, specifying their current employer, initially a [[sentinel value]] such as 0 indicating they are unemployed

*A two-dimensional array indexed by an applicant and an employer, specifying the position of that employer in the applicant's preference list

*A two-dimensional array indexed by an employer and a number <math>i</math> from 1 to <math>n</math>, naming the applicant who is each employer's {{nowrap|<math>i</math>th}} preference

Setting up these data structures takes <math>O(n^2)</math> time. With these structures it is possible to find an employer with an unfilled position, make an offer from that employer to their next applicant, determine whether the offer is accepted, and update all of the data structures to reflect the results of these steps, in constant time per offer. Once the algorithm terminates, the resulting matching can be read off from the array of employers for each applicant. There can be <math>O(n^2)</math> offers before each employer runs out of offers to make, so the total time is <math>O(n^2)</math>.{{r|kt}}

Although this time bound is [[Quadratic growth|quadratic]] in the number of participants, it may be considered as [[linear time]] when measured in terms of the size of the input, two matrices of preferences of size <math>O(n^2)</math>.{{sfnp|Gusfield|Irving|1989|p=182}}

The runtime complexity of this algorithm is <math>O(n^2)</math> where <math>n</math> is the number of men or women.<ref name="IwamaMiyazaki2008">{{cite journal|last1=Iwama|first1=Kazuo|author1-link=Kazuo Iwama (computer scientist)|last2=Miyazaki|first2=Shuichi|title=A Survey of the Stable Marriage Problem and Its Variants|year=2008|pages=131–136|doi=10.1109/ICKS.2008.7|journal=International Conference on Informatics Education and Research for Knowledge-Circulating Society (icks 2008)|isbn=978-0-7695-3128-1|hdl=2433/226940|url=https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/226940/1/ICKS.2008.7.pdf}}</ref>

===Correctness guarantees===

This algorithm guarantees that:

; Everyone gets matched: At the end, there cannot be an applicant and employer both unmatched. An employer left unmatched at the end of the process must have made an offer to all applicants. But an applicant who receives an offer remains employed for the rest of the process, so there can be no unemployed applicants. Since the numbers of applicants and job openings are equal, there can also be no open positions remaining.{{r|jeffe}}

; The matches are stable: No applicant ''X'' and employer ''Y'' can prefer each other over their final match. If ''Y'' makes an offer to ''X'', then ''X'' would only reject ''Y'' after receiving an even better offer, so ''X'' cannot prefer ''Y'' to their final match. And if ''Y'' stops making offers before reaching ''X'' in their preference list, ''Y'' cannot prefer ''X'' to their final match. In either case, ''X'' and ''Y'' do not form an unstable pair.{{r|jeffe}}

==Optimality of the solution==

; Everyone gets married : At the end, there cannot be a man and a woman both unengaged, as he must have proposed to her at some point (since a man will eventually propose to everyone, if necessary) and, being proposed to, she would necessarily be engaged (to someone) thereafter.

{{main|Lattice of stable matchings}}

; The marriages are stable : Let Alice and Bob both be engaged, but not to each other. Upon completion of the algorithm, it is not possible for both Alice and Bob to prefer each other over their current partners. If Bob prefers Alice to his current partner, he must have proposed to Alice before he proposed to his current partner. If Alice accepted his proposal, yet is not married to him at the end, she must have dumped him for someone she likes more, and therefore doesn't like Bob more than her current partner. If Alice rejected his proposal, she was already with someone she liked more than Bob.

There may be many stable matchings for the same system of preferences. This raises the question: which matching is returned by the Gale–Shapley algorithm? Is it the matching better for applicants, for employers, or an intermediate one? As it turns out, the Gale–Shapley algorithm in which employers make offers to applicants always yields the same stable matching (regardless of the order in which job offers are made), and its choice is the stable matching that is the ''best for all employers'' and ''worst for all applicants'' among all stable matchings.{{r|jeffe}}

In a reversed form of the algorithm, each round consists of unemployed applicants writing a single job application to their preferred employer, and the employer either accepting the application (possibly firing an existing employee to do so) or rejecting it. This produces a matching that is best for all applicants and worst for all employers among all stable matchings. These two matchings are the top and bottom elements of the [[lattice of stable matchings]].{{r|knuth}}

