## Contents

## Introduction

Have you ever had the experience when you were in elementary or middle school where you are asked to “find a partner you like to make a pair”?

In one form or another, I am sure you have heard something like this before, and each of you must have selected a partner with many thoughts running through your head. But, let us step back for a moment and recall what happened. Were you two truly satisfied with the pair? Our own standards of values are vague, and it is not realistic to think you share the same standard of values with others, so maybe it is hard to judge if you both were truly satisfied. However, maybe you could have paired with someone better. If you intend to pair up with someone very good, how could this be accomplished?

We often see the following examples in real life. For example, designating students to go to certain universities, or sending doctors or nurses to certain hospitals. Of course, with these examples we hope for those involved got their choice as much as possible.

These types of interests can be thought of mathematically. The above examples are the same as a group of people each making a pair. The more general these problems are, the harder it is to solve, while the more limited and specific the problems are, the easier they become to approach. Limited problems can be thought of as basic problems which can be expanded to tackle more general problems.

One such example of how these problems are reduced to a limited problem would be the situation where two groups of an equal number of people are asked to form duos comprising of one from each group. Since these two groups could be considered a male group and a female group, and pairing up can be seen as a marriage, this problem has been called the Stable Marriage Problem (SMP). Since SMP is a basic problem that has an interesting nature to it, it’s been extensively studied.