Vol. 107, No. 6 • February 2014 | MatheMatics teacher 447446 MatheMatics teacher | Vol. 107, No. 6 • February 2014

the Gale‑Shapley stable marriage theorem is a fascinating piece of twentieth‑century mathematics that has many practical appli‑cations—from labor markets to school

admissions—yet is accessible to secondary school mathematics students. David Gale and Lloyd Shap‑ley were both mathematicians and economists who published their work on the Stable Marriage problem in 1962 (Gale and Shapley 1962). Shapley received the 2012 Nobel Prize in Economics for his work on stable allocations. Because the Nobel Prize is not awarded posthumously, Gale, who died in 2008, did not.

The proof of the theorem is constructive: There exists an algorithm that will always solve the prob‑lem. Moreover, a class activity allows students to carry out the steps of the algorithm as a group and see how it works. This activity helps fulfill the fourth Standard for Mathematical Practice (Model with Mathematics): “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace” (CCSSI 2010, p. 7).

THE PROBLEMThe task of marrying n women to n men so that each marriage is stable is called the Stable Marriage problem. The problem may be stated as follows:

Imagine that you are the chief of an island on which there are exactly ten women and ten men of marriageable age. It is your job to choose a husband for each woman, taking into consideration the preferences of each prospective bride and groom. Although you recognize that it may not be possible for each person to have his or her first choice for a spouse, you would like to guarantee at least that each marriage is stable—that is, that no two people who are not married to each other would willingly leave their spouses to be with each other.

To make clear this notion of stability, sup‑pose that you

—marry Alice to Andrew, even though she pre‑fers Brian; and —marry Brian to Betty, even though he prefers Alice.

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she becomes engaged to him, breaking off the previ‑ous engagement. She rejects the rest of her suitors.

Subsequent RoundsProposals: Each man who is not engaged proposes to the woman whom he has ranked highest who has not already rejected him.Decisions: Each woman who receives one or more proposals chooses her favorite from among all her new suitors and the man she is engaged to, if any. She becomes (or remains) engaged to her favorite and rejects the rest.

The process stops when every man and every woman is engaged, at which point all engaged couples marry.

AN EXAMPLENow consider a group of four women (Abigail, Barbara, Charlotte, and Diane) and a group of four men (Will, Xavier, Young, and Zack) who are to be married. The preferences of the women and men are given in figure 1.

Suppose that we arbitrarily create the following pairing: (A, W), (B, X), (C, Y), and (D, Z). (Note that this pairing was not created using the algo‑rithm.) That is, Abigail marries Will; Barbara mar‑ries Xavier; Charlotte marries Young; and Diane marries Zack. Notice that Abigail, although married to Will, would rather be married to Young or Zack. Further, Young is married to Charlotte but would rather be married to Abigail. Thus, Abigail and Young would prefer to divorce their current partners and marry each other; this is an example of an un‑ stable pairing. Charlotte and Xavier would also pre‑fer to divorce their partners and marry each other; this is another example of an unstable pairing.

In contrast, the Gale‑Shapley algorithm always produces a stable pairing. The result of applying the algorithm, shown by round, is given in figure 2.

WHY THE ALGORITHM WORKSGale and Shapley proved that for any group of women and men and any possible preferences, this algorithm will eventually produce a stable set of marriages. We first need to see that everyone even‑tually marries. On any round where at least one woman and one man is not engaged, there will be a

proposal. Because no man can propose to the same woman twice, only finitely many proposals can be made. Thus, this sequence of proposals cannot go on forever; it must end at a point when everyone is engaged. When that happens, all couples marry.

Here is Gale and Shapley’s explanation of why these marriages are stable:

We assert that this set of marriages is stable. Namely, suppose John and Mary are not married to each other but John prefers Mary to his own wife. Then John must have proposed to Mary at some stage and subsequently been rejected in favor of someone that Mary liked better. It is now clear that Mary must prefer her husband to John and there is no instability. (Gale and Shapley 1962, p. 13)

CLASSROOM ACTIVITYWe have tried the following activity in a college liberal arts mathematics course. An equal number of male and female volunteers from the class are selected. (Four or five of each works well, although teachers may want to experiment with different numbers.) If too many students are selected, the activity may go through a large number of rounds.

