The structure of coalitional power under probabilistic voting procedures

We consider probabilistic voting procedures which map each feasible set of alternatives and each utility profile to a social choice lottery over the feasible set. It is shown that if we impose: (i) a probabilistic collective rationality condition known as regularity; (ii) probabilistic counterpart of Arrow's independence of irrelevant alternatives and citizens' sovereignty; (iii) a probabilistic positive association condition called monotonicity; then the coalitional power structure under a probabilistic voting procedure is characterized by weak random dictatorship.