Gillies and Miller’s subrelations of a relation over an infinite set of alternatives: General results and applications to voting games

In this paper, the authors have investigated two subrelations of a domination relation which is the classical collective relation in voting games. These two subrelations, due to Gillies and Miller, have some nice properties; in particular, they are transitive. From these subrelations one can define obvious solution concepts by taking their maximal elements. If the set of social states is infinite, however, the existence of maximal elements is far from obvious, due to the lack of continuity. The authors have assumed that the set of social states was a compact metric space. They have adopted a measure-theoretic analysis. In the case of Gillies’ subrelation, the authors have obtained very general existence theorems. In the case of Miller’s subrelation (the uncovered set), though the present theorem is still more general than what can be found in the literature, it is still far from being at the same level of generality.