Mathematical Modeling in the Social and Life Sciences

CHAPTER 8

Introduction to Utility Theory

Some reckon time by stars, And some by hours; Some measure days by Dreams, And some by flowers; My heart alone records My days and hours.

—Madison Cawein

I. Introduction

This chapter continues the axiomatic discussion, begun in Chapter 7, of certain aspects of measurement theory. We consider again the problem that motivated the development of the material in the preceding chapter from a new point of view. The problem is to construct a numerical measurement of “happiness”; in particular, to assign numbers that measure how happy a particular student would be if she were assigned various different courses by the college's registrar.

The point of view of this chapter is called utility theory. The theory dates back at least 200 years to a time when nobles of the French court asked mathematicians for advice on how to gamble. Quite a rich theory has been developed, and various aspects of it have been tested experimentally in situations requiring decision making with incomplete knowledge.

Consider the set S of possible choices of courses to which the student might be assigned. Using the mechanisms of Chapter 7, or some other scheme, it is determined that the student prefers course x over course y and course y over course z. Utility theory aims to assign numerical weights to these preferences.

Suppose we offer the student a choice: she may have course y, her intermediate choice, or she may flip a coin. If the coin comes up heads, she gets course ...