The Probability of Being Decisive

The probability of a voter being decisive (P_D), that is, of one vote affecting the outcome of an election, has generally been incorrectly calculated for the last twenty or more years. The method normally used is due to Banzhaf (1968) and generalised by Beck (1974). It assumes that voters know in advance how many people will vote for each candidate, which is clearly not the case. The correct formulation was given by Good and Mayer in 1975, but was ignored and has subsequently been all but forgotten since then. A simple explanation of these methods is given. Using the incorrect method, errors of magnitude of more than 10¹⁰⁰ in calculating P_D correctly can be made. The appropriateness of using a decision-theoretic formulation instead of a game-theoretic one is also discussed.