Coincidence probabilities for simple majority and proportional lottery rules

Abstract

Proportional lottery rules select the winner in a pairwise election by a random process, in which the probability that an alternative is selected is proportional to the number of votes that it receives in the election. This study develops a closed form representation for the expected likelihood that the proportional lottery rule selects the same winner as simple majority rule for a two candidate election. When three candidate elections are considered, pairwise comparisons are made on all pairs of alternatives to determine the winner. Closed form representations are developed for the expected likelihood that the proportional lottery rule has the same winner, and the same complete linear ranking on alternatives, as simple majority rule, given that there is a simple majority rule winner.