Giuseppe Munda
email: giuseppe.munda@jrc.ite_mail

Michela Nardo
email: michela.nardo@jrc.it

Linear Composite Indicators for Ranking Countries: A Defensible Setting

Keywords: Linear Composite Indicators for Ranking Countries: A Defensible Setting Index Numbers and Aggregation, Multi-Criteria Decision Analysis, Social Choice, Environmental Sustainability Index Index Numbers and Aggregation, Multi-Criteria Decision Analysis, Social Choice, Environmental Sustainability Index JEL Classification Numbers: C43, C82, Q01 Index Numbers and Aggregation, Multi-Criteria Decision Analysis, Social Choice, Environmental Sustainability Index JEL Classification Numbers: C43, C82, Q01

Introduction

Composite indicators are very common in fields such as economic and business statistics (e.g., the OECD Composite of Leading Indicators) and are used in a variety of policy domains such as industrial competitiveness, sustainable development, quality of life assessment, globalisation, innovation or academic performance (see Cox and others 1992[START_REF] Cribari-Neto | Research in econometric theory: quantitative and qualitative productivity rankings[END_REF], Färe et al. 1994, Griliches 1990, Forni et al. 2001, Huggins 2003, Grupp and Mogee 2004, Lovell et al. 1995, Author 2005, Author et al. 2005, Saisana and Tarantola 2002, and Wilson and Jones 2002, among others). The proliferation of these indicators is a clear symptom of their importance in policy-making, and operational relevance in economic statistics in general (see e.g. [START_REF] Granger | Macroeconometrics -Past and future[END_REF]. All the major international organizations such as OECD, the EU, the World Economic Forum or the IMF are producing composite indicators in a wide variety of fields (Author et al., 2005). A general objective of most of these indicators is the ranking of countries and their benchmarking according to some aggregated dimensions (see e.g. [START_REF] Cherchye | Using data envelopment analysis to assess macroeconomic policy performance[END_REF], Kleinknecht 2002and OECD, 2003) (Table 4 in the appendix supplies a sample of various types of composite indicators).

A recent report by OECD clearly states that "… Composite indicators are valued for their ability to integrate large amounts of information into easily understood formats for a general audience. However, composite indicators can be misleading, particularly when they are used to rank country performance on complex economic phenomena and even more so when country rankings are compared over time. They have many methodological difficulties which must be confronted and can be easily manipulated to produce desired outcomes… The proliferation of composite indicators in various policy domains raises questions regarding their accuracy and reliability. Given the seemingly ad hoc nature of their computation, the sensitivity of the results to different weighting and aggregation techniques, and continuing problems of missing data, composite indicators can result in distorted findings on country performance and incorrect policy prescriptions… Despite their many deficiencies, composite indicators will continue to be developed due to their usefulness as a communication tool and, on occasion, for analytical purposes" (OECD, 2003, p. 3).

As a consequence, the improvement of the way these indicators are constructed and used seems to be a very important research issue from both theoretical and operational points of view. Our main objective in this article is to contribute to the improvement of the overall quality of sustainability composite indicators (or indexes) by looking at one of their technical weaknesses, that is, the aggregation convention used for their construction. For this aim, we first try to clarify the axiomatic system of a linear aggregation rule. Then we check if this is compatible with the objective of a composite indicator using some results of multi-attribute utility theory and measurement theory literature. Finally, starting from concepts coming from multi-criteria decision theory and social choice, a non-compensatory aggregation rule is proposed and is corroborated with numerical examples.

On the Use of Linear Aggregation Rules

Although various functional forms for the underlying aggregation rules of a composite indicator have been developed in the literature (e.g. Diewert, 1976, Journal of Economic andSocial Measurement, 2002), in the standard practice, a composite indicator I, can be considered a weighted linear aggregation rule applied to a set of variables (OECD, 2003, p. 5):

1 N i i i I wx = =
, where i x is a scale adjusted variable (e.g. GDP per capita) normalized between zero and one, and i w a weight attached to i x , usually with The main research question to be answered is the following: under which axiomatic conditions a linear aggregation rule can be used?

A first answer to this question is given by the following theorem (Debreu, 1960;[START_REF] Keeney | Decision with multiple objectives: preferences and value tradeoffs[END_REF][START_REF] Krantz | Additive and polynomial representations[END_REF]: given the variables x 1 , x 2 ,..., preferentially independent if every subset Y of these variables is preferentially independent of its complementary set of evaluators.

Preferential independence is a very strong condition from both the epistemological and operational points of view, it implies that the trade-off ratio between two variables , x y S is independent of the values of the n-2 other variables, i.e. , 0

x y S q = , , x y Y q Q [START_REF] Ting | Aggregation of attributes for multiattributed utility assessment[END_REF].

(1)

From an operational point of view this means that an additive aggregation function permits the assessment of the marginal contribution of each variable separately (as a consequence of the preferential independence condition). The marginal contribution of each variable can then be added together to yield a total value. If, for example, environmental dimensions are involved, the use of a linear aggregation procedure then implies that among the different aspects of an ecosystem there are not phenomena of synergy or conflict. This appears to be quite an unrealistic assumption (Funtowicz et al., 1990). For example, "laboratory experiments made clear that the combined impact of the acidifying substances SO 2 , NO X , NH 3 and O 3 on plant growth is substantially more severe that the (linear) addition of the impacts of each of these substances alone would be." [START_REF] Dietz | Rethinking environmental economics: missing links between economic theory and environmental policy[END_REF]. 1 What happens if the linear aggregation is nevertheless done? The resulting indicator will be biased, i.e. it will not entirely and truthfully reflect the information of its components. The dimension and the direction of the error are not easily determined, thus the correction of the composite cannot be properly done. Summarising, we can conclude that the assumption of preference independence is essential for the existence of a linear

1 From an epistemological point of view, preferential independence implies the separability of values. This is quite an important issue in philosophy; for example, in ethics the thesis of the unity of the virtues is defended to different degrees by Plato, Aristotle and Aquinas. "... it is true that a virtue often cannot be treated apart from the company it keeps. Courage is not an excellence when it appears amid vices-for example as a disposition of the dedicated Nazi.... For example, one's appreciation of the value of freedom, "positive" or "negative", in a particular society, cannot be simply treated as separable from what individuals realise with that freedom. Different political and ethical values when applied in a particular context cannot be applied in isolation from one another" (O'Neill, 1993, p. 114 Another issue connected with the use of a linear aggregation rule is the one of weights. The common practice in attaching weights is well synthesised by a recent OECD document: "Greater weight should be given to components which are considered to be more significant in the context of the particular composite indicator" (OECD, 2003, p. 10). In the decision theory literature, this concept of weights is usually referred to as symmetrical importance, that is "… if we have two non-equal numbers to construct a vector in R 2 , then it is preferable to place the greatest number in the position corresponding to the most important criterion." (Podinovskii, 1994, p. 241).

Author (2006) illustrate that the concept of symmetrical importance is incompatible with a linear aggregation rule, given that in a linear aggregation rule, weights can only have the meaning of a trade-off ratio. As a consequence, since trade-offs always depend on the scales of measurement used, and since weights are connected to the values of trade-offs, weights also depend on the scales of measurement. As clearly shown by Anderson and Zalinski (1988), when weights depend on the range of variable scores, such as in the context of a linear aggregation rule, the interpretation of weights as a measurement of the (psychological) concept of importance is completely inappropriate. Summarizing the discussion we can state that the use of weights in combination with intensity of preference (given that variables are always supposed to be measured on an interval or ratio scale) within a linear aggregation rule originates compensatory aggregation conventions and gives the meaning of trade-offs to the weights 3 .

