Why composite should be non compensatory Giuseppe Munda Universitat Autonoma de Barcelona Dept.of Economics and Economic History
Index construction- Steps [OECD-JRC Handbook] Step 1. Developing a theoretical framework Step 2. Selecting indicators Step 3. Multivariate analysis Step 4. Imputation of missing data Step 5. Normalisation of data Step 6. Weighting and aggregation Step 7. Robustness and sensitivity Step 8. Association with other variables Step 9. Back to the details (indicators) Step 10. Presentation and dissemination
Structure of the presentation • Linear Aggregation Rule • Social Choice and Aggregation Rules • Non-Compensatory Aggregation Rules • Conclusions
Linear aggregation rules and preference independence The variables X1, X2,..., Xn are mutually preferentially independent if every subset Y of these variables is preferentially independent of its complementary set of evaluators. The following theorem holds: given the variables X1, X2, ..., Xn, an additive aggregation function exists if and only if these variables are mutually preferentially independent.
Preferential independence implies that the trade-off ratio between two variables is independent of the values of the n-2 other variables, i.e. Preferential independence is a very strong condition from both the operational and epistemological points of view.
The meaning of weights in linear aggregation rules • “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). • Weights as symmetrical importance, that is "… if we have two non-equal numbers to construct a vector in R2, then it is preferable to place the greatest number in the position corresponding to the most important criterion." (Podinovskii, 1994, p. 241).
The aggregation of several criteria implies taking a position on the fundamental issue of compensability. Compensability refers to the existence of trade-offs, i.e. the possibility of offsetting a disadvantage on some criteria by an advantage on another criterion. E.g. in the construction of a composite indicator of environmental sustainability a compensatory logic (using equal weighting) would imply that one is willing to accept, let’s say, 10% more CO2 emissions in exchange of a 3% increase in GDP.
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 ” (the greatest weight is placed on the most important “dimension”) non-compensatory aggregation procedures must be used to construct composite indicators.
Linear Aggregation is a correect rule iff Mutual preferential independence applies Compensability is desirable Weights are derived as trade-offs
Example (rearranged from Moulin, 1988, p. 228; 21 criteria and 4 alternatives ) Objective: find best country A first possibility: apply the plurality rule the country which is more often ranked in the first position is the winning one. Country a is the best. BUT Country a is also the one with the strongest opposition since 13 indicators put it into the last position! This paradox was the starting step of Borda’s and Condorcet’s research at the end of the 18th century, but the plurality rule corresponds to the most common electoral system in the 21st century!!!
Borda approach Frequency matrix (21 criteria 4 alternatives) Jean-Charles, chevalier de Borda
Borda approach Frequency matrix (21 criteria 4 alternatives) Borda score: Borda solution: bcad Now: country b is the best, not a(as in the case of the plurality rule) The plurality rule paradox has been solved.
Condorcet approach Outranking matrix (21 criteria 4 alternatives) Jean-Antoine-Nicolas de Caritat,Marquis de Condorcet
Condorcet approach Outranking matrix (21 criteria 4 alternatives) • For each pair of countries a concordance index is computed by counting how many individual indicators are in favour of each country (e.g. b better than a 13 times). • “constant sum property” in the outranking matrix (13+8=21)
Condorcet approach Outranking matrix (21 criteria 4 alternatives) • Pairs with concordance index > 50% of the indicators are selected. • Majority threshold = 11 (i.e. a number of individual indicators bigger than the 50% of the indicators considered) • Pairs with a concordance index ≥ 11: bPa= 13, bPd=21(=always), cPa=13, cPb=11, cPd=14, dPa=13. Condorcet solution: c b d a Yet, the derivation of a Condorcet ranking may sometimes be a long and complex computation process.
Which approach should one prefer? Both Borda and Condorcet approaches solve the plurality rule paradox. However, the solutions offered are different. Borda solution: b c a d Condorcet solution: c b d a In the framework of composite indicators, can we choose between Borda and Condorcet on some theoretical and/or practical grounds?
Example with 60 indicators [Condorcet, 1785] Borda approach Condorcet approach Frequency matrix Outranking matrix concordance threshold =31 aPb, bPc and cPa (cycle)???
From this example we might conclude that the Borda rule (or any scoring rule) is more effective since a country is always selected while the Condorcet one sometimes leads to an irreducible indecisiveness. However Borda rules have other drawbacks, too
Both social choice literature and multi-criteria decision theory agree that whenever the majority rule can be operationalized, it should be applied. However, the majority rule 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).
