Francesco Carlucci Stefano Pisani
The stochastic approach to the measurement of Well-being
1 From the Sen-Nussbaum approach to the stochastic one
The Nussbaum approach to well-being (and poverty), based on the
functionings/capabilities considered at the State’s level, has been made applicable by
characterizing each capability through a set of measurable attributes. For instance, the functioning/capability “escaping premature mortality” may be characterized by two attributes
- male expectation at birth,
- female expectation at birth,
but remains still abstract if each of these attributes is not measured, e.g. in years.
Analogously, the functioning/capability “being socially integrated” may be characterized by means of the attribute “social integration”, measured according to a qualitative scale.
After identifying functionings/capabilities and their attributes, we have to perform the task of transforming their measures into one overall indicator of well-being. This essay illustrates how to perform this task in a stochastic manner.
2 The attributes and their forms of measurement
Since value-objects constituting the well-being may be of very dissimilar types, attributes have to be able to be conceptually different and therefore measurable in forms also very general. The simplest form is the monetary one, which is also the most commonly used. Since attributes may be also monetarily measurable, the evaluation procedure of this chapter encompasses standard methods based on the monetary measurement.
The second form of measuring attributes, which then defines a new class of these, is the quantitative non monetary one. Constituent elements of well-being as “income
distribution” or “escaping premature mortality” belong to this class.
A third class of attributes is constituted by those representing the characteristics of well-being that are not numerically measurable. These can be, for instance, the “livability of
residence”, the “sentiment of safety”, the “hope of a better future”, the “social cohesion”, the “social justice”, and so on. The attributes of this form cannot be quantitatively measured but can be appraised by a qualitative evaluation of the type: “very high”, “high”, “more than sufficient”, “sufficient”, “less than sufficient”, “low”, “very low”.
3 A synthesis of procedure
An evaluation procedure based on attributes that are measurable according to those three forms has been developed by Carlucci and Pisani in 1995. It produces a measure that is defined in stochastic terms and possesses some properties necessary for a correct evaluation. We’ll illustrate them in the following.
The analytical foundations of procedure are not new but lie in the framework of the
1stochastic theory of utility, originally developed by Von Neumann and Morgenstern at the
end of the forties, subsequently improved by various authors and then extended to the case of several variables by Keeney and Raiffa (1976).
This analytical procedure, based on the construction of a multi-linear function of several variables (the attributes), has been transformed by Carlucci and Pisani by replacing the concept of utility with that of value and its measurement with the operation of evaluation (of the environmental situation (1989 and 1990) and of public goods (1993). Subsequently they have used it to evaluate the human development (1995), and improved it both extending its applicability and simplifying the part where parameters of evaluation functions are determined through a method (the DELPHI one) that harmonizes evaluators’ opinions.
Essentially, the procedure consists of the following steps:
- the situation of well-being in a country is characterized by means of a set of
(functionings/capabilities and then of) attributes that are measurable either
monetarily, or more in general quantitatively, or even qualitatively;
- for each attribute an univariate evaluation function is constructed, basing on
a stochastic option; this way, an adimensional value is related to each
attribute (measured in any form of the above-mentioned four);
- univariate functions are aggregated by two, three, four, ..., in multi-linear
functions; these in turn are aggregated in other multi-linear functions and
1 See J.Von Neumann and O.Morgenstern (1953).
the last one represents the evaluation of the overall well-being in the
Country (which is then related to an adimensional indicator of “value”).
Parameters of functions (univariate and multivariate in each aggregation step) are determined by iteratively harmonizing - by means of the DELPHI method or any other elicitation process - the opinions of a group of evaluators (the “evaluation maker”) who
express the thought of one institution (e.g. a State) or more. Obviously indeed, the overall evaluation cannot be objective, but necessarily reflects the opinions of some institutions which may change in the years or under new management.
At any rate, the effects of answers that define their assessments have to be well clear to evaluators (at the end of iterations in the elicitation method). Otherwise the aggregation of evaluations may lead to biased results.
The construction of this multi-attribute measure, the indicator of well-being, presents some methodological difficulties. A few of these are fundamental: the necessity of aggregating variables with meanings that may be very different; the possible non-linearity by which each variable enters the multivariate function; the possible presence of redundancies or synergies among attributes, which often affect the well-being in such a non independent manner that they produce over- or under-estimated measures. In the presence of these difficulties many researchers who have the task of measuring a social situation simplify the construction of the indicator and often measure the situation by means of a set of separate indices, each related to a particular aspect of it. Thus, the problems of aggregation and redundancy (or synergy) mentioned above are avoided, but an overall measure is not obtained.
