Property of (Strict/Weak) Monotony
This property concerns a derived indicator that aggregates a set of sub-indicators. Simplifying, if the increase/decrease of a specific sub-indicator is not associated to the increase/decrease of the derived indicator (ceteris paribus), then the derived indicator is not (strictly) monotonous with respect to that specific sub-indicator.
This definition implicitly entails that the symbolic manifestations of the sub-indicators are represented using ordinal or more powerful scales. When indicators are represented on scales with no order relation (i.e., nominal scales), the property of monotony loses its meaning.
We remark that monotony should always be expressed with respect to a specific sub-indicator: while it makes sense to state that a derived indicator is (or not) monotonous with respect to a certain sub-indicator, it makes no sense to state that a derived indicator is monotonous in general.
Going into more detail, the property of monotony is closely linked to the definition of monotonous function in Mathematical Analysis (Hazewinkel 2013). Precisely, a derived indicator f(x) is called strictly monotonically increasing with respect to a (sub)indicator, if—for all x and y values of this (sub)indicator such that x < y — one has f(x) < f( y) (i.e., f preserves the order). Likewise, a derived indicator is called strictly monotonically decreasing, if—for all x and y such that x < y — one has f(x) > f( y) (i.e., f reverses the order).
If the strict order “<” in the previous definition is replaced by the weak order “”, then one obtains a weaker requirement. A derived indicator that meets this requirement is called weakly monotonically increasing. Again, by inverting the order symbol, one finds a corresponding concept called weakly monotonically decreasing (see Fig. 4.18).
Derived indicators that are strictly monotonous with respect to a (sub)indicator are one-to-one (because for x 6¼ y, either x < y or x > y and so, by monotony, either f(x) < f( y) or f(x) > f( y), thus f(x) 6¼ f( y).
In general, (strict) monotony is a desirable property of derived indicators, since it proves their responsiveness with respect to variations in the relevant sub-indicators. Strict monotony is therefore preferred to weak monotony, which is in turn preferred
to non-monotony.
