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

• Published in: Mathematical social sciences 2012-07, Vol.64 (1), p.41-47
• Main Authors: Herzberg, Frederik, Eckert, Daniel
• Format: Article
• Language: English
• Subjects: Dictatorship; Electorate; Judgement; Mathematical analysis; Political power
• ISSN: 0165-4896; 1879-3118
• DOI: 10.1016/j.mathsocsci.2011.08.004

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. ► We use model theory for the analysis of abstract aggregation problems. ► The ultraproduct construction corresponds to Arrovian aggregation rules. ► In the finite case, dictatorship follows from the principality of the ultrafilter. ► In the infinite case, another source of impossibility is studied.