==Algorithm==

In both forms of the algorithm, one group of participants proposes matches, and the other group decides whether to accept or reject each proposal. The matching is always best for the group that makes the propositions, and worst for the group that decides how to handle each proposal.{{r|knuth}}

'''algorithm''' stable_matching '''is'''

Initialize all ''m'' ∈ M and ''w'' ∈ W to ''free''

'''while''' ∃ ''free'' man ''m'' who still has a woman w to propose to '''do'''

w := first woman on m's list to whom m has not yet proposed

'''if''' w is ''free'' '''then'''

(m, w) become ''engaged''

'''else''' some pair (m', w) already exists

'''if''' w prefers m to m' '''then'''

m' becomes ''free''

(m, w) become ''engaged''

'''else'''

(m', w) remain ''engaged''

'''end if'''

'''end if'''

'''repeat'''

== Strategic considerations ==

==Optimality of the solution==

The Gale–Shapley algorithm is a [[truthful mechanism]] from the point of view of the proposing side. This means that no proposer can get a better matching by misrepresenting their preferences. Moreover, the Gale–Shapley algorithm is even ''group-strategy proof'' for proposers, i.e., no coalition of proposers can coordinate a misrepresentation of their preferences such that all proposers in the coalition are strictly better-off.{{r|dubins-freedman}} However, it is possible for some coalition to misrepresent their preferences such that some proposers are better-off, and the others retain the same partner.{{r|huang}}

{{main|Lattice of stable matchings}}

[[File:DA-men-optimality.png|thumb|Proof that deferred acceptance is optimal for men]]

The existence of different stable matchings raises the question: which matching is returned by the Gale-Shapley algorithm? Is it the matching better for men, for women, or the intermediate one? In the above example, the algorithm converges in a single round on the men-optimal solution because each woman receives exactly one proposal, and therefore selects that proposal as her best choice, ensuring that each man has an accepted offer, ending the match.

The Gale–Shapley algorithm is non-truthful for the non-proposing participants. Each may be able to misrepresent their preferences and get a better match.{{r|sisterhood}} A particular form of manipulation is ''truncation'': presenting only the topmost alternatives, implying that the bottom alternatives are not acceptable at all. Under complete information, it is sufficient to consider misrepresentations of the form of truncation strategies. However, successful misrepresentation requires knowledge of the other agents' preferences; without such knowledge, misrepresentation can give an agent a worse assignment. Moreover, even after an agent sees the final matching, they cannot deduce a strategy that would guarantee a better outcome in hindsight. This makes the Gale–Shapley algorithm a [[Regret-free mechanism|''regret-free truth-telling mechanism'']]. Moreover, in the Gale–Shapley algorithm, truth-telling is the only strategy that guarantees no regret. The Gale–Shapley algorithm is the only regret-free mechanism in the class of quantile-stable matching mechanisms.<ref>{{Cite web |last=Fernandez |first=Marcelo Ariel |date=July 31, 2020 |title=Deferred acceptance and regret-free truth-telling |url=https://ideas.repec.org//p/jhu/papers/65832.html |type=Working paper|publisher=Johns Hopkins University Department of Economics}}</ref>

This is a general fact: the Gale-Shapley algorithm in which men propose to women ''always'' yields a stable matching that is the ''best for all men'' among all stable matchings. Similarly, if the women propose then the resulting matching is the ''best for all women'' among all stable matchings. These two matchings are the top and bottom elements of a larger structure on all stable matchings, the [[lattice of stable matchings]].

==Generalizations==

To see this, consider the illustration at the right. Let A be the matching generated by the men-proposing algorithm, and B an alternative stable matching that is better for at least one man, say ''m''₀. Suppose ''m''₀ is matched in B to a woman ''w''₁, which he prefers to his match in A. This means that in A, ''m''₀ has visited ''w''₁, but she rejected him (this is denoted by the green arrow from ''m''₀ to ''w''₁). ''w''₁ rejected him in favor of some man that she prefers, say ''m''₂. So in B, ''w''₁ is matched to ''m''₀ but "yearns" (wants but unmatched) for ''m''₂ (this is denoted by the red arrow from ''w''₁ to ''m''₂).