2 Often in practice the test of mutual preferential independence is not done because it is considered extremely time consuming. For M variables, there are M(M-1)/2 pairs that must be independent, thus the number of conditions to verify gets astronomically large as M gets even modestly large. However some results due to Leontieff (1947) can save much of the potential work to be done (see also Keeney and Raiffa, 1976, p. 112-114). 3 There is unanimous agreement in the literature that the only method where weights are computed as scaling constants and there is no ambiguous interpretation is the so-called trade-off method starting with revealed preferences. No weight importance judgment is required in this method. The trade-off method can be briefly described as follows. Vansnick (1990) showed that the two main approaches in multi-criteria aggregation procedures i.e., the compensatory and non-compensatory ones can be directly derived from the seminal work of Borda and Condorcet. If one wants the weights to be interpreted as "importance coefficients" (or equivalently symmetrical importance of variables) non-compensatory aggregation procedures must be used [START_REF] Bouyssou | Some remarks on the notion of compensation in MCDM[END_REF][START_REF] Bouyssou | Noncompensatory and generalized noncompensatory preference structures[END_REF]. From a social choice point of view, these non-compensatory rules are always Condorcet consistent rules (Author, 2006a); their use in the framework of composite indicators, can be corroborated by referring to a clear result of social choice literature 4 . The majority rule is theoretically the most desirable aggregation rule, but practically often produces undesirable intransitivities, thus "more limited ambitions are compulsory. The next highest ambition for an aggregation algorithm is to be Condorcet" (Arrow and Raynaud, 1986, p. 77).

Thus we can conclude that the use of non-compensatory aggregation rules to construct composite indicators is compulsory for reasons of theoretical consistency when weights with the meaning of importance coefficients are used or when the assumption of preferential independence does not hold. Moreover the ' ' ' ' ' ' 1 1 ( ) ( ) ( ,..., ,..., ,..., ) ( ,..., ,..., ,..., )

k t n k t n I A I B I x x x x I x x x x = = ' ' ' ' ' ' 1 1 , , N N i i k k t t i i k k t t i i i k t i k t w x w x w x w x w x w x = = + + = + + ' ' '' '' k k t t k k t t w x w x w x w x + = +
This last equation is an equation in the unknown w k and w t . To compute the N weights as tradeoffs, it is necessary to assess N-1 equivalence relations which together with the usual normalization constraint w i + … + w n =1 determine a linear system of N equations in the N unknown weights. Of course if some uncertainty on the variable scores exists, this method cannot be applied. As one can easily understand to assess weights as trade-offs, as it should be always done when using a linear aggregation rule, it is a much harder job than to use weights as importance coefficients. 4 Ebert and Welsch (2004) Nicoletti et al., 2000). On the contrary, by using

Condorcet aggregation rules no limitation on the measurement scale of the variable scores exists (Author, 2006).

For all these reasons, we think that in some applications the use of noncompensatory Condorcet consistent aggregation rules is desirable. Since this possibility has almost never been explored in the framework of composite indicators the following Section is devoted to this issue.

An Axiomatic Approach for the Construction of Non-

Compensatory/Non-Linear Condorcet Consistent Composite

Indicators

When various variables are used to evaluate two different countries, some of these variables may be in favour of country a while other variables may be in favour of country b. As a consequence a conflict among the variables exists. How this conflict can be treated at the light of a non-compensatory logic and taking into account the absence of preference independence? This is the classical multicriteria discrete problem (Author, 1995;Roy, 1996;Vincke, 1992). With this analogy in mind, we present an aggregation convention for (non-linear and noncompensatory) composite indicators able to rank different countries. For sake of clearness, some basic definitions are first given. These definitions are adapted to the context of composite indicators borrowing concepts from multi-criteria decision theory and complex system theory 5 .

Basic definitions

5 Some of these definitions were inspired by discussions with M. Giampietro. exposition, we assume that a higher value of an individual indicator is preferred to a lower one (i.e. the higher, the better), that is: The mathematical problem to be dealt with is then how to use this available information to rank in a complete pre-order (i.e. without any incomparability relation) all the countries from the best to the worst one. In doing so the following properties are desirable:

F

= > ) ( ) ( ) ( ) ( a g a g a I a a g a g a P a k m j m k j k m j m k j (2)
1. The sources of uncertainty and imprecise assessment should be reduced as much as possible.

2. The manipulation rules should be the more objective and as simple as possible, that is all ad hoc parameters should be avoided.

3. A theoretical guarantee that weights are used with the meaning of "symmetrical importance" must exist. As a consequence, complete compensability should be avoided. This entails that variables have to be used with an ordinal meaning. This is not a problem since no loss of information is implied [START_REF] Arrow | Social choice and multicriterion decision making[END_REF]. Moreover, given that often the measurement of variables is imprecise (see OECD, 2003, p.7), it seems even desirable to use indicator scores with an ordinal meaning.

4. Desirable ranking procedures using ordinal information are always of a Condorcet type [START_REF] Arrow | Social choice and individual values, 2d edition[END_REF]Raynaud, 1986, Moulin, 1988). A problem inherent to this family of algorithm is the presence of cycles, i.e. cases where aPb, bPc and cPa. This problem has been widely studied among others by Fishburn, 1973;Fishburn et al., 1979;Kemeny, 1959;Moulin, 1985;Truchon, 1995;Young andLevenglick, 1978, Vidu, 2002;Weber, 2002. The probability ( ) 5. Arrow's impossibility theorem [START_REF] Arrow | Social choice and individual values, 2d edition[END_REF] clearly shows that no perfect aggregation convention can exist. Then, it is essential to check not only which properties are respected by a given ranking procedure, but also if any essential property for the problem tackled is lost.

,

The proposed composite indicator

The mathematical aggregation convention we are proposing can be divided into two main steps:

1. Pair-wise comparison of countries according to the whole set of individual indicators used.

2. Ranking of countries in a complete pre-order.

For carrying out the pair-wise comparison of countries the following axiomatic system is needed (adapted from Arrow and Raynaud, 1986, p. 81-82).

Axiom 1: Diversity. Each individual indicator is a total order on the finite set A of countries to be ranked, and there is no restriction on the individual indicators; they can be any total order on A.

Axiom 2: Symmetry. Since individual indicators have incommensurable scales, the only preference information they provide is the ordinal pair-wise preferences they contain 6 .

Axiom 3: Positive Responsiveness. The degree of preference between two countries a and b is a strictly increasing function of the number and weights of individual indicators that rank a before b 7 . 6 In our case, this axiom is needed since the intensity of preference of individual indicators is not considered an useful preference information given that compensability has to be avoided and weights have to be symmetrical importance coefficients. Moreover, thanks to this axiom, a normalisation step is not needed. This causes a further reduction of the sources of uncertainty and imprecise assessment. 7 In social choice terms then the anonymity property (i.e. equal treatment of all individual indicators) is broken. Indeed, given that full decisiveness yields to dictatorship, Arrow's impossibility theorem forces us to make a trade-off between decisiveness (an alternative has to be chosen or a ranking has to be made) and anonymity. In our case the loss of anonymity in favour of decisiveness is even a positive property. In general, it is essential that no individual indicator Thanks to these three axioms a N× N matrix, E, called outranking matrix [START_REF] Arrow | Social choice and individual values, 2d edition[END_REF]Raynaud, 1986, Roy, 1996) can be built. Any generic element of E: e jk , j k is the result of the pair-wise comparison, according to all the M individual indicators, between countries j and k. Such a global pair-wise comparison is obtained by means of equation ( 2). preferences (with the principle one agent one vote). This analogy between a multi-criterion problem and a social choice one, as noted by [START_REF] Arrow | Social choice and multicriterion decision making[END_REF], is very useful for tackling the step of ranking the N countries in a consistent axiomatic framework.

The issue is now to exploit the information contained in the outranking matrix in order to rank all countries in a complete pre-order. A problem connected with the use of Condorcet consistent rules is that of cycles. A cycle breaking rule normally needs some arbitrary choices such as to delete the cycle with the lowest support. Now the question is: Is it possible to tackle the cycle issue in a more general way?

Condorcet himself was aware of the problem of cycles in his approach; he built examples to explain it and he was even close to find a consistent rule able to rank any number of alternatives when cycles are present. However, attempts to fully understand this part of Condorcet's voting theory have arrived at weight is more than 50% of the total weight; otherwise the aggregation procedure would become lexicographic in nature, and the indicator would become a dictator in Arrow's term. Nanson as quoted in Black, 1958, p. 175). Or "The obscurity and selfcontradiction are without any parallel, so far as our experience of mathematical works extends … no amount of examples can convey an adequate impression of the evils" (Todhunter, 1949, p. 352 as cited by Young, 1988Young, , p. 1234)).

