A practical multicriteria methodology for assessing risky
public investments
Michel Beuthe*, Louis Eeckhoudt, Giuseppe Scannella
GTM - Groupe Transport et MobiliteÂ, FaculteÂs Universitaires Catholiques de Mons, 151 chausseÂe de Binche, B-7000 Mons, Belgium
Abstract
This paper proposes a simple, practical but also realistic, methodology of multicriteria decision aid for the assessment of public investment projects which takes into account the uncertainty of projects' measures.
It follows the general framework of the expected utility developed by von Neumann and Morgenstern and its operational modelling by an additive non-linear function proposed by Keeney and Raia. However, it builds upon the linear goal programming model of Jacquet-LagreÁze and Siskos which oers several convenient computational procedures for building non-linear (albeit, piecewise linear) partial utility functions re¯ecting, as well as possible, the decision-maker's preferences.
In order to simplify as much as possible the process of decision making, the paper proposes an alternative method, named Quasi-UTA, whereby the decision maker chooses directly for each criterion a (piecewise linear) partial utility function with only two parameters. These are the respective relative weights and a curvature parameter that models risk aversion or proneness. The method substantially reduces the number of questions to be answered by the decision-maker.
In a context of budget constraint, the paper seeks to calibrate the utility function in equivalent money value, and shows how to compute such a value in multicriteria analysis. Money valuation has the advantage of permitting the use of standard ®nancial rules of assessment. The paper also shows how to compute the cost of a project's uncertainty, i.e. its risk premium, as proposed in the ®nancial theory of risk.
The method also suggests a procedure to generate and compute each criterion's random distribution as a triangular, rectangular or discrete function. An example illustrates the use of the methodology.#2000 Elsevier Science Ltd. All rights reserved.
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1. Introduction
The valuation of public projects faces at least two dicult problems. First, it has to integrate criteria that are not naturally measured in the same units. Second, most measures are not certain but stochastic.
As is well known, the ®rst problem can be solved by recourse to multicriteria analysis techniques, while the second is usually dealt with by a sensitivity analysis of some parameters and measures. However, sensitivity analysis, does not, in principle, completely solve the decision-maker's problem. It tells how the valuation can be aected by various stochastic elements, but abandons the decision maker (DM) with the question of how much weight should be given to the risk involved in a project.
The two problems are interlocked in many ways, as will be seen in this paper. Obviously, many analyses have previously attempted to deal with them. As their list would be rather long, we will just mention here one of the more important: the operational modeling by Keeney and Raia [2] of an additive non-linear utility function based on the expected utility theory of von Neumann and Morgenstern [3]. Its full implementation is rather lengthy and burdensome and, for that reason, not very often applied.
The present paper proposes a simpler multicriteria procedure. It follows the previous authors' leads by using an expected utility framework and the modeling of an additive non-linear function. The latter is built by borrowing from the UTA (Utility Additive) non-linear goal programming model of Jacquet-LagreÂze and Siskos [4,5] which oers several convenient computational procedures. Moreover, focusing on the valuation of uncertainty, we show how to derive equivalent money values from multicriteria utility functions and how to compute the cost of uncertainty aecting a particular project.
Section 2 begins by reviewing the reasons that favor a project's valuation in monetary units. Section 3 sets the general framework of uncertainty analysis applied in the current paper. Its subsection 3.3 focuses on computation of the risk premium (RP), while subsection 3.4 proposes a simple procedure for generating appropriate distribution functions for each criterion. Subsection 4.1 brie¯y reviews the UTA methodology while Subsection 4.2 proposes a simpli®ed ``Quasi-UTA'' procedure with two-parameter partial utility functions. Subsection 4.3 illustrates the methodology with an example. The conclusion will brie¯y draw some recommendations for public investment decision making, and outline directions for future research.
