# Ranked voting

(Redirected from Ranked voting system)

Ranked voting, also known as ranked-choice voting or preferential voting, refers to any voting system in which voters use a ranked (or preferential) ballot to select more than one candidate (or other alternative being voted on) and to rank these choices in a sequence on the ordinal scale of 1st, 2nd, 3rd, etc. Ranked voting is different from cardinal voting, where candidates are independently rated rather than ranked.[1]

Ovals
Names
Numbers
Various types of ranked voting ballot

The most important differences between ranked voting systems lie in the methods used to decide which candidate (or candidates) are elected from a given set of ballots. Some of the most significant methods are described below.

Another (more cosmetic) difference lies in the format of the ballot papers. Some jurisdictions require voters to rank all candidates; some limit the number who may be ranked; and some allow voters to rank as many as they see fit, with the rest being lumped together at the end. Other rules (sometimes entailed by the method of determining a winner) are imposed in different cases.

The subject of this article should not be confused with instant-runoff voting, a specific form of ranked voting to which the US organization FairVote has attached the name ‘ranked-choice voting’. Ranked voting is used in national or state elections in Australia, Ireland, two US states, New Zealand, Ireland, Malta, Slovenia[2] and Nauru.

## Principal ranked voting systems

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

In this section we summarise several of the most popular ranked voting systems, illustrating their operation in a simple election in which there are 100 voters and 3 candidates arranged from left to right, with 36 voters supporting A and voting A–B–C, 34 supporting C and voting C–B–A, and 30 supporting B split equally between B–A–C and B–C–A. The ballots cast are therefore as shown in the table.

We pay particular attention to whether methods satisfy the Condorcet criterion, which guarantees that any candidate who is preferred to every alternative by a majority of voters will be elected. In this case, since B is preferred to A by 64% and to C by 66%, B is the Condorcet winner, and will be elected by any system satisfying the Condorcet criterion. We can also recognise B as the consensus choice on account of being closest to the centre of the voter distribution. The two properties – being the Condorcet winner and being central – are related by the median voter theorem). Condocet-consistent methods differ in their results only in elections in which there is no Condorcet winner.

### Borda count

Candidate Score
A  87
B 130
C  83

The Borda count assigns a score to each candidate by adding a number of points awarded by each ballot. If there are n candidates, then the first-ranked candidate in a ballot receives n – 1 points, the second receives n – 2, and so on until the last-ranked candidate receives none. In the example B is elected with 130 of the total 300 points.

The Borda count is simple to implement but does not satisfy the Condorcet criterion. It has a particular weakness in that its result can be strongly influenced by the nomination of candidates who do not themselves stand any chance of being elected.

#### Other positional systems

Voting systems which award points in this way, but possibly using a different formula, are known as positional systems. Where the score vector (n – 1, n – 2,... ,0) corresponds to the Borda count, (1, ½, ⅓,... ,1/n ) defines the Dowdall system and (1, 0,... ,0) equates to FPTP.

### Alternative vote / instant-runoff voting (IRV)

Ballot
Count
1st round 2nd round 3rd round
36 A–B–C A–C A
15 B–A–C A–C A
15 B–C–A C–A A
34 C–B–A C–A A

IRV eliminates candidates in a series of rounds, emulating the effect of separate ballots on shrinking sets of candidates. The first round consists of the ballots as actually cast. The candidate with fewest first-place preferences is identified (in this case B) and deleted from the ballots for subsequent rounds. Thus in the second round the ballots express preferences between just 2 candidates (more generally m – 1). We stop at this point because A is identified as the winner on account of being the first preference of the majority of ballots; but if we constructed a third round A would be the sole candidate.

Elimination systems are relatively clumsy to implement, since each ballot needs to be re-examined on each round, rather than allowing computation from a simple table of derived statistics. IRV does not satisfy the Condorcet criterion. Unlike most ranked voting systems, it does not allow tied preferences except between a voter’s least preferred candidates.

#### Other elimination systems

Coombs' method is a simple modification of IRV in which the candidate eliminated in each round is the one with most last-place preferences rather than with fewest first-place preferences (so C rather than B is eliminated in the first round of the example and B is the winner). Coombs' method is not Condorcet-consistent but nonetheless satisfies the median voter theorem.[3] It has the drawback that it relies particularly on voters’ last-place preferences which may be chosen with less care than their first places.

Baldwin’s and Nanson’s methods use more complicated elimination rules based on the Borda count. They are Condorcet-consistent.

