# Ranked voting

Sample ballot of ranked voting using written numbers

Ranked voting is any election voting system in which voters use a ranked (or preferential) ballot to rank choices in a sequence on the ordinal scale: 1st, 2nd, 3rd, etc. There are multiple ways in which the rankings can be counted to determine which candidate (or candidates) is (or are) elected (and different methods may choose different winners from the same set of ballots). The other major branch of voting systems is cardinal voting, where candidates are independently rated, rather than ranked.[1]

The similar term "Ranked Choice Voting" (RCV) is used by the US organization FairVote to refer to the use of ranked ballots with specific counting methods: either instant-runoff voting for single-winner elections or single transferable vote for multi-winner elections. In some locations, the term "preferential voting" is used to refer to this combination of ballot type and counting method, while in other locations this term has various more-specialized meanings.[2]

A ranked voting system collects more information from voters compared to the single-mark ballots currently used in most governmental elections, many of which use First-Past-The-Post and Mixed-Member Proportional voting systems.

There are many types of ranked voting, with several used in governmental elections. Instant-runoff voting is used in Australian state and federal elections, in Ireland for its presidential elections, and by some jurisdictions in the United States, United Kingdom, and New Zealand. A type and classification of ranked voting is called the single transferable vote, which is used for national elections in Ireland and Malta, the Australian Senate, for regional and local elections in Northern Ireland, for all local elections in Scotland, and for some local elections in New Zealand and the United States. Borda count is used in Slovenia[3] and Nauru. Contingent vote and Supplementary vote are also used in a few locations. Condorcet methods are used by private organizations and minor parties, but currently are not used in governmental elections.

Arrow's impossibility theorem and Gibbard's theorem prove that all voting systems must make trade-offs between desirable properties, such as the preference between two candidates being unaffected by the popularity of a third candidate.[4][5] Accordingly there is no consensus among academics or public servants as to the "best" electoral system.[6]

Recently, an increasing number of authors, including David Farrell, Ian McAllister and Jurij Toplak, see preferentiality as one of the characteristics by which electoral systems can be evaluated.[2][7] According to this view, all electoral methods are preferential, but to different degrees and may even be classified according to their preferentiality.[2] By this logic, cardinal voting methods such as Score voting or STAR voting are also "preferential".

## Different types of systems

There are different preferential voting systems, so it is sometimes difficult to distinguish between them.

Selection of the Condorcet winner is generally considered by psephologists as the ideal election outcome for a ranked system,[8] so "Condorcet efficiency" is important when evaluating different methods of preferential voting.[9] The Condorcet winner is the one that would win every two-way contest against every other alternative.[4]

Another criterion used to gauge the effectiveness of a preferential voting system is its ability to withstand manipulative voting strategies, when voters cast ballots that do not reflect their preferences in the hope of electing their first choice. This can be rated on at least two dimensions—the number of voters needed to game the system, and the sophistication of the strategy necessary.[9]

### Instant-runoff voting

Sample ballot of ranked voting using column marks

Used in national elections in Australia, this system is said to simulate a series of runoff elections. If no candidate is the first choice of more than half of the voters, then all votes cast for the candidate with the lowest number of first choices are redistributed to the remaining candidates based on who is ranked next on each ballot. If this does not result in any candidate receiving a majority, further rounds of redistribution occur.[10][11]

This method is thought to be resistant to manipulative voting as the only strategies that work against it require voters to highly rank choices they actually want to see lose. At the same time, this system fails Condorcet criterion, meaning a candidate can win even if the voters preferred a different candidate, and fails the monotonicity criterion, where ranking a candidate higher can lessen the chances he or she will be elected and vice versa. Additionally, instant-runoff voting has a lower Condorcet efficiency than similar systems when there are more than four choices.[9]

### Single transferable vote

Sample ballot of ranked voting using written names

This is one of the preferential voting systems most used by countries and states. (See table below in "Use by politics".) It is used for electing multi-member constituencies. Any candidates that achieve the number of votes required for election (the "quota") are elected and their surplus votes are redistributed to the voter's next choice candidate. Once this is done, if not all places have been filled then the candidate with the lowest number of votes is eliminated, and their votes are also redistributed to the voter's next choice. This whole process is repeated until all seats are filled. This method is also called the Hare-Clark system.[12]

When STV is used for single-winner elections, it becomes equivalent to IRV.[13]

### Positional voting

Positional voting is a ranked voting electoral system in which the options receive points based on their rank position on each ballot and the option with the most points overall wins.[14]

#### Borda count

Borda is a positional system in which ballots are counted by assigning a point value to each place in each voter's ranking of the candidates, and the choice with the largest number of points overall is elected. This method is named after its inventor, French mathematician Jean-Charles de Borda.[4] Instead of selecting a Condorcet winner, this system may select a choice that reflects an average of the preferences of the constituency.[citation needed]

This system suffers from the fact that the outcome it selects is dependent on the other choices present.[clarification needed] That is, the Borda count does not exhibit independence of irrelevant alternatives[4] or independence of clones. The Borda count can be easily manipulated by adding candidates, called clones, whose views are similar to the preferred candidate's. An example of this strategy can be seen in Kiribati's 1991 presidential nomination contest.[15]

