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.
Highlights
In November, 2018, as the initial deadline for the United Kingdom’s departure from the European Union (“Brexit”) approached, YouGov conducted a survey in which over 20,000 British respondents were asked to rank three alternatives: remain in the EU (Remain), accept the withdrawal agreement promoted by Prime Minister1 3 Vol.:(0123456789) A
22 The system we describe is referred to as ranked-choice voting (RCV) in the US, the alternative vote system (AV) in the UK, preferential voting in Australia, single-winner STV in Ireland, and occasionally the Hare system
If there are three candidates a, b, and c who would be the last candidates standing in instant-runoff voting (IRV) or a similar elimination procedure, we can eliminate all other candidates from the ballots and use the diagram to represent the final rounds of the competition
Summary
In November, 2018, as the initial deadline for the United Kingdom’s departure from the European Union (“Brexit”) approached, YouGov conducted a survey in which over 20,000 British respondents were asked to rank three alternatives: remain in the EU (Remain), accept the withdrawal agreement promoted by Prime Minister. The fundamental reason why we need such a diagram is that in systems where voters’ second choices matter (as in e.g. Borda count or instant-runoff) the election outcome depends on more numbers than can be comfortably digested in a table or visualized in a straightforward way. This paper introduces a consistent approach to diagramming the result of any anonymous single-winner election method in which voters submit ranked preferences (i.e. ordinal voting systems). The diagram emphasizes how the winner in a given system depends on the row sums of Table 1 as a function of the column proportions within each row This approach accommodates incomplete rankings, which pose a problem for some previous graphical approaches, and it can be useful in IRV and Condorcet elections with more than three alternatives. Because the diagram summarizes five preference parameters in two dimensions, it can be viewed not just as a means of analyzing election methods and as a concise summary of preference profiles
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