Theory and Methodology
“Value” of additional information in multicriterion analysis under uncertainty

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Abstract

The expected value of information in classical (monocriterion) decision analysis has been well covered in the literature. One cannot say the same thing about the multicriterion analysis, particularly when one is in the presence of multicriterion aggregation procedures based on outranking relations for a ranking problematic. The objective of this paper is to try to extend the Bayesian approach to a multicriterion analysis in the context of uncertainty. After illustrating the a posteriori analysis, we shall mention some difficulties associated with the pre a posteriori analysis and the concepts of the “expected value” of perfect or imperfect information.

Introduction

It is usual, even when dealing with several criteria, to be confronted to a context of uncertainty. This uncertainty can originate from different sources (Rizzi, 1982). We suppose here that uncertainty is due to the fact that performance evaluations of alternatives on each or some of the criteria lead to random variables with subjective probability distributions a priori. For these criteria, we are then exactly in the same situation than in statistical decision analysis, except for the fact that we are in the presence of several criteria instead of one.

In such a situation, one is in search of additional information to integrate to the information a priori which can lead to elaborate better decisions. In our opinion, it seems that the Bayesian approach on which the statistical decision analysis is based, could also be appropriate to conduct multicriterion analysis in a similar context of uncertainty.

To achieve this, we shall use the procedure developed by Fishburn (1965) in a monocriterion context with a single or several sets of states of nature to modelize our context of uncertainty; next we shall use an outranking multicriterion aggregation procedure a priori and a posteriori. The use of an outranking approach instead of MAUT is motivated by the fact that we do not know the decision maker's utility function(s) and it is difficult to obtain explicitly this (or these) function(s). Through the comparison of the results obtained from the two multicriteria aggregations (a priori and a posteriori), we shall appreciate how the obtention of additional information on the states of nature and consequently on the consequences related to the occurrence of these states of nature has modified the recommendation a priori. Next, we discuss about ways to deal with the value of this additional information in a multicriterion context for a ranking problematic rather than a choice problematic (Roy, 1985). We want to adapt classical concepts of analysis in statistical decision theory to estimate the value of additional information.

A conclusion concerning the impact of additional information according to the recommendation on its opportunity is generally insufficient to clarify the decision-maker's mind. It is necessary to quantify this impact, not so much for already obtained information but more for additional information one wants to obtain through a study, an experiment,…, and to which is attached a cost (pre a posteriori analysis). In classical statistical decision analysis, two types of measures enable us to quantify the impact of additional information: the expected value of perfect information (EVPI) or the cost of uncertainty which is a measure of risk and gives an indication about the maximum amount a decision-maker may pay to obtain perfect information, and the expected value of imperfect information (EVII) obtained through a study (Raiffa and Schlaifer, 1961). These two measures are not associated to a particular result of the study.

In a multicriterion analysis context, in our opinion, the two measures cannot be used in this form for the following reasons:

(1) The type of multicriterion aggregation procedure based on the outranking approach we adopted. At the opposite of the monocriterion approach where a global “quotation” (expected utility in the MAUT (Keeney and Raiffa, 1976) is associated to each alternative, in an outranking approach an overall (global) index of outranking is associated to each alternatives' pair. With the monocriterion approach, the form of the recommendation consists generally in the choice of the best alternative (i.e., the one optimizing the global quotation), but with the outranking approach, the recommendation can take various forms according to the decision problematic retained and the use of various procedures;

(2) The nonmonetary and heterogeneous nature of the evaluations' (xik) of each alternative ai on each attribute k (the cost of the study being expressed generally in monetary terms);

(3) The sources of uncertainty are multiple (Samson et al., 1989), i.e., we can eventually be confronted to more than one set of states of nature.

Section snippets

Formalization of the decision situation

In the present situation, we suppose that one is in presence of a set A of m alternatives ai(i=1,2,…,m) which the decision-maker would like to compare locally and globally by building binary outranking relations on a set C of n attributes k(k=1,…,n). These alternatives will be evaluated on the set of attributes from the set E of evaluations which are, in our case, all (or at least some of them) characterized by random variables Xik with probability distributions fik unknown with certainty. Let

Aggregation procedure

Given cumulative probability functions Fik and Fjk for two alternatives ai and aj on an attribute k and a modality x of this attribute, we say that:

(1) Fik FSD Fjk if and only if FikFjk andH1(x)=Fik(x)−Fjk(x)⩽0,∀x∈xk,xk,where FSD is first degree stochastic dominance.

(2) Fik SSD Fjk if and only if FikFjk andH2(x)=xkxH1(y)dy⩽0,∀x∈xk,xk,where SSD is second degree stochastic dominance.

(3) Fik TSD Fjk if and only if FikFjk andH3(x)=xkxH2(y)dy⩽0,∀x∈xk,xk,where TSD is third degree

A posteriori and pre a posteriori aggregations

We assume here that an additional information y has been obtained from the source characterized by the likelihood matrix in Table 3. The result of such additional information leads to the computation of a distribution a posteriori of the states of nature which is obtained using the rule of Bayes:P(S̃=Sh/y)=P(Sh)=ph·P(y/Sh)Hh=1ph·P(y/Sh),where ph=P0(S̃=Sh) is probability a priori of occurrence of the state of nature Sh.

The additional information on the states of nature generates a

Measures of distance between two preorders

For reasons mentioned in the introduction, it is necessary to develop new concepts, new measures which can enable us to estimate the value of appropriate additional information in order to be able to make recommendation according to the problematic to be solved. For example, for a ranking problematic where the recommendation takes the form of a preorder (complete or partial), we have to develop a measure of the distance between two preorders and appreciate the cost of an experiment in relation

More than one source of uncertainty

If we are in the presence of a situation where more than one attribute k have alternatives evaluations depending on one or another set of the states of nature, and we get only subjective probabilities of occurrence of these states of nature, then we may wish to establish the value of additional information concerning these attributes.

Let us suppose for example, a second attribute k on which uncertainty is modelized as in Table 1, i.e., there is another set of states of natureŠk={Sk1,Sk2…,Sk

Conclusion

The present paper tried to state the value of additional information by comparing rankings a priori and a posteriori. The suggested methods (distances measure, multicriteria aggregation procedures) are not exhaustive.

The value of additional information is not expressed in monetary form. This is not surprising so far as we are in the presence of multiple attributes with different heterogeneous scales of measurement. The problem of converting (encoding) the value of the information expressed by a

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