A power analysis of linear games with consensus

Abstract

Linear simple games with consensus are considered. These games are obtained by intersecting a linear simple game (a game where the desirability order for players is total) and a symmetric ( k-out-of- n) game. We investigate the behavior of the Shapley–Shubik power index when passing from a linear game with consensus level q 1 to the same game with consensus level q 2> q 1. We also introduce a “range notion”, which intuitively represents the egalitarianism of the Shapley–Shubik index, obtain an upper bound on this measure, and characterize when it is achieved. Finally, the theory developed here is illustrated with several real-world voting systems.