Attempts of clarifying, fully understanding and axiomatizing Condorcet's approach for solving cycles have been mainly done by Kemeny (1959) who made the first intelligible description of the Condorcet approach, and by Young and Levenglick (1978) who made its clearest exposition and complete axiomatization.

In the version presented by Young and Levenglick (1978), the main methodological foundation is the maximum likelihood concept. In fact, "Condorcet's argument proceeds along the following lines. People differ in their opinions because they are imperfect judges of which decision really is best. If on balance each voter is more often right than wrong, however, then the majority view is very likely to identify the decision that is objectively best." (Young, 1988(Young, , p. 1232)).

The maximum likelihood principle selects as a final ranking the one with the maximum pair-wise support. This selected ranking is the one which involves the minimum number of pair-wise inversions. Since Kemeny (1959) proposes the number of pair-wise inversions as a distance to be minimized between the selected ranking and the other individual profiles, the two approaches are perfectly equivalent. A formal proof of this equivalence can be found in Truchon (1988b, pp. 6-10). The selected ranking is also a median ranking for those composing the profile (in multi-criteria terminology it is the "compromise ranking" among the various conflicting points of view), for this reason the corresponding ranking procedure is often known as the Kemeny median order. Arrow and Raynaud (1986, p. 77) arrive at the conclusion that the highest feasible ambition for an aggregation algorithm building a multi-criterion ranking is to be Condorcet. These authors discard the Kemeny median order, on the grounds that preference reversal phenomena may occur inside this approach (Arrow and Raynaud, 1986, p. 96). However, although the so-called Arrow-Raynaud's method does not present rank reversal, it is not applicable if cycles The acceptance of rank reversals phenomena implies that the famous axiom of independence of irrelevant alternatives of Arrow's theorem is not respected.

Anyway, Young (1988Young ( , p. 1241) ) claims that the maximum likelihood ranking procedure is the "only plausible ranking procedure that is locally stable". Where The final ranking ( r * ) is the one which maximises equation ( 6), which is: (6)

Other properties of this ranking procedure are the following (Young and Levenglick, 1978).

8 Anyway a Condorcet consistent rule always presents smaller probabilities of the occurrence of a rank reversal in comparison with any Borda consistent rule. This is again a strong argument in favour of a Condorcet's approach in our framework. reinforcement "… has definite ethical content and is therefore relevant to welfare economics and political science." (Arrow and Raynaud, 1986, p. 96). Moulin (1988, p. 312) clearly states that the Kemeny method (or equivalently the maximum likelihood approach) is "the correct method" for ranking alternatives, and that the "only drawback of this aggregation method is the difficulty in computing it when the number of candidates grows". In fact the number of permutations can easily become unmanageable; for example when 10 countries are present, it is 10!=3,628,800. Indeed this computational drawback is very serious since the Kemeny median order is NP-hard to compute. This NPhardness has discouraged the development of algorithms searching for exact solutions, thus the majority of the algorithms which have been proposed in the literature; are mainly heuristics based on artificial intelligence, branch and bound approaches and multi-stage techniques (see e.g., [START_REF] Barthelemy | Median linear orders: heuristics and a branch and bound algorithm[END_REF][START_REF] Charon | New results on the computation of median orders[END_REF][START_REF] Cohen | Learning to order things[END_REF]Davenport and Kalagnam, 2004;Dwork et Truchon, 1998b). Recently, a new numerical algorithm aimed at solving the computational problem connected to linear median orders by finding exact solutions has been developed too (Author, 2006b).

The computational problem

Thanks to the existence of all these computational algorithms, the maximum likelihood (or Kemeny) ranking procedure can always be applied in the context of composite indicators, where a high number of countries to be ranked is the normal state of affairs.

A sensitive issue: is information on intensity of preference complete lost in a Condorcet framework?

Given that the preference structure is based on equation ( 2), one might wonder if information on intensity of preference (when variables are measured on an interval or ratio scale) is completely lost in a Condorcet framework (since small and big intensities are treated equally). The problem is indeed very old. Its origins may be found in the famous bold paradox in Greek philosophy: how many hairs one has to cut off to transform a person with hairs to a bold one? Luce (1956) was the first one to discuss this issue formally in the framework of preference modelling. He introduced the idea of the existence of a sensibility threshold below which an agent either does not sense the difference between two elements, or refuses to declare a preference for one or the other. Mathematical characterisations of preference modelling with thresholds can be found in Roubens and Vincke (1985).

By introducing a positive indifference threshold q the resulting preference model is the so-called threshold model:

+ > q a g a g a I a q a g a g a P a k m j m k j k m j m k j ) ( ) ( ) ( ) ( (7)
where a j and a k belong to the set A of countries and g m to the set G of individual indicators. If one wishes to take into account the possible uncertainty around the value of the threshold q, sensitivity analysis and robustness analyses can be used (Saltelli et al., 2004), another possibility is the use of mathematical sophisticated concept such as the one of fuzzy preference modelling (Author, 1995). Yu, 1985;Zeleny, 1982). In this framework, to get a set of reference values, an "ideal point" can be defined by choosing the best values reached in any single indicator, and then mathematical distances are used.

F

An Illustrative numerical example

Let's take into consideration a simple hypothetical example where there are three countries (A, B, C) to be ranked according to a composite sustainability indicator. Let's assume that three dimensions have to be considered, i.e. the economic, the social and the environmental ones, and that each dimension should have the same weight, that is 0.3333.

The following individual indicators are used

Economic dimension

Indicator: GDP per capita. Weight: 0.165. Objective: maximisation of economic growth. Variable: US dollar per year.

Indicator: Unemployment rate. Weight: 0.165. Objective: minimisation of unemployed people. Variable: percentage of population.

Environmental dimension

Indicator: Solid waste generated per capita. Weight: 0.333. Objective: minimisation of environmental impact. Variable: tons per year.

Social dimension

Indicator The final ranking r * is then CAB.

A B C A E B C ! " # $ # $ = # $ # $ % &
Notice that using one of the standard ways to produce a composite indicator the result would be different. If for each country the composite indicator is calculated as difference from the group leader (which assigns 100 to the leading country and ranks the others as percentage points away from the leader), the impact matrix becomes:

Indicators

Conclusion

This article has firstly investigated the axiomatic system behind the mathematical modelling of standard composite indicators. The following main conclusions can be drawn:

1. The assumption of preference independence is essential for the existence of a linear aggregation rule. Given to its computational complexity, the test of mutual preferential independence is often not done in real-world practice. Moreover, this assumption has very strong consequences which often are not desirable in a composite indicator. assumed; this implies complete substitutability among the various components considered; this characteristics may not be desirable in many operational frameworks.

A Condorcet consistent non-compensatory/non-linear mathematical aggregation convention for the construction of composite indicators aimed at ranking countries has been developed here. This mathematical aggregation convention can be divided into two main steps:

• pair-wise comparison of countries to be ranked,

• ranking of countries in a complete pre-order.

Weights are never combined with intensities of preference, as a consequence the theoretical guarantee they are importance coefficients is assured. Since intensities of preference are not used the degree of compensability connected with the aggregation model is at the minimum possible level. Given that the summation of weights is equal to one, the pair-wise comparisons can be synthesised in an outranking matrix, which can be interpreted as a voting matrix.

The information contained in the voting matrix is exploited to rank all alternatives in a complete pre-order by using a Condorcet consistent rule. A problem connected with the use of Condorcet consistent rules is the one of cycles.

A cycle breaking rule normally needs some arbitrary choices such as to delete the cycle with the lowest support, and so on. A non-arbitrary cycle breaking rule is the so-called Kemeny median order which coincides with the maximum likelihood ranking proposed by Young and Levenglick.

