Obtaining a Product Structure

In this chapter we study a variant on the measurement problems we have considered so far. Specifically, we consider measurement when the underlying set can be expressed as a Cartesian product. In making choices, we often consider alternatives with a variety of attributes or dimensions or from several points of view. We talk about multidimensional or multiattributed alternatives. For example, in choosing a job, we might consider salary, job security, possibility for advancement, geographical location, and so on. In buying a house, we might consider price, location, school system, availability of transportation, and the like. In designing a rapid transit system, we might consider power source, vehicle design, right of way design, and so on. In such a situation, each alternative a in the set of alternatives can be thought of as an n-tuple (a1, a2, …, an), where ai is some rating of alternative a on the ith attribute or dimension. For example, in the case of a job, a1 might be salary, a2 might be fringe benefits, a3 might be some measure of job security (for example, amount of notice required), and so on.

Multidimensional alternatives arise in a different way in economics. If there are n commodities in consideration, at might be a quantity of the ith commodity, and (a1a2, …, an) then is a commodity bundle or market basket. Preferences are expressed among alternative market baskets.