In this paper we use rank information to construct a measure of preference satisfaction by collective decision rules and to compare the Borda rule with the plurality rule with respect to the inequality their outcome induces. While the construction of our measure of preference satisfaction is formally equivalent to Borda utilities, it turns out that, with respect to equality, the Borda rule is not necessarily superior to plurality rule for the case of more than three alternatives.

Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal result to key developments and concepts in the history of model theory, notably ultraproducts and preservation results.

Following the recent generalization of social choice in the literature on judgment aggregation, we extend the analysis of freedom of choice from sets of alternatives to sets of opinions. We establish the analogue of the cardinality based freedom of choice measure and suggest an alternative measure based on the Hamming distance.

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.

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.

Following Lauwers and Van Liedekerke (1995), this paper explores in a model-theoretic framework the relation between Arrovian aggregation rules and ultraproducts, in order to investigate a source of impossibility results for the case of an infinite number of individuals and an aggregation rule based on a free ultrafilter of decisive coalitions.

Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences which is closely related to the latter’s foundational contribution to model theory. Thus, Arrow’s celebrated impossibility theorem can be read as a model-theoretic preservation result "avant la lettre": The preservation of the …rst-order properties of weak orders under product formation is only possible in the case of ultraproducts, i.e. if the product is reduced over an ultra…lter on the index set of individuals. This perspective does not only shed new light on a well known source of dictatorship results in the …nite case, but allows to conceive of the problem of aggregation in general as a model-theoretic preservation problem.

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.

Ziel der vorliegenden Studie ist es, mithilfe einer explorativen Datenanalyse von OECD-Bildungsindikatoren (OECD 2005) nationale Bildungssysteme zu klassifizie­ren. In die vorliegende Analyse wurden neben den EU-15 Ländern zusätzlich jene europäischen Länder aufgenommen, für die die ausgewählten Bildungsindi­katoren vollständig verfügbar waren: Norwegen, Polen, Schweiz, Slowakische Republik, Tschechische Republik und Ungarn. Mithilfe des faktorenanalytischen Verfahrens wurden die vierzehn ausgewählten OECD-Bildungsindikatoren auf die drei Dimen­sionen (Gesellschaftliche Wertigkeit‚ Bildungsinvestitionen pro BildungsteilnehmerIn, Allgemeinen Bildungsstand) reduziert, und die Positionen der nationalen Bildungs­systeme im Raum dieser drei Faktoren berechnet und grafisch dargestellt. Für eine Klassifizierung der Länder wurde eine Clusteranalyse durchgeführt, mittels derer die nationalen Bildungssysteme zu sieben Ländergruppen verschmolzen wurden. 22.01.2007 | Erna Nairz-Wirth & Daniel Eckert (Wien)

The trade-off between equity and efficiency is analyzed in a geometric framework for the problem of committee selection, which has recently attracted interest in the social choice literature. It is shown that this trade-off can be maximal in the precise sense of the antipodality of the outcomes corresponding to the rules implementing the two normative principles. Following an approach in location theory, the minimization of the convex combination of the two criteria is presented as a compromise solution.

In a paper published in 1952, shortly after publication of Arrow's celebrated impossibility result, the French mathematicien Georges-Theodule Guilbaud has obtained a dictatorship result for the logical problem of aggregation, thus anticipating the literature on abstract aggregation theory and judgment aggregation. We reconstruct the proof of Guilbaud's theorem, which is also of technical interest, because it can be seen as the first use of ultrafilters in social choice theory.

Eliaz (2004) has established a for preference aggregation which implies both Arrow's Theorem (1963) and the Gibbard-Satterthwaite Theorem (1973, 1975). This theorem shows that the driving force behind impossibility theorems in preference aggregation is the mutual exclusiveness of Pareto optimality, individual responsiveness (preference reversal) and non-dictatorship. Recent work on judgment aggregation has obtained important generalizations of both Arrow's Theorem (List and Pettit 2003, Dietrich and List 2007a) and the Gibbard-Satterthwaite Theorem (Dietrich and List 2007b). One might ask, therefore, whether the impossibility results in judgment aggregation can be unified into a single theorem, a meta-theorem which entails the judgment-aggregation analogues of both Arrow's Theorem and the Gibbard-Satterthwaite Theorem. For this purpose, we study strong monotonicity properties (among them non-manipulability) and their mutual logical dependences. It turns out that all of these monotonicity concepts are equivalent for independent judgment aggregators, and the strongest monotonicity concept, individual responsiveness, implies independence. We prove the following meta-theorem: Every systematic non-trivial judgment aggregator is oligarchic in general and even dictatorial if the collective judgment set is complete. However, systematicity is equivalent to independence for blocked agendas. Hence, as a corollary, we obtain that every independent (in particular, every individually responsive) non-trivial judgment aggregator is oligarchic. This result is a mild generalization of a similar theorem of Dietrich and List (2008), obtained by very different methods. Whilst Eliaz (2004) and Dietrich and List (2008) use sophisticated combinatorial and logical arguments to prove their results, we utilize the filter method (cf. e.g. Dietrich and Mongin, unpublished) and obtain a much simpler and more intuitive derivation of our meta-theorem.

We show that systematic (complete) judgment aggregators can be viewed as both (2-valued) Boolean homomorphisms and as syntatic versions of reduced (ultra)products. Thereby, Arrovian judgment aggregators link the Boolean algebraic structures of (i) the set of coalitions (ii) the agenda, and (iii) the set of truth values of collective judgments. Since filters arise naturally in the context of Boolean algebras, our findings provide an explanation for the extraordinary effectiveness of the filter method in abstract aggregation theory.

We show how ultrafilters can be used to prove a central impossibility result in judgement aggregation introduced by Nehring and Puppe (2005), namely that for a logically strongly interconnected agenda, an independent and monotonic judgement aggregation rule which satisfies universal domain, collective rationality and sovereignty is necessarily dictatorial.

In a paper published in 1952, the French matematician Georges-Theodule Guilbaud has generalized Arrow's impossibility result to the logical problem of aggregation, thus anticipating the literature on abstract aggregation theory and judgment aggregation. We reconstruct the proof of Guilbaud's theorem, which is also of technical interest, because it can be seen as the first use of ultrafilters in social choice theory

In this paper we extend Saari's geometric approach to paradoxes of preference aggregation to the analysis of paradoxes of majority voting in a more general setting like Anscombe's paradox and paradoxes of judgment aggregation. In particular we use Saari's 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.