Plurality criterion

Plurality criterion is a voting system criterion devised by Douglas R. Woodall for ranked voting methods with incomplete ballots. It is stated as follows:

If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is given any preference, then A's probability of winning must be no less than B's.

This criterion is trivially satisfied by rank ballot methods which require voters to strictly rank all the candidates (and so do not allow truncation). The Borda count is usually defined in this way.

Woodall has called the Plurality criterion "a rather weak property that surely must hold in any real election" opining that "every reasonable electoral system seems to satisfy it." Most proposed methods do satisfy it, including Plurality voting, IRV, Bucklin voting, and approval voting.

Among Condorcet methods which permit truncation, whether the Plurality criterion is satisfied depends often on the measure of defeat strength. When winning votes is used as the measure of defeat strength in methods such as the Schulze method, Ranked Pairs, or Minimax, Plurality is satisfied. Plurality is failed when margins is used. Minimax using pairwise opposition also fails Plurality.

When truncation is permitted under Borda count, Plurality is satisfied when no points are scored to truncated candidates, and ranked candidates receive no fewer votes than if the truncated candidates had been ranked. If truncated candidates are instead scored the average number of points that would have been awarded to those candidates had they been strictly ranked, or if Nauru's modified Borda count is used, the Plurality criterion is failed.

References

D R Woodall, "Properties of Preferential Election Rules", Voting matters, Issue 3 (1994), pp. 8–15.
D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6 (1996), pp. 9–12.

Voting systems

Part of the Politics and Election series

Single-winner voting system	Approval voting Borda count Bucklin voting Contingent vote Coombs' method Copeland's method Exhaustive ballot First-past-the-post voting Instant-runoff voting Kemeny–Young method Simple majoritarianism Minimax Condorcet Nanson's method Plurality Positional voting system Range voting Ranked pairs Schulze method Two-round system

Proportional representation	Mixed-member Party-list Single transferable vote Schulze STV CPO-STV Highest averages method Sainte-Laguë D'Hondt Largest remainder method Alternative vote Plus Closed list Open list Overhang seat Underhang seat

Semi-proportional representation	Parallel voting Single non-transferable vote Cumulative voting Limited voting Proportional approval voting Sequential proportional approval voting Satisfaction approval voting

Usage	Table of voting systems by country

Voting system criterion	Condorcet criterion Condorcet loser criterion Consistency criterion Independence of clones Independence of irrelevant alternatives Independence of Smith-dominated alternatives Later-no-harm criterion Majority criterion Majority loser criterion Monotonicity criterion Mutual majority criterion Pareto efficiency Participation criterion Plurality criterion Resolvability criterion Reversal symmetry Smith criterion

Voting system quotas	Droop quota Hagenbach-Bischoff quota Hare quota Imperiali quota

Other	Ballot Election threshold First-preference votes Spoilt vote Sortition

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