Aggregation of non-binary evaluations

Abstract

We study an aggregation problem in which a society has to determine its position on each of several issues, based on the positions of the members of the society on those issues. There is a prescribed set of feasible evaluations, i.e., permissible combinations of positions on the issues. The binary case of this problem, where only two positions are allowed on each issue, is by now quite well understood. We consider arbitrary sets of conceivable positions on each issue. This general framework admits the modeling of aggregation of various types of evaluations, including: assignments of candidates to jobs, choice functions from sets of alternatives, judgments in many-valued logic, probability estimates for events, etc. We require that the aggregation be performed issue-by-issue, and that the social position on each issue be supported by at least one member of the society. The set of feasible evaluations is called an impossibility domain if these requirements are satisfied for it only by dictatorial aggregation; that is to say, if it gives rise to an analogue of Arrow's impossibility theorem for preference aggregation. We obtain a two-part sufficient condition for an impossibility domain, and show that the major part is a necessary condition. For the ternary case, where three positions are allowed on each issue, we get a full characterization of impossibility domains.