Marrying

So far in this book, we have considered the question of finding the best outcome for a single person under a fixed set of rules. What is the best way to bet, given the appropriate odds and probabilities? Should I take out an annuity? What is the shortest route from A to B?

Life becomes much more complicated when there are many people with different goals and the action of one person changes the rules for the others. In this chapter we shall see that, even in these circumstances, mathematics can sometimes provide insight. We shall also see that problems arise which lie outside the province of the mathematician.

We start by looking at problems of the following type. Suppose we wish to form 2n children into pairs. If we match Amber with Bertha and Caroline with Delia but Amber prefers Caroline to Bertha while Caroline prefers Amber to Delia, then the pairing is unstable since Amber and Caroline would both prefer to break up with their present partners and form a pair together. If, however, Amber prefers Caroline to Bertha but Caroline prefers Delia to Amber this particular event will not happen (though there may be other ways in which the pairing is unstable).

Our problem is the following.

The Kindergarten ProblemIs it always possible to arrange 2n children in stable pairs (i.e. so there are not two children in different pairs who would prefer each other to their present partner)?