There Is No Such Thing as a Free Lunch! Who Is Paying for Our Happiness?
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x is higher than the corresponding indicator value for y and no indicator for x is
lower than the corresponding indicator value for y. On the other hand, if r j (x) > r j (y)
for some indicator j and r i (x) < r i (y) for some other indicator i, x and y will be
called incomparable (notation: x || y) expressing the mathematical contradiction
due to conflicting indicator values. A set of mutual incomparable objects is called
an antichain. When all indicator values for x are equal to the corresponding indicator
values for y, i.e., r j (x) = r j (y) for all j, the two objects/nations will have identical
rank and will be considered as equivalent, i.e., x ~ y. The analysis of Equation 1
results in a graph, the Hasse diagram. Hasse diagrams are unique visualizations of
the order relations due to Equation 1.
2.2 The Hasse Diagram
The Eq. 1 is the basic for the Hasse diagram technique (HDT) (Bruggemann
and Carlsen 2006a, b; Bruggemann and Patil 2011). Hasse diagrams are visual
representation of the partial order. In the Hasse diagram comparable objects are
connected by a sequence of lines (Bruggemann and Carlsen 2006a, b; Bruggemann
and Patil 2011; Bruggemann and Münzer 1993; Bruggemann and Voigt 1995, 2008).
2.3 The More Elaborate Analyses
In addition to the basic partial ordering tools some more elaborate analyses have
been used including average ranks (Bruggemann and Annoni 2014; Morton et
al. 2009; De Loof et al. 2006; Lerche et al. 2003; Bruggemann et al. 2004;
Bruggemann and Carlsen 2011) and sensitivity analysis (Bruggemann and Patil
2011; Bruggemann et al. 2014), the latter gives an insight in the relative importance
of the included indicators (Bruggemann and Patil 2011; Bruggemann et al. 2014).
The average ranking is expressed as average height from bottom (min.
Height = 1) to the top (max height = n, i.e., the maximum number of objects)
(Bruggemann and Annoni 2014). The average rank is generated by calculating all
linear order preserving sequences (set LE), the “linear extensions of the original
partial order. From LE_0 the statistical characterization for each object is obtained.
For example the characterization is calculated as the average value an object has,
taken all positions of this object within LE_0, the averaged heights. It is clear that
this procedure is computationally extremely difficult. Hence, approximations were
developed.
For the sensitivity analysis (Bruggemann and Patil 2011; Bruggemann et al.
2014), let Q be the set of all indicators, then taken all indicators of Q leads to a
partial order, which is called PO_0. The corresponding set of linear extensions is
denoted by LE_0. Leaving out one indicator of Q, say r j , then another partial order
results, which is denoted as PO_j.
207
x is higher than the corresponding indicator value for y and no indicator for x is
lower than the corresponding indicator value for y. On the other hand, if r j (x) > r j (y)
for some indicator j and r i (x) < r i (y) for some other indicator i, x and y will be
called incomparable (notation: x || y) expressing the mathematical contradiction
due to conflicting indicator values. A set of mutual incomparable objects is called
an antichain. When all indicator values for x are equal to the corresponding indicator
values for y, i.e., r j (x) = r j (y) for all j, the two objects/nations will have identical
rank and will be considered as equivalent, i.e., x ~ y. The analysis of Equation 1
results in a graph, the Hasse diagram. Hasse diagrams are unique visualizations of
the order relations due to Equation 1.
2.2 The Hasse Diagram
The Eq. 1 is the basic for the Hasse diagram technique (HDT) (Bruggemann
and Carlsen 2006a, b; Bruggemann and Patil 2011). Hasse diagrams are visual
representation of the partial order. In the Hasse diagram comparable objects are
connected by a sequence of lines (Bruggemann and Carlsen 2006a, b; Bruggemann
and Patil 2011; Bruggemann and Münzer 1993; Bruggemann and Voigt 1995, 2008).
2.3 The More Elaborate Analyses
In addition to the basic partial ordering tools some more elaborate analyses have
been used including average ranks (Bruggemann and Annoni 2014; Morton et
al. 2009; De Loof et al. 2006; Lerche et al. 2003; Bruggemann et al. 2004;
Bruggemann and Carlsen 2011) and sensitivity analysis (Bruggemann and Patil
2011; Bruggemann et al. 2014), the latter gives an insight in the relative importance
of the included indicators (Bruggemann and Patil 2011; Bruggemann et al. 2014).
The average ranking is expressed as average height from bottom (min.
Height = 1) to the top (max height = n, i.e., the maximum number of objects)
(Bruggemann and Annoni 2014). The average rank is generated by calculating all
linear order preserving sequences (set LE), the “linear extensions of the original
partial order. From LE_0 the statistical characterization for each object is obtained.
For example the characterization is calculated as the average value an object has,
taken all positions of this object within LE_0, the averaged heights. It is clear that
this procedure is computationally extremely difficult. Hence, approximations were
developed.
For the sensitivity analysis (Bruggemann and Patil 2011; Bruggemann et al.
2014), let Q be the set of all indicators, then taken all indicators of Q leads to a
partial order, which is called PO_0. The corresponding set of linear extensions is
denoted by LE_0. Leaving out one indicator of Q, say r j , then another partial order
results, which is denoted as PO_j.