Introduction

The aggregation of different individual preferences on mutually exclusive alternatives to a single collective preference is an important theoretical problem with practical implications in many economic, engineering and political scenarios. Moreover, the aggregation of preferences may support some group decision-making problems. However, group decision-making problems embed much more than just the issue of preferences aggregation, since they represent a complex social process that involves negotiation, politics, problem structuring, etc.

The purpose of this paper is to propose an operational method for a particular case of aggregation of preferences that corresponds to the situation where the decision makers show their preferences through cardinal utility functions. For the location of this type of procedure within the context of a general group decision-making problem, see Belton and Pictet (1997).

The selection of a procedure to aggregate individual preferences is not an easy task. In fact, Arrow's (1951) impossibility theorem shows that there is no an aggregation procedure holding a set of sensible conditions or axioms. Moreover, impossibility remains even if these conditions are relaxed (see Sen, 1970). Hence, the search for acceptable aggregation procedures is the main subject of the specialized literature.

An aggregation procedure basically depends on how the decision-makers show their preferences towards the different alternatives (attributes) involved. Some ways for showing individual preferences are as follows:

1. As a complete preference ordering of the alternatives ('complete ranking'). Alternatives are ordered from best to worst using a linear order without any other additional information. Each alternative is comparable with each other.
2. As a partial preference ordering of alternatives ('partial rankings'). Alternatives are partially ordered. Alternatives may be incomparable. In this case, the notion of maximal substitutes the notion of best alternative.
3. As a binary preference relation. This case is related to procedures based on pairwise comparisons and can be represented by a matrix structure (pairwise comparison matrix). A crisp binary relation leads to the above cases 1 and 2. A valued binary relation reports some degree of credibility of preference for one alternative over another.
4. As a cardinal utility function. In this case, the decision-maker directly assigns a value for each alternative. This value represents the decision-maker's view of the performance of this alternative.

Kemeny and Snell (1962) spearheaded a new direction in preferences aggregation procedures by proposing a 'distance measure' (based on l¹ and l² metrics) between individual preferences. Further research on this distance measure has been carried out by many others, like Bogart (1973, 1975), Cook and Seiford (1978) or Cook and Kress (1985). Finally, Cook et al (1996) presented a general model for distance-based consensus.

Recently, González-Pachón and Romero (1999, 2001) have presented goal programming (GP) and interval GP approaches to aggregate complete and partial rankings (preference expression cases 1 and 2) in a distance-based framework, respectively. GP-based procedures for the aggregation of pairwise comparison matrices (preference expression case 3) have been proposed by Forman and Peniwati (1998) and Linares and Romero (2002).

This paper concerns a distance-based aggregation procedure based on GP for preference expression case 4; that is, for preference information based on cardinal utility functions.

Keeney (1976) demonstrated that if Arrow's conditions are stated in a cardinal fashion, then there will exist aggregate (social) cardinal utility functions holding these 'a priori' imposed sensible conditions. Moreover, Keeney and Kirkwood (1975) provided some representational theorems that show how the aggregate (social) utility function is restricted to a few special forms if a small number of sensible conditions hold. Basically, they justify, as an aggregate cardinal utility function, a multilinear decomposition that involves the additive and the multiplicative forms as special cases.

However, a key problem for implementing the procedure proposed by Keeney and Kirkwood (1975) remains unsolved. The problem is as follows: how to evaluate the different constants (parameters) involved in the aggregate cardinal utility function. Information about interpersonal comparisons is the cornerstone for answering this question. Two ways for gathering this kind of information have been explored. The first one is to state a 'benevolent dictator'; that is, someone who has a responsibility for the decision but wishes to take into account the views of others. A second way is stated by assuming that individuals are willing to perform interpersonal comparisons. This has been explored in Baucells and Sarin (2003), where bilateral agreements between pairs of individuals are sufficient to derive aggregate utility function parameters.

This paper proposes a general procedure that can be used to estimate the constant values involved in the aggregate utility function, through a consensus procedure. This procedure is stated in a distance-based framework. Moreover, unlike other procedures, Bodily (1979) or Nakayama et al (1979), the method is non-interactive, and the values of the constants are derived directly from the individual utility functions (prior individual preferences) without any additional interaction.

The paper is organized as follows. The following section states the analytical aspects of the proposed procedure. In this sense, it is shown how the consensus values of the respective constants can be obtained by minimizing a distance function model that, later on, is transformed into an Archimedean GP problem. The functioning of the model is explained in the next section with the help of a simple case involving three decision-makers. Finally, we present some conclusions derived from this work.

