THE “BANZHAF FALLACY ” FALLACY

In a recent paper, Howard Margolis (1982) disputes the conclusion — implied by application of the Banzhaf (or Shapley-Shubik) power index to Presidential elections — that the Electoral College system, and its winner-take-all or unit-rule feature in particular, favors voters in larger states. More precisely, such analyses have shown that the power of individual voters under the Electoral College system (or similar systems) is approximately proportional to the square root of the population of the states in which they vote. Thus a voter in a state with a population of 4 million has--according to these indices--about twice the voting power of a voter in a state with a population of 1 million. Margolis labels this conclusion the “Banzhaf Fallacy ” on the grounds that it is derived from an assumption about individual voting behavior that implies a distribution of aggregate election outcomes wildly at variance from reality. He argues that if the assumption about individual voting behavior is modified in a reasonable way the “Banzhaf effect ” evaporates and that individual voting power is then uniform across states of different population. I review these issues in this note and conclude that the “Banzhaf Fallacy ” argument of Margolis is itself fallacious. What Margolis shows is uniform across states (under his more reasonable voting behavior assumption) is the probability that an individual voter will cast a critica