Give each of the students a page listing his or her preferred partners in order. (Teachers may want to use the preferences from the fully worked example in fig. 2.) To minimize awkwardness, we recommend assigning each student a fictitious name rather than using students’ actual names. (Students should be given name tags with their fictitious names.) Even so, there may be some initial awkwardness, but with col‑lege students, at least, it disappears rapidly. Teachers of younger students may need to work harder at making the students feel comfortable. Have the female students stand around the room, not too close together. Group the male students together in another part of the room.

When the teacher announces the beginning of round 1, each male student goes to his first choice partner, according to his preference list. A female stu‑dent who has just one male suitor will temporarily be engaged to that male, who will then stand beside her. A female student who has more than one male suitor will select her top choice from among them according to her preference list; that male will then stand beside her.

The males who were not chosen in the first round regroup, and the teacher announces the beginning of round 2. This process continues until everyone is matched.

Allowing female students to stand too closely together can lead to confusion about which couples are engaged. Also, to enable students who do not directly participate to follow the algorithm, we hand out a sheet listing all preferences rather than list these preferences on the board, which was often obscured by standing couples. Students loved watch‑

ing the algorithm carried out and were quick to cheer on successful suitors and console the losers.

SWITCHING ROLESIf the idea of males doing all the proposing seems old‑fashioned, try repeating the process with the roles switched so that the female students propose to the males.

Regarding the example above, we leave it to readers to verify that the pairing when the females propose is (A, Z), (B, X), (C, W), and (D, Y), accom‑plished in just one round. Thus, this example has at least two solutions. Also, when the males propose, they do at least as well as when the females propose and sometimes better. Similarly, the females do bet‑ter when they propose. Gale and Shapley proved that every member of the group that does the pro‑posing will be at least as well off as he or she would be under any other stable assignment. (The proof by induction is too complicated to include here; inter‑ested readers should see Gale and Shapley [1962].) There are examples with only one stable pairing, in which case it does not matter who does the propos‑ing; either way leads to the only stable pairing.

These marriages are not stable, because Alice and Brian prefer each another to their spouses. Note that the marriage would not be unstable if Alice preferred Brian but Brian did not prefer Alice. It is unstable only if both partners prefer each other.

WHY DO WE CARE?The Stable Marriage problem arises naturally in school and university admissions and in matching medical students to hospitals for resident training—situations in which members of two different sets are paired. Suppose, for example, that a number of students are seeking admission to a state university system that has several campuses. Each student has ranked the cam‑puses from most to least desirable, and each campus has likewise ranked the students from most to least desirable. The problem is now to assign each student to a campus. Of course, if each campus is able to accept more than one student, the problem situation becomes a variation of the Stable Marriage problem—tanta‑mount to the case in which a woman is allowed to have more than one husband. Teachers may prefer to describe the problem to secondary school students as one of pairing each student with a partner to take to a school dance or to do a project with.

THE GALE-SHAPLEY ALGORITHMThe Gale‑Shapley algorithm solves the Stable Mar‑riage problem through a series of “proposals” made by the men, resulting in provisional “engagements” that the women are free to break later if they receive a better offer. When every man and every woman is engaged, all engaged couples marry.

Assume that we have an equal number of women and men, that each woman has ranked all the men in order of preference, and that each man has ranked all the women. Here are the steps of the algorithm.

Round 1Proposals: Each man proposes to the woman whom he has ranked first. (Some women may get more than one proposal, and some may get none.)Decisions: Each woman who receives one or more proposals rejects all her suitors except her favorite, to whom she becomes engaged. In particular, if she gets just one proposal, then she becomes engaged to that suitor.

Round 2Proposals: Each man who is not engaged proposes to the woman whom he has ranked second highest, even if she is already engaged. Decisions: Each woman who receives one or more proposals chooses her favorite from among all her new suitors and the man she is engaged to, if any. If she is already engaged to her favorite, she remains engaged to him. If she is not engaged to her favorite,

Abigail: Z, Y, W, X Will: A, B, C, DBarbara: X, Z, W, Y Xavier: A, D, C, BCharlotte: W, Z, X, Y Young: B, A, C, DDiane: Y, X, W, Z Zack: D, B, C, A

Fig. 1 the marriage preferences of the four women and

four men are shown here.