The approach

Notations and definitions

Let us consider a decision-making problem where there are q attributes (social states/electoral candidates, etc) involved and where n decision-makers (social groups/electoral committees, etc) have shown their preferences by means of n multiattribute utility functions. The problem is to find an aggregate utility function that represents the preferences of the n decision-makers or social groups as a whole. The following elements will be used throughout the paper to introduce the analytical aspects of the approach:

1. x_i is the value achieved by the ith attribute (i=1, 2, ..., q).
2. T(x₁, ..., x_i, ..., x_q)=K represents the hypersurface that defines the efficient and feasible set of solutions.
3. U_j denotes the multiattribute utility function for the jth decision-maker or social group (j=1, 2, ..., n). U_j will be a function of the value achieved by the q attributes, that is:
   
   
   
   
4. x^*=(x₁^*, ..., x_i*, ..., x_q^*) represents the ideal vector, that is, the unfeasible point for which each attribute achieves the best (optimum) value.
5. x_^*=(x_1^*, ..., x_i^*, ..., x_q^*) represents the anti-ideal vector, that is, the feasible and non-efficient point for which each attribute achieves the worst value.
   For the ideal and the anti-ideal vector, the following usual conventions are used:
   
   
   
   Thus, the possible values achieved by the n individual multi attribute utility functions are bounded by 0 and 1.
6. U=g(U₁, ..., U_j, ..., U_n) is the aggregate or group utility function that measures the preferences of the group as a whole and represents the outcome to be provided by the proposed approach.

The model

The first step for implementing our approach will involve extracting m observations (sample) from the feasible and efficient set T(x₁, ..., x_i, ..., x_q)=K. These observations will represent our training set. Thus, we have a set of m points in ^q

such that T(x^k)=K ∀k{1, 2, ..., m); that is, each vector x^k represents a hyperpoint of the feasible set.

The m values of the training set are substituted into the respective individual multiattribute utility functions U_j to get the following m vectors:

The above m vectors represent the values achieved by the n individual multiattribute utility functions for each of the m elements (sample) of the previously established training set.

The problem now is to determine the m scalars U^k (k{1, 2, ..., m}) that represent the respective utility consensus values. These utility consensus values represent the first set of unknowns of our aggregation problem and can be obtained by formulating a distance function model as a measure of disagreement between the utility consensus U^k and individual utility values U_j^k. This function can be formulated for a general metric p as follows (Cook et al, 1996):

Given the non-linear and non-differentiable character of function (1), its minimization is not an easy computational task. However, it has been demonstrated elsewhere how the optimization problem underlying (1) can be transformed into the following Archimedean GP problem (Romero, 1991; González-Pachón and Romero, 1999):

Achievement function:

where _kj and _kj are the negative and positive deviation variables that measure the under-achievements and over-achievements, respectively, from the utility values U_j^k derived from the kth observation of the training set. Different utility consensus values U^k are obtained for different values of metric p. Thus, for p=1, model (1) becomes a linear weighted GP model. For this particular metric, the structure of preferences underlying the GP model represents a solution for which the sum of individual disagreements is minimized (González-Pachón and Romero, 1999). Moreover, the utility consensus values attached to this solution are statistically defined by the median weight (Cook and Seiford, 1978). These utility consensus values can be problematic. In fact, let us assume that the number of decision-makers is odd and the preferences of one of the decision-makers are exactly in the middle of all the judgement preferences. In such a situation, the aggregate utility function U will coincide with the utility function of this particular decision-maker. Such a paradox is known in social choice literature as the 'median voter theorem' (Downs, 1957). In short, the intermediate points of view ('kingmaker' parties) are clearly favoured in this context for small values of metric p.

On the contrary, as the value of metric p increases, more importance is given to the utility function of the decision-maker (social group) with more displaced points of view with respect to the average preferences. Therefore, it is tempting to increase the value of p up to . This converts the GP model into a MINIMAX (Chebyshev) formulation, where the disagreement of the most displaced decision-maker (social group) is minimized. Thus, the following GP formulation is proposed (Ignizio and Cavalier, 1994; Romero, 2001):

Achievement function:

where variable D represents the disagreement of the decision-maker (social group) whose utility function provides the most significantly different judgement with respect to the utility consensus values U^k obtained.

Once the m utility consensus values (U^k, k=1, ..., m) have been obtained, the next phase in the proposed procedure will be to obtain the aggregate or group utility function U that better explains these m values. This is a statistical fitting problem that can be accomplished in the utility functions space. With this purpose in mind, let us consider the following general group multilinear utility function:

where U and the U_j are scaled from zero to one, and 0_j1 for every j.

Keeney and Kirkwood (1975) have proposed the above utility decomposition as a suitable cardinal social welfare function. It should be noted that the multilinear form involves the multiplicative and additive decomposition as special cases (see Keeney and Raiffa, 1993, pp 293–294).