Of course the approach we are proposing is not the "first best" possible approach for the construction of composite indicators. It is a "second best" approach based on theoretical and empirical grounds, which makes assumptions explicit and thus easier to be evaluated in relation with a particular use. This increases both the coherence and the transparency of the ranking obtained. One has to remember, however, that Arrow's theorem implies that some useful and desirable properties in an aggregation convention are always lost. Here the properties lost are anonymity (which does not constitute a problem since weights are explicitly allowed and decisiveness is gained) and independence of irrelevant alternatives (which on the contrary is a serious loss since rank reversals may appear). Furthermore, information on intensity of preference of variables is never used. This loss of information is a necessary price to pay for guaranteeing that compensability is reduced and that weights can be considered as symmetrical importance coefficients. However, information on the intensity of preference can be partly used if indifference thresholds are used. Moreover, ranking procedures can be complemented with benchmarking exercises which are fully based on the use of intensity of preference (of course when variables are measured on an interval or ratio scale).

Finally, one has to note that the overall quality of any composite indicator depends on the following elements:

1. information available (often data to measure variables are unreliable or simply missing; thus garbage in, garbage out phenomena may easily occur (Funtowicz and Ravetz, 1990));

2. individual indicators and variable chosen (i.e. which representation of reality is used). This, to a large extent, determines interpretability of an index.

3. direction of each indicator (i.e. the bigger the better or vice versa, sometimes this decision is not that easy); The quality of the aggregation convention is, yet, an important ingredient to guarantee the consistency between the assumptions used and the ranking obtained. Indeed, the overall quality of a composite indicator depends crucially on the way this mathematical model is embedded in the social, political and technical structuring process [START_REF] Munda | Social multi-criteria evaluation (SMCE)": methodological foundations and operational consequences[END_REF]. This is especially true for the choice of weights that remains the most important source of uncertainty and debate. We have the firm conviction that weights are and must be context-dependent in that they reflect political, social and economic priorities and depend on the development model a country (or a group of countries) wants to pursue. Often we have seen policymakers using the umbrella of "science" to disguise lobbies, The proliferation of composite indicators is a clear symptom of their importance in policy-making, and operational relevance in economic statistics in general (see e.g. [START_REF] Granger | Macroeconometrics -Past and future[END_REF]. All the major international organizations such as OECD, the EU, the World Economic Forum or the IMF are producing composite indicators in a wide variety of fields [START_REF] Nardo | Handbook on constructing composite indicators: methodology and user guide[END_REF]. A general objective of most of these indicators is the ranking of countries and their benchmarking according to some aggregated dimensions (see e.g. [START_REF] Cherchye | Using data envelopment analysis to assess macroeconomic policy performance[END_REF], Kleinknecht 2002and OECD, 2003).

F
The improvement of the way composite indicators are constructed and used seems to be a very important research issue from both theoretical and operational points of view. Our main objective in this article is to contribute to the improvement of the overall quality of composite indicators by looking at one of their technical weaknesses, that is, the aggregation convention used for their construction. For this aim, we build upon concepts coming from multi-criteria decision analysis, measurement theory and social choice.

Although various functional forms for the underlying aggregation rules of a composite indicator have been developed in the literature (e.g. Diewert, 1976, Journal of Economic andSocial Measurement, 2002) [START_REF] Munda | Constructing Consistent Composite Indicators: the Issue of Weights[END_REF] analyse the formal axioms behind the linear aggregation rule and their operational implications and conclude that the use of non-linear aggregation rules to construct composite indicators is compulsory for reasons of theoretical consistency when weights with the meaning of importance coefficients1 are used or when the assumption of preferential independence does not hold. Moreover, in standard linear composite indicators, compensability among the different individual indicators is always assumed; this implies complete substitutability among the various components considered. For example, in a hypothetical sustainability index, economic growth can always substitute any environmental destruction or inside e.g., the environmental dimension, clean air can compensate for a loss of potable water. From a normative point of view, such a complete compensability is often not desirable. Vansnick (1990) showed that the two main approaches in multi-criteria aggregation procedures i.e., the compensatory and non-compensatory ones can be directly derived from the seminal work in social choice of Borda (father of the compensatory approach) and Condorcet (father of the non-compensatory approach)2 . If one wants the weights to be interpreted as "importance coefficients" (or equivalently symmetrical importance of variables) noncompensatory aggregation procedures must be used [START_REF] Bouyssou | Some remarks on the notion of compensation in MCDM[END_REF][START_REF] Bouyssou | Noncompensatory and generalized noncompensatory preference structures[END_REF]. The majority rule is theoretically the most desirable aggregation rule, but practically often produces undesirable intransitivities, thus "more limited ambitions are compulsory. The next highest ambition for an aggregation algorithm is to be Condorcet" (Arrow and Raynaud, 1986, p. 77).

Finally, one should note that to use a linear aggregation rule, the assumption that the variable scores are measured on an interval or ratio scale of measurement and no uncertainty exists must always apply. Rarely this happens in the practice of composite indicators, where for instance, sometimes quantitative scores are arbitrarily given to variable scores originally measured on an ordinal measurement scale (see e.g. Nicoletti et al., 2000). On the contrary, by using Condorcet aggregation rules no limitation on the measurement scale of the variable scores exists.

For all these reasons, we think that in some applications the use of nonlinear/non-compensatory Condorcet consistent aggregation rules is desirable.

Since this possibility has almost never been explored in the framework of composite indicators the following Section is devoted to this issue.

An Axiomatic Approach for the Construction of Non-Compensatory/Non-Linear Condorcet Consistent Composite

Indicators

For the sake of clearness, some basic definitions are first given. These definitions are adapted to the context of composite indicators borrowing concepts from multi-criteria decision theory and complex system theory 3 .

Basic definitions

Dimension: is the highest hierarchical level of analysis and indicates the scope of objectives, individual indicators and variables. For example, a sustainability composite indicator can include economic, social and environmental dimensions.

Objective: an objective indicates the direction of change desired. For example, within the economic dimension GDP has to be maximised; within the social dimension social exclusion has to be minimised; within the environmental dimension CO 2 emissions have to be minimised. For example, GDP, saving rate and inflation rate inside the objective "growth maximisation".

Variable: is a constructed measure stemming from a process that represents, at a given point in space and time, a shared perception of a real-world state of affairs consistent with a given individual indicator. To give an example, in comparing two countries, inside the economic dimension, one objective can be "maximisation of economic growth"; the individual indicator might be R&D performance, the indicator score or variable can be "number of patents per million of inhabitants". Another example: an objective connected with the social dimension can be "maximisation of the residential attractiveness". A possible individual indicator is then "residential density". The variable providing the individual indicator score might be the ratio persons per hectare.

A composite indicator or synthetic index is an aggregate of all dimensions, objectives, individual indicators and variables used. This implies that what formally defines a composite indicator is the set of properties underlying its aggregation convention. The rest of this section deals with this issue.

When various individual indicators are used to evaluate two different countries, some of these individual indicators may be in favour of country a while other variables may be in favour of country b. As a consequence a conflict among the individual indicators exists. How this conflict can be treated at the light of a non-linear/non-compensatory logic? This is the classical multi-criteria discrete problem [START_REF] Munda | Multicriteria evaluation in a fuzzy environment[END_REF]Roy, 1996;Vincke, 1992). With this analogy in mind, we present an aggregation convention for (non-linear and non-compensatory) composite indicators able to rank different countries (or regions, cities and so on).

Problem definition

Given a set of individual indicators G={g m }, m=1,2,..., M, and a finite set A={a n }, n=1, 2,..., N of countries, let's assume that the variable (i.e. the individual indicator score) of each country a n with respect to an individual indicator g m is based on an ordinal, interval or ratio scale of measurement. For simplicity of exposition, we assume that a higher value of a variable is preferred to a lower one (i.e. the higher, the better), that is:

= > ) ( ) ( ) ( ) ( a g a g a I a a g a g a P a k m j m k j k m j m k j (2)
Where, P and I indicate a preference and an indifference relation respectively, both fulfilling the transitive property. The mathematical problem to be dealt with is then how to use this available information to rank in a complete pre-order (i.e. without any incomparability relation) all the countries from the best to the worst one. In doing so the following operational properties are desirable:

1. The sources of uncertainty and imprecise assessment should be reduced as much as possible.

2. The manipulation rules should be the more objective and as simple as possible, that is all ad hoc parameters should be avoided.