In their original work on the problem, Gale and Shapley considered a more general form of the stable matching problem, suitable for [[university and college admission]]. In this problem, each university or college may have its own ''quota'', a target number of students to admit, and the number of students applying for admission may differ from the sum of the quotas, necessarily causing either some students to remain unmatched or some quotas to remain unfilled. Additionally, preference lists may be incomplete: if a university omits a student from their list, it means they would prefer to leave their quota unfilled than to admit that student, and if a student omits a university from their list, it means they would prefer to remain unadmitted than to go to that university. Nevertheless, it is possible to define stable matchings for this more general problem, to prove that stable matchings always exist, and to apply the same algorithm to find one.{{r|gale-shapley}}

A form of the Gale–Shapley algorithm, performed through a real-world protocol rather than calculated on computers, has been used for coordinating higher education admissions in France since 2018, through the [[Parcoursup]] system. In this process, over the course of the summer before the start of school, applicants receive offers of admission, and must choose in each round of the process whether to accept any new offer (and if so turn down any previous offer that they accepted). The method is complicated by additional constraints that make the problem it solves not exactly the stable matching problem. It has the advantage that the students do not need to commit to their preferences at the start of the process, but rather can determine their own preferences as the algorithm progresses, on the basis of head-to-head comparisons between offers that they have received. It is important that this process performs a small number of rounds of proposals, so that it terminates before the start date of the schools, but although high numbers of rounds can occur in theory, they tend not to occur in practice.{{r|mathieu}} It has been shown theoretically that, if the Gale–Shapley algorithm needs to be terminated early, after a small number of rounds in which every vacant position makes a new offer, it nevertheless produces matchings that have a high ratio of matched participants to unstable pairs.{{r|almost}}

Since B is a stable matching, ''m''₂ must be matched in B to some woman he prefers to ''w''₁, say ''w''₃. This means that in A, ''m''₂ has visited ''w''₃ before arriving at ''w''₁, which means that ''w''₃ has rejected him. By similar considerations, and since the graph is finite, we must eventually have a directed cycle in which each man was rejected in A by the next woman in the cycle, who rejected him in favor of the next man in the cycle. But this is impossible since such "cycle of rejections" cannot start anywhere: suppose by contradiction that it starts at e.g. ''m''₀ - the first man rejected by his adjacent woman (''w''₁). By assumption, this rejection happens only after ''m''₂ comes to ''w''₁. But this can happen only after ''w''₃ rejects ''m''₂ - contradiction to ''m''₀ being the first.

==Recognition==

== Strategic considerations ==

Shapley and Roth were awarded the 2012 [[Nobel Memorial Prize in Economic Sciences]] "for the theory of stable allocations and the practice of [[market design]]". Gale had died in 2008, making him ineligible for the prize.{{r|nobel}}

The GS algorithm is a [[truthful mechanism]] from the point of view of men (the proposing side). I.e, no man can get a better matching for himself by misrepresenting his preferences. Moreover, the GS algorithm is even ''group-strategy proof'' for men, i.e., no coalition of men can coordinate a misrepresentation of their preferences such that all men in the coalition are strictly better-off.<ref>{{citation

==See also==

*[[Deferred-acceptance auction]]

*[[Stable roommates problem]]

*[[Kolkata Paise Restaurant Problem]]

==References==

{{reflist|refs=

<ref name=almost>{{cite journal

| last1 = Floréen | first1 = Patrik

| last2 = Kaski | first2 = Petteri

| last3 = Polishchuk | first3 = Valentin

| last4 = Suomela | first4 = Jukka

| date = August 2009

| doi = 10.1007/s00453-009-9353-9

| issue = 1

| journal = Algorithmica

| pages = 102–118

| title = Almost stable matchings by truncating the Gale–Shapley algorithm

| volume = 58| arxiv = 0812.4893

}}</ref>

<ref name=bergstrom>{{cite journal|last=Bergstrom|first=Theodore C.|date=June 1992|issue=2|journal=[[Journal of Economic Literature]]|jstor=2727713|pages=896–898|title=Review of ''Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis'' by A. E. Roth and M. A. O. Sotomayor|volume=30}}</ref>

<ref name=carter-price>{{cite book

| last1 = Carter | first1 = Michael W.

| last2 = Price | first2 = Camille C.

