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{{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
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}}
In other words, a matching is stable when there does not exist any
The stable marriage problem has been stated as follows:
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==Applications==
Algorithms for finding solutions to the stable marriage problem have applications in a variety of real-world situations, perhaps the best known of these being in the assignment of graduating medical students to their first hospital appointments.<ref>[http://www.dcs.gla.ac.uk/research/algorithms/stable/ Stable Matching Algorithms]</ref> In 2012, the [[Nobel Memorial Prize in Economic Sciences]] was awarded to [[Lloyd S. Shapley]] and [[Alvin E. Roth]] "for the theory of stable allocations and the practice of market design."<ref>{{cite web|url=https://www.nobelprize.org/nobel_prizes/economics/laureates/2012/ |title=The Prize in Economic Sciences 2012 |publisher=Nobelprize.org |access-date
An important and large-scale application of stable marriage is in assigning users to servers in a large distributed Internet service.<ref name=nuggets>{{cite journal | author=
The [[Gale–Shapley algorithm]] for stable matching is used to assign rabbis who graduate from [[Hebrew Union College – Jewish Institute of Religion|Hebrew Union College]] to Jewish congregations.<ref>{{Cite journal |last1=Bodin |first1=Lawrence |last2=Panken |first2=Aaron |date=June 2003 |title=High Tech for a Higher Authority: The Placement of Graduating Rabbis from Hebrew Union College—Jewish Institute of Religion |url=https://pubsonline.informs.org/doi/10.1287/inte.33.3.1.16013 |journal=Interfaces |language=en |volume=33 |issue=3 |pages=1–11 |doi=10.1287/inte.33.3.1.16013 |issn=0092-2102|url-access=subscription }}</ref>
==Different stable matchings==
{{
In general, there may be many different stable matchings. For example, suppose there are three men (A, B, C) and three women (X, Y, Z) which have preferences of:
: A: YXZ B: ZYX C: XZY
: X: BAC Y: CBA Z: ACB
There are three stable solutions to this matching arrangement:
* men get their first choice and women their third
* all participants get their second choice
* women get their first choice and men their third
All three are stable, because instability requires both of the participants to be happier with an alternative match. Giving one group their first choices ensures that the matches are stable because they would be unhappy with any other proposed match. Giving everyone their second choice ensures that any other match would be disliked by one of the parties. In general, the family of solutions to any instance of the stable marriage problem can be given the structure of a finite [[distributive lattice]],
and this structure leads to efficient algorithms for several problems on stable marriages.<ref>{{cite journal
| last = Gusfield | first = Dan |
| doi = 10.1137/0216010
| issue = 1
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| title = Proceedings of the 50th Symposium on Theory of Computing (STOC 2018)
| year = 2018| arxiv = 1711.01032
| isbn = 978-1-4503-5559-9 }}</ref>
Counting the number of stable matchings in a given instance is [[♯P-complete|#P-complete]].<ref>{{cite journal
| last1 = Irving | first1 = Robert W.
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==Algorithmic solution==
{{
[[File:Gale-Shapley.gif|thumb|right|Animation showing an example of the Gale–Shapley algorithm]]
In 1962, [[David Gale]] and [[Lloyd Shapley]] proved that, for any equal number in different groups, in the context of
The [[Gale–Shapley algorithm]] (also known as the deferred acceptance algorithm) involves a number of "rounds" (or "[[iteration]]s"):
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* This process is repeated until everyone is engaged.
