Ranked voting

Typical counting process of a single-seat ranked choice voting election, developed by William Robert Ware

The earliest mention of ranked voting can be traced back to the works of Ramon Llull in the late 13th century. His interpretations, however, are not universally clear, and he is often believed to have favored Copeland's method, which used a series of pairwise elections instead of ranked-choice ballots. His ideas eventually caught the attention of Nicholas of Cusa in the early 15th century, who appeared to develop independently what is now known as the Borda count method that utilizes ranked ballots. The writings of both Llull and Nicholas were lost, only to be rediscovered in the 20th century.

The modern analysis of ranked voting commenced when Jean-Charles de Borda published a paper in 1781, advocating for the method that now bears his name. This methodology drew criticism from the Marquis de Condorcet, who developed his own method to determine collective preference, asserting that Borda's approach did not always reflect the group's preference. However, his example remains a point of contention (see Comparison of electoral systems).[6]

Interest in ranked voting was renewed in the 19th century. Danish pioneer Carl Andræ formulated the Single Transferable Vote (STV) system, which was adopted by his native Denmark in 1855. Simultaneously, Thomas Hare in the UK independently invented STV in 1857. The single-winner variant of STV, known as instant-runoff voting (IRV), was proposed by William Robert Ware around 1870. While Ware might have been unaware, this system was mentioned earlier by Condorcet, albeit only to criticize it.[7][8]

Ranked voting first saw governmental use in the 1890s in Tasmania, deploying the STV system. Its broader adoption in Australia began in the 1910s.[9] By the 1920s, ranked voting had expanded globally. It was used in Ireland, South Africa, and approximately 20 cities each in Canada and the U.S. Instant-runoff voting was utilized for single official elections, like mayors, while STV was employed for multiple official elections, such as councillors or school board trustees. Moreover, ranked voting was also adopted for governmental elections in Ireland, Malta, and Canada (specifically Alberta and Manitoba) beginning in the 1920s.

Theoretical exploration of electoral processes was initiated by a 1948 paper from Duncan Black,[10] which was soon followed by Kenneth Arrow's research on the consistency of voting criteria. This subject has since continued to receive scholarly interest under social choice theory, typically falling within the economics discipline.

In more recent years, ranked choice voting has seen increased implementation. In November 2016, the voters of Maine passed Question 5, instigating a ranked choice voting system 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.[11][12]

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 Borda count is simple to administer, it does not meet the Condorcet criterion. It has a significant limitation in that its result can be heavily influenced by the entry of candidates who themselves have no real possibility of winning.

The alternative vote (AV), also known as instant-runoff voting and as ranked-choice voting in the United States, is a voting method that sequentially eliminates candidates in a series of rounds until a winner emerges. In this system, voters rank candidates in order of preference. If a candidate does not receive a majority of first-choice votes, the candidate with the fewest votes is eliminated, and their votes are redistributed according to the voters' next preferences. This process continues until one candidate receives a majority of the remaining votes.

In the given example, Candidate A is declared the winner in the second 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.

This method, unlike party-list proportional representation (PR), examines each transferred vote to respect the voter's ranking rather than allocating seats from a derived statistical table. Thus, it ensures a more accurate representation of voters' preferences.

IRV is notable in that it does not fulfill the Condorcet winner criterion, a property held by many other ranked voting systems. Furthermore, unlike its counterparts, IRV generally does not accommodate tied preferences, except occasionally among a voter's least preferred candidates. An adapted version of the single transferable vote (STV) system, intended for party ranking, was proposed for use in German elections in 2013.[14] This proposed system is referred to as the spare vote.[15]

An example of an STV election:

There are three positions to be elected, with a total of 100 votes. The quota, determined by the Droop method, is 26 votes.

