How Often Are You Decisive: an Enquiry About the Pivotality of Voting Rules

The probability that an individual is decisive in an election is an important criterion for the evaluation of voting rules. It depends on factors such as the number and the behaviour of the other voters, the available alternatives and, of course, also on the voting rule itself. Classical power indices like the Banzhafor the Shapley-Shubik-Index are only applicable in special cases. In this paper, an approach is proposed that can be viewed as a natural extension of the Banzhaf-Index for more than two alternatives and for different stochastic assumptions. The approach is applied to plurality voting with two and more alternatives and computed for variations of the number of voters and alternatives. For three alternatives the pivotality is also computed for the Borda-Rule. The comparison of the computations for three alternatives shows that the probability of being decisive under the Borda-Rule is uniformly larger than under plurality voting.