# Comparing Actions and Developing Criteria

• Bernard Roy
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 12)

## Summary

In Section 9.1.1 we review the different meanings of the term “criterion” and mention that criterion and criterion function are usually synonymous in the field of decision aiding. We then define (Def. 9.1.1) and generalize this concept in Section 9.1.2 and add five remarks.

We address the development of criteria from action consequences that are modeled by state and dispersion indicators along various dimensions in Section 9.2. In Section 9.2.1 we discuss the case when criterion g is associated with a single dimension i that leads to a point evaluation. In this case, we show (Res. 9.2.1) that developing such a criterion leads to a state indicator encoding (Def. 9.2.1) and describe several examples. In Section 9.2.2, we discuss the case when the criterion g is associated with a single dimension i that leads to a nonpoint evaluation. Here, we distinguish between two cases. In the first case (Section 9.2.2.1), g is the only criterion that affects the evaluation along dimension i, and we say that there is point reduction in the dimension. We discuss and illustrate the principal point reduction techniques, especially the technique based on utility theory. In the second case (Section 9.2.2.2), g is not the only criterion that affects the evaluation along dimension i, and we say that the criteria split dimension i. We discuss the reasons for such splitting. Finally, in Section 9.2.3 we consider the case when a criterion g, affects all the dimensions in a subset I. When gI, is conceived so as to contain all the evaluation information contained in this subset of dimensions, we say that there is subaggregation of these dimensions. We explore when developing such criteria makes sense and clarify terminology.

In Section 9.3 we investigate the limits of using the values of a criterion to form indifference or strict preference relations over pairs of actions. This leads us to propose the notion of discriminating power of a criterion in Section 9.3.1. We provide an operational meaning to this notion in Section 9.3.2 when we introduce the concepts of indifference and preference thresholds. In Section 9.3.3 we define the important concept of pseudo-criterion as a criterion function to which discrimination thresholds are added (Def. 9.3.2). We introduce specific cases where at least one of the two thresholds is empty (Def. 9.3.3). The structures of the resulting systems of preference relations correspond to structures already seen in Chapter 7 (Res. 9.3.1). We finish this subsection by discussing practical ways to determine discrimination thresholds.

In Section 9.4, we examine when the difference g(a′) — g(a) can be used to reflect the qualitative importance of the difference in actions a′ and a according to the criterion g. After motivating the general problem in Section 9.4.1, we present the two important definitions of gradation (Def. 9.4.1) and gradable criterion (Def. 9.4.2) in Section 9.4.2. We devote Sections 9.4.3 and 9.4.4 to criteria that can be called measures. In Section 9.4.3, we present the basic definition (Def. 9.4.3), provide examples, and discuss properties (Res. 9.4.2). In Section 9.4.4, we discuss the special case of the von Neumann-Morgenstern expected utility criterion. We specify the axiomatic basis (Axioms 9.1–9.4) that defines this criterion up to a positive affine transformation (Res. 9.4.3) and discuss the conditions under which it is a measure. These conditions are closely related (Axiom 9.5 or 9.6) to those that allow differences in criterion values to reflect importance.

## Keywords

Criterion Function Elementary Consequence Discrimination Threshold Ideal Action Significance Axis