2. The money yardstick in multicriteria analysis
Beyond the well known problems associated with the adjustment of market prices under non-competitive conditions, it remains to be seen whether all impacts can be valued in money terms. Actually, the choices to be made between dierent types of consumption, investment and saving, implicitly de®ne the importance and the monetary value attached to dierent objectives: safety, gains of time, employment, industrial policy, etc. Given the presence of a budget constraint, there is an implicit price for everything, a price that translates the importance given by the DM to the objectives, services or results he/she wishes to obtain. These prices are known as opportunity costs or shadow prices, to distinguish them from the market prices. Since there is an opportunity cost while there may not be a market price, we recommend that, as far as possible, prices and money values be the yardstick used for valuing a project's impacts.
Since, in many public infrastructure projects, impacts are not marketed, there is the issue of how to evaluate these impacts in money values. Actually, this is what social cost±bene®t analysis (SCBA) attempts as it strives to value time and congestion, injuries, pollution, etc. in money values. The basis for such valuation is the willingness to pay or to accept compensation as derived from observed choices and behaviors, or deduced from statements of value declared by those interviewed. Despite the potential strengths of this approach, SCBA is unable to provide a full account of selected impacts in public decision making. Examples include industrial, land use and regional policies, which are generally beyond the private citizen's direct responsibility and control. To the extent that such objectives of a political nature are not taken into account, there is a need for going beyond SCBA.
Nevertheless, this does not imply that the monetary yardstick should be discarded. Multicriteria analysis (MCA) provides the framework and tools to estimate the opportunity costs of interest here, i.e. values imputed to selected qualitative or political objectives in terms of forgone investments. The multicriteria arena oers various methods for eliciting a DM's preferences among projects based on the relative importance is given to the various criteria. In doing so, it builds an index of preference or utility via a weighted aggregation of several criteria. The index can also be calibrated so that the valuation is made in money by reference to those criteria that are naturally de®ned in such terms. Thus, as in cost±bene®t analysis (CBA), MCA can provide an estimation of a project's net present money value with all impacts being taken into account.
With such a money valuation of projects, it is then possible to keep unchanged the basic rules for accepting a project in CBA: a project can be accepted if its net present value, all impacts' values included, is positive. If a choice must be made between several projects, the ratio of the net present value to the budgetary cost of each project should be used as the rank ordering index. This would guarantee that the total net present value of all projects implementable with the available total budget will be maximized. An additional advantage of a money valuation of all impacts is that it permits the computation of a project's risk in such terms.
3. Uncertainty
3.1. Importance in decision making
main reasons explain the presence of risk or uncertainty in decision making: the inevitable imperfection of statistical estimations, the diculty of forecasting within incomplete and/or dynamic systems, and the basic ignorance of the future. If the DM is risk averse, he/she should certainly identify and seek to account for risk in decision making. In the context of a public project, the DM represents the state and its citizens. His/her attitude towards risk should thus account for the size of the project compared to the country's gross income.
For small projects that are not correlated with the country's gross income, Arrow and Lind [6] have shown that the risk could be neglected if it is actually shared by a large number of citizens. In that case, the state, which plays a role similar to a mutual insurance fund, can make decisions without attention to risk, i.e. it can take a risk-neutral attitude in decision-making. However, this neglects the fact that, beside the usual ®nancial costs and bene®ts, there may be substantial external eects that are not as well distributed over the total population, or are not distributed at all but aect everyone in the same way [7]. These are also components of the global risk of a project that should be taken into account.
On the other hand, the risk of a large-scale project cannot be neglected: its risk cannot be spread over a sucient number of people in many cases, particularly when some impacts are not evenly distributed over the population. Moreover, the project may aect the entire economy as well as other projects, so that it generates additional risk through interdependence.
The distribution of impacts across segments of the population and over regions is a matter of equity in distribution in the income sense of the word. It also determines the level of risk of a project as well as who bears it. This should be a matter of serious concern for the DM.
3.2. Risk aversion, cost of uncertainty and non-linearity
The expected utility theory of von Neumann and Morgenstern [3] is still the reference model for risk analysis in decision making, and is used here as the framework for the proposed methodology. It assumes that the utility function is strictly concave if there is any degree of risk aversion. It follows that the expected utility, taken as a measure of a risky project's utility, has a lower value than the utility of a certain project with outcome equal to the weighted average outcome of the project. Hence, there is a loss of utility resulting from the project's risk.