### Minimax

2nd
1st
A B C
A 36:64 51:49
B 64:36 66:34
C 49:51 34:66

The minimax system determines a result by constructing a results table in which there is an entry for every pair of distinct candidates showing how often the first is preferred to the second. Thus since 51 voters prefer A to C and 49 have the opposite preference, the (A,C) entry reads ‘51:49’. In each row we identify the least favourable (i.e. minimal) result for the first candidate (shown in bold), and the winning candidate is the one whose least favourable result is most favourable (i.e. maximal). In the example the winner is B, whose least favourable result is a win while the other candidates’ least favourable results are slightly different losses.

Determining the minimax winner from a set of ballots is a particularly simple operation. The method satisfies the Condorcet criterion, and can be seen as electing the Condorcet winner if there is one, and as electing the candidate who comes closest to being a Condorcet winner (under a simple metric) otherwise.

### Llull's method / Copeland's method

Candidate Score
A 1
B 2
C 0

Copeland's method (actually due to Ramon Llull) assigns each candidate a score derived from the results table as shown above for minimax. The score is simply the number of favourable results in the candidate’s row, i.e. the number of other candidates a particular candidate was preferred to by a majority of voters. The candidate with the highest score (in this case B) wins.

Copeland’s method is simple and Condorcet-consistent but has the drawback that for certain patterns of voter preferences with no Condorcet winner it will yield a tie however large the electorate. Its advocates therefore generally recommend its use in conjunction with a tie-break. Suitable rules for this purpose are minimax, IRV, and the Borda count, the last of which gives the Dasgupta-Maskin system.

### Condorcet completions

A Condorcet completion elects the Condorcet winner if there is one, and otherwise falls back on a separate procedure for determining the result. If the Borda count is the fallback we get Black's method; if we use IRV we get Tideman’s ‘Condorcet-Hare’.[4]

### Other methods

• The Kemeny-Young method is complex but Condorcet-consistent.
• Smith’s method reduces the set of candidates to the Smith set, which is a singleton comprising the Condorcet winner if there is one, and is otherwise usually smaller than the original set. It is normally advocated for use in conjuction with a tie-break, with IRV and minimax[5] the commonest. It is computationally simple though not intuitive to most voters.
• The contingent vote is a 2-round version of IRV, and the supplementary vote is a restricted form of contingent vote.
• Bucklin's method exists in several forms, some of which are Condorcet-consistent.
• The ranked pairs method, Schulze method and Split cycle method[6] are Condorcet-consistent methods of medium computational complexity based on analysing the cycle structure of ballots.
• Dodgson's method is famous chiefly for having been devised by Lewis Carroll. It is Condorcet-consistent but computationally complex.
• Single transferable vote (STV) is a multiwinner version of IRV.

## Pros and cons of RCV

### Pros

Advocates of instant run-off and single transferable vote argue that IRV promotes majority support: the voting process continues until the winner is selected using a majority of votes, thus gaining support and favor over a greater majority of people.[7] Subsequently, RCV provides more choice for voters over candidates they choose, potentially, minimizing tactical voting whereby a voter would support another candidate more strongly than their honest preference, for the purpose to prevent an undesirable outcome.[8]

Advocates frequently claim candidates that run a negative campaign strategy may see a decline in support as first or second choices. This has never been thoroughly investigated, and empirical evidence supporting the claim is absent. In Australia, which uses instant runoff voting, negative campaigning is commonplace and was seen as particularly problematic in 2016.[9][10][11] In 2021, party supporters called for the Labor Party to fully embrace negative campaigning as a necessary tactic to win elections.[12]

Claims are also made that a ranked choice voting system may cost less to run due to the requirement of only one election, rather than multiple primaries or run-off elections to narrow down the field.[13]

Supporters add that new, diverse voices will emerge by providing candidates a starting ground for those with a lack of name recognition. Diversification of results is a mathematical guarantee under single transferable vote due to its voter proportionality. Previously, it would be difficult for women and people of color to share their voice because of this lack of name recognition that their challengers may have, providing a more equal and fair competition ground for all.[14]

### Cons

Critics of a ranked choice voting system argue that the concept is new and a subset of voters dislike change, possibly causing them to dislike the system and not participate. Among other arguments is the fear that the ballots and counting processes will be more expensive and prone to user error. For many RCV systems, counting ballots by hand is more complex; many can be done by hand or aggregated from counts done at polling centers before transporting the ballots to a central location, particularly systems which only use pairwise victories, and the Borda count; others, such as Meek-STV, are only feasible with a computerized counting system. While utilizing a computerized counting system, critics of ranked voting argue it is still necessary to hold on to the paper ballots so that election recounts can still be performed, minimizing error and holding a greater validity of results.[15]

Some critics find that a single-election implementation of ranked voting makes it harder to vet and critique candidates without a primary election that winnows the candidate field. Ranked choice voting does not require the elimination of primaries, however, and systems such as final five use a non-partisan nominating election to reduce the number of candidates voters must understand and rank in the general election.

