An analysis of simple two-stage voting systems

Abstract

This paper examines two-stage voting procedures [x,k,y] at the level either of a local governmental organization or of a national governmental system in which x is the number of candidates a voter is to vote for on the first ballot, k is the number of candidates to be placed on the second ballot according to the greatest vote totals from the first ballot, and y is the number of the k candidates on the second ballot that a voter is to vote for on the second ballot. The winner is the candidate with the most votes on the second ballot. These systems are practical in many situations and correspond to many two-stage voting systems in current use. The main purpose of the paper is to assess the propensities of alternative two-stage systems to elect the candidate which would be elected by a theoretically more sophisticated but less practical norm procedure such as simple majority of the Borda sum-of-ranks method. Extensive computer simulation, restricted to three to ten candidates but allowing a variety of methods by which voter preferences are randomly generated, reveals that the best propensity maximizing two-stage system has the form [x,2,1] in which exactly two candidates appear on the second ballot. Under the restriction that a voter is to vote for only one candidate on each ballot, the best system is common double plurality [1,2,1]. Without this restriction the best system usually has x = 2, with x approximately equal to the optimal number of candidates to vote for on the single ballot in a one-stage election in which the winner is determined by the greatest number of votes from the single ballot.