In the following we show how the methodology presented in 1995 is able to overcome these difficulties and how relevant the subjective elements are in the elicitation of evaluations.
4 The aggregation of attributes and the problem of non-linearity
Let’s first tackle the issue of aggregating attributes with different units of measurement. Supposing that the social situation in a country is characterized, among others, by means of two attributes, for instance the “adult literacy ratio” (x) and the “male life 1
expectancy at birth” (x), the construction of the simple, but frequently used, overall indicator 2
V(x, x) given by the linear combination 12
V(x, x) = Kx + Kx(4.1)121 12 2
involves the solution of at least three problems. First, that of aggregating xmeasured for 1
example as a percentage (as in the case of the “adult literacy ratio”) with x measured in 2
different terms (in years as in the case of the “male life expectancy at birth”). Actually this problem does not stem only from the qualitative difference of attributes, but also from their different variability.
One of the most common ways of solving it lies basically in (subjectively) evaluating parameters Kand K after determining their right dimensions, and replacing x and xwith 1 2 ,12
particular transformations that equalize their variability. As for as the reduction of variability is concerned, several proposals have been put forward. They can be classified into the following two types: The ones that in substance standardize attributes, and the ones that
2impose some functional forms determined a priori on the transformations. Both types of
proposals are only partially satisfactory. The first because it eliminates a relevant amount of information; the second because it uses analytical tools (the functional forms) that are not necessarily consistent with the empirical reality.
To determine the right dimensions of parameters Kand K the rules of dimensional 1 2
analysis can be used. Since indicator V(x, x) is adimensional, the terms Kx and Kx must 121 12 2
be too, so that the dimension of Kis 1
i.e. a pure number (made equal to 1 only by convention), and that of K is 2
-1 [time] (4.3)
The second problem to solve is the evaluation of parameters K and Kthat are 12
conditional upon dimensions (4.2) and (4.3). This assessment can be achieved by eliciting the opinions of an evaluation maker, which however raises the difficult problem of determining the level of the adult literacy ratio that, for him, is equivalent to a given “male life expectancy at birth”. A third problem that originates from formulation (4.1) lies in the fact that often the
evaluation maker reaches the conclusion that the functional (linear in the V(x)s) form (4.1) is ii
not correct and that it should be replaced by a non-linear one. The determination of the type of non-linearity increases the difficulties related to aggregation.
The proposed methodology constructs indicator V(x, x) by means of a different 12
approach. First of all, x and xare replaced by univariate evaluation functions V(x), i=1,2, 12 ii
2 See UNDP(various years), where a particular function is used to get decreasing increments of human development while the GDP is increasing.
which transform attributes x, i=1,2, into adimensional indices. Indicator (4.1) is then i
replaced by the other
V(x, x) = KV(x) + KV(x) (4.4)121 112 22
The main disadvantage in using (4.4) instead of (4.1) consists in the fact that the
evaluation maker, besides determining K and K , now has to construct the univariate 12
functions V(x) too. In other words, whilst in the case of (4.1) he had to perform only one ii
task, in the case of (4.4) he has to perform two. The advantages, however, are remarkable, since the above-indicated difficulties are overcome. Indeed, formulation (4.4) aggregates attributes that have the same dimension (in fact, they are adimensional) and therefore may be easily summed. Second, the assessment of K with respect to Kis performed by the 12
evaluation maker in a simpler and therefore more accurate manner, as he has to compare the value of an adimensional index, V(x), with that of V(x), which is also adimensional. Since 1122
the evaluation maker does not have to compare dimensions, he can make a better comparison of x and xin terms of their meanings only. Third, the linearity that makes aggregation (4.1) 12
scarcely flexible and that the normal evaluation maker would replace with a non-linear form constitutes a problem that in the case of quantitative-dimensioned attributes is solved during the construction of the univariate functions V(x), which may be non-linear. ii
Furthermore, function (4.4) itself, linear in the univariate functions V(x), is only a ii
particular case of the multilinear specification that is used in the procedure, which for two attributes is
V(x, x) = KV(x) + KV(x) + K( V(x) V(x) (4.5)12111222121122
where the mixed term represents the possible interaction between x and x. Specification (4.5) 12
captures a possible non-linearity affecting functions V(x) and V(x) together. 1122
If attributes are n, the general formulation of the multi-linear function is
；K(V(x)(V(x)(....(V(x) +…. (4.6) 123...n112nn
consisting of products of only univariate functions V(x), and all the possible interactions ii
between V(x)s are shown explicitly. The determination of V(x)s and Ks will be illustrated in ii
the following Section.
