Ranked voting

The term ranked voting, also known as preferential voting or ranked choice voting, pertains to any voting system where voters use a rank to order candidates or options—in a sequence from first, second, third, and onwards—on their ballots. Ranked voting systems vary based on the ballot marking process, how preferences are tabulated and counted, the number of seats available for election, and whether voters are allowed to rank candidates equally. An electoral system that utilizes ranked voting employs one of numerous counting methods to determine the winning candidate or candidates. Additionally, in some ranked voting systems, officials mandate voters to rank a specific number of candidates, sometimes all; while in others, voters may rank as many candidates as they desire.

Ovals

Names

Numbers

Various types of ranked voting ballot

The instant-runoff voting method is commonly employed for the election of single members using ranked votes. Conversely, the election of multiple members using ranked votes typically utilizes single transferable voting (STV). There exist other systems applicable for single-member selection or multi-member elections. Ranked voting in multi-member districts plays a part in national elections in Australia, Ireland, Malta, and the United Kingdom—specifically, the Scottish and Welsh Parliaments.^[1] On the other hand, single-winner ranked voting is used in the election of national-level politicians from the states of Maine^[2] and Alaska^[3] within the United States. Ranked voting is also utilized in Slovenia,^[4] and Nauru. For certain local elections in New Zealand, ranked voting is used.^[5] Moreover, in the United States, some cities, counties, and federal primaries across 16 states, as well as 5 additional states' overseas voters for federal elections, employ ranked voting.

History of ranked voting

 

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

Main articles: Electoral system and Social choice theory

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]

Theoretical foundations of ranked voting

Condorcet criterion

Main article: Condorcet winner criterion

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 preferred over all others by a majority of voters. A voting system that consistently elects this candidate, if one exists, is known as Condorcet consistent or as satisfying the Condorcet criterion. Such systems are referred to as Condorcet methods.

However, in elections where no Condorcet winner exists, a Condorcet cycle is likely to occur, best explained through an example. Suppose an election involves three candidates - A, B, and C, with 30 voters such that ten vote C–B–A, ten vote B–A–C, and ten vote A–C–B. In this case, no Condorcet winner exists. Specifically, 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 cycle where the search for a Condorcet winner fails to find one.

Spatial voting models

Main article: Median voter theorem

 

Spatial model diagram

Ballot	Count
A–B–C	36
B–A–C	15
B–C–A	15
C–B–A	34

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 typically the candidate closest to the median of the voter distribution.

Applying these principles to the illustrated diagram reveals a Condorcet winner, candidate B, who is preferred over candidate A by 64% of voters and over candidate C by 66%. Consistent with the median voter theorem, the Condorcet winner aligns closely with the median of the voter distribution.

Other theorems

Main articles: Arrow's impossibility theorem and Gibbard's theorem

The implications of Arrow's impossibility theorem cast a shadow on the promise of ranked voting. While the median voter theorem demonstrates the feasibility of creating a voting method that functions flawlessly for a broad spectrum of voter preferences, Arrow's theorem posits the impossibility of devising a system that operates without fail in every conceivable scenario.

The true reflection of electoral behavior—whether it aligns more closely with the optimism posited by Black or the pessimism suggested by Arrow—is a question best answered through empirical investigation. Certain studies, including a notable publication by Tideman and Plassman,^[13] propose that simplistic spatial models that adhere to the median voter theorem accurately emulate observed voter behavior. Adding to the discourse of pessimistic views, Gibbard's theorem, put forth by Allan Gibbard, asserts the inevitability of susceptibility to tactical voting in any voting system.

Borda count

Main article: Borda count

Candidate	Score
A	87
B	130
C	83

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.

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.

Alternative vote (instant-runoff voting)

Main article: Instant-runoff voting

Candidate	Count
	1st round	2nd round	3rd round
A (votes marked A-B)	36	51	51
B (15 votes marked B-A; 15 marked B-C)	30	0	0
C (votes marked C-B)	34	49	49

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]

Single transferable vote

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.

Candidate	Count
	1st Round	2nd Round	3rd Round	4th Round
A (votes marked A-B)	26	elected	elected
B (votes marked B-E)	18	26 (elected)	elected
C (votes marked C-D)	17	17	17	elected (as last remaining candidate to fill final seat)
D (votes marked D-C)	16	16	16	eliminated
E (votes marked E-B)	15	15	eliminated
F (votes marked F-B)	8	eliminated	0

Main article: Single transferable vote

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.

Minimax

Main article: Minimax Condorcet method

Candidate Preference Comparison		Second candidate
		A	B	C
First candidate	A	–	36:64	51:49
	B	64:36	–	66:34
	C	49:51	34:66	–

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.

Llull's Method / Copeland's Method

Main article: Copeland's method

Candidate	Score
A	1
B	2
C	0

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.

