INSTITUTE OF MATHEMATICAL ECONOMICS

Wulf Albers

Working-Paper No. 269

July 1997

Abstract

The paper introduces the present state of the applications of the theory of prominence of ALBERS-ALBERS (1983) to the perception of numerical information. Basic elements of the theory can be found in the WEBER-FECHNER law (1834, 1860) concerning the psychophysical perception of physical stimuli as brightness, loudness, or weight. The rules are (1) evaluation of the intensity of stimuli is logarithmic, (2) stimuli are perceived with a constant relative exactness, and (3) there is a smallest absolute intensity that can be perceived. The same rules can be applied to the perception of stimuli which are presented in a numerical way, as prices, quantities, percentages, or time. The perception of all these different kinds of stimuli is ruled by identical basic laws concerning the perception of numbers, here presented for the decimal system.
Basic elements of the theory is a system of numbers which are most easily perceived, the full step numbers {a*10^i: a Î {1,2,5}, i integer}. Comparison of numerical stimuli happens on a scale, on which the full step number define the full steps. Half steps, quarters, etc. can be defined. The difference of numerical stimuli is given by the difference measured in steps on this scale. Every number is perceived as a sum of full step numbers, where the coefficients are +1, -1, or 0, i.e. one obtains a number by starting with some high full step number and refines the number stepwise by adding or subtracting finer full step numbers, for instance 17=20-5+2. The exactness of such a presentation is the finest full step number needed in the presentation, the exactness of a number is given by the crudest exactness over all possible presentations of the number, the relative exactness of a number is its exactness divided by the number.
The system of half steps, quarter steps, etc. corresponds to perception with decreasing exactness. As in the WEBER-FECHNER laws (2) and (3) it turns out that - depending only on situation, person, and task - relative exactness and absolute exactness of perception are constants. These rules are insofar different from the WEBER-FECHNER law that 1. rule (3) becomes important since it enables to compare positive and negative payoffs (while the variables of the WEBER-FECHNER approach are allways positive), and 2. the constants of exactness do now also depend on the task. While spontaneous perception usually happens at a half step level, the absolute exactness of perception, more precisely the finest perceived full step essentially depends on the task. For the evaluation of money the finest perceived full step is roughly 20% of the largest absolute money value involved in the task.
Accordingly, it is not possible to present the obtained rules of perception of numerical differences by a universal perception function, specificly, it is not possible to give an universal utility function describing the perception of monetary payoffs (as KAHNEMAN-TVERSKY do in their v-function). The nonexistence of such a function creates for instance the possibility that the same prospect can be evaluated differently depending on other prospects with which it is compared. This is the reason for a kind of preference reversal which could be predicted by the theory here, and afterwards verified in the experiment.
The generalized perception function of our model is compared with the evaluation functions of KAHNEMAN-TVERSKY (1992). The new approach could be successfully used to modify traditional fairness concepts for different types of bargaining situations (KALAI-SMORODINSKY's equal concession solution, HARSANYI-SELTEN's risk dominance, NASH's bargaining solution) in a way that they now seem to be the best predictors for the related experimental behavior.
It may be mentioned that the modificaton of traditional concepts follows certain simple rules, and that in the obtained solutions the variables are only treated on an additive level (no products, no quotients), and all coefficients of involved variables are either +1 or -1. This simplicity might be a general phenomenon of certain boundedly rational models, so that the decision maker has only to decide, whether to apply a given variable or not, and in which direction it works.

Further results that confirm the given approach are presented in Part IV.