5 The construction of the univariate evaluation functions
Since each univariate evaluation function V(x) is adimensional, we can take as
function co-domain the interval [0,1], where 1 corresponds to the largest assessment of x and 0 to the smallest.
* ** Given then the attribute x, with xits minimum (in evaluation) and x its maximum
(again in evaluation), we can relate an assessment to it that lets us build an ordering so that, for each pair (x’, x’’), we can affirm the truth of only one of the following sentences:
- x’ is more highly evaluated than x’’ (in formal terms x’>x’’),
- x’ is less highly evaluated than x’’ ( x’< x’’),
- x’ is evaluated in the same manner as x’’ (x’x’’). ，
Such sentences are essentially preference relations and satisfy the transitive property. As soon as the ordering has been determined, we can construct a monotonic increasing function V(x), which has x as an attribute and establishes a complete numerical representation in the co-domain [0,1], of the “transparent kind”, so that it is
V(x’) > V(x’’) only and only if x’>x’’.
In order to formally define function V(x) we use a probabilistic argument by supposing
that the evaluation maker considers situation x, which is certain, to be perfectly equivalent to
*** the uncertain situation (lottery) consisting of x with probability p and of xwith probability
* **1-p. If we denote such an uncertain situation by (x, x), we formally assert that p
* ** x (x, x) (5.1) ，p
* ** Equation (5.1) defines x as equivalent (in evaluation) to the lottery (x,x)in the p
* ** x px+(1-p)x (5.2) ，
and determines the correspondence between the situation x and the probability p. Obviously,
** ***** ** x (x,x) ; x (x,x) (5.3) ，，1.000.00
Probability p then can be considered as an evaluation of the attribute x if we define V(x)
3in the following way
V(x) = 1-p (5.4)
To interpolate V(x) by means of a finite number of equivalences (5.1) we can exploit the evaluation maker’s opinions by asking him, for instance, to indicate the values of x’, x’’
and x’’’ so that the following relations hold
* *** *** ** x’ (x,x) ; x’’ (x,x) ; x’’’ (x,x) (5.5) ，，，0.750.50 0.25
or, conversely, if modalities for x are few, the values of p’, p’’ and p’’’ so that the following
* *** *** ** x’ (x,x) ; x’’ (x,x) ; x’’’ (x,x) (5.6) ，，，p’p’’p’’’
for fixed x’, x’’ and x’’’.
These relations are a formalization of the questions that evaluation maker has to answer in order to construct the univariate evaluations functions V(x). In practice, using (5.5)
***or (5.6) each evaluation maker faces two attributes: one desirable x and one disagreeable x,
for instance a “male life expectancy at birth” equal to 75 years against one equal to 65 years. Firstly, the disagreeable alternative is associated with a high probability (equal to 0.75) and
the desirable one with a low probability (0.25), so that an evaluation maker has to point out
***the certain value x he would like to obtain instead of getting x with probability 0.75 or x
3*** By virtue of this definition we get from (5.3) that V(x) = 0 and V(x) = 1, so that V(x) = 1-p = ***pV(x) + (1-p)V(x), a transform of (5.2) by function V(x).
with probability 0.25. Afterwards, evaluation maker has to answer two additional questions, with the same alternatives but different probabilities. As in previous example, “male life expectancy at birth” equal to 65 years would be presented as less likely (with probabilities 0.50 and 0.25, respectively), against situations with higher probability of “male life expectancy at birth” equal to 75 years (probabilities 0.50 and 0.75).
As soon as values x’, x’’ and x’’’ have been assessed, they are interpolated by means
of the two extremes determined by (5.3) and thus function V(x) is constructed.
It is worth noting that, by virtue of (5.4), equations (5.5) are equivalent to the three following equalities
V(x’) = 0.25 , V(x’’) = 0.50 , V(x’’’) = 0.75 (5.7)
and the evaluation maker can be asked to give the values of x’, x’’ and x’’’ by using either
4(5.5) or (5.7) indifferently. Conversely, equations (5.6) are equivalent to the other equalities
V(x’) = p’ , V(x’’) = p’’ , V(x’’’) =p’’’ (5.8)
and the evaluation maker can be asked to give the values of p’, p’’ and p’’’ by using either
(5.6) or (5.8) indifferently.