3. A theoretical guarantee that weights are used with the meaning of "symmetrical importance" must exist. As a consequence, complete compensability should be avoided. This entails that variables have to be used with an ordinal meaning. This is not a problem since no loss of information is implied [START_REF] Arrow | Social choice and multicriterion decision making[END_REF]. Moreover, given that often the measurement of variables is imprecise (see OECD, 2003, p.7), it seems even desirable to use indicator scores with an ordinal meaning.

4. Desirable ranking procedures using ordinal information are always of a Condorcet type [START_REF] Arrow | Social choice and individual values, 2d edition[END_REF]Raynaud, 1986, Moulin, 1988). A problem inherent to this family of algorithm is the presence of cycles, i.e. cases where aPb, bPc and cPa. This problem has been widely studied among others by Fishburn, 1973;Fishburn et al., 1979;Kemeny, 1959;Moulin, 1985;Truchon, 1995;Young andLevenglick, 1978, Vidu, 2002;Weber, 2002. The probability ( ) 5. Arrow's impossibility theorem [START_REF] Arrow | Social choice and individual values, 2d edition[END_REF] clearly shows that no perfect aggregation convention can exist. Then, it is essential to check not only which properties are respected by a given ranking procedure, but also if any essential property for the problem tackled is lost.

,

The proposed composite indicator

The mathematical aggregation convention we are proposing can be divided into two main steps:

1. Pair-wise comparison of countries according to the whole set of individual indicators used.

2. Ranking of countries in a complete pre-order.

For carrying out the pair-wise comparison of countries the following axiomatic system is needed (adapted from Arrow and Raynaud, 1986, p. 81-82).

Axiom 1: Diversity. Each individual indicator is a total order on the finite set A of countries to be ranked, and there is no restriction on the individual indicators; they can be any total order on A. Thanks to these three axioms a N× N matrix, E, called outranking matrix [START_REF] Arrow | Social choice and individual values, 2d edition[END_REF]Raynaud, 1986, Roy, 1996) can be built. Any generic element of the matrix E, e jk , j k is the result of the pair-wise comparison, according to all the M individual indicators, between countries j and k. Such a global pair-wise comparison is obtained by means of equation ( 2). (with the principle one agent one vote). This analogy between a multi-criterion problem and a social choice one, as noted by [START_REF] Arrow | Social choice and multicriterion decision making[END_REF], is very useful for tackling the step of ranking the N countries in a consistent axiomatic framework.

The issue is now to exploit the information contained in the outranking matrix in order to rank all countries in a complete pre-order. A problem connected with the use of Condorcet consistent rules is that of cycles. A cycle breaking rule normally needs some arbitrary choices such as to delete the cycle with the lowest support. Now the question is: Is it possible to tackle the cycle issue in a more general way?

5 In social choice terms then the anonymity property (i.e. equal treatment of all individual indicators) is broken. Indeed, given that full decisiveness yields to dictatorship, Arrow's impossibility theorem forces us to make a trade-off between decisiveness (an alternative has to be chosen or a ranking has to be made) and anonymity. In our case the loss of anonymity in favour of decisiveness is even a positive property. In general, it is essential that no individual indicator weight is more than 50% of the total weight; otherwise the aggregation procedure would become lexicographic in nature, and the indicator would become a dictator in Arrow's term. Nanson as quoted in Black, 1958, p. 175). Or "The obscurity and selfcontradiction are without any parallel, so far as our experience of mathematical works extends … no amount of examples can convey an adequate impression of the evils" (Todhunter, 1949, p. 352 as cited by Young, 1988Young, , p. 1234)).

Attempts of clarifying, fully understanding and axiomatizing Condorcet's approach for solving cycles have been mainly done by Kemeny (1959) who made the first intelligible description of the Condorcet approach, and by Young and Levenglick (1978) who made its clearest exposition and complete axiomatization.

In the version presented by Young and Levenglick (1978), the main methodological foundation is the maximum likelihood concept. In fact, "Condorcet's argument proceeds along the following lines. People differ in their opinions because they are imperfect judges of which decision really is best. If on balance each voter is more often right than wrong, however, then the majority view is very likely to identify the decision that is objectively best." (Young, 1988(Young, , p. 1232)).

The maximum likelihood principle selects as a final ranking the one with the maximum pair-wise support. This selected ranking is the one which involves the minimum number of pair-wise inversions. Since Kemeny (1959) proposes the number of pair-wise inversions as a distance to be minimized between the selected ranking and the other individual profiles, the two approaches are perfectly equivalent. A formal proof of this equivalence can be found in Truchon (1988b, pp. 6-10). The selected ranking is also a median ranking for those composing the profile (in multi-criteria terminology it is the "compromise ranking" among the various conflicting points of view), for this reason the corresponding ranking procedure is often known as the Kemeny median order. Arrow and Raynaud (1986, p. 77) arrive at the conclusion that the highest feasible ambition for an aggregation algorithm building a multi-criterion ranking is to be Condorcet. These authors discard the Kemeny median order, on the grounds that preference reversal phenomena may occur inside this approach (Arrow and Raynaud, 1986, p. 96). However, although the so-called Arrow-Raynaud's method does not present rank reversal, it is not applicable if cycles exist. Since in the context where composite indicators are built, cycles are very probable to occur, here the only solution is to follow Kemeny rule (or the equivalent maximum likelihood ranking procedure), thus accepting that rank reversals might appear 6 .

The acceptance of rank reversals phenomena implies that the famous axiom of independence of irrelevant alternatives of Arrow's theorem is not respected.

Anyway, Young (1988Young ( , p. 1241) ) claims that the maximum likelihood ranking procedure is the "only plausible ranking procedure that is locally stable". Where local stability means that the ranking of alternatives does not change if only an interval of the full ranking is considered.

The adaptation of the maximum likelihood ranking procedure to the ranking problem we are dealing with is reasonably simple [START_REF] Munda | Multi-Criteria Decision Analysis and Sustainable Development[END_REF]. The final ranking ( r * ) is the one which maximises equation (6), which is:

6 Anyway a Condorcet consistent rule always presents smaller probabilities of the occurrence of a rank reversal in comparison with any Borda consistent rule. This is again a strong argument in favour of a Condorcet's approach in our framework. (6)

Other properties of this ranking procedure are the following (Young and Levenglick, 1978).

• Neutrality: it does not depend on the name of any country, all countries are equally treated. reinforcement "… has definite ethical content and is therefore relevant to welfare economics and political science." (Arrow and Raynaud, 1986, p. 96). Moulin (1988, p. 312) clearly states that the Kemeny method (or equivalently the maximum likelihood approach) is "the correct method" for ranking alternatives, and that the "only drawback of this aggregation method is the difficulty in computing it when the number of candidates grows". In fact the number of permutations can easily become unmanageable; for example when 10 countries are present, it is 10!=3,628,800. Indeed this computational drawback is very serious since the Kemeny median order is NP-hard to compute. This NPhardness has discouraged the development of algorithms searching for exact solutions, thus the majority of the algorithms which have been proposed in the literature; are mainly heuristics based on artificial intelligence, branch and bound approaches and multi-stage techniques (see e.g., [START_REF] Barthelemy | Median linear orders: heuristics and a branch and bound algorithm[END_REF][START_REF] Charon | New results on the computation of median orders[END_REF][START_REF] Cohen | Learning to order things[END_REF]Davenport and Kalagnam, 2004;Dwork et al., 2001;Truchon, 1998b). Recently, a new numerical algorithm aimed at solving the computational problem connected to linear median orders by looking for exact solutions has been developed too [START_REF] Munda | A Solution to the Computational Problem of linear median orders[END_REF].

The computational problem

Thanks to the existence of all these efficient computational algorithms, the maximum likelihood (or Kemeny) ranking procedure can always be applied in the context of composite indicators, where a high number of countries to be ranked is the normal state of affairs.

A sensitive issue: is information on intensity of preference complete lost in a Condorcet framework?

