Power Mean

The power mean is not, in itself, a single aggregation operator, but a framework that generates a family of them. It is not an independent decision method on its own; it is a building block showing that operators introduced separately, such as WAM, WGM and WHM, are in fact special cases of one formula. The formula carries a single power parameter (r); changing this parameter, with the same criterion scores and weights, produces an entirely different aggregation result. Its output is a single number on the same scale as the inputs.

The classical mathematical foundation of the power mean rests on Hardy, Littlewood and Pólya's 1934 book Inequalities. That book gave a definitive proof that the mean grows as the power parameter grows (the monotonicity property), and established which classical mean the result converges to as the parameter tends to zero or to plus or minus infinity. Bullen, Mitrinović and Vasić's comprehensive 1988 study extended these inequalities to weighted forms and to a broader class.

The power mean's philosophy is to adjust the degree of compensation with a single dial. Setting the power parameter to minus one yields the harmonic mean (the most punishing extreme); as it approaches zero it yields the geometric mean; set to one it yields the arithmetic mean (full compensation); raised to two it yields the quadratic mean (more sensitive to high values). By choosing a single number, r, the analyst answers directly the question "how much do I want to penalise weakness on a criterion in this decision problem."

This sets the power mean apart from the rest of the family: WAM, WGM and WHM each offer a fixed idea of compensation on its own, whereas the power mean places that idea along a continuous line. The same three criterion scores produce a sequence of results that rises from small to large as r changes; this rise never reverses, meaning the result never shrinks as the power grows.

The power mean requires three inputs: criterion scores, a power parameter (r), and weights summing to 1.

First, choose the power parameter and the weights. Whether criterion scores must be positive depends on the sign of r; for negative values of r, the scores must be greater than zero. The weights are checked to confirm they sum to 1.

Second, take the weighted power mean when r is non-zero. Each criterion score is raised to the power r, a weighted average of these values is taken, and then the 1/r-th root of that average is taken.

Third, handle the special case as r tends to zero. As r approaches zero the formula becomes undefined; in this limit the result equals the weighted geometric mean, and is calculated via logarithms for numerical stability.

Fourth, use the monotonicity property. As r grows, the result never shrinks; this property serves to order results obtained with different r values into a single chain, and to confirm the classical inequalities within the family (harmonic ≤ geometric ≤ arithmetic).

The formula behind this step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The number the power mean produces can equal, or sit close to, the number WAM, WGM or WHM would give, depending on the r value chosen; at r = 1 it equals WAM exactly, as r approaches zero it equals WGM, and at r = minus one it equals WHM. The result's meaning therefore cannot be considered apart from which r was chosen.

A notable feature of this number is the width of the range that different r values can produce from the same data. If the criterion scores differ sharply from one another (one very low, one very high), low r values keep the result low by giving prominence to the low score, while high r values push the result up by giving prominence to the high score. The same three numbers can yield results differing by nearly a factor of two depending on the r chosen.

Thus instead of writing:

"The power-mean score came out as such, and this is the one correct result"

the report should read:

"This score depends on the power parameter (r) chosen; a different choice of r produces a markedly different result from the same data, and this choice must be justified in the report"

The power mean works with crisp data; where negative values of r are used, the criterion scores must be greater than zero. DecisionMind holds no separate fuzzy, grey or intuitionistic extension of this building block; it works with crisp numbers.

You need at least two criteria' scores, on the same scale and comparable with one another; a choice for the power parameter r; and weights summing to 1. The choice of r comes not from the data but from the analyst's own view of compensation, and must be stated in the report. Choosing r = 1 reproduces WAM, r approaching zero reproduces WGM, and r = −1 reproduces WHM; the power mean is thus the most general form, containing all three of these building blocks.

The power mean is a suitable choice if the degree of compensation needs to be adjustable through a single parameter, or if you need to compare how different views of compensation (WAM, WGM, WHM) yield different results on the same data. In sensitivity analysis it is a natural tool for answering directly "how much does the result depend on the choice of r."

If there is no information to justify a particular choice of r, and a single fixed view of compensation (full compensation, say, at all times) suffices for the decision-maker, using WAM directly is a simpler choice that is easier to explain. If capturing an interaction between criteria (a weakness on one criterion combining with a weakness on another) is wanted, the power mean cannot do this; Bonferroni or Heronian means are needed for that.

The degree of compensation should be adjustable through a single parameter, sensitivity analysis is wanted → Power mean

A fixed, full compensation suffices, no extra parameter needed → WAM (equivalent to r = 1)

Penalising scores near zero is wanted → WGM (equivalent as r approaches zero)

The weakest criterion should dominate → WHM (equivalent to r = −1)

Capturing an interaction between criteria is wanted → Bonferroni or Heronian mean

The power mean's greatest strength is its unifying power: it allows WAM, WGM and WHM to be seen as special cases of one formula rather than learned separately. Thanks to the monotonicity property proved by Hardy, Littlewood and Pólya, a strict ordering among results obtained with different r values is guaranteed; this makes sensitivity analysis highly reliable. As Bullen, Mitrinović and Vasić showed, these inequalities also hold in the weighted case (Bullen, Mitrinović and Vasić, 1988).

The power mean's weakness is that it introduces an extra parameter (r); the choice of this parameter generally comes not from the data but from the analyst's own assumption, and if left unjustified the report looks arbitrary. In addition, as r approaches zero the formula becomes numerically unstable and requires a careful mode of calculation (via logarithms). Finally, the power mean is frequently confused, through the similarity of its name, with Yager's 2001 "power average" operator; yet the two operators rest on entirely different ideas and cannot be substituted for one another (Yager, 2001).

The most common mistake is confusing the power mean with Yager's "power average" operator; although the names are alike, in Yager's operator the weights vary according to the support between data points (how close they are to one another), whereas in the power mean the weights are fixed and only the power parameter r varies. A second mistake is choosing and reporting an r value without offering any justification; a different choice of r can produce a markedly different result from the same data. A third mistake is failing to check, when setting r to a negative value, whether any criterion score is zero; in that case the result becomes undefined.

The governing principle is this:

A power-mean result is a function of the power parameter chosen; a power-mean result reported without justifying that parameter is an incomplete result.