Now we are able to show how parameters in (4.6) can be determined. Indeed the following relationships hold
********K？V(x,x,x,;i,j;h！i,h！j)？K？KK？V(x,x,;j;j！i) , , … (5.9) iijijijhij
for which, furthermore, the sum of all K , K , …, is equal to one. i ij
6 Subjective elements in the elicitation of evaluations
Thus the construction of function (4.4), according to the present procedure, follows two steps of intervention for the evaluation maker; the first in the construction of univariate functions V(x), and the second in the comparative assessment of parameters K . If more than iii
4 Such an evaluation procedure reflects the utility measurement in stochastic terms used in the decision theory under uncertainty: see, for instance, R.L.Keeney and H.Raiffa (1976).
two attributes were aggregated into one indicator, the generalization would be straightforward and would again involve two steps of intervention.
The way of eliciting an evaluation maker’s opinions is shown in the work of 1995 and therefore is not repeated here. We believe it necessary, however, to stress a point already emphasized. In general, the construction of the multi-attribute evaluation function V(x), where
is the vector of attributes, contains subjective elements that depend on the evaluation x
maker’s opinions. On the one hand this fact is unavoidable as this subjectivity includes at least two aspects of the evaluations of socio-economic situations that are intrinsically subjective:
a) the functional form of V(x) in the transformation of every attribute i
b) the comparison of the value of every attribute with respect to each other.
But on the other hand such subjectivity is also desirable, because aspects a) and b) have necessarily to reflect not an absolute opinion, valid for any institution whatever and at any time, but an opinion related to a particular complex organism (political, social, associative,…), probably variable in time according to changes in human conditions, in the
sensibility of institutions and in the socio-economic policies they decide to adopt.
As a consequence, two aspects of the procedure are fundamental. It must be the same, in both its functional forms and its parameter values, in a given evaluation context. Furthermore, it has to be constructed by an evaluation maker who is not one person alone, but rather a panel of people, possibly representative of different organizations and/or social needs, who can express a balanced consensus that may be obtained after negotiations or successive compromises. The achievement of this consensus, essential for the efficient working of procedure, can be reached by means of any good method of opinion harmonization. The
5DELPHI method has been used here quite satisfactorily.
7 Redundancies and synergies in attributes
The example of the two social characteristics, the “adult literacy ratio” and the “male life expectancy at birth”, shown in Section 4, can also be used to deal with the aspect of
redundancy in attributes, which cannot be taken into account if linear indicators of type (4.1)
5 In general, the DELPHI method is used to achieve consensus in forecasting. We have been using it to reach consensus in evaluation for a long time.
are constructed. The higher the “adult literacy ratio” is, then generally the higher the “male life expectancy” is. This means that a part of the positive value in the overall indicator V(x, x) 12
produced by (a high value of) x,corresponds to a part of the positive value produced by (a 1
high value of) x. In fact, this redundancy also appears in indicator (4.4), but here it can be 2
eliminated by the term K(V(x)(V(x), to be subtracted on the right-hand side, making the 121122
evaluation function V(x, x) multi-linear, of type (4.5). 12
If attributes are redundant, i.e. their total contribution to V(x, x) is less than the sum 12
of their separate contributions, the sign of parameter Kis negative. But in other situations the 12
opposite case may exist, where the total contribution of attributes is greater than the sum of
6separate contributions. In this event a synergy of attributes emerges and Kis positive. Even 12
the value of Kis evaluated through a consensus procedure, in the same way as for 12
parameters K and K, and with advantages and disadvantages similar to those expounded in 12
the previous Section.
It is quite straightforward to generalize and extend the handling of redundancies and synergies to the case of more than two attributes. The general case is given by (4.6) and the particular one with three attributes, used in the subsequent application , is
V(x, x, x) = KV(x) + KV(x) + KV(x) + 1 2 3111222333;;;
+ K( V(x) V(x) + K( V(x) V(x) + K ( V(x) V(x) + 121122131133232233
+ K ( V(x) V(x)V(x) (7.1) 123222233
8 Some comments on the procedure used to elicit opinions
The proposed procedure to determine parameters in the evaluation functions is in substance the one illustrated in the work of 1995, but since then it has been expanded with some innovations aimed at simplifying the questions the evaluation maker has to answer and rendering the link between his answers and the evaluation results more explicit.
In general, the “evaluation maker” is in fact a number of people, representative of
institutions, who harmonize their opinions by means of a particular procedure, for instance the DELPHI method. The evaluation maker's opinions are used to determine parameters in the
6 The problem of redundancies, or double counting, has been already faced in the specialised literature; see, for instance, J. G. Hirschberg et al. (1991). The general formalization proposed in
this paper permits to resolve also the problems related to synergies in the attributes.