As will be explained in subsection 3.3, if one of the outcomes is valued in monetary terms, it is possible to compute the money equivalent values of both the project and the loss of utility, i.e. cost of the risk in the project. It amounts to the dierence between the money values of the uncertain outcome of the project and the certain project which has a utility equal to the expected utility of the uncertain project. In the ®nancial literature, this dierence is called the RP since it is the maximum that one would pay for insurance against the risk involved. This
RP1 is a measure of the willingness to pay for averting the risk. It is what we need for
adjusting the equivalent money value (EMV) of an uncertain project in ®nding its certain money equivalent.
1
A linear utility function is the most convenient form for a multicriteria analysis and is thus often assumed. But, in order to avoid the risk neutrality associated with such a function, a non-linear form must be used. We will here follow Keeney and Raia [2] in proposing an additive non-linear speci®cation.
It is necessary to note, though, that there are basically two types of utility functions. The usual one, which is applied in most texts of micro-economic theory, is derived under conditions of certainty. It expresses the preferences of the DM when confronted with certain outcomes. In the more advanced theoretical literature, it is called a ``value function''. When it is used for computing an expected utility, it implies that the utility of the uncertain outcome is the weighted average of the utilities that would be obtained from certain outcomes. This approach was ®rst proposed by Bernoulli to handle the problem of risk [9]. From an operational point of view it means, concretely, that to estimate such a value function we must focus on the DM's preferences between projects with certain outcomes, i.e. without consideration of a range of outcomes for any speci®c project.
In the context of uncertainty, a ``utility function'' incorporates whatever attitude towards risk taking the DM may have in mind: risk aversion, risk neutrality or risk proneness. Actually, it can be thought of as a transformation of the value function that accounts for attitude towards risk in the assessment of the alternatives confronted by the DM. It implies that, when estimating the utility function, questions addressed by the analyst to the DM must refer to uncertain outcomes, or lotteries, as they are more generally referred to in the literature. This is obviously a much more dicult task, particularly in an analysis with several criteria.
A possible procedure could be as follows: ®rst, build a value function of some form; then, transform it into a utility function by questioning the DM on the utility he/she attaches to dierent lotteries of values. One may wonder, though, whether this might be too burdensome for most DMs. Moreover, this procedure neglects the possibility that the DM's risk attitude varies over criteria.
If an attempt is made to incorporate some risk analysis by way of computing an expected utility, we do not believe that it is necessary to go as far as generating a full-¯edged utility function with the long and dicult questionnaires it involves. Given the current practice of public project valuation, we feel that an important step forward would already be achieved if an attempt was made to simply obtain non-linear value functions.
While the methodological distinction between a value and a utility function is assumed in order to avoid theoretical confusion and controversy, it will not be necessary to maintain such a distinction throughout our discussion. In what follows, then, only the more common terminology of a utility function will be used.
as much on the project's impact on regional development as it does on the level itself. Also, it can be argued that the utility provided by a project can be seen as a utility dierential, which is de®ned mathematically as a weighted addition of marginal utilities, i.e. as impacts' utilities2.
Two approaches for generating additive multiattribute utility functions can be distinguished. A number of methods separately build each partial utility function, and then, in a second step, estimate the weights that link the functions. This approach was initially proposed by Keeney and Raia [2], but can be implemented in dierent ways. The more recent MACBETH method by Bana e Costa and Vansnick [13,14] is probably one of the most convenient as it applies a questionnaire methodology with verbal propositions that seeks a good approximation of interval-scaled preferences in certainty. Such a value function measuring strength of preference cannot, however, be taken as a von Neumann±Morgenstern utility function for valuing risky actions. While a utility function measures the strength of preference, asking questions in certainty based on strength of preference cannot extract risk attitudes3.
Other methods proceed from stated global preferences between projects with linear goal programming models. Among these, the UTA method proposed by Jacquet-LagreÁze and Siskos [4,5], is probably the most developed and useful. For a number of reasons, to be explained in Section 4, this is the approach that will be followed in the proposed assessment methodology.
