INSTITUTE OF MATHEMATICAL ECONOMICS

Wulf Albers

Working-Paper No. 265

July 1997

Abstract

Starting point of the theory of prominence is the observation that the selection of a numerical response is performed by a process of stepwise refinement of a reasonable answer until the available information does not permit a further specification. The procedure starts with a sufficiently high number, and stepwise decides whether to add, subtract or not use the next finer of the set of full step numbers for the presentation, where the full step numbers are {a*10^i: a Î {1,2,5}, i integer}.
The result is the presentation of a number as sum of full step numbers with coefficients +1, -1, or 0, where every full step number is 'used' at most once. For instance 17=20-5+2, or 24=20+5-1. This presentation is not necessarily unique. Important is the finest full step number used by a presentation. It is denoted as the exactness of the presentation. The exactness of a number is the finest exactness among all presentations of the number. It informs about the crudest level of exactness on which the number can be perceived, i.e. constructed by a response process. - Central tools for the analysis of numerical responses are two types of scales. S(r,a)-scales are based on the observation that subjects adjust relative exactness r, and absolute exactness a to a given type problem or situation.
M(i,a)-scales are constructed by starting with the full step numbers, and stepwise inserting the respective 'crudest number' as 'midpoint' between any two neighboured numbers of the preceding scale. Accordingly one obtains scales on the full step, half step, ... level. Both types of scales permit to define perception functions by assuming that the distances of any two neighboured numbers of a scale are equal, i. e. by applying usual interpolation principles. Several lemmata concerning the structure of scales are given.