A probabilistic synopsis of binary decision rules

Decision rules for Yes–No voting systems are placed in a probabilistic framework. Selfdual and permutationally invariant distributions are introduced. Under such distributions, the mean success margin of the majority rule and of the unanimity rule are shown to bound the mean success margin of all other decision rules. For bloc decision rules in the Penrose/Banzhaf model, a product formula for the voters’ influence probabilities is derived. Other indices and the Shapley/Shubik model are also discussed.