BoundedRationality_TheAdaptiveToolbox.pdf

306 WulfAlbers
S(r, a) can contain at most two elements, namely (20,30) or (70,80) or the multiples
of these pairs with integer powers often.
PERCEPTION OF UTILITY
Perception of Utility and Evaluation of Prospects
The basic idea of this approach is that decision makers measure utility by means
of scales with a step structure in such a way that the differences between any two
neighboring steps of the scale are perceived as equal, and interpolation of the
distance between neighboring steps is possible. For scales on the money and on
the probability space, interpolation is modeled as linear with respect to the respective
space.
Steps of these scales are related to the full-step numbers: for a given task there
is a task-dependent "finest perceived full-step number" (FPF) with the property
that all full-step numbers that are greater than or equal to FPF are steps of the
scale, while all finer full-step numbers are not steps of the scale. The idea is that
full-step numbers that are finer than FPF are not perceived as steps. Below FPF
the next step of the scale is zero.
If negative values are possible, the steps in the range of negative numbers are
obtained from the steps in the range of positive numbers by multiplication with
—1. For the money scale, steps in the range of negative numbers are valued twice.
The probability scale is a union of two parts. Part A has steps in the range between
0% and 50% (probabilities in the narrower sense). They are constructed
as above with task-dependent FPF and zero, but they are restricted from above
by 50% as maximal element. Part B has the steps in the range between 50% and
100% (counterprobabilities in the narrower sense). These steps are obtained by
subtracting the steps of Part A from 100%.
For monetary amounts, the finest perceived full step is the crudest full-step
number that is more than one full step below the maximal absolute value of the
numbers given in the task. For probabilities, the finest perceived full step is the
crudest full step that is finer than or equal to all probabilities and
counterprobabilities given in the task.
Examples: When 10 is the finest perceived full step, one obtains the steps ...,
-100, -50, -20, -10, 0,10,20, 50,100,... This permits us to measure the difference
between-50 and 10 as seven steps (valuing steps in the range of negative
numbers twice), or to obtain the distance of-50 and 15 as 7.5 steps by using linear
interpretation between full steps. When 1% is the finest perceived probability,
we obtain the probability scale 0%, 1%, 2%, 5%, 10%, 20%, 50%, 80%,
90%, 95%, 98%, 99%, 100%. On this scale, 1% is one step above zero, and the
total distance from 0% to 100% is 12 steps. Accordingly, 1% is perceived as 1 of
12 steps, i.e., as 1/12, whereas 99% is perceived 11/12. If 10% is the finest perceived
full step of probability, we obtain the scale 0%, 10%, 20%, 50%, 80%,