Distribution of coalitional power under probabilistic voting procedures (Q1575092)

This paper represents an interesting extension of the results in an important earlier paper by \textit{P. K. Pattanaik} and \textit{B. Peleg} on the power structure determined by probabilistic voting procedures (PVPs) [Econometrica 54, 909-921 (1986; Zbl 0596.90009)]. In this paper, the author also considers PVPs with linear individual preference orderings, but satisfying different axioms. Pattanaik and Peleg required that PVPs satisfy axioms of independence of irrelevant alternatives (IIA), ex-post Pareto optimality and regularity. They showed that the coalitional power structure determined by a PVP satisfying the three axioms is characterized be a weighted random dictatorship, subject to the additional conditions that the universal set contain at least four elements and also that if the feasible set is the universal set, then it must contain at least two more alternatives than the number of voters. In the paper under review, the author retains the regularity and IIA axioms, but replaces ex-post Pareto optimality by axioms of citizens' sovereignity and strong monotonicity. The author shows that a PVP satisfying these axioms is completely characterized by a random dictatorship so long as the universal set contains at least three elements. The restriction of Pattanaik and Peleg on the case when the feasible set is the universal set is removed. Additionally, the author shows that the result still holds if IIA is replaced by a binary IIA condition and strong monotonicity is replaced by a complete strong monotonicity axiom.