Stable Marriages and Designing Markets

A few days ago Jeremy Kun with the Math ∩ Programming blog wrote about the problem of stable marriages, by which here is meant pairing off people so that everyone is happy with their pairing. Put like that it almost sounds like the sort of thing people used to complain about in letters to Ann Landers about mathematicians doing foolish things — don’t mathematicians know that feelings matter in this, and, how does this help them teach kids to do arithmetic.

But the problem is just put that way because it’s one convenient representation of a difficult problem. Given a number of agents that can be paired up, and some way of measuring the collection of pairings, how can you select the best pairing? And what do you mean by best? Do you mean the one that maximizes whatever it is you’re measuring? The one that minimizes it (if you’re measuring, say, unhappiness, or cost, or something else you’d want as little of)? Jeremy Kun describes the search for a pairing that’s stable, which requires, in part, coming up with a definition of just what “stable” means.

The work can be put to describe any two-party interaction, which can be marriages, or can be the choice of people where to work and employers who to hire, or can be people deciding what to buy or where to live, all sorts of things where people have preferences and good fits. Once the model’s developed it has more applications than what it was originally meant for, which is part of what makes this a good question. Kun also write a bit bout how to expand the problem so as to handle some more complicated cases, and shows how the problem can be put onto a computer.

Here is a fun puzzle. Suppose we have a group of 10 men and 10 women, and each of the men has sorted the women in order of their preference for marriage (that is, a man prefers to marry a woman earlier in his list over a woman later in the list). Likewise, each of the women has sorted the men in order of marriageability. We might ask if there is any way that we, the omniscient cupids of love, can decide who should marry to make everyone happy.

Of course, the word happy is entirely imprecise. The mathematician balks at the prospect of leaving such terms undefined! In this case, it’s quite obvious that not everyone will get their first pick. Indeed, if even two women prefer the same man someone will have to settle for less than their top choice. So if we define happiness in this naive way…

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