Models of preferences

In Chapter 1 (Example 1.21), we mentioned that the classic utility function of economists that represents the preferences of a consumer on a set of commodity bundles (bundle x is preferred to bundle y if u(y) < u(x)) defines a particular (strict) order, called a weak order. In this modeling of preferences by a utility function, two bundles with the same utility are indifferent for the consumer. Then his indifference relation is transitive. Yet, it was observed long ago that this assumption is not necessarily satisfied. This observation has led us to define other preference ordinal models allowing a numerical representation of the preference along with a non-transitive indifference relation, namely interval orders and semiorders. The orders of these two classes have been studied extensively. In this section, we concentrate on their basic properties and their numerical representations obtained in the frameworks of psychophysics and preference modeling. First, let us observe or specify several points.

The order relations studied in this section are in particular used in the many areas where one needs to modelize preferences, i.e., not only in microeconomics but more generally in the normative or descriptive decision theories (preferences of a decisionmaker over alternatives, preferences of a player on lotteries) or in voting theory (preferences of a voter over candidates).

In these models, one can modelize either the so-called strict preference (interpreted as “object x is better than object y”) or the so-called weak preference (interpreted as “object x is at least as good as object y”).