Optimization of Public Decisions: New Results in the Theory of Aggregation

Abstract

The theory of aggregation of consumer preferences is at the heart of the discussion of public decision-making and our ultimate goal, as we stated at the beginning of this study, has been to offer several aggregation procedures that can be used to reach collective decisions. If we look back for a moment it appears that our search for a global optimum was successful in the sense that we were able to characterize uniquely the subset of Pareto-optimal states for a public economy, as we have shown in Chapter III above. At the time, however, we remarked that more sophisticated methods were called for to isolate a single point among all Pareto-optimal states. The idea of using a pattern recognition approach had then to be refined and specialized to make this problem tractable. For this reason we limited ourselves to a finite alternative set A — which can be viewed as the set of Pareto-optimal states since we now have a way of distinguishing them from all feasible states. We also limited ourselves to binary preference patterns although we noted that a straightforward generalization allowing for various preference strengths was readily available. Our discussion of the algebraic foundations of aggregation theory has provided us with a set of very effective tools to be used in devising aggregation procedures. Generally speaking these procedures can rely on any one of the many structures we used to describe consumer preference patterns e.g. the finite, symmetric lattice structure of the set of transitive patterns, the convex polyhedral structure of the set of tournament matrices, or the group structure of the set of permutations of all the alternatives A.