Other methods

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

Comparison and evaluation of ranked voting systems

Main article: Comparison of electoral systems

Analysis using examples

Michael Gallagher points out that the strengths and weaknesses of different ranked voting methods can be best elucidated through specific examples.^[21] A practical example can be provided by considering a voter's ranking of candidates: "If A is eliminated, the voter's vote is transferred to B. If B is then eliminated, the voter's vote is transferred to C. This process continues until one candidate has a majority of the votes," explains Steven Brams and Peter Fishburn.^[22]

Applying logical voting criteria

Logical voting criteria play a crucial role in evaluating voting methods. The Condorcet criterion, as defined by Duncan Black, states that "the candidate who would win in a head-to-head race against all other candidates should be the winner of the election."^[23] Moreover, the independence of irrelevant alternatives (IIA) criterion, explained by Kenneth Arrow, implies that "the ranking of candidates should not change if the order of the candidates is changed."^[24]

Empirical approach: simulated elections

Steven Brams and Peter Fishburn highlight the utility of simulated elections as a means to study the properties of ranked voting methods.^[25] These simulations can estimate the efficacy of each voting method, which is defined by how frequently it elects the candidate closest to the center of the voter distribution. In terms of these simulations, Andrew Reynolds indicates that "Condorcet methods and Coombs' method tend to deliver optimal outcomes, followed by the Borda count. Instant-runoff voting comes further behind, with first-past-the-post voting trailing."^[26]

Drawbacks of ranked voting

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]: §8.1}If 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]

See also

• Approval Voting
• Arrow's impossibility theorem
• Cardinal voting
• Comparison of electoral systems
• Condorcet winner criterion
• Election
• Gibbard's theorem
• Group voting ticket
• History and use of instant-runoff voting
• Implicit utilitarian voting
• Instant-runoff voting
• Category:Preferential electoral systems
• List of electoral systems by country
• Matrix vote
• Median voter theorem
• Multiwinner voting
• Preferential block voting
• Ranked-choice voting in the United States
• Social choice theory
• Spoiler effect
• Voting methods in deliberative assemblies

References

1. Wiki: Politics of U.K.
2. "Ranked Choice Voting in Maine". Maine State Legislature. Retrieved 21 October 2021.
3. "Alaska Better Elections Implementation". Alaska Division of Elections. Retrieved 21 October 2021.
4. Toplak, Jurij (2006). "The parliamentary election in Slovenia, October 2004". Electoral Studies. 25 (4): 825–831. doi:10.1016/j.electstud.2005.12.006.
5. "New Zealand Cities Voting to Implement Ranked Choice Voting". 19 September 2017.
6. George G. Szpiro, "Numbers Rule" (2010).
7. Nanson, E. J. (1882). "Methods of election: Ware's Method". Transactions and Proceedings of the Royal Society of Victoria. 19: 206. The method was, however, mentioned by Condorcet, but only to be condemned.
8. Condorcet, Jean-Antoine-Nicolas de Caritat (1788). On the Constitution and the Functions of Provincial Assemblies. p. 243. En effet, lorsqu'il y a plus de trois concurrents, le véritable vœu de la pluralité peut être pour un candidat qui n'ait eu aucune des voix dans le premier scrutin. {{cite book}}: |work= ignored (help)
9. Farrell and McAllister, The Australian Electoral System, p. 17
10. Duncan Black, "On the Rationale of Group Decision-making" (1948).
11. "Ranked Choice Voting in Maine". legislature.maine.gov. State of Maine. 2022-08-23. Retrieved 2022-11-20.
12. Piper, Kelsey (2020-11-19). "Alaska voters adopt ranked-choice voting in ballot initiative". vox.com. Vox Media. Retrieved 2022-11-20.
13. T. N. Tideman and F. Plassman, "Modeling the Outcomes of Vote-Casting in Actual Elections" (2012).
14. Alternative II.: Einführung einer Ersatzstimme (in German)
15. What is Dual Level Voting?
16. Riker, William Harrison (1982). Liberalism against populism: a confrontation between the theory of democracy and the theory of social choice. Waveland Pr. pp. 29–30. ISBN 0881333670. OCLC 316034736. Ordinal utility is a measure of preferences in terms of rank orders—that is, first, second, etc. ... Cardinal utility is a measure of preferences on a scale of cardinal numbers, such as the scale from zero to one or the scale from one to ten.
17. J. Green-Armytage, T. N. Tideman and R. Cosman, "Statistical Evaluation of Voting Rules" (2015).
18. B. Grofman and S. L. Feld, "If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule" (2004)
19. R. B. Darlington, "Are Condorcet and Minimax Voting Systems the Best?" (v8, 2021).
20. W. H. Holliday and E. Pacuit, "Split Cycle: A New Condorcet Consistent Voting Method Independent of Clones and Immune to Spoilers" (2021).
21. Gallagher, Michael (2011). Comparison of Electoral Systems. Oxford University Press. p. 25.
22. Brams, Steven; Fishburn, Peter (2007). The Mathematics of Voting. Princeton University Press. p. 13.
23. Black, Duncan (1958). The Theory of Voting. Cambridge University Press. p. 12.
24. Arrow, Kenneth (1963). Social Choice and Individual Values. Wiley. p. 10.
25. Brams, Steven; Fishburn, Peter (2007). The Mathematics of Voting. Princeton University Press. p. 14.
26. Reynolds, Andrew (2011). Electoral Systems: The Key to Power. Routledge. p. 12.
27. February 1, 2004
28. 2004
29. complete table of votes cast Dublin North (zip file)

External links

• Explainer video: How does ranked-choice voting work?, Minnesota Public Radio MPR.org
• Ballotpedia: "Ranked Choice Voting"
• Examples from RCV123.org
  • US State of Maine, 2020 Ballot
  • US State of Alaska, 2021 Ballot
• FairVote.Org