3.3. Risk premium computation in a multicriteria framework
Let U(A) be an additive utility function of a project de®ned by A=(M, X), a vector of two
criteria, whereM is the present value of the project as computed by a classical CBA, andX is
its environmental impact measured in physical units:
U A u M v X:
The EMV ofA, M(A) is de®ned as:
M A uÿ1 u M v X,
where uÿ1 is the inverse of the partial utility function of M. In Fig. 1, M(A) corresponds to
the intercept of indierence curveU8(A) on the Maxis4.
In a situation of uncertainty, with random MÄ and XÄ, the certain EMV of project AÄ, MÃ is
de®ned by the condition that the expected utility of M(AÄ) is equal to the utility of MÃ:
Euuÿ1ÿu M~ v X~ Eu M~ v X~u M^:
Indeed, the DM would then be indierent between the amountMÃ and the random project AÄ.
The RP, which corresponds to the cost of uncertainty, is then equal to the dierence
2
For a more detailed discussion, see Chapter 6 of the APAS/Road/3 report [10]. It is interesting to note that, in the similar ®eld of conjoint analysis, Green and Srinivasan [11,12] argue that the additive compensatory model is likely to predict well even if the decision process is more complex or non-compensatory.
3
On this topic, see Bell and Raia [15] and Bouyssou and Vansnick [16] who discuss under which restricted con-ditions equivalence of both types of functions can be found.
4
between M(A-), i.e. EMV computed at the average value of AÄ, and the certain MÃ of the project, i.e. RP=M(A-)-MÃ. It is thus the dierence between the money value of the project for
M- and X- and the money value MÃ. RP can be interpreted as the maximum amount that the
DM would be ready to pay in order to escape the uncertain situation. It is worth noting that this multicriteria RP is a coherent concept since:
. MÃ and RP are unique;
. if bothu(M) andv(X) are linear so that there is no risk aversion, then RP = 0 ; and
. if u(M) or/and v(X) are strictly concave, and neither strictly convex, then RP > 0, because u(M-) >E[u(MÄ)] or/andv(X-) >E[v(XÄ)], and the inverse inequalities are excluded.
An illustration of an EMV and RP computation can be found in Section 4.3, where a numerical example of the proposed Quasi-UTA method is given. As will be seen in that case, it is possible that the computedM(A)s fall outside the variation interval of the scale used for M
over which u(M) is de®ned in a particular MCA. Rather than computing these values by
simple extrapolation of u(M) beyond this range, we propose to modify the theoretical method
ofM(A)'s computation as follows:
^
Muÿ1ÿEu M~ v X~ and M A uÿ1ÿu M v X,
whereuÿ1MmaxÿMmin
u Mmax , and Mmin andMmax are, respectively, the minimum and maximum
values on the scale of MÄ. Hence, u-ÿ1 is the inverse of the average slope of u(MÄ) over its
variation interval.
The use of the average slope u-, rather than slope ofu(M) atM-, is made in order to obtain a
risk aversion measure independent of a particular project's M-. This RP retains the above
characteristics.
3.4. Distribution
Regardless of the method used to account for the risks inherent in a project, a proper analysis requires systematic estimation of the uncertainty aecting all relevant variables and parameters including their range of variation, and the probability of their realization.
If statistical estimates are not available, these elements should be assessed by the analyst, in some cases with the help of outside experts, on a subjective basis. In order to facilitate this collection of information, a questionnaire, to be addressed to the promoters and analysts of a project, should be constructed. The outline of such a questionnaire is proposed in Appendix A. Since the multicriteria function is taken to be additive in the present case, each distribution is assumed to be independent. However, account should be taken, as much as possible, of all those factors that aect their distribution and, particularly, of the so-called systematic risk. The latter, which corresponds, for instance, to the possibility of an earthquake or of an economic recession, is as much a part of the overall risk as any other factor speci®c to a project. Considering the diculty of the task, it would be wise to limit the requested information to a minimum. On the basis of the overall assessment framework and the modeling corresponding to it, the requested information should focus solely on the criteria or their additive components.