The Single transferable vote (STV) is a multi-winner method and a proportional version of IRV. Like IRV, STV allows for contingent voting based on secondary preferences, which only come into play if the first preference does not lead to an effective result. With STV, voters cast one vote with the possibility to mark back-up preferences. Initially, a voter's ballot is allocated to their most preferred candidate. After a candidate reaches the quota and is elected, surplus votes are transferred from this winner to remaining hopefuls based on the voter's preference order. If there are still vacant seats, the least performing candidates are eliminated, and their votes are transferred to the remaining contenders. Different versions of STV might use varying methodologies for eliminating candidates and transferring votes.

The Minimax voting system determines the winner by constructing a comparison table of candidate preferences, as shown above. This table illustrates the preferences between each pair of distinct candidates (A, B, C). The cell value x:y represents that x voters prefer the first candidate over the second, while y voters have the opposite preference. This data is derived from the ranked votes provided in the Spatial Model section above.

Each row identifies the least satisfactory outcome (i.e., minimal preference) for the first candidate, highlighted in bold. The winning candidate is the one whose least satisfactory outcome is most favorable.

In this specific example, candidate B is the winner. Despite B's worst performance being a 64:36 victory margin over candidate A, this is the most favorable least satisfactory result when compared to those of the other candidates.

The Minimax method, which is straightforward in determining the winner from a set of ballots, satisfies the Condorcet criterion. It elects the Condorcet winner, if one exists. Otherwise, it selects the candidate who comes closest to being a Condorcet winner under a simple metric.

Copeland's method assigns a score to each candidate derived from the results table, as shown above in the minimax section. The score is simply the number of favourable results in a candidate's row, i.e., the number of other candidates to whom a specific candidate is preferred by a majority of voters. The candidate with the highest score (in this case B) is the winner.

Copeland's method is Condorcet-consistent and straightforward, but for certain patterns of voter preferences (with no Condorcet winner), it will yield a tie regardless of the size of the electorate. Therefore, its proponents typically recommend its use in conjunction with a tie-breaker. Suitable rules for this purpose include minimax, IRV, and the Borda count, the latter giving rise to the Dasgupta-Maskin method.

Several other voting methods provide alternatives to the methods described above, each with unique characteristics:

• Ranked voting: Different from cardinal voting, candidates are independently rated rather than ranked.[16]

• Condorcet completions: Elects the Condorcet winner if there is one and otherwise falls back on a separate procedure for determining the result. Examples include Black's method when using the Borda count as the fallback and the Nicolaus Tideman Condorcet-Hare when using IRV.[17]

• Bottom-Two-Runoff IRV or BTR-IRV: A Condorcet-consistent variant of IRV.

• Coombs' method: A variant of IRV where the candidate eliminated in each round has the most last-place preferences rather than the fewest first-place choices. Not Condorcet-consistent but satisfies the median voter theorem.[18]

• Baldwin's and Nanson's methods: Use more complicated elimination rules based on the Borda count. They are Condorcet-consistent.

• Kemeny-Young method: Complex but Condorcet-consistent.

• Smith's method: Reduces the set of candidates to the Smith set, which contains the Condorcet winner if there is one. Advocated for use with a tie-break, like IRV and minimax.[19]

• contingent vote and supplementary vote: Variants of IRV.

• Bucklin's method: Exists in several forms, some of which are Condorcet-consistent.

• Ranked Pairs and Related Methods: The Ranked Pairs method, including the Schulze method, Schulze STV, and Split cycle method,[20] are Condorcet-consistent methods of medium computational complexity based on analyzing the cycle structure of ballots.

• Dodgson's method: Devised by Lewis Carroll. It is Condorcet-consistent but computationally complex.

• Expanding Approvals Rule

Ranked voting allows voter to select candidates that more accurately represent their preferences than first-past-the-post ballots. It allows that one candidate will get 50%+1 of the vote. However, they are more complicated.[19]^: §8.1If there are a large number of candidates, which is quite common in single transferable vote elections, then it is likely that many preference voting patterns will be unique to individual voters, which could allow voters to identify themselves in a context of corruption or intimidation, undermining the secrecy of ballots.[27][28] In the 2002 Irish general election, the electronic votes were published for the Dublin North constituency.[29]