Of the different procedures for implementing statistical fitting, the following l¹ regression procedure appears to be suited for minimizing the importance of the possible outliers. Thus, the minimization of the sum of the absolute deviation leads to the following linear weighted GP formulation (Charnes et al, 1955):

Achievement function:

By substituting the optimum values of parameters derived from model (5) in the aggregate utility function U given by (4), the procedure ends by providing the social utility function that surrogates the preferences of the n decision-makers or social groups as a whole.

An illustration

Let us illustrate the proposed aggregation procedure with the help of a simple example. A public forest is managed according to two attributes of the type 'more is better': x₁, timber production (m³/year) and x₂, recreational services (visits/year). The following production-transformation function explains the technological domain of the joint production timber/recreational services (see Romero, 1997 for details):

The following ideal and anti-ideal vectors are easily obtained from equation (6):

It is assumed that there are three social groups involved in the management of the forest. They have different perceptions or points of view with respect to the two attributes considered. The following three utility functions have been elicited and represent the preferences of each social group with respect to the attributes timber and recreational services, respectively:

It is easy to check that the three utility functions are bounded by 0 and 1, thus we have:

Table 1 shows the training set used for our exercise; that is, an arbitrary seven-point number has been derived from the transformation curve given by (6), as well as the respective values for the three individual utility functions.

Table 1 - Training set in the attribute and utility spaces.

Full table

Taking the data from Table 1, the MINIMAX Chebyshev GP model (3) is formulated as follows:

Achievement function:

The U^k utility consensus values obtained by solving model (7) were as follows:

This set of values represents a discrete approximation of the social utility function U. To get a continuous representation of U, the above consensus values, as well as the values achieved for the three individual utility functions from the training set (see Table 1), are introduced into an l¹ regression model for a group multilinear utility function with three decision-makers (see model (5)). Thus, the following linear GP formulation was obtained:

Achievement function:

The solution obtained by solving model (8) was as follows:

From the above optimum values of the scaling factors, the following final social/group utility function U was obtained:

The above social utility function U, as well as the three individual utility functions U₁, U₂, and U₃, were maximized over the feasible set given by the production-transformation function represented by Equation (6). The results are shown in the pay-off matrix presented in Table 2. It is interesting to note that utility functions U, U₂ and U₃ lead to very similar solutions in the utility space, as well as in the attribute space. It is not surprising to find that several utility functions lead to the same optimum. Indeed, Köksalan and Sagala (1995, pp 200–201) have shown how different utility functions often lead to the same optimum solution.

Table 2 - Pay-off matrix obtained when the social utility function U, as well as the three individual utility functions U₁, U₂ and U₃ are maximised over the feasible set.

Full table

Summary and conclusions

This paper proposes a non-interactive procedure for aggregating individual utility functions. In fact, the interactive process with the decision-maker is only necessary in order to obtain the individual utility functions, but after that no more interaction is required during the aggregation process. Hence, it does not seem to anticipate serious challenges in implementing the method in practice. The potential problems can derive from the degree of acceptation by the individual decision-makers of the aggregate consensus obtained. However, it is not bold to assume that the optimum utility consensus obtained could be employed as an initial point for initiating a negotiation process.

From a theoretical and practical point of view, the proposed method for aggregating individual utility functions appears to have some good properties like the following:

1. It is a method for working with evaluations based upon consensus, for which the utility values obtained can be straightforwardly interpreted. Thus, the consensus values obtained are compatible with the minimization of the disagreement of the most displaced decision-maker's judgement values with respect to the consensus obtained. Moreover, some implementation problems concerning 'median voter' responses are avoided by using the Chebyshev metric as the aggregation distance. This type of implementation problems could appear in a general l^p distance based framework. In this general framework, a kind of sensitivity analysis based on the p parameter value could be implemented. However, for computational reason, this would not be easy to do.
2. From a computational point of view, only two linear programming problems need to be solved along the process: one to obtain the utility consensus values and the other to obtain the constants (weights) of the aggregate utility function. Hence, the computational burden of the proposed procedure is very low.
3. The procedure has been applied to a general group multiattribute utility function (ie, a multilinear utility decomposition). Once the utility consensus values have been obtained, however, the next step in the procedure is just a statistical fitting problem. Hence, it can be easily applied to any type of mathematical decomposition of the aggregate utility function.

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Acknowledgements

A preliminary version of this paper was presented at the XVII MCDM Conference (Whistler, Canada, August, 2004). This research was funded by the Spanish 'Ministerio de Educación y Ciencia' under grant SEY2005-04392. Comments raised by a reviewer are highly appreciated. We would like to thank Mrs Rachel Elliott for editing the English.