Round 1

W proposes to A (W’s first choice)

Accepted, because A has no other suitors yet; A and W become engaged

X proposes to A (X’s first choice)

Rejected, because A is already engaged to W, and A prefers W to X

Y proposes to B (Y’s first choice)

Accepted; B and Y become engaged

Z proposes to D (Z’s first choice)

Accepted; D and Z become engaged

Round 2

X proposes to D (X’s second choice)

Accepted, because D is engaged to Z, but D prefers X to D; D and X become engaged; engagement of D and Z is broken off

Round 3

Z proposes to BAccepted; B and Z become engaged; engagement of B and Y is broken off

Round 4

Y proposes to AAccepted; A and Y become engaged; engagement of A and W is broken off

Round 5

W proposes to B Rejected

Round 6

W proposes to C Accepted; C and W become engaged

Fig. 2 We use the Gale-shapley algorithm to create the four pairs: (a, Y), (B, Z), (c, W),

and (D, X). that is, abigail marries Young, Barbara marries Zack, charlotte marries Will,

and Diane marries Xavier.

450 MatheMatics teacher | Vol. 107, No. 6 • February 2014

For additional exercises with their answers, download one of the free apps for your smartphone and then scan this tag to access www.nctm.org/mt050.

The optimality of the procedure for the suitors led to a lawsuit in 2002 by three physicians against the National Resident Matching Program, which uses the Gale‑Shapley algorithm, with the hospitals playing the role of the suitors (Robinson 2003). The physicians claimed that the procedure is anti‑competitive, but the lawsuit was eventually dis‑missed. The U.S. Supreme Court declined to hear the appeal of the dismissal (NRMP 2007).

HOW MANY PROPOSALS ARE REQUIRED?With n men and n women, the procedure requires at most n2 – n + 1 proposals. The basic idea of the proof, outlined in Gusfield and Irving (1989) and Knuth (1997), is that the algorithm stops when the last woman receives her first proposal. In the worst case, this could involve each of the other n – 1 women receiving n proposals and the last woman receiving 1, for a total of n(n – 1) + 1 = n2 – n + 1. Gusfield and Irving (1989) give an example with n couples showing that this upper bound can be achieved. In their example, n proposals are made on the first round, and on each subsequent round one of the remaining n2 – 2n + 1 proposals are made, taking a total of n2 – 2n + 2 rounds. This is the maxi‑mum number of rounds, as stated (without proof) by Gale and Shapley. Thus, if there are n = 4 students, as in the previous example, then 13 rounds could be required. If there are 5 students of each gender, 21 rounds could be required. Teachers would be well advised to test any preference lists in advance to see how many rounds are actually required.

This activity also allows for a student exploration to find a set of preference schedules that require the maximum number of rounds. (The minimum num‑ber of rounds required is one, and students should easily be able to figure out a set of preference lists that require just one round.) For additional exercises, see the appendix at www.nctm.org/mt050.

FURTHER EXPLORATIONSThe Stable Marriage problem allows numerous avenues for further exploration As mentioned ear‑lier, the Hospitals‑Residents problem, also known as the College Admissions problem, is similar to the Stable Marriage problem except that a hospital can accept more than one resident and a college can accept more than one student. Gale and Shap‑ley show how their algorithm can always produce a stable pairing in this situation. Gusfield and Irving (1989) and Knuth (1997) list open problems that could serve as student research projects. For interested teachers, the website http://mathsite .math.berkeley.edu/smp/smp.html provides an explanation and demonstration of the Stable Mar‑riage problem, including experiments that students may carry out online.

BIBLIOGRAPHYCommon Core State Standards Initiative (CCSSI). 2010.

Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Gale, David, and Lloyd S. Shapley. 1962. “College Admissions and the Stability of Marriage.” The American Mathematical Monthly 69 (1): 9–15. Reprinted in 2013 in The American Mathematical Monthly 120 (5): 386–91.

