A diagram for analyzing ordinal voting systems

Abstract

I introduce a diagram for describing and analyzing single-winner elections in which voters rank the candidates—a class of voting systems including positional methods (e.g. plurality, Borda count, anti-plurality), Condorcet methods, and instant-runoff voting (i.e. ranked-choice voting or the alternative vote). The diagram shows how the outcome of an election depends on each candidate’s share of top rankings as a function of the voting system and the pattern of lower rankings. Using as examples two Brexit polls, a mayoral election in San Francisco, and the US’s first instant-runoff congressional election (all since 2018), I show how the diagram can concisely present preferences and results under different voting systems, identify Condorcet cycles, highlight system properties such as join-inconsistency and the no-show paradox, and illuminate strategic voting incentives.