Uncertainty in Weights for Composite Indicators Generated by Weighted Sums
59
Fig. 10 The development of U, within the same evolution from s = 0 to s = 1, however with
different starting points for the weights.Abscissa: s starting with 0 and ending at 1.Abscissa:
describes the s-evolution
Indeed the values of U directly taken from all the posets possible in the series
of env(g,s) as well of such, following Eq. 12 are coinciding and show the expected
curve (Fig. 10).
It remains to check the behavior of the starting point g. Starting with g1 = 0.1
the left curve, starting with g1 = 0.5 the right curve in Fig. 10 results.
4 Discussion
Generally Eq. 9 appears as a good starting point to check, how many incomparabilities are to be expected, when the uncertainty in the numerical values of the weights
is described by the parameter s and s is evolving from 0 (sharp knowledge) to 1
(every weight between 0 and 1) is possible. The actual study goes beyond Eq. 9 and
analyzes why deviations from the straight line, represented by Eq. 9, are possible
and how they differ from the straight line. In order to get an idea the most simple
system, still allowing incomparabilities was studied: The system with m = 2. It
becomes clear that the crucial influence is given by those weights, for which the
composite indicator values of any two incomparable objects become nearly equal
values. Furthermore, it becomes clear that the larger the number of U0 the more
dense is the distribution of gc-values along the weight axis, as more evenly is
the distribution of jumps and it can be expected that then the deviations from the
U = s*U0 – line becomes less striking.
Clearly most of multi-indicator-systems will have by far more than 2 indicators.
Can the study above provide us with an advise? No and yes! No, because an explicit
formula for crucial weights in case of m > 2 leads to m-2-dimensional manifolds,
and even if they can be algebraically determined, it is rather difficult to derive
information about crucial weights in general. Yes, because following the device
59
Fig. 10 The development of U, within the same evolution from s = 0 to s = 1, however with
different starting points for the weights.Abscissa: s starting with 0 and ending at 1.Abscissa:
describes the s-evolution
Indeed the values of U directly taken from all the posets possible in the series
of env(g,s) as well of such, following Eq. 12 are coinciding and show the expected
curve (Fig. 10).
It remains to check the behavior of the starting point g. Starting with g1 = 0.1
the left curve, starting with g1 = 0.5 the right curve in Fig. 10 results.
4 Discussion
Generally Eq. 9 appears as a good starting point to check, how many incomparabilities are to be expected, when the uncertainty in the numerical values of the weights
is described by the parameter s and s is evolving from 0 (sharp knowledge) to 1
(every weight between 0 and 1) is possible. The actual study goes beyond Eq. 9 and
analyzes why deviations from the straight line, represented by Eq. 9, are possible
and how they differ from the straight line. In order to get an idea the most simple
system, still allowing incomparabilities was studied: The system with m = 2. It
becomes clear that the crucial influence is given by those weights, for which the
composite indicator values of any two incomparable objects become nearly equal
values. Furthermore, it becomes clear that the larger the number of U0 the more
dense is the distribution of gc-values along the weight axis, as more evenly is
the distribution of jumps and it can be expected that then the deviations from the
U = s*U0 – line becomes less striking.
Clearly most of multi-indicator-systems will have by far more than 2 indicators.
Can the study above provide us with an advise? No and yes! No, because an explicit
formula for crucial weights in case of m > 2 leads to m-2-dimensional manifolds,
and even if they can be algebraically determined, it is rather difficult to derive
information about crucial weights in general. Yes, because following the device