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.[16] The Irish government's Commission on Electronic Voting reported that publication of STV ballot votes in full but in random order, as in 2002, would aid in ensuring the accuracy of the counting of results, since anyone would be free to recount these for themselves, but this raised the possibility of revealing deliberate and distinctive voter signatures of low-preference votes, allowing voters to identify themselves in a context of corruption or intimidation.[17] For example, in the 2002 Irish general election, the electronic votes were published for the Dublin North constituency.[18] There were 12 candidates and almost 44,000 votes cast. The most common pattern (for the three candidates from one party in a particular order) was chosen by only 800 voters, and more than 16,000 patterns were chosen by just one voter each.

The number of possible complete rankings with no ties is the factorial of the number of candidates, N; but if ties are allowed freely, it is equal to the corresponding ordered Bell number and is asymptotic to

${\displaystyle {\frac {N!}{2(\ln 2)^{N+1}}}}$[19]

In the case common to instant-runoff voting in which no ties are allowed, except for unranked candidates who are tied for last place on a ballot, the number of possible rankings for N candidates is precisely

${\displaystyle \sum _{n=1}^{N-1}{\frac {N!}{n!}}=\lfloor (e-1)N!-1\rfloor =\mathrm {floor} \left((e-1)N!-1\right)}$[20]

## Use by politics

### Countries and regions

Country Years in use Type Notes
Australia 1918–present[21][22] Single transferable vote, instant-runoff voting From 1949, the single transferable vote method has been used for upper house legislative elections.[23] Instant-runoff voting is used for lower house elections.[24]
Canada Instant-runoff voting Used in whole or in part to elect the leaders of the three largest federal political parties in Canada: the Liberal Party of Canada,[25] the Conservative Party of Canada, and the New Democratic Party, albeit the New Democratic Party uses a mixture of IRV and exhaustive voting, allowing each member to choose one format or the other for their vote.
Estonia 1990–c. 2001 Single transferable vote As of 2001, single transferable vote had been in use since 1990 to decide legislative elections.[23] This is no longer the case.[26]
Fiji 1998–present Instant-runoff voting [27]
Hong Kong 1998–present[28] Instant-runoff voting[29] Instant-runoff voting is only used in the 4 smallest of Hong Kong's 29 functional constituencies.[30] Officially called preferential elimination voting, the system is identical to the instant-runoff voting.[29]
Ireland 1922–present Single transferable vote[31] Single transferable vote is prescribed by Constitution or statute for all public elections.[31] In single-winner cases (presidential, most Dáil by-elections) this reduces to instant-runoff voting. Referendums to abolish STV for Dáil elections failed in 1958 and 1968.[31]
Malta 1921–present Single transferable vote [23]
Nauru 1968–present[23] Borda count Nauru uses the Dowdall system, a variant of the Borda count that behaves more like FPTP.[32][33]
New Zealand 2004–present[34] Single transferable vote[35] Instant-runoff voting is used in only some single-seat elections, such as district health boards as well as some city and district councils.[35]
Northern Ireland 1973–present[23] Single transferable vote[36] Used for local government, European Parliament and the regional legislature, but not elections to Westminster.
Papua New Guinea 2007–present[37] Instant-runoff voting[38] Between 1964 and 1975, Papua New Guinea used a system that allowed voters the option of ranking candidates.[23] Currently, voters can rank only their top three choices.[39]
Slovenia 2000–present[40] Borda count Only two seats, which are reserved for Hungarian and Italian minorities, are decided using a Borda count.[41]
Sri Lanka 1978–present Contingent vote and open list Contingent vote is used for presidential elections,[23] and open list for legislative elections.[42]
United States 2020 Limited instant-runoff voting In their 2020 primaries, several states used a form of instant run-off in Democratic Party primaries.
Zimbabwe[43] 1979–1985 Instant-runoff voting Was only used for white candidates

