# Stable marriage problem

In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) 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 mapping from the elements of one set to the elements of the other set. A matching is not stable if:

1. 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
2. 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 match (A, B) which both prefer each other to their current partner under the matching.

The stable marriage problem has been stated as follows:

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.

Note that the existence of two classes that need to be paired with each other (men and women in this example), distinguishes this problem from the stable roommates problem.

## 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.[1] In 2012, The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel was awarded to Lloyd S. Shapley and Alvin E. Roth "for the theory of stable allocations and the practice of market design."[2]

An important and large-scale application of stable marriage is in assigning users to servers in a large distributed Internet service.[3] Billions of users access web pages, videos, and other services on the Internet, requiring each user to be matched to one of (potentially) hundreds of thousands of servers around the world that offer that service. A user prefers servers that are proximal enough to provide a faster response time for the requested service, resulting in a (partial) preferential ordering of the servers for each user. Each server prefers to serve users that it can with a lower cost, resulting in a (partial) preferential ordering of users for each server. Content delivery networks that distribute much of the world's content and services solve this large and complex stable marriage problem between users and servers every tens of seconds to enable billions of users to be matched up with their respective servers that can provide the requested web pages, videos, or other services.[3]

## Solution

Animation showing an example of the Gale–Shapley algorithm

In 1962, David Gale and Lloyd Shapley proved that, for any equal number of men and women, it is always possible to solve the SMP and make all marriages stable. They presented an algorithm to do so.[4][5]

The Gale–Shapley algorithm (also known as the Deferred Acceptance algorithm) involves a number of "rounds" (or "iterations"):

• 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.
• 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).
• This process is repeated until everyone is engaged.

The runtime complexity of this algorithm is ${\displaystyle O(n^{2})}$ where ${\displaystyle n}$ is the number of men or women.[6]

This algorithm guarantees that:

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.
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.

## Algorithm

function stableMatching {
Initialize all m ∈ M and w ∈ W to free
while ∃ free man m who still has a woman w to propose to {
w = first woman on m's list to whom m has not yet proposed
if w is free
(m, w) become engaged
else some pair (m', w) already exists
if w prefers m to m'
m' becomes free
(m, w) become engaged
else
(m', w) remain engaged
}
}


## 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 - (AY, BZ, CX);
• all participants get their second choice - (AX, BY, CZ);
• women get their first choice and men their third - (AZ, BX, CY).

All three are stable, because instability requires one 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.

## Optimality of the solution

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.

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.

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 m0. Suppose m0 is matched in B to a woman w1, which he prefers to his match in A. This means that in A, m0 has visited w1, but she rejected him (this is denoted by the green arrow from m0 to w1). w1 rejected him in favor of some man that she prefers, say m2. So in B, w1 is matched to m0 but "yearns" (wants but unmatched) to m2 (this is denoted by the red arrow from w1 to m2).

Since B is a stable matching, m2 must be matched in B to some woman he prefers to w1, say w3. This means that in A, m2 has visited w3 before arriving at w1, which means that w3 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. m0 - the first man rejected by his adjacent woman (w1). By assumption, this rejection happens only after m2 comes to w1. But this can happen only after w3 rejects m2 - contradiction to m0 being the first.

## Strategic considerations

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.[7] 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.[8]

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.

## Counting the stable matchings

In a uniformly-random instance of the stable marriage problem with n men and n women, the average number of stable matchings is asymptotically ${\displaystyle e^{-1}n\ln n}$.[9]

The maximum grows at least exponentially with n, and counting the number of stable matchings in a given instance is ♯P-complete.[10]

## Implementation in software packages

• R: The Gale–Shapley algorithm (also referred to as deferred-acceptance algorithm) for the stable marriage and the hospitals/residents problem is available as part of the matchingMarkets[11][12] and matchingR[13] packages.
• API: The MatchingTools API provides a free application programming interface for the Gale–Shapley algorithm.[14]
• Python: The Gale–Shapley algorithm is included along with several others for generalized matching problems in the QuantEcon/MatchingMarkets.py package[15]
• Matlab: The Gale–Shapley algorithm is implemented in the assign2DStable function that is part of the United States Naval Research Laboratory's free Tracker Component Library.[16]

## Similar problems

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 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 NRMP was before 1995)[17] or resident-oriented. This problem was solved, with an algorithm, in the same original paper by Gale and Shapley, in which the stable marriage problem was solved.[18]

The 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.[19]

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.

