# Ranked voting

Ranked voting is any voting system that uses voters' rankings of candidates to choose a single winner or multiple winners. More formally, a ranked system is one that depends only on which of two candidates is preferred by a voter, and as such does not incorporate any information about intensity of preferences. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them very different properties. In instant-runoff voting (IRV) and the single transferable vote system (STV), lower preferences are used as contingencies (back-up preferences) and are only applied when all higher-ranked preferences on a ballot have been eliminated or when the vote has been cast for a candidate who has been elected and surplus votes need to be transferred.
Some ranked vote systems use ranks as weights; this type of system is called positional voting. In the Borda method, the 1st, 2nd, 3rd... candidates on each ballot receive 1, 2, 3... points, and the candidate with the fewest points is elected. Thus intensity of preference is assumed to be at ratios of 1 to 2, 2 to 3, etc. Although not typically described as such, the well-known plurality rule can be seen as a ranked voting system where a voter gives a single point to the candidate marked as their choice and zero points to all others, and the candidate with the most points is elected. Taking the ranked ballots of instant-runoff voting and the single transferable vote system as indicating one choice at a time (that is, giving one point to the preference in use and zero points to all others), instant-runoff voting can be seen as a non-degenerate ranked voting systems. It operates as a staged variant of the plurality system that repeatedly eliminate last-place plurality winners if necessary to determine a majority winner.
In the United States and Australia, the terms ranked-choice voting and preferential voting, respectively, almost always refer to instant-runoff voting; however, because these terms have also been used to mean ranked systems in general, many social choice theorists recommend the use of instant-runoff voting in contexts where it could cause confusion. Ranked voting systems, such as Borda count, 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 to 10). Ranked vote systems produce more information than X voting systems such as first-past-the-post voting. Rated voting systems produce more information than ordinal ballots; as a result, some common results like Arrow's theorem do not directly apply to them.

## History of ranked voting

The earliest known proposals for a ranked voting system 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, which is similar to Condorcet'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. Carl Andræ formulated a version of the single transferable vote election system, which was adopted by his country, Denmark, in 1855. This used contingent ranked votes to transfer votes of eliminated un-popular candidates but did not include transfers of eliminated candidates.
In the 1700s, Condorcet considered a single-winner version of STV, the instant-runoff system, but rejected it as pathological. In 1864, Edward J. Nanson, while noting Condorcet's concerns, said the IRV method is "a perfectly feasible and practicable one for elections on any scale".
Professor W.R. Ware held a demonstration STV election in 1871, by having 150 M.I.T. students elect four English authors as their favorites. The field of candidates held the names of 35 authors in the first count, and eventually concentrated a large proportion of the votes behind just four. Ware went on to surmise how a casual vacancy might be handled and found that countback would produce a winner. This proved that ranked transferable votes could be used to produce a single winner, despite the qualms of Condorcet and others.
Single transferable voting (multi-seat contests using contingent ranked transferable votes) later found common use in cities and to elect legislators in North America, Ireland and other parts of the English-speaking world, with single-winner versions, alternative voting (now known as instant-runoff voting) and contingent voting (also known as supplementary voting), being its companions for single-winner contests of mayors and others.
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. As plurality voting has exhibited weaknesses from its start, especially as soon as a third party joins the race, some individuals turned to transferable votes (facilitated by contingent ranked ballots) to reduce the incidence of wasted votes and unrepresentative election results.
A form of the single transferable vote system was invented by Carl Andræ in Denmark, where it was used briefly before being abandoned in favour of open list list PR, but still carried on for indirect election of the upper house until 1953.
At approximately the same time, the single transferable vote system was independently devised by British lawyer Thomas Hare, whose writings soon spread the method throughout the British Empire. Tasmania used STV (called the Hare system) in government elections in the 1890s. STV began permanent and wider adoption throughout Australia beginning in 1907 and the 1910s. The single transferable vote system, using contingent ranked votes, has been adopted in Ireland, South Africa, Malta, and approximately 40 cities in the United States and Canada. The single transferable vote system has also been used to elect legislators in Canada, South Africa and India.
In more recent years, the use of contingent ranked votes has seen a comeback in the United States. Single-winner ranked voting (specifically instant-runoff voting), is used to elect politicians in Maine and Alaska. In November 2016, the voters of Maine narrowly passed Question 5, approving ranked-choice voting (instant-runoff voting) for all elections. This was first put to use in 2018, marking the inaugural use of ranked votes in a statewide election in the United States (when Maine's statewide vote tally was used to determine the two at-large electoral college seats).
In November 2020, Alaska voters passed Measure 2, bringing ranked choice voting (instant-runoff voting) into effect from 2022. 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. However, a referendum on the issue in 2024 saw a narrow majority in favour of retention of IRV.
Some local elections in New Zealand and in the U.S. use the multi-winner single transferable vote. STV is also used to elect local authorities in Scotland and Ireland. Nauru uses a rank-weighted positional method called the Dowdall system.

## Equal-ranked ballots

In voting with ranked ballots, a tied or equal-rank ballot is one where multiple candidates receive the same rank or rating. In instant runoff and first-preference plurality, such ballots are generally rejected; however, in social choice theory some election systems assume equal-ranked ballots are "split" evenly between all equal-ranked candidates (e.g. in a two-way tie, each candidate receives half a vote). Meanwhile, other election systems, the Borda count and the Condorcet method, can use different rules for handling equal-rank ballots. These rules produce different mathematical properties and behaviors, particularly under strategic voting.

## Theoretical foundations of ranked voting
### 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, a candidate who would win against any other candidate in a two-way race. 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 > C > B, one votes B > A > C, and one votes C > B > 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.

## Examples
### Borda count

The Borda count is a weighted-rank 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. Also, 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 voting

Instant-runoff voting, often conflated with ranked-choice voting in general, is a contingent ranked-vote voting method that recursively eliminates the plurality loser of an election until one candidate has the majority of the remaining votes. 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. Instant-runoff voting does not fulfill the Condorcet winner criterion.

### Single transferable voting

Single transferable voting is a contingent ranked-vote voting method that elects multiple members. It elects any candidates who achieve quota, and if necessary recursively eliminates the plurality loser at each stage of the vote count and transfers surplus votes of winners until enough are elected by quota or by still being in the running when the field of candidates is thinned to the number of remaining open seats. Because elected members are elected with the same or about the same number of votes, each party popular enough for representation receives a number of seats appropriate to the vote tallies of its candidates. The transfers reduce waste to about one quota - which in a five-seat district is about 17 percent of valid votes; in districts with more members than five, the waste is smaller. All but one quota of votes approximately are used to actually elect someone in the district so the percentage of effective votes is dependably about 80 to 90 percent of valid votes.

### 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 the Minimax Condorcet method, 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).