This algorithm is guaranteed to produce a stable marriage for all participants in [[Big O notation|time]] <math>O(n^2)</math> where <math>n</math> is the number of men or women.<ref name="IwamaMiyazaki2008">{{cite conference|last1=Iwama|first1=Kazuo|author1-link=Kazuo Iwama (computer scientist)|last2=Miyazaki|first2=Shuichi|contribution=A Survey of the Stable Marriage Problem and Its Variants|year=2008|pages=131–136|doi=10.1109/ICKS.2008.7|title=International Conference on Informatics Education and Research for Knowledge-Circulating Society (ICKS 2008)|publisher=IEEE|isbn=978-0-7695-3128-1|hdl=2433/226940|hdl-access=free}}</ref>
Among all possible different stable matchings, it always yields the one that is best for all men among all stable matchings, and worst for all women.<ref>{{cite book
| last = Erickson | first = Jeff
It 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>{{cite journal▼
| 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>
▲It is a [[truthful mechanism]] from the point of view of men (the proposing side)
| last1 = Dubins | first1 = L. E. | author1-link = Lester Dubins
| last2 = Freedman | first2 = D. A. | author2-link = David A. Freedman
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| editor1-last = Azar | editor1-first = Yossi
| editor2-last = Erlebach | editor2-first = Thomas
| contribution = Cheating by men in the
| doi = 10.1007/11841036_39
| mr = 2347162
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| publisher = Springer
| series = Lecture Notes in Computer Science
| title = Algorithms
| volume = 4168
| year = 2006
}}</ref>
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.
==
{{
The
* Each participant may only be willing to be matched to a subset of the participants on the other side of the matching.
* The participants on one side of the matching (the hospitals) may have a numerical capacity, specifying the number of doctors they are willing to hire.
* The total number of participants on one side might not equal the total capacity to which they are to be matched on the other side.
* The resulting matching might not match all of the participants.
In this case, the condition of stability is that no unmatched pair prefer each other to their situation in the matching (whether that situation is another partner or being unmatched). With this condition, a stable matching will still exist, and can still be found by the Gale–Shapley algorithm.
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* Any hospital that has some empty positions in some stable matching, receives exactly the same set of doctors in ''all'' stable matchings.
==
In '''[[stable matching with indifference]]''', some men might be indifferent between two or more women and vice versa.
The '''[[stable roommates problem]]''' is similar to the stable marriage problem, but differs in that all participants belong to a single pool (instead of being divided into equal numbers of "men" and "women").
The '''[[Hospital resident|hospitals/residents problem]]''' – also known as the '''college admissions problem''' – differs from the stable marriage problem in that a hospital can take multiple residents, or a college can take an incoming class of more than one student. Algorithms to solve the hospitals/residents problem can be ''hospital-oriented'' (as the [[National Resident Matching Program|NRMP]] was before 1995)<ref name="Robinson">{{cite journal|last=Robinson|first=Sara|date=April 2003|title=Are Medical Students Meeting Their (Best Possible) Match?|url=http://www.siam.org/pdf/news/305.pdf|journal=SIAM News|issue=3|page=36|
The '''[[Hospital resident|hospitals/residents problem with couples]]''' allows the set of residents to include couples who must be assigned together, either to the same hospital or to a specific pair of hospitals chosen by the couple (e.g., a married couple want to ensure that they will stay together and not be stuck in programs that are far away from each other). The addition of couples to the hospitals/residents problem renders the problem [[NP-complete]].<ref>{{cite book|title=The Stable Marriage Problem: Structure and Algorithms|last1=Gusfield|first1=D.|last2=Irving|first2=R. W.|publisher=MIT Press|year=1989|isbn=0-262-07118-5
The '''[[assignment problem]]''' seeks to find a matching in a weighted [[bipartite graph]] that has maximum weight. Maximum weighted matchings do not have to be stable, but in some applications a maximum weighted matching is better than a stable one.
