BoundedRationality_TheAdaptiveToolbox.pdf

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WulfAlbers
and only if their difference is \x-y\ > a. The level of relative exactness is defined
correspondingly:
A decision maker who gives a response has a level of relative exactness,
r, if she can distinguish any two numbers (x, y) if and only if
|x-j|/max(|x|,|y|)>K
Assume the decision maker gives the response 20. This is possible, for instance,
in a situation where the decision maker can distinguish between 20 and 10, but
not between 20 and 15 (exactness level 10), or in a situation in which she can distinguish
between 20 and 18 but not between 20 and 19 (exactness level 2). The
response does not provide information concerning the level of exactness on
which it was produced.
Conclusions regarding the exactness level of the decision maker (or of a
group of decision makers) can be drawn when sets of responses are available.
For instance, when the responses 30, 35,40, and 45 are given with nearly equal
frequency, we may conclude that the level of exactness was 5. When 30 and 40
are about three times as frequent as 35 and 45, we may conclude that half of the
subjects had an exactness level of 5 (i.e., they provided responses of 30, 35,40,
and 45 with equal frequency) and half of the subjects had an exactness level of
10 (i.e., they provided responses of 30 and 40 with equal frequency and each
number was selected by each person twice as often).
For certain data sets, part of the subjects responded with relatively fine exactness,
while most selected a cruder level of exactness. Therefore, we define the
level of exactness of a data set as the crudest level of exactness that is fulfilled by
80% of the data. (A less ad hoc measure is presented in Selten [1987].)
Natural Scales
A frequently used refinement of the structure of full-step numbers is represented
by spontaneous numbers. They insert one additional step (i.e., a "midpoint") between
any two consecutive full-step numbers. The most prominent number between
the two full steps is selected as the midpoint. This criterion immediately
selects 15 as the midpoint between 10 and 20, and30 as the midpoint between 20
and 50 (as the numbers with the crudest level of exactness within the respective
intervals; note that 30 = 50 - 20 is the crudest number between 20 and 50, since
40 = 50 - 10). Between 50 and 100 there are two candidates with the crudest
level of exactness: 70 and 80.70 = 100 - 50 + 20 is cruder in that it has a slightly
The idea behind this example can be generalized using the following rules of thumb
for the relations of frequencies of numbers with different exactness: numbers with
exactness 5 are twice as frequent as numbers with exactness 10 (last digits are 5, 0
versus 0); numbers with exactness 2 are three times as frequent as numbers with
exactness 5 (last digits 2, 3, 5, 7, 8, 0 versus 5, 0); numbers with exactness 2 are
three-fifths as frequent as numbers with exactness 2 (last digits 1,2,3,4,5 vs. 2,3,5).