Weighted Voting Systems - W.H. Freeman

394 PART III Voting and Social ChoiceSPOTLIGHT11.2Power IndicesThe first widely accepted numerical index forassessing power in voting systems was theShapley–Shubik power index, developed in1954 by a mathematician, Lloyd S. Shapley,and an economist, Martin Shubik. A particularvoter’s power as measured by this index isproportional to the number of differentpermutations (or orderings) of the voters inwhich he or she has the potential to cast thepivotal vote—the vote that first turns fromlosing to winning.The Banzhaf power index was introducedin 1965 by John F. Banzhaf III, a law professorwho is also well-known as the founder of theantismoking organization ASH (Action onSmoking and Health). The Banzhaf index is theone most often cited in court rulings, perhapsbecause Banzhaf brought several cases to courtand continues to file amicus curiae briefs whencourts evaluate weighted voting systems. Avoter’s Banzhaf index is the number of differentpossible voting combinations in which he orshe casts a swing vote—a vote in favor of amotion that is necessary for the motion to pass,or a vote against a motion that is essential forits defeat.Lloyd S. Shapley John F. Banzhaf III Martin ShubikThe voters in the system [6 : 5, 3, 1] are not equally powerful—the weight-5voter has veto power and the other two don’t—and yet none of the voters aredummies. We can’t compare power by comparing the voting weights because theweight-3 voter has the same voting power as the weight-1 voter. Together, theycan stop the weight-5 voter from passing a motion, and either one can combinewith the weight-5 voter to pass a motion. A power index gives a way to measurethe share of power that each participant in a voting system (weighted or otherwise)has. Spotlight 11.2 is a brief history of power indices.11.1 The Shapley–Shubik Power IndexWhen an election looms, politicians focus on “moderate voters.” These are peoplewho could be convinced to favor one side or the other. Moderate voters can