Ranked voting

Ranked voting is any voting system that uses voters' orderings (rankings) of candidates to choose a single winner. For example, Dowdall's method assigns $1, 1/2, 1/3$... points to the 1st, 2nd, 3rd... candidates on each ballot, then elects the candidate with the most points. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives each one very different properties.

Ranked voting systems are usually contrasted with rated voting methods, which allow voters to indicate how strongly they support different candidates (e.g. on a scale from 0-10). Rated voting systems use more information than ordinal ballots; as a result, they are not subject to many of the problems with ranked voting (including results like Arrow's theorem).

Although not usually described as such, the most common ranked voting system is the well-known plurality rule, where each voter gives a single point to the candidate ranked first and zero points to all others. The most common non-degenerate ranked voting rule is the closely-related alternative vote, a staged variant of the plurality system that repeatedly eliminates last-place plurality winners.

In the United States and Australia, the terms ranked-choice voting and preferential voting are usually used to refer to the alternative or single transferable vote by way of conflation. However, terms these have also been used to refer to ranked voting systems in general.

History of ranked voting
The earliest known proposals for a ranked voting system other than plurality can be traced to the works of Ramon Llull in the late 13th century, who developed what would later be known as Copeland's method. Copeland's method was devised by Ramon Llull in his 1299 treatise Ars Electionis, which was discussed by Nicholas of Cusa in the fifteenth century.

A second wave of analysis began when Jean-Charles de Borda published a paper in 1781, advocating for the Borda count, which he called the "order of merit". This methodology drew criticism from the Marquis de Condorcet, who developed his own methods after arguing Borda's approach did not accurately reflect group preferences, because it was vulnerable to spoiler effects and did not always elect the majority-preferred candidate.

Interest in ranked voting continued throughout the 19th century. Danish pioneer Carl Andræ formulated the single transferable vote (STV), which was adopted by his native Denmark in 1855. Condorcet had previously considered the single-winner version of it, the instant-runoff system, but immediately rejected it as pathological.

Theoretical exploration of electoral processes was revived by a 1948 paper from Duncan Black and Kenneth Arrow's investigations into social choice theory, a branch of welfare economics that extends rational choice to include community decision-making processes.

Adoption
Plurality voting is the most common voting system, and has been in widespread use since the earliest democracies.

The single transferable vote (STV) system was first invented by Carl Andræ in Denmark, where it was used briefly before being abandoned. It was later rediscovered by British lawyer Thomas Hare, whose writings soon spread the method throughout the British Empire. Tasmania adopted the method in the 1890s, with broader adoption throughout Australia beginning in the 1910s and 1920s. It has been adopted in Ireland, South Africa, Malta, and approximately 20 cities each in the United States and Canada.

In more recent years, STV has seen a comeback the United States. In November 2016, the voters of Maine narrowly passed Question 5, approving ranked-choice voting for all elections. This was first put to use in 2018, marking the inaugural use of a ranked choice voting system in a statewide election in the United States. Later, in November 2020, Alaska voters passed Measure 2, bringing ranked choice voting into effect from 2022. However, as before, the system has faced strong opposition. After a series of electoral pathologies in Alaska's 2022 congressional special election, a poll found 54% of Alaskans supported a repeal of the system; this included a third of the voters who had supported Peltola, the ultimate winner in the election.

In the United States, single-winner ranked voting is used to elect politicians in Maine and Alaska. Nauru uses a positional method called the Dowdall system. Some local elections in New Zealand use the single transferable vote.

Majority-rule
Many concepts formulated by the Marquis de Condorcet in the 18th century continue to significantly impact the field. One of these concepts is the Condorcet winner, the candidate preferred over all others by a majority of voters. A voting system that always elects this candidate is called a Condorcet method.