Given that the preference structure is based on equation ( 2), one might wonder if information on intensity of preference (when variables are measured on an interval or ratio scale) is completely lost in a Condorcet framework (since small and big intensities are treated equally). The problem is indeed very old. Its origins may be found in the famous bold paradox in Greek philosophy: how many hairs one has to cut off to transform a person with hairs to a bold one? Luce (1956) was the first one to discuss this issue formally in the framework of preference modelling. He introduced the idea of the existence of a sensibility threshold below which an agent either does not sense the difference between two elements, or refuses to declare a preference for one or the other. Mathematical characterisations of preference modelling with thresholds can be found in Roubens and Vincke (1985).

By introducing a positive indifference threshold q the resulting preference model is the so-called threshold model: where a j and a k belong to the set A of countries and g m to the set G of individual indicators. If one wishes to take into account the possible uncertainty around the value of the threshold q, sensitivity analysis and robustness analyses can be used (Saltelli et al., 2004), another possibility is the use of mathematical sophisticated concept such as the one of fuzzy preference modelling [START_REF] Munda | Multicriteria evaluation in a fuzzy environment[END_REF].

+ > q a g a g a I a q a g a g a P a k m j m k j k m j m k j ) ( ) ( ) ( ) ( (7)
Finally, one should note that the intensity of preference can easily be used in the benchmarking step, where is not the ranking but the distance from a reference point what matters. For the majority of indicators used in assessment exercises no clear reference point is available, for instance, when GDP is used nobody knows the ideal value of a Country GDP, thus it is quite common to compare with other

Countries GDP, e.g. the USA one. A first very simple benchmarking procedure can be the application of a normalisation rule known as "distance from the group leader", which assigns 100 to the leading country in that particular individual indicator and other countries are ranked as percentage points away from the leader (OECD, 2003). More elegant approaches can be based on the so-called ideal point approaches, which is a well established technique in multi-criteria evaluation literature (see e.g. Yu, 1985;Zeleny, 1982). In this framework, to get a set of reference values, an "ideal point" can be defined by choosing the best values reached in any single indicator, and then mathematical distances are used.

An Illustrative numerical example

Let's take into consideration a simple hypothetical example where there are three countries (A, B, C) to be ranked according to a composite sustainability indicator. Let's assume that three dimensions have to be considered, i.e. the economic, the social and the environmental ones, and that each dimension should have the same weight, that is 0.3333.

The following individual indicators are used

Economic dimension

Indicator: GDP per capita. Weight: 0. Although the official ESI ranking is based upon the linear aggregation of 21 equally weighted individual indicators, in the methodological appendix, an attempt has been made to apply the non-compensatory approach presented here, in order to tackle the issue of weights as "importance measure" and the compensability of different and crucial dimension of environmental sustainability (see the Methodological Appendix in Esty et al., 2005). It is important to underline that although both aggregation rules seem to produce consistent rankings (the R 2 is 0.92), those rankings do not nevertheless coincide.

Using the non-linear/non-compensatory approach, 43 out of 146 countries display a change in rank higher than 10 positions (none before the 30th ESI2005 rank).

When compensability among indicators is not allowed, countries having very poor performance in some indicators, such as Indonesia or Armenia worsen their rank with respect to the linear yardstick, whereas countries that have less extreme values improve their situation, such as Azerbaijan or Spain.

Conclusion

This article starts from the assumption that, in some real-world applications, the linear aggregation rule is not the correct one for the building of relevant composite indicators. For this reasons a Condorcet consistent non-linear/noncompensatory mathematical aggregation rule for the construction of composite indicators aimed at ranking countries has been developed here. This mathematical aggregation rule can be divided into two main steps:

• pair-wise comparison of countries to be ranked, • ranking of countries in a complete pre-order.

Weights are never combined with intensities of preference, as a consequence the theoretical guarantee they are importance coefficients is assured. Since intensities of preference are not used the degree of compensability connected with the aggregation model is at the minimum possible level. Given that the summation of weights is equal to one, the pair-wise comparisons can be synthesised in an outranking matrix, which can be interpreted as a voting matrix. The information contained in the voting matrix is exploited to rank all alternatives in a complete pre-order by using a Condorcet consistent rule. A problem connected with the use of Condorcet consistent rules is the one of cycles.

A cycle breaking rule normally needs some arbitrary choices such as to delete the cycle with the lowest support, and so on. A non-arbitrary cycle breaking rule is the so-called Kemeny median order which coincides with the maximum likelihood ranking proposed by Young and Levenglick.

Of course the approach we are proposing is not the "first best" possible approach for the construction of composite indicators. It is a "second best" approach based on theoretical and empirical grounds, which makes assumptions explicit and thus easier to be evaluated in relation with a particular use. This increases both the coherence and the transparency of the ranking obtained. One has to remember, however, that Arrow's theorem implies that some useful and desirable properties in an aggregation convention are always lost. Here the properties lost are anonymity (which does not constitute a problem since weights are explicitly allowed and decisiveness is gained) and independence of irrelevant alternatives (which on the contrary is a serious loss since rank reversals may appear). Furthermore, information on intensity of preference of variables is never used. This loss of information is a necessary price to pay for guaranteeing that compensability is reduced and that weights can be considered as symmetrical importance coefficients. However, information on the intensity of preference can be partly used if indifference thresholds are used. Moreover, ranking procedures can be complemented with benchmarking exercises which are fully based on the use of intensity of preference (of course when variables are measured on an interval or ratio scale).

Finally, one has to note that the overall quality of any composite indicator depends on the following elements:

1. information available (often data to measure variables are unreliable or simply missing; thus garbage in, garbage out phenomena may easily occur (Funtowicz and Ravetz, 1990));

2. individual indicators and variable chosen (i.e. which representation of reality is used). This, to a large extent, determines interpretability of an index.

3. direction of each indicator (i.e. the bigger the better or vice versa, sometimes this decision is not that easy); The quality of the aggregation convention is, yet, an important ingredient to guarantee the consistency between the assumptions used and the ranking obtained. Indeed, the overall quality of a composite indicator depends crucially on the way this mathematical model is embedded in the social, political and technical structuring process [START_REF] Munda | Social multi-criteria evaluation (SMCE)": methodological foundations and operational consequences[END_REF]. This is especially true for the choice of weights that remains the most important source of uncertainty and debate. We have the firm conviction that weights are and must be context-dependent since they reflect political, social and economic priorities and depend on the development path a country (or a group of countries) wants to pursue. Often we have seen policymakers using the umbrella of "science" to disguise lobbies, individual interests, or incompetence. Again, whatever aggregations procedure is used, we think that transparency must remain one of the main ingredients: without transparency the interpretability and coherence of indicators are difficult to achieve.

ACKNOWLEDGEMENTS:

  x n , an additive aggregation function exists if and only if these variables are mutually preferentially independent. A subset of indicators Y is preferentially independent of Y c =Q (the complement of Y) only if any conditional preference among elements of Y, holding all elements of Q fixed, remain the same, regardless of the levels at which Q are held. The variables x 1 , x 2 ,..., x n are mutually

  Where, P and I indicate a preference and an indifference relation respectively, both fulfilling the transitive property.Let's also assume the existence of a set of individual indicator weights W={w m }, m=1,2,...,M, with as importance coefficients.

  N M of obtain a cycle with N countries and M with N as well as with the number of indicators. With many countries and individual indicators, cycles occur with an extremely high frequency. Therefore, the ranking procedure used has to deal with the cycle issue properly.

  of individual indicators presenting a preference and an indifference relation respectively. It clearly holds e jk + e kj = 1. (4) Property (4), although obvious, is very important since it allows us to consider the outranking matrix E as a voting matrix i.e., a matrix where instead of using individual indicators, alternatives are compared by means of voters'

  like "… the general rules for the case of any number of candidates as given by Condorcet are stated so briefly as to be hardly intelligible … and as no examples are given it is quite hopeless to find out what Condorcet meant" (E.J.

  in the context where composite indicators are built, cycles are very probable to occur, here the only solution is to follow Kemeny rule (or the equivalent maximum likelihood ranking procedure), thus accepting that rank reversals might appear 8 .

  local stability means that the ranking of alternatives does not change if only an interval of the full ranking is considered.The adaptation of the maximum likelihood ranking procedure to the ranking problem we are dealing with is very simple(Author, 2005). The maximum likelihood ranking of countries is the ranking supported by the maximum number of individual indicators for each pair-wise comparison, summed over all pairs of countries considered. More formally, all the N(N-1) pair-wise comparisons compose the outranking matrix E, where e jk + e kj = 1, with j k. Call R the set of all N! possible complete rankings of alternatives, R={r s }, s=1,2,..., N!. For each r s , compute the corresponding score s as the summation of e jk over all the 2 N pairs j,k of alternatives, i.e.