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The '''matching with contracts''' problem is a generalization of matching problem, in which participants can be matched with different terms of contracts.<ref>{{cite journal |first1=John William |last1=Hatfield |first2=Paul |last2=Milgrom |title=Matching with Contracts |journal=[[American Economic Review]] |volume=95 |issue=4 |year=2005 |pages=913–935 |jstor=4132699 |doi=10.1257/0002828054825466}}</ref> An important special case of contracts is matching with flexible wages.<ref>{{cite journal |first1=Vincent |last1=Crawford |first2=Elsie Marie |last2=Knoer |title=Job Matching with Heterogeneous Firms and Workers |year=1981 |journal=[[Econometrica]] |volume=49 |issue=2 |pages=437–450 |jstor=1913320 |doi=10.2307/1913320}}</ref>
==
* [[Matching (graph theory)]]
* [[Envy-free matching]] – a relaxation of stable matching for many-to-one matching problems
▲*[[Matching (graph theory)]] - matching between different vertices of the graph; usually unrelated to preference-ordering.
* [[
* [[
* [[Lattice of stable matchings]]
* [[Secretary problem]] (also called '''marriage problem''') – deciding when to stop to obtain the best reward in a sequence of options
==References==
{{
==Further reading==
* Kleinberg, J., and Tardos, E. (2005) ''Algorithm Design'', Chapter 1, pp 1–12. See companion website for the Text [http://www.aw-bc.com/info/kleinberg/] {{Webarchive|url=https://web.archive.org/web/20110514100625/http://www.aw-bc.com/info/kleinberg/ |date=2011-05-14 }}.
* {{cite book |
* {{cite journal |last=Pittel |first=B. |year=1992 |title=On likely solutions of a stable marriage problem |journal=[[The Annals of Applied Probability]] |volume=2 |issue=2 |pages=358–401 |doi=10.1214/aoap/1177005708 |jstor=2959755 |doi-access=free }}
* {{cite journal |last=Roth |first=A. E. |year=1984 |title=The evolution of the labor market for medical interns and residents: A case study in game theory |journal=[[Journal of Political Economy]] |volume=92 |issue=6 |pages=991–1016 |doi=10.1086/261272 |s2cid=1360205 |url=http://dash.harvard.edu/bitstream/handle/1/29410143/evolut.pdf }}
* {{cite book | last1=Roth |first1=A. E. |last2=Sotomayor |first2=M. A. O. |year=1990 |title=Two-sided matching: A study in game-theoretic modeling and analysis |publisher=[[Cambridge University Press]] |title-link= Two-Sided Matching}}
* {{cite book | last1=Shoham | first1=Yoav | last2=Leyton-Brown | first2=Kevin | title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations | publisher=[[Cambridge University Press]] | isbn=978-0-521-89943-7 | url=http://www.masfoundations.org | year=2009 | ___location=New York}} See Section 10.6.4; [http://www.masfoundations.org/download.html downloadable free online].
* {{cite book|author1=Schummer, J.|author2=Vohra, R. V.| isbn = 978-0521872829 |chapter=Mechanism design without money| title = Algorithmic Game Theory | editor1-last = Nisan | editor1-first= Noam |year=2007 | chapter-url = http://www.cambridge.org/journals/nisan/downloads/Nisan_Non-printable.pdf | editor2-last = Roughgarden | editor2-first= Tim | editor3-last = Tardos | editor3-first= Eva | editor4-last = Vazirani | editor4-first= Vijay | pages = 255–262}}
* {{cite book | last1=Gusfield | first1=D. | last2=Irving | first2= R.W. | title=The Stable Marriage Problem: Structure and Algorithms. | publisher=[[MIT Press]] | year=1989 | url=https://archive.org/details/stablemarriagepr0000gusf/mode/2up | isbn=0-262-07118-5}}
==External links==
*[http://mathsite.math.berkeley.edu/smp/smp.html Interactive
*https://web.archive.org/web/20080512150525/http://kuznets.fas.harvard.edu/~aroth/alroth.html#NRMP
*http://www.dcs.gla.ac.uk/research/algorithms/stable/EGSapplet/EGS.html
*[http://www.csee.wvu.edu/~ksmani/courses/fa01/random/lecnotes/lecture5.pdf
{{game theory}}
{{Authority control}}
[[Category:Combinatorics]]
[[Category:Game theory game classes]]
[[Category:Cooperative games]]
[[Category:Mathematical problems]]
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