The questions should be constructed on the basis of a triangular density function, a rectangular one, or a discrete distribution. Under the two ®rst hypotheses, only a few questions, that appear not too dicult, are necessary to generate the full distribution. They should concern (1) a lower limit value under which the variable would not be likely to fall, (2) the value of the mode (or the mean) of the distribution, and (3) the upper limit value above which the variable would not be likely to rise. The likelihood that the variable would go beyond the two limits should also be indicated.
A triangular density function may be taken as a convenient approximation of the normal density, but it may also handle cases where the distribution is not symmetrical. Thus, it can be a useful approximation of many continuous distributions, with the rectangular one as a limit case. The option of a discrete distribution should also be oered as there may be qualitative criteria with but a few values.
4. UTA methods for building a non-linear additive utility function
4.1. UTA general method
The UTA method aims at building a preference index or utility function of the form
Ui u1 Xi1 u2 Xi2 . . .un Xin,
where Xij is the value taken by the jth criterion in the ith project. Each ui(Xij) is a partial
utility function that can be non-linear in a piecewise linear approximation over a set of intervals. Actually, the method ®nds the values taken by the uj(Xij)'s at a number of equally
between two successive end-points, Xij(k) and Xij(k + 1), then correspond to a linear
combination of the values taken byui(Xij ) at these end-points.
The estimation made by UTA is based on simple statements by the DM of global preferences between projects in a sample. The problem is set as a linear programming model, where the constraints indicate the order of preference between projects. For instance, if project i is preferred to projectk, the following constraint would hold:
Sj
uj Xij ÿuj Xkj eiÿeked a
where: ei and ek, which cannot be negative, are error terms resulting from estimation of the
DM's rank ordering; andd is a small positive number.
This is a goal-programming type constraint such that the constraint is not absolute, but embodies a goal that the additive utility function should try to achieve: in the case of (1), it is the respect of the preference order between projectsi and k. Another set of constraints requires that the uj(Xij) functions be monotonically increasing. The objective of the program is then to
minimize theSieiover all projects in the sample.
Like in all goal-programming models, it is possible that the solution will not be unique. It is thus recommended that one use the average of the optimal utility functions. Further, there is no problem extrapolating the function to additional projects of the same nature, if the sample is representative. Finally, the sample may be partly composed of ®ctitious, but realistic, projects, in order to elicit from the DM more complete information about project preferences, as it is almost always done in conjoint analysis [11,12].
Key advantages of the UTA method are that it assumes a utility function of a more general form, and that it requires a limited amount of information from the DM. Moreover, since all partial utilities are generated simultaneously on the basis of global preference statements, the method accounts for the interdependence that may exist between various criteria. Furthermore, the method, sometimes described as an ordinal regression method, produces weight coecients. Like in a linear regression, these are aected by correlation between criteria and biased by the absence of some variables, for instance a product of two criteria. Thus, if the preferences are in¯uenced by some interdependence between criteria, this phenomenon is taken into account by the weights. This advantageous ``ad hoc'' feature of the UTA method is illustrated through a series of simulations in [17]. On the other hand, it must be recognized that the method does not probe the DM about each partial function.
Finally, calibration in terms of the money value of the SCBA result, say Xi1, should not
present any diculty, as it is easy to impose that u1(Xi1)=Xi1 if (1) it can be assumed that there is no risk aversion with respect toXi1, or (2) it is possible to derive EMVs on the basis of the utility function, as explained in Section 3.3.
This is just an outline of the general approach, of which there are several versions. For instance, it is possible to introduce additional constraints about the strength of preferences between projects or the concavity of the functions. Detailed analyses of the method and its main variants are provided in Beuthe and Scannella [17,18]. Other useful references include Vincke [19] and Pomerol and Barba-Romero [20].
software provides a number of features that help both the DM and the analyst explore possible adjustments to the utility function and to run sensitivity analyses. In this regard, it speci®es a concave utility function, if that is seen as useful. There also exists a UTA-based Multicriteria Decision Aiding System, named MINORA (Multicriteria INteractive Ordinal Regression Analysis) by Siskos et al. [23]. This system oers a feedback procedure in case of inconsistencies between the DM and the preference model provided by UTA. Actually, these software programs are decision support systems that permit an interactive approach to the building of a utility function.
