An Introduction to Decision Theory

It is hard to think of any minimally reasonable decision rule that totally ignores how much utility an act brings about. However, the concept of utility has many different technical meanings, and it is important to keep these different meanings apart. In Chapter 2 we distinguished three fundamentally different kinds of measurement scales. All scales are numerical, i.e. utility is represented by real numbers, but the information conveyed by the numbers depends on which type of scale is being used.

In ordinal scales, better objects are assigned higher numbers. However, the numbers do not reflect any information about differences or ratios between objects. If we wish to be entitled to say, for instance, that the difference between ten and five units is exactly as great as the difference between five and zero units, then utility has to be measured on an interval scale. Furthermore, to be entitled to say that ten units of utility is twice as much as five, utility must be measured on a ratio scale.

Arguably, utility cannot be directly revealed by introspection. We could of course ask people to estimate their utilities, but answers gathered by this method would most certainly be arbitrary. Some more sophisticated method is needed. So how on earth is it possible to assign precise numbers to outcomes and acts that accurately reflect their value? And how can this process be a scientific one, rather than a process that is merely arbitrary and at best based on educated guesses?

How to Construct an Ordinal Scale

Let us first show how to construct an ordinal utility scale. To make the discussion concrete, imagine that you have a collection of crime novels, and that you wish to assign ordinal utilities to each book in your collection. To avoid a number of purely mathematical obstacles we assume that the number of books in the collection is finite.