Scoring rule and majority agreements for large electorates with arbitrary preferences

Abstract

This paper examines elections among three candidates when the electorate is large and voters can have any of the 26 nontrivial asymmetric binary relations on the candidates as their preference relations. Comparisons are made between rule-λ rankings based on rank-order ballots and simple majorities based on the preference relations. The rule-λ ranking is the decreasing point total order obtained when 1, λ and 0 points are assigned to the candidates ranked first, second and third on each voter's ballot, with 0 ⩽ λ ⩽ 1. Limit probabilities as the number of voters gets large are computed for events such as ‘the first-ranked rule-λ candidate has a majority over the second-ranked rule-λ candidate’ and ‘the rule-λ winner is the Condorcet candidate, given that there is a Condorcet candidate’. The probabilities are expressed as functions of λ and the distribution of voters over types of preference relations. In general, they are maximized at λ = 1/2 (Borda) and minimized at λ = 0 (plurality) and at λ = 1 for any fixed distribution of voters over preference types. The effects of more indifference and increased intransitivity in voter's preference relations are analyzed when λ is fixed.