| isbn = 9780849322563

| page = 102

| publisher = CRC Press

| title = Operations Research: A Practical Introduction

|url=https://books.google.com/books?id=CbFwJLHpX7sC&pg=PA102

| year = 2000}}</ref>

<ref name=dubins-freedman>{{cite journal

| last1 = Dubins | first1 = L. E. | author1-link = Lester Dubins

| last2 = Freedman | first2 = D. A. | author2-link = David A. Freedman

| doi = 10.2307/2321753

| issue = 7

| journal = [[The American Mathematical Monthly]]

| mr = 628016

| pages = 485–494

| title = Machiavelli and the Gale–Shapley algorithm

| volume = 88

| year = 1981| jstor = 2321753 }}</ref>

| year = 1981}}</ref> However, it is possible for some coalition to misrepresent their preferences such that some men are better-off and the other men retain the same partner.<ref>{{cite conference

<ref name=giagkousi>{{cite thesis |last=Giagkousi |first=Kyriaki |date=March 2021 |title=Gender and Computing Algorithms: The case of Stable Matching |url=https://pergamos.lib.uoa.gr/uoa/dl/object/2940468/file.pdf |degree=Master's |publisher=National and Kapodistrian University of Athens, Department of History and Philosophy of Science and Department of Informatics and Telecommunications |access-date=2023-12-20}}</ref>

<ref name=gale-shapley>{{cite journal |first1=D. |last1=Gale|author1-link=David Gale |first2=L. S. |last2=Shapley|author2-link=Lloyd Shapley |title=College admissions and the stability of marriage |journal=[[The American Mathematical Monthly]] |volume=69 |issue=1 |pages=9–14 |year=1962 |jstor=2312726 |doi=10.2307/2312726 |url=http://www.dtic.mil/get-tr-doc/pdf?AD=AD0251958 |access-date=2019-11-20 |archive-date=2017-09-25 |archive-url=https://web.archive.org/web/20170925172517/http://www.dtic.mil/get-tr-doc/pdf?AD=AD0251958 |url-status=dead }}</ref>

<ref name=gusfield-irving>{{cite book

| last1 = Gusfield | first1 = Dan | author1-link = Dan Gusfield

| last2 = Irving | first2 = Robert W.

| isbn = 9780262515528

| page = 6

| publisher = MIT Press

| title = The Stable Marriage Problem: Structure and Algorithms

| year = 1989}}</ref>

<ref name=huang>{{cite conference

| last = Huang | first = Chien-Chung

| editor1-last = Azar | editor1-first = Yossi

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| publisher = Springer

| series = Lecture Notes in Computer Science

| title = Algorithms -– ESA 2006, 14th Annual European Symposium, Zurich, Switzerland, September 11-1311–13, 2006, Proceedings

| volume = 4168

| year = 2006}}</ref>

<ref name=jeffe>{{cite book

The GS algorithm is non-truthful for the women (the reviewing side): each woman may be able to misrepresent her preferences and get a better match.

| last = Erickson | first = Jeff

| contribution = 4.5 Stable matching

| contribution-url=https://jeffe.cs.illinois.edu/teaching/algorithms/book/04-greedy.pdf

| access-date = 2023-12-19

| date = June 2019

| pages = 170–176

| publisher = University of Illinois

| title = Algorithms}}</ref>

<ref name=knuth>{{cite book

==Implementation in software packages==

| last = Knuth | first = Donald E. | authorlink = Donald Knuth

| isbn = 0-8405-0342-3

| language = French

| mr = 0488980

| publisher = Les Presses de l'Université de Montréal

| ___location = Montréal, Quebec

| title = Mariages stables et leurs relations avec d'autres problèmes combinatoires

|url= https://www-cs-faculty.stanford.edu/~knuth/mariages-stables.pdf

| year = 1976}} See in particular Problem 6, pp. 87–94.</ref>

<ref name=kt>{{cite book

*[[R (programming language)|R]]: The Gale–Shapley algorithm (also referred to as deferred-acceptance algorithm) for the stable marriage and the [[Hospital resident|hospitals/residents problem]] is available as part of the <code>matchingMarkets</code><ref>{{cite journal|last=Klein|first=T.|year=2015|title=Analysis of Stable Matchings in R: Package matchingMarkets|url=http://cran.at.r-project.org/web/packages/matchingMarkets/vignettes/matching.pdf|journal=Vignette to R Package MatchingMarkets}}</ref><ref>{{cite web|url=http://cran.at.r-project.org/web/packages/matchingMarkets/|title=matchingMarkets: Analysis of Stable Matchings|date=|work=R Project}}</ref> and <code>matchingR</code><ref>{{cite web|url=https://cran.r-project.org/web/packages/matchingR/|title=matchingR: Matching Algorithms in R and C++|date=|work=R Project}}</ref> packages.