4.2. A more practical approach: Quasi-UTA
Use of the full-¯edged UTA method is recommended but, like other complete multicriteria techniques, it implies a somewhat cumbersome process of eliciting preferences between criteria and projects from the DM. An alternative, more practical approach, would be to let the DM directly choose the partial utility function relative to each criterion from a set of functions. Such a methodology is used in the decision system MIIDAS (Multicriteria Interactive Intelligent Decision Aiding System) by Siskos et al. [24], where dierent types of curves are visually presented to the DM. However, their mathematical expression involves three parameters, plus the weight found by running the UTA model.
Here, a more convenient mathematical speci®cation is proposed, a sort of recursive `exponential' function, that involves only two parameters to de®ne the relative weights with respect to each other and their curvature. The formula is ¯exible enough to represent very dierent attitudes towards risk, from strong risk aversion (strongly concave function) to strong risk love (strongly convex function). Naturally, the choice of parameters can be presented graphically with typical values associated with meaningful verbal statements.
Using the same notation as above, let us consider a criterion Xi de®ned over a total interval
betweenXi(1) and Xi(n), which is divided in (nÿ1) equal smaller intervals by n bounds, and a
partial utility functionv(Xi(k)) de®ned at thekth bound of the scale. Then,v is de®ned at each
bound by the following ®nite geometric series:
v Xi 1 0,
v Xi 2 v Xi 1 g1g,
v Xi 3 v Xi 2 g2gg2g 1g,
. . .
v Xi n v Xi nÿ1 gnÿ1gg2g3. . .gnÿ1g 1gg2. . .gnÿ2,
8
For 1 <kEn, this ®nite geometric sequence is given by:
v Xi k g we can take the limit of the function forg41 and apply L'HoÃpital's rule [25], to obtain
limg
This leads to the following de®nition of v:
8
where ui is the UTA's scaling constant which is equal to the utility of the last bound on Xi
and therefore equal to the relative weight of Xi multiplied by [Xi(n)ÿXi(1)]. Graphical
examples of this function are given in Fig. 2.
However, if one needs to change the number of bounds dividing the variation interval of a
given criterion Xi, the formula must be adjusted in order to maintain the same curvature with
a giveng over the same interval as previously. This is because the number of bounds is used as
a variable in the function in place ofXi itself.
Thus, for two dierent numbers of bounds,n and m, over a given interval ofXi, [Xi,Xi], the
relation between the bounds' indices, k and p, must be such that k=p= 1 for Xi, and k=n,
k nÿ1
mÿ1p mÿn
mÿ1 or p mÿ1
nÿ1k nÿm
nÿ1:
In fact, switching from the scale with n bounds to the scale with m bounds, ui(Xi(p)) must be
rede®ned as:
8 > > > > > > <
> > > > > > :
ui Xi p
gÿgmnÿÿ11pmmÿÿn1
gÿgn u
i, 1EpEmand 8g61
ui Xi p
nÿ1 mÿ1p
mÿn mÿ1
ÿ1
nÿ1 , 1EpEmand g1:
This transformation is illustrated in Fig. 3.
4.3. Example of Quasi-UTA
In this section, we present an application of the Quasi-UTA model to two simple projects with only two impacts, the net present value and a environmental impact. They do not correspond to any real project, but they permit a sucient illustration of the method.
Table 1 gives the minimum and the maximum values of the physical scale on each impact, as well as the two parameters' values which must be chosen by the DM. We note that, in this
case, with g=0.25, the DM appears to exhibit more risk aversion with respect to the net
present value result.
Table 2 gives the impacts' data, which are needed to compute their triangular distribution. They illustrate typical answers given to the questionnaire outlined in Appendix A.
The corresponding results of the MCA are presented in Table 3. They are computed by a program that uses the data of Tables 1 and 2 to set the partial utility functions and to estimate the triangular distributions.
The expected utilities, E[U(A)], give a valuation of the projects that takes into account their
Fig. 3. Utility function with dierent numbers of breakpoints.
