The design of socially optimal decisions in a consensus scenario

Abstract

Several situations of conflict between basic social principles can crop up during a consensus searching process. The majority principle and respect for minority groups is a possible example of a conflictive situation between two social principles. In this paper, we outline a specific consensus searching scenario, where individual preferences are expressed by “pairwise” comparison matrices. The set of compromise consensuses between the majority and minority principles is determined using a procedure based upon an adaptation of Yu's p-metric distances. Finally, we use three different theoretical approaches – utility theory, p-metric distance functions and bargaining theory – to obtain the social optimum from the set of compromise consensuses. The links and differences among the three approaches are analysed. Finally, the working of the proposed theoretical framework is illustrated with the help of a forestry case study.

Introduction

The basic problem in the design of socially optimal decisions is how to aggregate individual preferences on mutually exclusive alternatives into a single collective preference. In other words, all group decision-making problems require a collective choice rule that possesses good properties and is able to aggregate individual preferences.

“Possessing good properties” implies very complex problems of aggregation. The degree of complexity heavily depends on how the different decision-makers state their individual preferences. It is a well-known fact that there is no aggregation procedure satisfying a set of sensible conditions for preferences that are stated ordinally, i.e. Arrow's famous impossibility theorem [1]. The complications attached to the aggregation of preferences are considerably attenuated, however, if the individual decision-makers express their preferences cardinally. Thus, Keeney [2] and Keeney and Kirkwood [3] demonstrated that when individual preferences are defined by cardinal utility functions, then there are aggregation rules (collective choice rules) that hold for the conditions or axioms imposed by Arrow.

The above results are important because individual preferences are, as a matter of fact, expressed in a cardinal fashion in many real scenarios. In finance, a recent application is Avkiran and Morita [4]. On the other hand, the broad field of natural resources management is a good example of decision-making problems involving several stakeholders that express their individual preferences towards several criteria according to different types of cardinality. In many cases, cardinality is expressed in a “pairwise” comparison format (e.g., [5], [6], [7]).

Other problems tend to crop up, though, even after the cardinal information has been aggregated according to collective choice rules possessing good properties. In fact, in some such applications, a conflict develops between the point of view of the majority and the point of view of the individual or social group with more displaced views with respect to the majority consensus. In short, the “Majority Principle” and the “Minority Principle” clash.

González-Pachón and Romero [8], [9] adapted Yu's p-metric distance function framework (1973, 1985) to propose a method for cardinally aggregating given individual preferences through utility functions and through “pairwise” comparison matrices, respectively. Within this framework, the p-metric is interpreted as a compensatory parameter between the “best social optimum” from the point of view of the majority, which is equivalent to metric p=1, and the “best social optimum” from the point of view of the minority, which is equivalent to metric p=∞.

The main purpose of this paper is to propose a theoretical framework for formalising the conflict that sometimes materializes between two desirable social principles: majority and minority principles. It should be noted that the aim of this research differs from other research concerned with supporting consensus formation using indicators (or similar measures) based on the idea of agreement and disagreement. Key works in this alternative direction are Bryson [10] and Mobolurin and Bryson [11] for the ordinal preferences scenario, and Bryson [12] for the cardinal preferences scenario.

We will show how the formalisation process leads initially to a non-computable bi-objective programming problem. However, it is shown how this type of problem can be straightforwardly transformed into a goal programming (GP) formulation, requiring the solution of simple linear programming (LP) problems only. In this way, the parametric set of social compromise consensuses between the above two basic principles is easily reached. Finally, we apply three approaches – utility optimisation, p-metric distance optimisation and bargaining theory – on this parametric set to search the social optimum or “best solution” for the group as a whole. The links and differences among the three approaches are analysed.

To do this, we state in this paper a particular scenario for searching the compromise consensus: individual preferences are cardinally expressed by means of “pairwise” comparisons. Note that the information contained in a “pairwise” comparison matrix is local; i.e., a possible global ranking of alternatives is implicit, but it is not explicit. This differs from some other works where the information about individual preferences is global; i.e., a global ranking of the alternatives is the explicit information, see Mobolurin and Bryson [11].

In this auxiliary scenario, intensity of preferences is quantified on a ratio scale. Note that decision makers in this scenario could be irrational. However, our analysis could incorporate rationality conditions using results published in González-Pachón and Romero [13], [14].

In short, the proposed procedure encompasses two phases. In the first phase, we obtain the whole set of compromise consensuses between the majority and the minority principles (the λ-consensus set). In the second phase, the social or collective optimum point is determined from the λ-consensus set according to several well-known theoretical frameworks.

The paper is organized as follows. In Section 2, the opposition between the majority and minority principles is modelled in the context of consensus between “pairwise” comparison matrices. Section 3 focuses on the determination of the social or group optimum with the help of three different analytical frameworks: utility theory, p-metric distance functions and bargaining theory. The above ideas are illustrated with the help of a case study in Section 4. Finally, Section 5 reviews the main conclusions derived from this research.