| last1 = Kleinberg | first1 = Jon | author1-link = Jon Kleinberg

*[[Application programming interface|API]]: The MatchingTools API provides a free application programming interface for the Gale–Shapley algorithm.<ref>{{cite web|url=https://matchingtools.com|title=MatchingTools API}}</ref>

| last2 = Tardos | first2 = Éva | author2-link = Éva Tardos

*[[Python (programming language)|Python]]: The Gale–Shapley algorithm is included along with several other well-known matching game algorithms in the <code>matching</code> library,<ref>{{cite journal | first=H. | last=Wilde | first2=V. | last2=Knight | first3=J. | last3=Gillard | title=Matching: A Python library for solving matching games | journal=Journal of Open Source Software | year=2020 | doi=10.21105/joss.02169| doi-access=free }}</ref> and <code>QuantEcon/MatchingMarkets.py</code> package<ref>{{cite web|url=https://github.com/QuantEcon/MatchingMarkets.py|title=matchingMarkets.py|date=|work=Python package}}</ref> includes several others for generalized matching games.

| contribution = 2.3 Implementing the stable matching algorithm using lists and arrays

*[[MATLAB]]: The Gale–Shapley algorithm is implemented in the <code>assign2DStable</code> function that is part of the [[United States Naval Research Laboratory |United States Naval Research Laboratory's]] free Tracker Component Library.<ref>{{cite web|url=https://github.com/USNavalResearchLaboratory/TrackerComponentLibrary|title=Tracker Component Library|work=Matlab Repository|access-date=January 5, 2019}}</ref>

| pages = 42–47

| publisher = Addison-Wesley

| title = Algorithm Design

| year = 2006}}</ref>

<ref name=mairson>{{cite journal

==See also==

| last = Mairson | first = Harry | author-link = Harry Mairson

*[[Deferred-acceptance auction]]

| journal = The Brandeis Review

| title = The stable marriage problem

|url= https://www1.cs.columbia.edu/~evs/intro/stable/writeup.html

| volume = 12

| year = 1992}}</ref>

<ref name=mathieu>{{cite web|url=https://www.youtube.com/watch?v=wQuXE3wB4s8|first=Claire|last=Mathieu|author-link=Claire Mathieu|title=College admission algorithms in the real world|type=Invited lecture at the European Symposium of Algorithms|year=2018|publisher=Aalto University}}</ref>

==References==

{{reflist}}

<ref name=nobel>{{cite journal

==External links==

| last = Bhattacharjee | first = Yudhijit

*[http://www.sephlietz.com/gale-shapley/ Gale–Shapley JavaScript Demonstration]

| date = October 15, 2012

| journal = [[Science (journal)|Science]]

| title = Economics Nobel honors matchmaking finesse | volume = 338

| issue = 6105

| page = 314

| doi = 10.1126/science.338.6105.314

| pmid = 23087221}}</ref>

<ref name=roth>{{cite journal

| last = Roth | first = Alvin E. | author-link = Alvin E. Roth

| date = February 2003

| doi = 10.1001/jama.289.7.909

| issue = 7

| journal = [[JAMA]]

| pages = 909–912

| title = The origins, history, and design of the resident match

| volume = 289}}</ref>

<ref name=sisterhood>{{cite journal

| last1 = Gonczarowski | first1 = Yannai A.

| last2 = Friedgut | first2 = Ehud

| date = April 2013

| doi = 10.37236/3267

| issue = 2

| journal = [[Electronic Journal of Combinatorics]]

| pages = P12:1–P12:18

| title = Sisterhood in the Gale–Shapley matching algorithm

| volume = 20| doi-access = free

| arxiv = 1104.2217

}}</ref>

<ref name=wagner>{{cite journal

| last = Wagner | first = Roy

| date = April 2009

| doi = 10.1177/0306312708099443

| issue = 2

| journal = [[Social Studies of Science]]

| pages = 289–308

| title = Mathematical marriages: Intercourse between mathematics and semiotic choice

| volume = 39| hdl = 20.500.11850/121579

| hdl-access = free

}}</ref>

}}

{{DEFAULTSORT:Gale-Shapley algorithm}}

[[Category:Stable matching]]

[[Category:Lloyd Shapley]]

[[Category:AlgorithmsCombinatorial algorithms]]