### Federal provinces or states

Province/state Country Years in use Type Notes
Alaska[44] United States 2022 Instant-runoff voting Approved by Alaska voters in 2020 via ballot measure.
Australian Capital Territory[23] Australia 1993–present Single transferable vote
British Columbia[23] Canada 1926–1955 Instant-runoff voting
Maine[45] United States 2018–present Instant-runoff voting Originally approved by Maine voters as a 2016 ballot referendum to replace the First Past The Post system statewide, a 2017 state law sought to delay implementation of ranked-choice voting until 2021, to allow time for amending the state constitution. Supporters overrode the delay with a 2018 people's veto referendum that received a majority of votes, ensuring that ranked-choice voting would be used for future primary and federal elections.
New South Wales[23] Australia 1918–present Single transferable vote (1918–1926, 1978–present), contingent vote (1926–1928), instant-runoff voting (1929–present) Since 1978, NSW has used the single transferable vote method to decide upper house legislative elections only. Optional preferential voting for lower house since 1981. This means a voter can number all candidates with their preference, or stop at just one candidate.
North Carolina[46] United States 2006–2013 Instant-runoff voting A state law in 2006 established instant-runoff voting for certain judicial elections, until a 2013 law repealed the practice.
Northern Territory[23] Australia 1980 only
Ontario Canada 2018–present Instant-runoff voting (municipal elections only) In 2016, the provincial government passed Bill 181, the Municipal Elections Modernization Act, which permitted municipalities to adopt ranked balloting in municipal elections.[47] In the 2018 elections, the first ones conducted under the new legislation, the city of London used ranked balloting,[48] while the cities of Kingston and Cambridge held referendums on whether to adopt ranked ballots for the next municipal elections in 2022.[49]
Queensland[23] Australia 1892–1942, 1962–present Contingent vote (1892–1942), instant-runoff voting (1962–present) Full preferential voting used 1962–1992 and since 2016.
South Australia[23] Australia 1929–present, 1982–present Instant-runoff voting (1929–present), single transferable vote (1982–present) Instant runoff for the lower house, single transferable for the upper house.
Tasmania[23] Australia 1907–present Single transferable vote (1907–present), instant-runoff voting (1909–present) Single transferable for the lower house, instant runoff for the upper house.
Victoria[23] Australia 1911–present Instant-runoff voting (1911–present), single transferable vote (2006–present) Prior to 1916, Victoria did not use any preferential voting method to decide upper house legislative elections. Instant runoff for the lower house, single transferable for the upper house. Full preferential voting for lower house since 1916.
Western Australia[23] Australia 1907–present Instant-runoff voting (1907–present), single transferable vote (1989–present) Instant runoff for the lower house, single transferable for the upper house. Full preferential voting for lower house since 1912.

### Municipalities

City/town Years in use Type Notes
Ann Arbor, MI[50] 1975 only Instant-runoff voting
Aspen, CO[51] 2009 only Instant-runoff voting
Berkeley, CA[52] 2010–present Instant-runoff voting
Burlington, VT[53][54][55] 2005–2010; 2021–present Instant-runoff voting Repealed for mayoral elections after the 2009 election; in 2021 a referendum reinstated it for the city council elections.
Cambridge, MA[56] 1941–present Single transferable vote
Hendersonville, NC[57] 2007–2013 Instant-runoff voting part of a statewide pilot program,[58] deauthorized in 2013[59]
London 2000–present[60] Supplementary vote[61]
London, Ontario[62] 2018 – present Instant-runoff voting
Memphis, TN[63] 2011–present Instant-runoff voting
Minneapolis, MN[64] 2009–present Instant-runoff voting
New York City, NY[65][66] 2021–present Instant-runoff voting Only applies to primaries and special elections for municipal offices
Oakland, CA[52] 2010–present Instant-runoff voting
Portland, ME[63] 2011–present Instant-runoff voting
San Francisco, CA 2004–present[67] Instant-runoff voting[63]
San Leandro, CA[52] 2010–present Instant-runoff voting
Santa Fe, NM 2018–present Instant-runoff voting
St. Paul, MN 2011–present[68] Instant-runoff voting[69]
Takoma Park, MD[70] 2006–present Instant-runoff voting
Telluride, CO[71] 2011–present Instant-runoff voting

### International organizations

Organization Years in use Type Notes
European Union[72] option to use single transferable vote Member countries can use either party-list proportional representation (not a type of preferential voting)[citation needed] or single transferable vote to elect MEPs

## Use outside of politics

The winner of the Eurovision Song Contest is selected by a positional voting system. The most recent system was implemented in the 2016 contest, and sees each participating country award two sets of 12, 10, 8–1 points to their 10 favourite songs: one set from their professional jury and the other from tele-voting.[73]

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