Scale construction from a decisional viewpoint

• Authors: Michael Smithson
• Venue: Minds and Machines
• Year: 2006

Many quantitative scales are constructed using cutoffs on a continuum with scores assigned to the cutoffs. This paper develops a framework for using or constructing such scales from a decision-making standpoint. It addresses questions such as: How many distinct thresholds or cutoffs on a scale (i.e., what levels of granularity) are useful for a rational agent? Where should these thresholds be placed given a rational agent's preferences and risk-orientation? Do scale score assignments have any bearing on decision-making and if so, how should scores be assigned? Given two possible states of nature $$\{A, \sim A\}$$ , an ordered collection of alternatives $$\{R_{0}, R_{1},{\ldots}, R_{K}\}$$ from which one is to be selected depending on the probability that A is the case, a simple expected utility condition stipulates when adjacent alternatives are distinguishable and determines the threshold odds separating them. Threshold odds and utilities are mapped onto scale scores via a simple distance model. The placement of the thresholds reflects relative concern over decisional consequences given A versus consequences given ~ A. Likewise, it is shown that scale scores reflect risk-aversion or risk-seeking not only with respect to A versus ~ A but also with respect to the rank of the R j . Connections are drawn between this framework and rank-dependent expected utility (RDEU) theory. Implications are adumbrated for both machine and human decision-making.