Distribution of coalitional power under probabilistic voting procedures

Pattanaik and Peleg (1986) imposed regularity, ex-post Pareto optimality and independence of irrelevant alternatives on a probabilistic voting procedure and showed that: (i) the distribution of coalitional power for decisiveness in two-alternative feasible sets is subadditive in general, but additive if the universal set has at least four alternatives; and (ii) the distribution of coalitional power in an arbitrary feasible set is almost complete random dictatorship, and becomes complete random dictatorship under certain additional conditions. This paper formulates the problem in terms of citizens’ sovereignty and monotonicity conditions (which are in line with Arrow’s original work) instead of ex-post Pareto optimality and proves that: (i) Pattanaik and Peleg’s coalitional weights for decisiveness in two-alternative feasible sets become additive even with only three alternatives in the universal set; and (ii) the distribution of coalitional power in an arbitrary feasible set is completely characterized by random dictatorship without the additional conditions of Pattanaik and Peleg.  2000 Elsevier Science B.V. All rights reserved.