Choosing Aggregation Rules for Composite Indicators

Given the set of individual indicators G, 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 indicators g ₁ , g ₂ ,…, g _m are mutually preferentially independent if every subset Y of these indicators is preferentially independent of its complementary set of indicators. From an operational point of view, this means that an additive aggregation function permits the assessment of the marginal contribution of each indicator separately (as a consequence of the preferential independence condition). The marginal contribution of each indicator can then be added together to yield a total value, no phenomena of synergy or conflict can be taken into account.

The original Arrow’s impossibility theorem (Arrow 1963) is slightly different, above all with respect to the independence of irrelevant alternatives axiom. In the social choice literature formulation, it is called the axiom of binary independence, i.e., the social ranking of each pair of alternatives depends only on the preferences of each voter on that specific pair of alternatives. The ranking of any other alternative is irrelevant for this social ranking. Indeed in the version proposed by Arrow and Raynaud (1986) the axiom of independence of irrelevant alternatives is closer to the definition given by Chernoff (1954), which is derived from Nash’s bargaining theory. For a deep discussion on the independence of irrelevant alternatives axiom and its various definitions see e.g., Ray (1973) and Bordes and Tideman (1991).

Often this search for clear properties characterizing an algorithm is indicated as the axiomatic approach. However, one should note that properties or assumptions are NOT axioms. As perfectly synthesized by Saari (2006, p. 110) “Many, if not most, results in this area are merely properties that happen to uniquely identify a particular procedure. But unless these properties can be used to construct, or be identified with all properties of the procedure (such as in the development of utility functions in the individual choice literature), they are not building blocks and they most surely are not axioms: they are properties that just happen to identify but not characterize, a procedure. As an example, the two properties (1) Finnish-American heritage (2) a particular DNA structure, uniquely identify me, but they most surely do not characterize me”.

It is to be noted that an important key implicit assumption underlying a Borda rule has to be accepted, i.e. the arbitrary transition from an ordinal scale of measurement to an interval or ratio scale one (according to the scoring rule adopted).

The term “outranking matrix” was invented by B. Roy in the so-called ELECTRE methods.

Arrow and Raynaud (1986, pp. 77–78) arrive at the conclusion that a Condorcet aggregation algorithm has always to be preferred in a multi-criterion framework. On the complete opposite side one can find Saari (1989, 2000, 2006). His main criticism against Condorcet based approaches are based on two arguments: (1) if one wants to preserve relationships among pairs (e.g., to impose a side constraint to protect some relationship-balanced gender for candidates in a public concourse) then it is impossible to use pair-wise voting rules, a Borda count should be used necessarily. It is important to note that, although desirable in some cases, to preserve a relationship among pairs implies the loss of neutrality; this is not desirable on general grounds. (2) The individual rationality property (i.e., transitivity) has necessarily to be weakened if one wishes to adopt a Condorcet based voting rule.

However, one should note that this voting rule is normally referred in the literature as the Kemeny’s method or Kemeny’s rule.

One should note that in general the opportunity cost for decisiveness is the loss of one of the basic requirements of a social choice rule, i.e. anonymity, neutrality or monotonicity. Saari (2006) defence of the Borda rule is based on the fact that it is less dangerous, or even sometimes desirable, to eliminate neutrality and if one eliminates neutrality, then only a Borda rule can be adopted. But, if one wishes to keep neutrality and eliminate anonymity, then a Condorcet voting rule is appropriate.

Simon (1983) notes that humans have at their disposal neither the facts nor the consistent structure of values nor the reasoning power needed to apply the principles of utility theory. In microeconomics, where the assumption that an economic agent is always a utility maximize is a fundamental one, it is generally admitted that this behavioural assumption has a predictive meaning and not a descriptive one (see Friedman 1953 for the most forceful defence of this non-descriptive meaning of the axioms of ordinal utility theory). A corroboration of this criticism in the framework of social choice can be found in Kelsey (1986), where it is stated that because of social choice problems, an individual with multiple objectives may find it impossible to construct a transitive ordering. Recent analyses of the concept of rational agent can also be found in Bykvist (2010) and Sugden (2010).

Arrow and Raynaud (1986, pp. 95–96) took into consideration the paper by Young and Levenglick (1978), but they arrive at the conclusion that reinforcement “… has definite ethical content and is therefore relevant to welfare economics and political science. But here our aim is operations research, of use to businessmen. We are unable to see why the “consistency” criterion has any compelling justification when efficiency is the prime consideration.” (Arrow and Raynaud 1986, p. 96).

The complexity class of decision problems that are intrinsically harder than those that can be solved by a nondeterministic Turing machine in polynomial time. When a decision version of a combinatorial optimization problem is proved to belong to the class of NP-complete problems, then the optimization version is NP-hard (definition given by the National Institute of Standards and Technology, http://​www.​nist.​gov/​dads/​HTML/​nphard.​html).