Gale–Shapley algorithm

In mathematics, economics, and computer science, the Gale–Shapley algorithm (also known as the deferred acceptance algorithm or propose-and-reject algorithm) is an algorithm for finding a solution to the stable matching problem, named for David Gale and Lloyd Shapley who had described it as solving both the college admission problem and the stable marriage problem. It takes polynomial time, and the time is linear in the size of the input to the algorithm. It is a truthful mechanism from the point of view of the proposing participants, for whom the solution will always be optimal.

Background

Main article: stable matching problem

The stable matching problem, in its most basic form, takes as input equal numbers of two types of participants (n men and n women, or n medical students and n internships, 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 stable matching always exists, and the algorithmic problem solved by the Gale–Shapley algorithm is to find one. 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 is no pair (A, B) where both participants prefer each other to their matched partners.

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. In 1984, Alvin E. Roth observed that essentially the same algorithm had already been in practical use since the early 1950s, in the National Resident Matching Program.

The Gale–Shapley 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 $O(n^{2})$ where $n$ is the number of men or women. Since the input preference lists also have size proportional to $n^{2}$, the runtime is linear in the input size.

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

algorithm stable_matching is
    Initialize m ∈ M and w ∈ W to free
    while ∃ free man m who has a woman w to propose to do
        w := first woman on m's list to whom m has not yet proposed
        if ∃ some pair (m', w) then
            if w prefers m to m' then
                m' becomes free
                (m, w) become engaged
            end if
        else
            (m, w) become engaged
        end if
    repeat

Optimality of the solution

Main article: Lattice of stable matchings

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 chooses that proposal, 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 and worst for all women among all stable matchings. Similarly, if the women propose then the resulting matching is the best for all women and worst for all men among all stable matchings. These two matchings are the top and bottom elements of the lattice of 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) for 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. 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.

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 and matchingR packages.
• R: Implementation in an R Shiny tool designed for student placement in university programs with limited enrollment (does not use the packages above)
• API: The MatchingTools API provides a free application programming interface for the Gale–Shapley algorithm.
• Python: The Gale–Shapley algorithm is included along with several other well-known matching game algorithms in the matching library, and QuantEcon/MatchingMarkets.py package includes several others for generalized matching games.
• 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.

Recognition

Shapley and Roth were awarded 2012 Nobel Memorial Prize in Economic Sciences "for the theory of stable allocations and the practice of market design"; Gale had died in 2008.

See also

• Deferred-acceptance auction

External links

• Gale–Shapley JavaScript Demonstration