2. Background. In three articles that appeared in American law journals in the mid-1960's, a lawyer named John Banzhaf III proposed to evaluate representation systems in terms of the extent to which they allocated fairly [1], [2], [3]. Banzhaf's analysis makes use of game-theoretic notions in which power is equated with the ability to affect outcomes. Consider a group of citizens choosing between two opposing candidates. To calculate the power of the individual voter, we generate the set of all possible voting coalitions among the district's electorate. If there are N voters in the district, then there will be 2N possible coalitions. Then we ask, for each of these possible coalitions, whether a change in an individual voter's choice from candidate A to candidate B (or from candidate B to candidate A) would alter the electoral outcome. If so, that voter's ballot is said to be decisive. The (absolute) Banzhaf index of a voter's power is defined as the number of the voter's decisive votes divided by 2X. The higher the percentage of voter coalitions in which a voter's vote is decisive, the higher that voter's power score. The Banzhaf index has considerable intuitive appeal; power is based on ability to affect outcome. For single-member district systems (smds) whose districts are of equal population, all voters have identical power. But what about the case of multiple-member district systems (mmds), with districts of more than one size? Here, since the voters who elect k representatives have k times as much impact as voters who can elect only one representative, we might think that to equalize voter power we should assign to each district a number of representatives proportional to the size of the district's population since, intuitively, we would expect a voter's ability to decisively affect outcomes should be inversely proportional to district population. Banzhaf [2] pointed out that this argument is mathematically incorrect. In a two-candidate/party contest where all voters have equal weight, in order for a voter to be decisive in a district of size N the rest of the voters (who are N- 1 in number) must split half for one candidate/party and half against. A straightforward combinatoric analysis reveals ([2], [7], [6]; Whitcomb v. Chavis (1970) 403 U.S. at 145 n. 23) that, if all combinations of vote outcomes are equally likely (i.e., if each voter is equally likely to vote for either candidate/party), then the number of each member's decisive votes, b, is given by:
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