The literature of composite indicators offers several examples of aggregation techniques. The most used are additive techniques that range from summing up country ranking in each indicator to aggregating weighted normalised indicators. Yet, additive aggregations imply requirements and properties, both of the indicators and of the associated weights, which are often not desirable and at times difficult to meet or burdensome to verify. To overcome these difficulties the literature proposes other, and less widespread, aggregation methods such as multiplicative (e.g. geometric) aggregations or non-compensatory aggregations, such as the multi-criteria analysis.

1. Additive methods
2. Geometric Aggregation
3. Non compensatory multi-criteria analysis

1. Additive methods

The simplest additive aggregation method is the sum of a country's rank in each of the indicators. This method is therefore based on ordinal information. Its advantages are simplicity and the independence to outliers. Its disadvantage is that the method loses the absolute value information.

A second method uses nominal scores for each indicator to calculate the difference between the number of indicators that are above or below an arbitrarily defined threshold around the mean. The pros and cons of this method are similar to the previous method.

By large, the most widespread additive aggregation is the linear summation of weighted and normalized indicators. Although widely used, this aggregation entails restrictions on the nature of indicators and the interpretation of the weights. A condition on the nature of indicators is that there should be no phenomena of conflict or synergy among the indicators. Furthermore, the indicators have to be preferentially independent, i.e. any subset of the indicators is preferentially independent of its complementary set of indicators. Preferential independence is a very strong condition since it implies that the trade-off ratio between two indicators is independent of the values of the remaining indicators.

2. Geometric aggregation

As discussed previously, an undesirable feature of additive aggregations is the full compensability they imply: poor performance in some indicators can be compensated by sufficiently high values of other indicators. For example if an hypothetical composite indicator was formed by inequality, environmental degradation, GDP per capita and unemployment, two countries, one with values (21,1,1,1) and the other with (6,6,6,6) would have equal composite indicator value (=6) under an additive aggregation. Obviously the two countries would represent very different social conditions that would not be reflected in the composite.

Geometric aggregation (i.e. the product of weighted indicators) is a less compensatory approach. In our simple example the first country would have a much lower composite indicator value (=2.14) than the second country (=6.00) under the geometric aggregation. The use of geometric aggregation can also be justified on the grounds of the different incentives they apply to countries in a benchmarking exercise. Countries with low scores in some indicators would prefer a linear rather than a geometric aggregation (as explained previously). On the other hand, an increase in an indicator value would have higher marginal utility on the composite indicator if the indicator value is low: let us assume an increase of 1 unit for the second indicator, then the first country would increase its composite from 2.14 to 2.54, while the second country would score from 6.00 to 6.23. In other terms the first country would increase its composite by 19% while the second only by 4%. The lesson is that a country should be more interested in increasing those sectors/activities/alternatives with the lowest score in order to have the highest chance to improve its position in the ranking if the aggregation is geometric rather than linear.

3. Non compensatory multi-criteria analysis

In additive or geometric aggregations the substitution rates among indicators are equal to the weights of the indicators up to a multiplicative coefficient (Munda & Nardo, 2003). As a consequence, weights in those aggregation schemes necessarily have the meaning of substitution rates and do not indicate the importance of the indicator associated. This implies a compensatory logic. Compensability refers to the existence of trade-offs, i.e. the possibility of offsetting a disadvantage on some indicators by a sufficiently large advantage on other indicators. The implication is the existence of a theoretical inconsistency in the way weights are actually used and their real theoretical meaning. For the weights to be interpreted as “importance coefficients” non-compensatory aggregation procedures must be used to construct the composite indicators (Podinovskii, 1994). This can be done using a non-compensatory multi-criteria approach.

The multi-criteria procedure tries to resolve the conflict arising in countries comparisons as some indicators are in favour of one country while other indicators are in favour of another. This conflict can be treated at the light of a non-compensatory logic and taking into account the absence of preference independence within a discrete multi-criteria approach (Munda, 1995). The approach employs a mathematical formulation (Condorcet-type of ranking procedure) to rank in a complete pre-order (i.e. without any incomparability relation) all the countries from the best to the worst after a pair-wise comparison of countries across the whole set of the available indicators.

Let us attempt to provide a ‘hand waiving’ description of the algorithm’. Assume that we have three countries, A, B and C and we aim at ranking their overall performance according to N indicators. We build to this effect an ‘outranking matrix’ whose entries eij tell us how much country ‘i’ does better than country ‘j’. eij is in fact the sum of all weights of all indicators for which country ‘i’ does better than country ‘j’. eji will likewise be the sum of all weights for which the reverse is true. If the two countries do equally well on one indicator, its weight is split between eij and eji. As a result eij + eji =1 if weights have been scaled to unity. We now write down all possible permutations of country order (ABC,ACB,BAC,BCA,CAB,CBA) and compute for each of them the ordered sum of the scores, e.g. for ABC we compute Y=eAB+eAC+eBC. We do this for all permutations and take as the multicriteria country ranking the one with the highest total score Y. Note that this ordering is only based on the weights, and on the sign of the difference between countries values for a given indicator, the magnitude of the difference being ignored. With this approach no compensation occurs. This means that a country that does marginally better on many indicators ranks higher than a country that does a lot better on a few indicators because it cannot compensate deficiencies in some dimensions with outstanding performances in others.

This aggregation method has the advantage to overcome some of the problems raised by additive or multiplicative aggregations: preference dependence, the use of different ratio or interval scale to express the same indicator and the meaning of trade-offs given to the weights. With this method, moreover, qualitative and quantitative information can be jointly treated. In addition, it does not need any manipulation or normalization to assure the comparability of sub-indicators.

The drawbacks, instead, include the dependence of irrelevant alternatives, i.e. the possible presence of cycles/rank reversal in which in the final ranking, country a is preferred to b, b is preferred to c but c is preferred to a (the same problem highlighted for Analytic Hierarchy Process with indicators). Furthermore, information on intensity of preference of variables is never used: if one indicator for country a is much less than the same indicator for country b produces the same ranking as the case in which this difference is very small. Notice that with this method the focal point is shifted to the determination of weights, which becomes crucial for the result.
 

Next