Table 1
Physical scales and partial utility functions
Impact X Min Valuea X Max Valuea Weight Curvature parameter
Cost±Bene®t NPV 0 4 0.55 0.25
Environmental impact 0 4 0.45 0.50
a
uncertainty. Thus, we see that, as a result of the greater risk aecting A1's NPV, A2 is preferred to A1. In, contrast, if they were valued in terms of their utilities at the average, i.e. U(A-)'s, without paying any attention to uncertainty, A1would be preferred toA2.
Likewise, the multicriteria EMV of A2, MÃ1=6.844 billion ECU is greater that MÃ1=6.377
billion ECU, even though the average M(A-1)=7.090 billion is greater than M(A
-2)=6.982
billion. Again, the rank order of the two projects is reversed when their uncertainty is considered. Indeed,A1's RP (0.753) is much greater than A2's (0.137).
The EMVs allow the use of standard ®nancial criteria. Obviously, both projects have
positive values and could be recommended. However, the budget of A1 being smaller, its
®nancial ratio (0.253) is greater than that of A2 (0.201). Hence, it should be ranked ®rst.
Nevertheless, if the total available budget permits the implementation of only one project, the second project could very well be chosen since its equivalent value is larger.
5. Conclusion
Decision-making on public investments is a convoluted process with many interacting actors. Even if we set aside problems related to the dynamics of the political process, as well as to group decision making, the assessment of a project by one DM remains a dicult matter both at the theoretical and empirical levels.
Table 2
Impact data in uncertainty
Project A1 Project A2
Cost±bene®t NPVa Environmental impact Cost±bene®t NPVa Environmental impact
Min 1 3 1 2.5
Mode 2.5 3.5 3 3.5
Max 4.5 4.5 3.5 4
Average 2.66 3.66 2.5 3.31
a 0 0 0 0.025
b 0 0 0 0.025
aNote: Cost±Bene®t NPV in Billion ECU.
Table 3
Results of the multicriteria risk analysis
Project Utilities Multicriteria EMVs (billion ECU)
U(A-) E[U(A)] Cost±Bene®t NPV (A-) M(A-) MÃ RP MÃ/Budget Ranking
A1 0.975 0.871 2.666 7.090 6.337 0.753 0.253 1
Within a CBA, the direct valuation of some impacts remains problematic. Further, the aggregation of all costs and bene®ts raises the sensitive question of the outcomes' distribution across individuals. In many cases, CBA is unable to encompass all the impacts a project can have on society, where some impacts may be so important to the DM that they ought to be weighed separately. These considerations suggest then an assessment with several criteria, which does not simplify the task. If most MCAs provide an ordering, or at least a classi®cation of projects, they generally lack a clear-cut decision rule for project acceptance, in contrast with CBA. Furthermore, eliciting a DM's preferences across criteria is a delicate task, particularly when considering his/her attitude towards risk.
Uncertainty about some outcomes also creates a problem, particularly when the risk involved in a project cannot be suciently distributed to be neglected. A sensitivity analysis of the results generally does not completely solve the problem, since it leaves the DM with the task of somehow assessing how much risk is worth taking.
In the present paper, we sought some answers to these questions by proposing a methodological framework that is both practical and in accordance with basic theoretical standards. Following the expected utility approach, we thus proposed a speci®cation of partial utility functions, actually ``value'' functions, with only one curvature parameter. This speci®cation permits a parsimonious approach in terms of the number of questions needed for the DM to reveal his/her preferences. It also proposed a simple method for obtaining workable approximations of the outcomes' distributions. Further, it showed how to properly compute the EMV of multicriteria utility functions, so that, in a context of budget constraint, well known ®nancial decision rules could be applied to better order and select projects. Importantly, transforming utilities into EMV allows one to estimate a project's RP, i.e. the cost of the risk involved as valued by the DM.
The proposed methodology implies some strong assumptions, but opens a practical and reasonable approach to analyse the risks involved in projects. Providing a convenient framework for estimating both distributions and partial utility functions, it should ease the task of assessing risk and allow a deeper testing of what should be a crucial, but easily forgotten, factor in public investment decision making. Actually, we submit that this methodology is suciently accessible that analysts need not neglect a formal analysis of risks taken by the DM in selecting projects. We are aware, nevertheless, that some ``side actors'' in a decision process may be rather reluctant to provide necessary information about the uncertain character of some impacts. This is perhaps the main obstacle to such an assessment procedure.