•

  Neutrality: it does not depend on the name of any country, all countries are equally treated. • Unanimity (sometimes called Pareto Optimality): if all individual indicators prefer country a to country b than b should not be chosen. • Monotonicity: if country a is chosen in any pair-wise comparison and only the individual indicator scores (i.e. the variables) of a are improved, then a should be still the winning country. • Reinforcement: if the set A of countries is ranked by 2 subsets G 1 and G 2 of the individual indicator set G, such that the ranking is the same for both G 1 and G 2 , then G the same ranking. This general consistency requirement is very important in the framework of composite indicators, since one may wish to apply the individual indicators belonging to each single dimension first and then pool them in the general model. It has to be noted that the maximum likelihood ranking procedure is the only Condorcet consistent rule which holds the reinforcement property and as noted by Arrow and Raynaud,

  By applying the C-K-Y-L rule to the 3! possible rankings it is: + 0.333 + 0.666 = 1.3339 Comparing A with B it turns out that A stands out according to the indicators Solid wastes (weight 0.333), Income disparity (weight 0.165) and Crime rate (weight (0.165).

  As a consequence, these sources of uncertainty and imprecision assessment are reduced at the minimum possible level and a sensitivity analysis of the results is much easier to carry out than any other composite indicator. Sustainability Index" (ESI) for 2005 is produced by a team of environmental researchers from the Yale and Columbia Universities, in co-operation with the World Economic Forum and the EU's Joint Research Centre.The aim of the ESI is that of benchmarking the ability of 146 nations to protect the environment over the next decades, by integrating 76 data sets into 21 indicators of environmental sustainability (seeEsty at al., 2005). The data base used to construct the ESI covers a wide range of aspects of environmental sustainability ranging from variables measuring the physical state and stress of the environmental systems (like natural resource depletion, pollution, ecosystem destruction), to the more general social and institutional capacity to respond to environmental challenges. Poverty, short-term thinking and lack of investment in capacity and infrastructure committed to pollution control and ecosystem protection thus compete in determining the measure of a country's sustainability.Although the official ESI ranking is based upon the linear aggregation of 21 equally weighted indicators, in the methodological appendix, an attempt has been made to apply the non-compensatory approach presented here, in order to tackle the issue of weights as "importance measure" and the compensability of different and crucial dimension of environmental sustainability (see the Methodological Appendix inEsty et al., 2005).

Figure 1 Figure 1 .

 11 Figure1compares the ranking obtained by means of the non-compensatory aggregation rule with the one of the ESI2005. In both cases all 21 indicators are equally weighted. From this figure it is clear that the aggregation method used affects principally the middle-of-the-road and, to a lesser extent, the leader and the laggard countries. Overall for the set of 146 countries, the assumption on the aggregation scheme has an average impact of 8 ranks and a rank-order correlation coefficient of 0.962. In particular, while the top 50 countries only move on

2.

  Weights in linear aggregation rules have always the meaning of trade-off ratio. The specification of weights as trade-off is a very complex procedure, for this reason often in practice they are treated as importance coefficients. If weights with the meaning of importance coefficients are used, then a Condorcet approach is needed for axiomatic consistency reasons. 3. In standard composite indicators based on the linear aggregation rule, compensability among the different individual indicators is always

4.

  relative importance of these indicators (weights attached which are one of the main sources of technical uncertainty of the results provided; methodological discussion and sensitivity analysis are necessary to tackle this uncertainty by making it explicit); 5. mathematical aggregation convention used (the present paper has dealt with this step).

3

  Some of these definitions were inspired by discussions with M. Giampietro.Individual indicator: it is the basis for evaluation in relation to a given objective (any objective may imply a number of different individual indicators). It is a function that associates each single country with a variable indicating its desirability according to expected consequences related to the same objective.

  Let's also assume the existence of a set of individual indicator weights W={w m }, m=1,2,...,M, with as importance coefficients.

  N M of obtain a cycle with N countries and M individual indicators increases with N as well as with the number of indicators. With many countries and individual indicators, cycles occur with an extremely high frequency. Therefore, the ranking procedure used has to deal with the cycle issue properly.

Axiom 2 :

 2 Symmetry. Since individual indicators have incommensurable scales, the only preference information they provide is the ordinal pair-wise preferences they contain 4 . Axiom 3: Positive Responsiveness. The degree of preference between two countries a and b is a strictly increasing function of the number and weights of individual indicators that rank a before b 5 .

  of individual indicators presenting a preference and an indifference relation respectively. It clearly holds e jk + e kj = 1. (4) Property (4), although obvious, is very important since it allows us to consider the outranking matrix E as a voting matrix i.e., a matrix where instead of using individual indicators, alternatives are compared by means of voters' preferences

  Condorcet himself was aware of the problem of cycles in his approach; he built examples to explain it and he was even close to find a consistent rule able to rank any number of alternatives when cycles are present. However, attempts to fully understand this part of Condorcet's voting theory have arrived at conclusions like "… the general rules for the case of any number of candidates as given by Condorcet are stated so briefly as to be hardly intelligible … and as no examples are given it is quite hopeless to find out what Condorcet meant" (E.J.

  The maximum likelihood ranking of countries is the ranking supported by the maximum number of individual indicators for each pair-wise comparison, summed over all pairs of countries considered. More formally, all the N(N-1) pair-wise comparisons compose the outranking matrix E, where e jk + e kj = 1, with j k. Call R the set of all N! possible complete rankings of alternatives, R={r s }, s=1,2,..., N!. For each r s , compute the corresponding score s as the summation of e jk over all the 2 N pairs j,k of alternatives, i.e.

•

  Unanimity (sometimes called Pareto Optimality): if all individual indicators prefer country a to country b than b should not be chosen. • Monotonicity: if country a is chosen in any pair-wise comparison and only the individual indicator scores (i.e. the variables) of a are improved, then a should be still the winning country. • Reinforcement: if the set A of countries is ranked by 2 subsets G 1 and G 2 of the individual indicator set G, such that the ranking is the same for both G the same ranking. This general consistency requirement is very important in the framework of composite indicators, since one may wish to apply the individual indicators belonging to each single dimension first and then pool them in the general model. It has to be noted that the maximum likelihood ranking procedure is the only Condorcet consistent rule which holds the reinforcement property and as noted by Arrow and Raynaud,

Figure 1 Figure 1 .

 11 Figure 1 compares the ranking obtained by means of the non-linear/noncompensatory aggregation rule with the one of the ESI2005. In both cases all 21 individual indicators are equally weighted. From this figure it is clear that the aggregation method used affects mainly the middle-of-the-road and, to a lesser extent, the leader and the laggard countries. Overall for the set of 146 countries,

  importance of these indicators (weights attached which are one of the main sources of technical uncertainty of the results provided; methodological discussion and sensitivity analysis are necessary to tackle this uncertainty by making it explicit); 5. mathematical aggregation convention used (the present paper has dealt with this step).

  ).

	aggregation rule. Unfortunately, it is usually never tested whether preference
	independence applies to a given composite indicator, although this assumption
	has very strong consequences which often are not desirable in an index 2 . When
	preference independence cannot be advocated, non-linear aggregation rules are
	then needed.
	F o r
	P
	e
	e r
	R e v
	i e
	w

  use of Condorcet consistent rules is also desirable in general as advised by social choice literature. Finally, one should note that to use a linear aggregation rule, the assumption that the variable scores are measured on an interval or ratio scale of measurement and no uncertainty exists must always apply. Rarely this happens in the practice of composite indicators, where for instance, sometimes quantitative scores are arbitrarily given to variable scores originally measured on an ordinal measurement scale (see e.g.

