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.

Aggregation of binary evaluations: a Borda-like approach

“…It places the most able and the least able students together into S H which we take to be undesirable for ability grouping. 7 This example motivates the imposition of a natural monotonicity constraint to our partitioning problem. We will require that the lowest level of ability among the students in S H be at least as great as the greatest level of ability in S L .…”

“…Zwicker's technique allows us to decompose a standard preference profile into a portion that tends towards a majority cycle, and a portion that tends towards no cycle in majority preference. 10 He then shows that the Borda ranking at 7 We are grateful to Nicholas Houy for proposing the first example of this kind. 8 In a planned sequel to this paper (Social Dichotomy Functions) we show that a measure of external displacement, related to total tension, is also optimized by "cutting" at the mean in this way, with no need to impose a monotonicity condition.…”

Approximate Judgement Aggregation

“…We say that f satisfies the Pareto criterion if f (0) = 0 and f (1) = 1 18 . I.e., when all the individuals voted unanimously 0 then f should return 0 and similarly for the case of 1.…”

“…17 An equivalent definition is: ∀x, y : f (x)+f (y) = f (x+y) when the addition is in Z2 and Z n 2 , respectively. 18 In the literature this criterion is sometimes referred to as Unanimity, e.g., in [32]. We choose to follow [17] and refer to it as Pareto to distinguish between it and the unanimity function which is the oligarchy of {1, 2, .…”

Aggregation theory and the relevance of some issues to others

“…In other words, we merge people's judgments on propositions of the sort 'politician k is of quality v' for pairs of a politician k and a possible grade v. The last two examples are versions of the evaluation aggregation problem, in which we merge people's positions on some matters: people's estimates of variables, people's grades given to politicians, people's degrees of belief in some events, etc. (e.g., Rubinstein and Fishburn 1986, Dietrich and List 2010b, Dokow and Holzman 2010b.…”