Further research on this methodology is certainly warranted. First, we are presently developing software for its convenient and interactive application5. Second, even if the use of a value function, rather than a strictly de®ned cardinal utility function, seems to be a reasonable option in public investment decision making, we feel that the proposed speci®cation opens the way to a more straightforward estimation of utilities under uncertainty. Currently, we are preparing experimental and comparative validation of the proposed methodology.
5
Acknowledgements
This paper is one output of the research program EUNET of the European Commission on Socio-Economic and Spatial Impacts of Transport (see [1]). We wish to thank the French Community of Belgium (FSRIU grant) and the Belgian `Services FeÂdeÂraux des Aaires Scienti®ques, Techniques et Culturelles' for granting ®nancial support to a part of this research.
Appendix A. Questionnaire about risk identi®cation and measurement
A.1. Introductory remark
The purpose of this questionnaire is to determine the degree to which measures of each criterion introduced in the valuation of a project are aected by uncertainty. The information collected will be used to design a sensitivity analysis of the global results, and to assess the extent to which the risk characterizing a project changes its valuation in terms of its expected value.
The questionnaire is sent to the leader of each research team working on the estimation of a particular project's impact. If the team is working on several impacts or variables that will be additively combined to form a single, or part of a single, criterion, responses are requested for each component separately. An obvious example of this situation could be the usual components of a CBA criterion: fuel savings, time savings, accidents' material savings, etc. When the assessment framework and its various elements are well de®ned, additional questions relating to each particular item may have to be added to the questionnaire.
When responding to such a questionnaire, the analyst should account for all factors that might in¯uence the value of a criterion. Those considered beyond the DM's control should also be included, e.g. the possible in¯uences of an economic recession.
A.2. Identi®cation of the variables
1. For which criterion (or impact) will your estimation output be used? (Give simply the name of the criterion)
2. What is the output?
(Give its name in concordance with the assessment framework)
3. Is it a sub-criterion that will be added to other sub-criteria, all being components of the same main criterion?
(Answer Yes or No)
4. Should it be taken as a stochastic variable because (1) it is the result of a statistical estimation, or (2) because of the future's uncertainty ?
(Answer No, 1, or 2 )
5. Should it be taken as a non-stochastic decision variable for which dierent values could be considered, like a social discount rate or a legal prescription ?
(Answer Yes or No)
6. Should it be taken as a constant, because it corresponds to a physical description devoid of any signi®cant uncertainty ?
(Answer Yes or No)
A.3. Distribution analysis
1. If it is a constant, give its value:
2. If it is a continuous variable for which you have a statistical con®dence interval, give:
* its mode (m), i.e. its most likely value:
* its mean (not necessary):
* the lower limit (a) of the interval:
* the upper limit (b) of the interval:
* the probability level of the con®dence interval in % at each tail (a and b):
(note: usuallya and bare taken as equal)
3. If you do not have such statistically estimated information, or if the variable's uncertain value cannot be ascertained by a statistical procedure, give, on a subjective basis:
* its mode (m) , i.e. its most likely value (if any):
(note : if there is no mode, the density function will be taken as rectangular; if the distribution is symmetrical, mode and mean are equal)
* its mean (not necessary):
* the likely lower bound (a) of the variable:
* the subjective likelihood (1-a) that the variable will have a value larger than a: (note: for
a rectangular density, a=b=0):
* the likely upper bound (b) of the variable:
* the subjective likelihood (1-b) that the variable will have a value lower than b:
4. If it is a discrete variable, indicate the numerical values it can take and their probabilities. 5. If you are concerned with a non-stochastic decision variable, could you suggest values that
would deserve investigation in a sensitivity analysis?
A.4. Scenario analysis
1. If the variable (criterion or sub-criterion) is correlated with another variable of the valuation
framework, give, to the best of your knowledge, the correlation (from ÿ1 to +1) between
them.
2. In words, suggest critical or likely scenarios involving speci®c values of the variables, and which you feel would be interesting to investigate.
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