	F o r
	P
	e
	e r
	R e v
	i e
	w

also propose the use social choice to improve the theoretical framework of environmental indexes.

  al.,

	Manuscript submitted to Applied Economics
	2001;
	F o r
	P
	e
	e r
	R e v
	i e
	w
	15

  Finally, one should note that the intensity of preference can easily used in the benchmarking step, where is not the ranking but the distance from a reference point what matters. For the majority of indicators used in assessment exercises no clear reference point is available, for instance, when GDP is used nobody knows the ideal value of a Country GDP, thus it is quite common to compare with other

	Countries GDP, e.g. the USA one. A first very simple benchmarking procedure
	can be the application of a normalisation rule known as "distance from the group
	leader", which assigns 100 to the leading country in that particular individual
	indicator and other countries are ranked as percentage points away from the leader
	(OECD, 2003). More elegant approaches can be based on the so-called ideal point
	approaches, which is a well established technique in multi-criteria evaluation
	literature (see e.g.	r o
		P
		e
		e r
		R e v
		i e
		w

Table 1 . Impact matrix of the illustrative numerical example

 1 Indicator: Crime rate. Weight: 0.165. Objective: minimisation of criminality.Variable: robberies per 1000 inhabitants.The impact matrix described in Table1can then be constructed.

	Indicators	GDP		Unemp. rate	Solid waste	Inc. dispar.	Crime rate
	Countries
	A		22,000	0.17	0.4	10.5	40
	B		45,000	0.09	0,45	11.0	45
	C		20,000	0.08	0.35	5.3	80
	F Considering axioms 1, 2 and 3, and equation (3), the following pair-wise o r comparisons hold:
	AB=0.333+0.165+0.165=0.666 9 P BA=0.165+0.165=0.333 e AC=0.165+0.165=0.333 e CA=0.165+0.333+0.165=0.666 r BC=0.165+0.165=0.333
	CB=0.165+0.333+0.165=0.666 These results can be synthesised in the outranking matrix: R e v
						i e
	0 0.333	0.666 0.333 0 0.333		w
	0.666 0.666	0

: Income disparity. Weight: 0.165. Objective: minimisation of distributional inequity. Variable: Q5/Q1.

Table 2 . Impact matrix: distance from the leader

 2

		GDP	Unemp. rate	Solid waste	Inc. dispar.	Crime rate
	Countries
	A	48.9	47.05	87.5	50.5	100
	B	100	88.9	77.8	48.2	88.9
	C	44.4	100	100	100	50
	The index will be calculated averaging each indicator (with the same weights as
	in the multi-criterion matrix) obtaining A I =69.8, B I = 79.7, and C I =81.9. The
	ranking would be CBA, different from what found with our algorithm.
	It is interesting to note that the only sources of uncertainty and imprecision
	assessment left in the mathematical modelling of the composite indicator
	proposed are:
	1. The scale adjustment technique used (neither normalisation nor a common
	measurement unit are needed).
	2. The values of weights attached to dimensions and individual indicators.

Table 3 . ESI rankings obtained by linear aggregation (LIN) and non- compensatory rule (NCMC): countries that largely improve or worsen their rank position

 3 Table3shows the countries displaying the largest variation in their ranks.

		Aggregation	ESI rank	rank with	Change
			with LIN	NCMC	in Rank
		Azerbaijan	99	61	38
	Improvement	Spain Nigeria South Africa	76 98 93	45 69 68	31 29 25
		Burundi	130	107	23
		Indonesia	75	114	39
	Deterioration	Armenia Ecuador Turkey Sri Lanka	44 51 91 79	79 78 115 101	35 27 24 22
		Average change over 146 countries		8
		F o r
			P
				e
				e r
					R e v
					i e
					w

  Grupp and Mogee 2004, Lovell et al. 1995[START_REF] Munda | Multi-Criteria Decision Analysis and Sustainable Development[END_REF][START_REF] Nardo | Handbook on constructing composite indicators: methodology and user guide[END_REF], Saisana and Tarantola 2002, and Wilson and Jones 2002, among others).

	individual interests, or incompetence. Again, whatever aggregations procedure is
	used, we think that transparency must remain one of the main ingredients: without 1. Introduction
	transparency the interpretability and coherence of indicators are difficult to Composite indicators (along all this paper the term composite indicator is used as
	achieve. a synonymous of index) are very common in fields such as economic and business
	statistics (e.g., Öcal 2006, Binner et al. 1999 and 2006 or the OECD Composite
	of Leading Indicators) and are used in a variety of policy domains such as
	industrial competitiveness, sustainable development, quality of life assessment,
	globalisation, innovation or academic performance (see Cox and others 1992,
	Cribari-Neto et al 1999, Färe et al. 1994, Griliches 1990, Forni et al. 2001,
	Huggins 2003,
	o r F o r F o r F o r F o r F o r
	P P P P P P
	e e e e e e
	e r e r e r e r e r e r
	R e v R e v R e v R e v R e v R e v
	i e i e i e i e i e i e
	w w w w w w

Table 1 . Impact matrix of the illustrative numerical example

 1 Indicator: Solid waste generated per capita. Weight: 0.333. Objective: minimisation of environmental impact. Variable: tons per year.The impact matrix described in Table1can then be constructed.By applying the Kemeny rule to the 3! possible rankings it is:

	165. Objective: maximisation of economic

7

Comparing A with B it turns out that A stands out according to the indicators Solid wastes (weight 0.333), Income disparity (weight 0.165) and Crime rate (weight (0.165). * is then CAB.

Notice that using one of the standard ways to produce a composite indicator the result would be different. If for each country the composite indicator is calculated as difference from the group leader (which assigns 100 to the leading country and ranks the others as percentage points away from the leader), the impact matrix becomes:

Table 2 . Impact matrix: distance from the leader

 2 The values of weights attached to dimensions and individual indicators.As a consequence, these sources of uncertainty and imprecision assessment are reduced at the minimum possible level and a sensitivity analysis of the results is much easier to carry out than any other composite indicator.

	The index will be calculated averaging each indicator (with the same weights as
	in the multi-criterion matrix) obtaining A I =69.8, B I = 79.7, and C I =81.9. The
	ranking would be CBA, different from what found with our non-linear/non-
	compensatory proposal.
	It is interesting to note that the only sources of uncertainty and imprecision
	assessment left in the mathematical modelling of the composite indicator
	proposed are:
	1. The scale adjustment technique used (neither normalisation nor a common
	measurement unit are needed).

Table 3 . ESI rankings obtained by linear aggregation (LIN) and non- linear/non-compensatory (NCMC) rules: countries that largely improve or worsen their rank position

 3 Table3shows the countries displaying the largest variation in their ranks.

		Aggregation	ESI rank	rank with	Change
			with LIN	NCMC	in Rank
		Azerbaijan	99	61	38
	Improvement	Spain Nigeria South Africa	76 98 93	45 69 68	31 29 25
		Burundi	130	107	23
		Indonesia	75	114	39
	Deterioration	Armenia Ecuador Turkey	44 51 91	79 78 115	35 27 24
		Sri Lanka	79	101	22
		Average change over 146 countries		8
		F o r
			P
				e
				e r
					R e v
					i e
					w

  This research has been partially developed in the framework of the European Commission financially supported project "Knowledge Economy Indicators: Development of Innovative and Reliable Indicator Systems (KEI)" (KEI SCS8 2004-502529).

	F o r F o r F o r
	P P P
	e e e
	e r e r e r
	R e v R e v R e v
	i e i e i e
	w w w

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

In the decision theory literature, this concept of weights as importance coefficients is usually referred to as symmetrical importance, that is "… if we have two non-equal numbers to construct a vector in R

, then it is preferable to place the greatest number in the position corresponding to the most important criterion."(Podinovskii, 1994, p. 241). 2Ebert and Welsch (2004) also propose the use social choice to improve the theoretical framework of environmental indexes.

In our case, this axiom is needed since the intensity of preference (i.e. the difference between 2 variables measured on an interval or ratio scale) of individual indicators is not considered useful preference information given that compensability has to be avoided and weights have to be symmetrical importance coefficients. Moreover, thanks to this axiom, a normalisation step is not needed. This causes a further reduction of the sources of uncertainty and imprecise assessment.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

 Saisana andTarantola, 2002, cited in OECD 2003, p. 4