# Posts Tagged ‘game theory’

## Matching Theory

Matching theory is an active field in mathematics, economics, and computer science. It ensured a [Nobel memorial prize][1] for [Alvin E. Roth][2] and [Lloyd S. Shapley][3] in 2012. The theory is applied in the real world to match students and colleges, doctors and hospitals, and to organize the allocation of donor organs. It all started with a beautiful paper by [David Gale][4] and Lloyd Shapley in 1962: [College Admissions and the Stability of Marriage][5] (read it!). In this post, we study the simplest version of what is known as the *stable marriage problem* and take a look at inherent conflicts. The exposition follows chapter 22 of the wonderful book [Game Theory][6] by Michael Maschler, Eilon Solan, and Shmuel Zamir.

**What are stable matchings?**

There is a set \( G \) of \( n \) girls and a set \( B \) of \( n \) boys. All girls and boys are heterosexual and desperate enough to prefer every member of the opposite sex to staying single. Each girl \( g \) has preferences over the boys, represented by a [total order][7] \( \succeq_g \) on \( B \). Similarly, each boy \( b \) has preferences over the girls, represented by a total order \(\succeq_b\) on \( G \).

We want to pair up the girls and the boys. Formally, a *matching* is simply a [bijection][8] \( M \) from \(G\) to \(B\) and if \( f(g)=b \), we say that \( g \) and \( b \) are *matched* (under \( M\)). Now, nobody can force the girls and boys to be together, so we have to look at matchings in which the boys and girls are relatively content with whom they get. To be precise, a matching is *stable* if we cannot find a girl and a boy who prefer each other under their preference ordering to whomever they are matched with.

**Are there any?**

Before we go into a deep study of stable matchings, we ought to make sure that there are stable matchings. We do so by giving an explicit algorithm for finding a stable matching, the *boy courtship algorithm*.