Daniel Eckert

A distance-based comparison of basic voting rules

In this paper we provide a comparison of different voting rules in a distance-based framework with the help of computer simulations. Taking into account the informational requirements to operate such voting rules and the outcomes of two well-known reference rules, we identify the Copeland rule as a good compromise between these two reference rules. It will be shown that the outcome of the Copeland rule is “close” to the outcomes of the reference rules, but it requires less informational input and has lower computational complexity.

The model-theoretic approach to aggregation: Impossibility results for finite and infinite electorates ☆

It is well known that the literature on judgement aggregation inherits the impossibility results from the aggregation of preferences that it generalises. This is due to the fact that the typical judgement aggregation problem induces an ultrafilter on the set of individuals. We propose a model-theoretic framework for the analysis of judgement aggregation and show that the conditions typically imposed on aggregators induce an ultrafilter on the set of individuals, thus establishing a generalised version of the Kirman–Sondermann correspondence. In the finite case, dictatorship then immediately follows from the principality of an ultrafilter on a finite set. This is not the case for an infinite set of individuals, where there exist free ultrafilters, as Fishburn already stressed in 1970. Following Lauwers and Van Liedekerke’s (1995) seminal paper, we investigate another source of impossibility results for free ultrafilters: the domain of an ultraproduct over a free ultrafilter extends the individual factor domains, such that the preservation of the truth value of some sentences by the aggregate model–if this is as usual to be restricted to the original domain–may again require the exclusion of free ultrafilters, leading to dictatorship once again.

Distance-Based Aggregation Theory

The problem of aggregating several objects into an object that represents them is a central problem in disciplines as diverse as economics, sociology, political science, statistics and biology (for a survey on aggregation theory in various fields see Day and McMorris [17]). It has been extensively dealt with in the theory of social choice (see Arrow et al. [5]), which analyses the aggregation of individual preferences into a collective preference. In this context, the idea of a consensus is normatively particularly appealing. A natural way to operationalize the consensus among a group of individuals is by means of a distance function that measures the disagreement between them. Thus, in particular, the construction of aggregation rules based on the minimization of distance functions inherits the normative appeal of consensus.

Anonymity, ordinal preference proximity and imposed social choices

Abstract. Extending on an impossibility result by Baigent [1], it is shown that an anonymous social choice procedure which preserves preference proximity cannot satisfy the weakest possible form of non-imposition.

A Geometric Approach to Paradoxes of Majority Voting: From Anscombe's Paradox to the Discursive Dilemma with Saari and Nurmi

Among many other topics, Hannu Nurmi has worked on voting paradoxes and how to deal with them. In his work he often uses a geometric approach developed by Don Saari for the analysis of paradoxes of preference aggregation such as the Condorcet paradox or Arrow’s general possibility theorem. In this chapter this approach is extended to other paradoxes analysed by Nurmi and the recent work in judgment aggregation. In particular we use Saari’s representation cubes to provide a geometric representation of profiles and majority outcomes.

A Geometric Approach to Paradoxes of Majority Voting in Abstract Aggregation Theory

In this paper we extend Saaris geometric approach to paradoxes of preference aggregation to the analysis of paradoxes of majority voting in a more general setting like Anscombes paradox and paradoxes of judgment aggregation. In particular we use Saaris representation cubes to provide a geometric representation of profiles and majority outcomes. Within this geometric framework, we show how profile decompositions can be used to derive restrictions on profiles that avoid the paradoxes of majority voting.

The Problem of Judgment Aggregation in the Framework of Boolean-Valued Models

A framework for boolean-valued judgment aggregation is described. The simple (im)possibility results in this paper highlight the role of the set of truth values and its algebraic structure. In particular, it is shown that central properties of aggregation rules can be formulated as homomorphy or order-preservation conditions on the mapping between the power-set algebra over the set of individuals and the algebra of truth values. This is further evidence that the problems in aggregation theory are driven by information loss, which in our framework is given by a coarsening of the algebra of truth values.