A power index in weighted voting systems (Q1395033)

To a finite number of players are given integral weights and a measure passes if the total weight of votes meets or exceeds an integral, the majority quota. The authors begin with the observation that while the set of minimal winning coalitions is determined by the set of weights and the quota, the converse is not true. The authors define an equivalence relation on weighted voting systems, called structural isomorphism, where two weighted voting systems are structurally isomorphic if they have the same set of minimal winning coalitions (up to suitable identification of the players and ignoring dummies). Two players are considered equal with respect to the set of minimal winning coalitions if one may be substituted for the other without affecting whether the coalitions are winning. The authors replace a weighted voting system with a structurally isomorphic by one where equal players have equal weights. Next, the authors apply a minimization procedure to allocate weights to collections of unequal players. The resulting set of weights can be used as a power index, up to a scalar multiple if normalization is desired. This result is not true in general for the widely-used Banzhaf and Shapley-Shubik power indices.