Gusfield, Dan, and Robert W. Irving. 1989. The Stable Marriage Problem: Structure and Algorithms. Cam‑bridge, MA: MIT Press.

Irving, Robert W. 1985. “An Efficient Algorithm for the ‘Stable Roommates’ Problem.” Journal of Algorithms 6: 577–95. doi:http://dx.doi.org/ 10.1016/0196‑6774(85)90033‑1

Knuth, Donald E., 1997. Stable Marriage and Its Rela-tion to Other Combinatorial Problems: An Introduc-tion to the Mathematical Analysis of Algorithms. Providence, RI: American Mathematical Society.

MathSite Presents: Stable Marriage Problem. http://mathsite.math.berkeley.edu/smp/smp.html

National Resident Matching Program (NRMP). 2007. NRMP Statement on Litigation. http://www.nrmp .org/memo_litigation.html

Robinson, Sara. 2003. “Are Medical Students Meeting Their (Best Possible) Match?” SIAM News 36 (3): 1, 8–9.

Tanenbaum, Peter. 2010. Excursions in Modern Math-ematics. Upper Saddle River, NJ: Pearson.

RAYMOND N. GREENWELL, raymond.n .greenwell@hofstra.edu, is a mathematics professor at hofstra University in hemp-stead, New York. he is interested in vari-ous areas of applied mathematics and has coauthored textbooks in finite math-ematics and calculus with applications. DANIEL E. SEABOLD, daniel.seabold@

hofstra.edu, is also a mathematics professor at hofs-tra University. he has published in the fields of logic, set theory, and parliamentary law.

The 2015 Annual Perspectives in Mathematics Education (APME) Volume: Assessment to Enhance Learning and Teaching will consist of chapters that represent current thinking in assessment in the context of mathematics education. To that end, the focus will be on the important role that assessment plays in informing teacher practice and encouraging student learning. Chapters should make strong links between research and practice and should highlight important assessment issues as they relate to informing mathematics teaching and learning. Each chapter should appeal to a broad audience that includes mathematics educators in a variety of capacities, such as teachers, teacher leaders, professional development leaders, mathematics teacher educators, and researchers.

Suggested topics include, but are not limited to, the following:

◆ Assessment, progress monitoring, evidence-based assessment, effective feedback, peer- and self-assessment, using assessment data

◆ Critical examination of assessment practices in regard to issues of access and equity to support teaching and learning of all students

◆ Teacher knowledge and professional development with regard to assessment practices to promote student learning and inform teaching

◆ Issues and examples related to the enactment of classroom practices that reflect current thinking and research in assessment and mathematics education (e.g., the use of assessment for learning, as learning, and of learning in mathematics classrooms)

◆ Critical examination of benefits, limitations, and/or uses of current high-stakes assessment and their influence on teaching and learning

◆ The design of alternative modes of assessment in mathematics (e.g., incorporating technology, investigations, various forms of formative assessment)

◆ The development of assessment tasks and criteria that reflect the complexity of mathematical thinking, problem solving, and other competencies

◆ Synthesis of research, theory, purposes, and/or issues related to assessment to inform teaching and learning

◆ Challenges to and opportunities for blending formative and summative assessment productively in the mathematics classroom

◆ Design of assessments for national, state, or provincial standards, such as the Common Core State Standards, including targeting specific Standards for Mathematical Practice and/or Content Standards

◆ Use of assessments as part of differentiated instruction and/or programs aimed at assessing and meeting individual student’s learning needs

◆ Assessment as a vehicle for reform

◆ Alignment and coherence of curriculum, instruction, and assessment

◆ Considerations and issues regarding assessment of teaching

Details for Submission

Prospective authors must fill out an Intention to Submit form, found at www.nctm.org/APME2015, and send to Christine.Suurtamm@uottawa.ca by March 1, 2014.

The full chapter is to be submitted electronically by May 15, 2014, to the same e-mail address. Late or partial manuscripts will not be considered. All chapter submissions will be blind peer-reviewed, and authors will receive feedback within 8 weeks.

Complete details regarding full-chapter submission requirements will be sent once the Intention to Submit form is received.

Annual Perspectives in Mathematics Education

Assessment to Enhance Learning and Teaching

APME 2015 Call for Chapters