Algebraic Foundations of the Theory of Aggregation

Abstract

Our discussion of social decision rules in the preceding chapter has been carried out at a rather high level of generality. Unfortunately the price we must pay for it should not be understated. In effect we have been able to assign a meaning to the intuitive idea of clusters of consumer preference patterns and this has led us to characterize a certain subset of the feature space, viz. the set of Pareto-optimal patterns. But as we said when we defined the notion of the feature space F _N the actual interpretation of each of its N dimensions was left voluntarily vague by saying that they could represent any public issue. For the sake of generality we then assumed that each issue was measurable in some appropriate unit and we further took as our feature space F _N the N-dimensional Euclidean space ℝ^N. As a first approximation this approach has proved fruitful but it is still far removed from the possibility of applications. Consequently we would now like to extend our previous discussion in a way that would provide some actual tools for public decision-making. This discussion, however, will still be cast in the general framework of pattern recognition, as stated earlier. The main difference lies in the fact that we shall restrict our concepts of pattern and feature space to a more limited but also more applicable case. Generally speaking we are still concerned with the problem of representing, grouping and aggregating the preferences of a set of consumers (citizens) over a set of public issues, the alternative set A=a ₁, a ₂,… a _m , where m is an arbitrary but finite integer. More specifically the N dimensions of our feature space F _N can now be thought of individually as binary comparisons of the form a _i versus a _k , so that there will be exactly C ² _m such comparisons i.e. N = m(m−1)/2.