The matching with contracts problem is a generalization of matching problem, in which participants can be matched with different terms of contracts.[20] An important special case of contracts is matching with flexible wages.[21]

## References

1. ^ Stable Matching Algorithms
2. ^ "The Prize in Economic Sciences 2012". Nobelprize.org. Retrieved 2013-09-09.
3. ^ a b Bruce Maggs and Ramesh Sitaraman (2015). "Algorithmic nuggets in content delivery" (PDF). ACM SIGCOMM Computer Communication Review. 45 (3).
4. ^ Gale, D.; Shapley, L. S. (1962). "College Admissions and the Stability of Marriage". American Mathematical Monthly. 69 (1): 9–14. doi:10.2307/2312726. JSTOR 2312726.
5. ^ Harry Mairson: "The Stable Marriage Problem", The Brandeis Review 12, 1992 (online).
6. ^ Iwama, Kazuo; Miyazaki, Shuichi (2008). "A Survey of the Stable Marriage Problem and Its Variants". International Conference on Informatics Education and Research for Knowledge-Circulating Society (icks 2008): 131–136. doi:10.1109/ICKS.2008.7. hdl:2433/226940. ISBN 978-0-7695-3128-1.
7. ^ Dubins, L. E.; Freedman, D. A. (1981-08-01). "Machiavelli and the Gale-Shapley Algorithm". The American Mathematical Monthly. 88 (7): 485–494. doi:10.1080/00029890.1981.11995301. ISSN 0002-9890.
8. ^ Huang, Chien-Chung (2006). Azar, Yossi; Erlebach, Thomas (eds.). "Cheating by Men in the Gale-Shapley Stable Matching Algorithm". Algorithms – ESA 2006. Lecture Notes in Computer Science. Springer Berlin Heidelberg. 4168: 418–431. doi:10.1007/11841036_39. ISBN 9783540388760.
9. ^ "SIAM (Society for Industrial and Applied Mathematics)". doi:10.1137/0402048. Cite journal requires |journal= (help)
10. ^ "SIAM (Society for Industrial and Applied Mathematics)". doi:10.1137/0215048. Cite journal requires |journal= (help)
11. ^ Klein, T. (2015). "Analysis of Stable Matchings in R: Package matchingMarkets" (PDF). Vignette to R Package MatchingMarkets.
12. ^ "matchingMarkets: Analysis of Stable Matchings". R Project.
13. ^ "matchingR: Matching Algorithms in R and C++". R Project.
14. ^
15. ^ "matchingMarkets.py". Python package.
16. ^ "Tracker Component Library". Matlab Repository. Retrieved January 5, 2019.
17. ^ Robinson, Sara (April 2003). "Are Medical Students Meeting Their (Best Possible) Match?" (PDF). SIAM News (3): 36. Retrieved 2 January 2018.
18. ^ Gale, D.; Shapley, L. S. (1962). "College Admissions and the Stability of Marriage". American Mathematical Monthly. 69 (1): 9–14. doi:10.2307/2312726. JSTOR 2312726.
19. ^ Gusfield, D.; Irving, R. W. (1989). The Stable Marriage Problem: Structure and Algorithms. MIT Press. p. 54. ISBN 0-262-07118-5.
20. ^ Hatfield, John William; Milgrom, Paul (2005). "Matching with Contracts". American Economic Review. 95 (4): 913–935. doi:10.1257/0002828054825466. JSTOR 4132699.
21. ^ Crawford, Vincent; Knoer, Elsie Marie (1981). "Job Matching with Heterogeneous Firms and Workers". Econometrica. 49 (2): 437–450. doi:10.2307/1913320. JSTOR 1913320.