Pairing where no unchosen pair prefers each other over their choice
In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem) 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:
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:
 ;templatestyles src="Template:Blockquote/styles.css" / ;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.
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. 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."
An important and large-scale application of stable marriage is in assigning users to servers in a large distributed Internet service. 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.
The Gale-Shapley algorithm for stable matching is used to assign rabbis who graduate from Hebrew Union College to Jewish congregations.
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:
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.
In a uniformly-random instance of the stable marriage problem with n men and n women, the average number of stable matchings is asymptotically formula_1.
In a stable marriage instance chosen to maximize the number of different stable matchings, this number is an exponential function of n.
Counting the number of stable matchings in a given instance is #P-complete.
Algorithmic solution.
In 1962, David Gale and Lloyd Shapley proved that, for any equal number of men and women, it is always possible to solve the stable marriage problem and make all marriages stable. They presented an algorithm to do so.
The Gale–Shapley algorithm (also known as the deferred acceptance algorithm) involves a number of "rounds" (or "iterations"):
This algorithm is guaranteed to produce a stable marriage for all participants in time formula_2 where formula_3 is the number of men or women.
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.
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. 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.
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.
Rural hospitals theorem.
The rural hospitals theorem concerns a more general variant of the stable matching problem, like that applying in the problem of matching doctors to positions at hospitals, differing in the following ways from the basic n-to-n form of the stable marriage problem:
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.
For this kind of stable matching problem, the rural hospitals theorem states that:
Related 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) 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.
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.
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. An important special case of contracts is matching with flexible wages.
References.
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