The stability set of voting games: Classification and genericity results

In 1980 Rubinstein introduced a new solution concept for voting games called the stability set which incorporates the idea that before entering into a possibly winning coalition with respect to some pair of alternatives, a voter will consider what might happen in the future. He showed that if the voters' preferences are given by linear orders the stability set is non-empty for a large class of voting games with finite sets of alternatives. We consider the case where indifference is allowed (preferences are then complete preorders) and show that the picture is then quite different. First, in the finite case, we obtain classification results for the non-emptiness of the stability set which are based on Nakamura's number. When preferences are continuous, we prove a general non-emptiness theorem and show that the set of profiles for which the stability set is non-empty is dense in the set of profiles.We are most grateful to the French Commissariat Général du Plan for financial support and to Georges Bordes, Prasanta K. Pattanaik, Hervé Moulin, Bezalel Peleg, Ariel Rubinstein, Norman Schofield and two anonymous referees of this journal for helpful comments. In particular, Bezalel Peleg has called our attention to the similarities of the stability set with the Gillies set and the uncovered set. A previous draft of this paper has been presented at the conference on Economic Models and Distributive Justice in Brussels, at the European Meeting of the Econometric Society in Copenhagen and at seminars in Europe and the United States. We thank the participants for their remarks.