Generalized Ultraproduct and Kirman-Sondermann Correspondence for Vote Abstention

The present paper refines Herzberg and Eckert’s model-theoretic approach to aggregation. The proposed refinement is aimed at naturally accounting for vote abstention, and is technically based on a more general notion of ultraproduct than the one standardly occurring in model theory textbooks. Unlike the standard ultraproduct construction, which yields the empty model as soon as any one single coordinate features the empty model, this generalized ultraproduct construction faithfully reflects the indication of ‘large sets’. Thus, our proposed refinement naturally accounts for those situations in which e.g. a voting round is non-null if and only if a ‘large set’ of voters actually participate in the vote. In the present setting, Arrow’s impossibility theorem also covers ‘elections with only two candidates’.