However, it is possible for an election to have no Condorcet winner, a situation called a Condorcet cycle: suppose an election with 3 candidates A, B, and C has 3 voters. One votes A–B–C, one votes B–C–A, and one votes B–C–A. In this case, no Condorcet winner exists: A cannot be a Condorcet winner as two-thirds of voters prefer B over A. Similarly, B cannot be the winner as two-thirds prefer C over B, and C cannot win as two-thirds prefer A over C. This forms a rock-paper-scissors style cycle with no Condorcet winner.

Social well-being
Voting systems can also be judged on their ability to deliver results that maximize the overall well-being of society, i.e. to choose the best candidate for society as a whole.

Spatial voting models
Spatial voting models, initially proposed by Duncan Black and further developed by Anthony Downs, provide a theoretical framework for understanding electoral behavior. In these models, each voter and candidate is positioned within an ideological space that can span multiple dimensions. It is assumed that voters tend to favor candidates who closely align with their ideological position over those more distant. A political spectrum is an example of a one-dimensional spatial model. The accompanying diagram presents a simple one-dimensional spatial model, illustrating the voting methods discussed in subsequent sections of this article. It is assumed that supporters of candidate A cast their votes in the order of A-B-C, while candidate C's supporters vote in the sequence of C-B-A. Supporters of candidate B are equally divided between listing A or C as their second preference. From the data in the accompanying table, if there are 100 voters, the distribution of ballots will reflect the positioning of voters and candidates along the ideological spectrum.

Spatial models offer significant insights because they provide an intuitive visualization of voter preferences. These models give rise to an influential theorem—the median voter theorem—attributed to Duncan Black. This theorem stipulates that within a broad range of spatial models, including all one-dimensional models and all symmetric models across multiple dimensions, a Condorcet winner is guaranteed to exist. Moreover, this winner is the candidate closest to the median of the voter distribution.

Empirical research has generally found that spatial voting models give a highly accurate explanation of most voting behavior.

Other theorems
Arrow's impossibility theorem is a generalization of Condorcet's result on the impossibility of majority rule. It demonstrates that every ranked voting algorithm is susceptible to the spoiler effect. Gibbard's theorem provides a closely-related corollary, that no voting rule can have a single, always-best strategy that does not depend on other voters' ballots.

Borda count
The Borda count is a ranking system that assigns scores to each candidate based on their position in each ballot. If m is the total number of candidates, the candidate ranked first on a ballot receives m - 1 points, the second receives m - 2, and so on, until the last-ranked candidate who receives zero. In the given example, candidate B emerges as the winner with 130 out of a total 300 points. While the Borda count is simple to administer, it does not meet the Condorcet criterion. It is heavily affected by the entry of candidates who have no real chance of winning.

Other positional systems
Systems that award points in a similar way but possibly with a different formula are called positional systems. The score vector (m - 1, m - 2,..., 0) is associated with the Borda count, (1, 1/2, 1/3,..., 1/m) defines the Dowdall system and (1, 0,... 0) equates to first-past-the-post.

Instant-runoff (Ranked-choice) voting
Instant-runoff voting, often conflated with ranked-choice voting in general, is a voting method that recursively eliminates the plurality loser of an election until only one candidate is left.

In the given example, Candidate A is declared winner in the third round, having received a majority of votes through the accumulation of first-choice votes and redistributed votes from Candidate B. This system embodies the voters' preferences between the final candidates, stopping when a candidate garners the preference of a majority of voters.

IRV is notable in that it does not fulfill the Condorcet winner criterion, and as a result will not always elect majority-preferred candidate.

Defeat-dropping Condorcet
The defeat-dropping Condorcet methods all look for a Condorcet winner, i.e. a candidate who is not defeated by any other candidate in a one-on-one majority vote. If there is no Condorcet winner, they repeatedly drop (set the margin to zero) for the one-on-one matchups that are closest to being tied, until there is a Condorcet winner. How "closest to being tied" is defined depends on the specific rule. For minimax, the elections with the smallest margin of victory are dropped, whereas in ranked pairs only elections that create a cycle are eligible to be dropped (with defeats being dropped based on the margin of victory).