The last couple of months we have been breathing, eating, sleeping and dreaming of Economics. Very quickly words and phrases such as optimization, consistency, “my function of studying has a negative first derivation since monday”, “there is no Nash equilibrium in our restaurant choices” entered our every day communication. Most popular Google searches and old bookmarks were replaced by blogs about econometrics and game theory, and reading the news came down to skipping to the Economics and Finance section. There was nothing to complain about, as Bukowski put it “if you are going to try, go all the way”, and there was no other way to do a Masters in Economics. But when it came down to economic models trying to explain love, I was not going to give in so easily.
For Economics followers, the Nobel Prize in Economics is a bit of an Oscar’s. We wake up quite excited in the morning to find out who was awarded this year and to have a browse through their papers with our first morning coffee. And at least for me, this year was no different,. While enjoying my first coffee, I was pleasantly intrigued by this year’s choice of Nobel laureates having learnt that their point of view of economics is beyond the one that evolved around money and finance, but the one of using game theory principles to ensure a more efficient allocation of goods outside markets such as the allocation of kidneys, graduate school applicants to programs and future spouses. Wait.. did I just read there is a model explaining how to choose a future spouse to maximize the stability of marriage? Trying extremely hard to restrain myself from criticizing right away, I decided to give Gale and Shapley’s explanation on how to achieve a stable marriage, unbiased of my own perception of what I consider a stable marriage or love for that matter, a fair chance.
Gale and Shapley have centered their research on studying the problem of bilateral choices under uncertainty. Their model on how to achieve the maximum number of stable marriages originally stems from the model on how graduate schools decide on how many graduate applicants to admit above their acceptance quota (assuming applicants will also make a choice whether to accept the offer). Uncertainty is created from the fact that it is not known whether a given applicant has also applied somewhere else and how he ranks the colleges he has applied to and of those colleges which ones he will be accepted to. In the original model, the first requirement of an assignment is that it does not exhibit instability[1], but the optimal assignment will be the one that is stable and in which every applicant is at least as well off under it as under any other stable assignment. By construction, this optimal assignment will be unique (a “no-tie” rule within rankings of applicants and graduate schools is assumed).
When Gale and Shapley apply this model to the problem of marriage, they suppose a certain community consisting of “N women and N men who rank those of the opposite sex in accordance with his/her preferences for a marriage partner” (Gale, Shapley, 1992). The goal is to seek a satisfactory way of marrying off all members of the community- a set of marriages is unstable if, among the set of chosen individuals, there are a man and a woman who are not married to each other but prefer each other to their actual mates.
Gale and Shapley argue that no matter the initial set of preferences there will always exist a stable set of marriages. To prove this they use an iterative procedure for finding a stable set of marriages: first, every boy proposes to his favorite girl. Each girl who receives more than one proposal rejects all but her favorite among those who have proposed to her. However, she does not accept her favorite proposal just yet, but keeps him on a string to allow for the possibility that someone better may come along later[2]. In the second stage, the boys who were rejected now propose to their second choices. Each girl receiving proposals chooses her favorite from the group consisting of the new proposers and the boy on her string, if there was one. She rejects all the rest and again keeps the favorite in suspense. We proceed in the same manner in all the sequent stages (at most n2-2n+2 stages) until every girl will have received a proposal, because for as long as any girl has not been proposed to there will be rejections and new proposals, but since no boy can propose to the same girl more than once, every girl is sure to get a proposal at one point. So as soon as the last girl gets her proposal, the matching phase will have ended and each girl is now required to accept the boy on her string. This iterative procedure ensures that the final set of marriages is stable.
For example, if we suppose Brad and Angelina are not married to each other but Brad prefers Angelina to his own wife. Then Brad must have proposed to Angelina at some stage and subsequently been rejected in favor to someone that Angelina liked better. It is clear that Angelina prefers her husband to Brad implying there is no instability. The application of this theory is visible if we assume a set of four men (A,B,C,D) and four women (1,2,3,4)[3]: if the procedure is repeated ten times, it will result in one set of stable marriages indicated by the shadowed entries in the matrix (where the first number is the ranking of women by the men). Moreover, Gale and Shapley prove that this final set of marriages is not just stable, but also optimal for each person involved (the proof is by induction, and might be an entertaining way to practice you theorem-proofing skills!)
|
M/F |
1 |
2 |
3 |
5 |
|
A |
1,3 |
2,3 |
3,2 |
4,3 |
|
B |
1,4 |
4,1 |
3,3 |
2,2 |
|
C |
2,2 |
1,4 |
3,4 |
4,1 |
|
D |
4,1 |
2,2 |
3,1 |
1,4 |
Quite clearly even after reading the article, I was not a bit convinced that any “model in economics” and “marriage and love” should even be mentioned in the same sentence. The only thing I have to say in defense of Gale and Shapley is their admittance in the conclusion “that they have abandoned reality altogether”, although I still have trouble understanding how one can assume that the graduate admission process – specifically intended to be emotionless and unbiased to ensure fairness – could be applied to the stability of love and marriage, as I can’t think of many things more based on emotions and irrationality. So,dear (future) economists, please at least stay away from the topic of love (and marriage). Let’s leave a bit of spontaneity, a bit of emotions , a bit of mystery in our lives. Let’s minimize our R-squared in the topic of love. If we regress and forecast everything starting from our future monthly earnings to how intelligent our second child will be, shouldn’t we leave at least this one thing to chance? And since I cannot find a better way to say this, I will rephrase Einstein…”It would be possible to describe everything scientifically, but it would make no sense; it would be without meaning, as if you described a Beethoven symphony as a variation of wave pressure.”
Reference: Gale, D., Shapley, L.S. (1962): College admission and the Stability of Marriage, The American Mathematical Monthly Review, Vol. 69, 9-15
[1]Definition. An assignment of applicants to colleges will be called unstable if there are two applicants a and b who are assigned to colleges A and B, respectively, although b prefers A to B and A prefers b to a.
[2] (Gale and Shapley do must think women are quite cruel!)
[3] It is not a necessary condition that there is an equal number of women and men. See the paper for further on this.
The “marriage” story in the Gale-Shapley model is just a phrasing. There’s no intention to say that this is the way we should marry people (albeit, dating agencies do proceed a bit like in the Gale-Shapley model). There is however a (very large) literature in economics on marriage, that started with Gary Becker (also a Nobel laureate). This literature is often referred to as “family economics”. Economists do not want to enter into the sentimental aspect of marriage, but they do observe that, in the aggregate, people tend to marry individuals with similar levels of education, revenue, etc. Some recent works also show how the “stability of marriage” (i.e., divorce) depend on specific regulations such as how the household’s wealth is split in case of divorce. At the individual level there are sentiments, but in the aggregate we do observe some “economic” regularities.
Professor Haeringer thank you very much for your comprehensive comment, it feels both exciting and intimidating being read by professors at such an early stage of the blog. Although being very aware that Gale and Shapley in their article use “the marriage story” only as a phrase to explain their example, I still cannot run away from my personal opinion that particularly when it comes to marriage (if we can naively adopt the perhaps traditional concept of marriage as one being based on love), economic explanations should be let aside. However, that is not to say that family economics is not a relevant area of economic research, for example understanding fertility is of crucial value today both in the developed and developing world. If you have time and interest, it would be very interesting if you could, from an expert’s point of view, summarize recent developments in this area through your own personal view, or introduce readers of this blog to your current research in the areas of matching and game theory. The blog is indeed intended to serve as an interactive platform for students and experts.
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