Minimizing the mean majority deficit : the second square-root rule

Let W be a composite (two-tier) simple voting game (SVG) consisting of a council, making yes/no decisions, whose members are delegates, each voting according to the majority view in his/her district. The council’s decision rule is an arbitrary SVG V. The mean majority deficit ∆[W] is the mean difference between the size of the majority camp among all citizens and the number of citizens who agree with the council’s decision. Minimizing ∆[W] is equivalent to maximizing the sum of the voting powers of all the citizens, as measured by the (absolute) Banzhaf index β'. We determine the V which minimize ∆[W]. We discuss the difference between majoritarianism and equalization of the voting powers of all citizens.