Stable marriage problem

In mathematics and computer science, the stable marriage problem (SMP) is the problem of finding a stable matching between two sets of elements given a set 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 stable whenever it is not the case that both:

some given element A of the first matched set 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 alternative pairing (A, B) in which both A and B are individually better off than they would be with the element to which they are currently matched.

The stable marriage problem is commonly stated as:

Given n men and n women, where each person has ranked all members of the opposite sex with a unique number between 1 and n 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. If there are no such people, all the marriages are "stable".

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]

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.^[2]^[3]

The Gale-Shapley algorithm involves a number of "rounds" (or "iterations") where each unengaged man "proposes" to the most-preferred woman to whom he has not yet proposed. Each woman then considers all her suitors and tells the one she most prefers "Maybe" and all the rest of them "No". She is then provisionally "engaged" to the suitor she most prefers so far, and that suitor is likewise provisionally engaged to her. 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. 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" to her suitor she most prefers (whether her existing provisional partner or someone else) and rejects the rest (again, perhaps including her current provisional partner). 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 algorithm guarantees that:

Everyone gets married: Once a woman becomes engaged, she is always engaged to someone. So, 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 unengaged, she would have had to have said yes.
The marriages are stable: Let Alice be a woman and Bob be a man who are both 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 = m's highest ranked such woman to whom he 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, w) become engaged
           m' becomes free
         else
           (m', w) remain engaged
    }
}

Optimality of the solution

While the solution is stable, it is not necessarily optimal from all individuals' points of view. The traditional form of the algorithm is optimal for the initiator of the proposals and the stable, suitor-optimal solution may or may not be optimal for the reviewer of the proposals. An example is as follows:

There are three suitors (A,B,C) and three reviewers (X,Y,Z) which have preferences of:

 A: YXZ  B: ZYX  C: XZY  X: BAC  Y: CBA  Z: ACB

There are 3 stable solutions to this matching arrangement:

 suitors get their first choice and reviewers their third (AY, BZ, CX)
 all participants get their second choice (AX, BY, CZ)
 reviewers get their first choice and suitors their third (AZ, BX, CY)

All three are stable because instability requires both 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. The algorithm converges in a single round on the suitor-optimal solution because each reviewer receives exactly one proposal, and therefore selects that proposal as its best choice, ensuring that each suitor has an accepted offer, ending the match. This asymmetry of optimality is driven by the fact that the suitors have the entire set to choose from, but reviewers choose between a limited subset of the suitors at any one time.

References

^ Stable Matching Algorithms
^ D. Gale and L. S. Shapley: "College Admissions and the Stability of Marriage", American Mathematical Monthly 69, 9-14, 1962.
^ Harry Mairson: "The Stable Marriage Problem", The Brandeis Review 12, 1992 (online).
^ D. Gusfield and R. W. Irving, The Stable Marriage Problem: Structure and Algorithms, p. 54; MIT Press, 1989.

Textbooks and other important references not cited in the text

Dubins, L., and Freedman, D. (1981). Machiavelli and the Gale-Shapley algorithm. The American Mathematical Monthly, 88(7), 485–494.
Kleinberg, J., and Tardos, E. (2005) Algorithm Design, Chapter 1, pp 1–12. See companion website for the Text [1].
Knuth, D. E. (1976). Marriages stables. Montreal: Les Presses de I'Universite de Montreal.
Roth, A. E. (1984). The evolution of the labor market for medical interns and residents: A case study in game theory. Journal of Political Economy, 92, 991–1016.
Roth, A. E., and Sotomayor, M. A. O. (1990). Two-sided matching: A study in game-theoretic modeling and analysis. Cambridge University Press.
Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7. See Section 10.6.4; downloadable free online.
Schummer, J., and Vohra, R. V. (2007). Mechanism design without money. In Nisan, N., Roughgarden, T., Tardos, E., and Vazirani, V. (Eds.). (2007). Algorithmic game theory. Cambridge, UK: Cambridge University Press, chapter 10, 243–265.

External links

[1] Stable Matching Algorithms

[2] D. Gale and L. S. Shapley: "College Admissions and the Stability of Marriage", American Mathematical Monthly 69, 9-14, 1962.

[3] Harry Mairson: "The Stable Marriage Problem", The Brandeis Review 12, 1992 (online).

[4] D. Gusfield and R. W. Irving, The Stable Marriage Problem: Structure and Algorithms, p. 54; MIT Press, 1989.

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