Condorcet approach Basic problem: presence of cycles, i.e. aPb, bPc and cPa The probability of obtaining a cycle increases with both N (indicators) and M (countries) [Fishburn (1973, p. 95)]
Condorcet approach Condorcet himself was aware of this problem (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… … but Maximilien deRobespierre
Condorcet approach Basic problem: presence of cycles, i.e. aPb, bPc and cPa Furter attempts made by Kemeny (1959) and by Young and Levenglick (1978) … led to: Condorcet-Kemeny-Young-Levenglick (C-K-Y-L) ranking procedure
C-K-Y-L ranking procedure Main methodological foundation: maximum likelihood concept. The maximum likelihood principle selects as a final ranking the one with the maximum pair-wise support. What does this mean and how does it work?
C-K-Y-L ranking procedure AB = 0.333+0.166+0.166=0.666 BA = 0.166+0.166=0.333 AC = 0.166+0.166=0.333 CA = 0.166+0.333+0.166=0.666 BC = 0.166+0.166=0.333 CB = 0.166+0.333+0.166=0.666
C-K-Y-L ranking procedure AB = 0.333+0.166+0.166=0.666 A B C BA = 0.166+0.166=0.333 A B C 0 0.6660.333 AC = 0.166+0.166=0.333 0.333 0 0.333 CA = 0.166+0.333+0.166=0.666 0.666 0.666 0 BC = 0.166+0.166=0.333 CB = 0.166+0.333+0.166=0.666
C-K-Y-L ranking procedure BCA = 0.333 + 0.666 + 0.333 = 1.333 A B C ABC = 0.666 + 0.333 + 0.333 = 1.333 A B C 0 0.6660.333 CAB = 0.666 + 0.666 + 0.666 = 2 0.333 0 0.333 ACB = 0.333 + 0.666 + 0.666 = 1.666 0.666 0.666 0 CBA = 0.666 + 0.333 + 0.666 = 1.666 BAC = 0.333 + 0.333 + 0.333 = 1
The Computational problem C-K-Y-L ranking procedure The only drawback of this aggregation method is the difficulty in computing it when the number of candidates grows. With only 10 countries 10! = 3,628,800 permutations (instead of 3!=6 of the example) To solve this problem of course there is a need to use numerical algorithms
Compensabiliity is a fundamental concept in choosing an aggregation rule
Using Borda too The KEI example …
The frequency matrix (Borda type approach) of a country’s rank in each of the seven dimensions and the overall KEI was calculated across the ~2,000 scenarios.
REFERENCES • Borda J.C. de (1784) – Mémoire sur les élections au scrutin, in Histoire de l’ Académie Royale des Sciences, Paris. • Brand, D.A., Saisana, M., Rynn, L.A., Pennoni, F., Lowenfels, A.B. (2007) Comparative Analysis of Alcohol Control Policies in 30 Countries, PLoS Medicine, 0759 April 2007, Volume 4, Issue 4, e151, p. 0752-0759, www.plosmedicine.org • Condorcet, Marquis de (1785) – Essai sur l’application de l’analyse à la probabilité des décisions rendues à la probabilité des voix, De l’ Imprimerie Royale, Paris. • Fishburn P.C. (1973) – The theory of social choice, Princeton University Press, Princeton. • Fishburn P.C. (1984) – Discrete mathematics in voting and group choice, SIAM Journal of Algebraic and Discrete Methods, 5, pp. 263-275.
Munda G. (1995), Multicriteria evaluation in a fuzzy environment, Physica-Verlag, Contributions to Economics Series, Heidelberg. • Munda G. (2004) – “Social multi-criteria evaluation (SMCE)”: methodological foundations and operational consequences, European Journal of Operational Research,vol. 158/3, pp. 662-677. • Munda G. (2005a) – Multi-Criteria Decision Analysis and Sustainable Development, in J. Figueira, S. Greco and M. Ehrgott (eds.) –Multiple-criteria decision analysis. State of the art surveys,Springer International Series in Operations Research and Management Science, New York, pp. 953-986. • Munda G. (2005b) – “Measuring sustainability”: a multi-criterion framework, Environment, Development and Sustainability Vol 7, No. 1, pp. 117-134. • Munda G. and Nardo M. (2005) - Constructing Consistent Composite Indicators: the Issue of Weights, EUR 21834 EN, Joint Research Centre, Ispra. • Munda G. and Nardo M. (2007) - Non-compensatory/Non-Linear composite indicators for ranking countries: a defensible setting, Forthcoming in Applied Economics. • Munda G. (2008) - Social multi-criteria evaluation for a sustainable economy, Springer-Verlag, Heidelberg, New York.