### Path Voting (Schulze Method)

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][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[27] 1998–present Instant-runoff voting
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 Instant-runoff voting, single transferable vote Single transferable vote is used to decide legislative elections only. Since 1937 Ireland has used instant-runoff voting to decide presidential elections.[23]
Malta[23] 1921–present Single transferable vote
Nauru 1968–present[23] Borda count Nauru uses the Dowdall system, a variant of the Borda count that behaves more like FPTP.[31][32]
New Zealand 2004–present[33] Single transferable vote[34] Instant-runoff voting is used in only some single-seat elections, such as district health boards as well as some city and district councils.[34]
Northern Ireland 1973–present[23] Single transferable vote[35] Used for local government, European Parliament and the regional legislature, but not elections to Westminster.
Papua New Guinea 2007–present[36] Instant-runoff voting[9] 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.[37]
Slovenia 2000–present[38] Borda count Only two seats, which are reserved for Hungarian and Italian minorities, are decided using a Borda count.[39]
Sri Lanka 1978–present Contingent vote and open list Contingent vote is used for presidential elections,[23] and open list for legislative elections.[40]
United States 2020 Limited instant-runoff voting In their 2020 primaries, the Democratic Party permitted voters who could not be physically present at the Iowa and Nevada caucus to early vote using a preferential ballot.
Zimbabwe[41] 1979–1985 Instant-runoff voting Was only used for white candidates

### Federal provinces or states

Province/state Country Years in use Type Notes
Australian Capital Territory[23] Australia 1993–present Single transferable vote
British Columbia[23] Canada 1926–1955 Instant-runoff voting
Maine[42] United States 2018–present Instant-runoff voting ("Ranked-choice 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. Full preferential voting for lower house since 1981.
North Carolina[43] 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.[44] In the 2018 elections, the first ones conducted under the new legislation, the city of London used ranked balloting,[45] while the cities of Kingston and Cambridge held referendums on whether to adopt ranked ballots for the next municipal elections in 2022.[46]
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[47] 1975 only Instant-runoff voting
Aspen, CO[48] 2009 only Instant-runoff voting
Berkeley, CA[49] 2010–present Instant-runoff voting
Burlington, VT[50] 2005–2010 Instant-runoff voting ended after the Burlington mayoral election of 2009
Cambridge, MA[51] 1941–present Single transferable vote
Hendersonville, NC[52] 2007–2013 Instant-runoff voting part of a statewide pilot program,[53] deauthorized in 2013[54]
London 2000–present[55] Supplementary vote[56]
London, Ontario[57] 2018 – present Instant-runoff voting
Memphis, TN[10] 2011–present Instant-runoff voting
Minneapolis, MN[58] 2009–present Instant-runoff voting
Oakland, CA[49] 2010–present Instant-runoff voting
Portland, ME[10] 2011–present Instant-runoff voting
San Francisco, CA 2004–present[59] Instant-runoff voting[10]
San Leandro, CA[49] 2010–present Instant-runoff voting
Santa Fe, NM 2018–present Instant-runoff voting
St. Paul, MN 2011–present[60] Instant-runoff voting[61]
Takoma Park, MD[62] 2006–present Instant-runoff voting
Telluride, CO[63] 2011–present Instant-runoff voting

### International organizations

Organization Years in use Type Notes
European Union[64] option to use single transferable vote Member countries can use either 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.[65]

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1. ^ 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.
2. ^ a b c Toplak, Jurij (2017). "Preferential Voting: Definition and Classification". Lex Localis – Journal of Local Self-Government. 15 (4): 737–761. doi:10.4335/15.4.737-761(2017).
3. ^ Toplak, Jurij (2006). "The parliamentary election in Slovenia, October 2004". Electoral Studies. 25 (4): 825–831. doi:10.1016/j.electstud.2005.12.006.
4. ^ a b c d Mankiw, Gregory (2012). Principles of Microeconomics (6th ed.). South-Western Cengage Learning. pp. 475–479. ISBN 978-0538453042.
5. ^ Hamlin, Aaron (October 6, 2012). "Interview with Dr. Kenneth Arrow". The Center for Election Science. Center for Election Science. CES: you mention that your theorem applies to preferential systems or ranking systems. ... But the system that you're just referring to, Approval Voting, falls within a class called cardinal systems. ... Dr. Arrow: And as I said, that in effect implies more information. ... I’m a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.
6. ^ "Electoral Systems in Europe: An Overview". Brussels: European Centre for Parliamentary Research and Documentation. October 2000. Retrieved November 7, 2019.
7. ^ Farrell, David M.; McAllister, Ian (2004-02-20). "Voter Satisfaction and Electoral Systems: Does Preferential Voting in Candidate-Centered Systems Make A Difference". Cite journal requires |journal= (help)
8. ^ Saari, Donald (1995). Basic Geometry of Voting. Springer. p. 46. ISBN 9783540600640.
9. ^ a b c d Grofman, Bernard; Feld, Scott L. (2004). "If you like the alternative vote (a.k.a. the instant runoﬀ), then you ought to know about the Coombs rule" (PDF). Electoral Studies. 23 (4): 641–659. doi:10.1016/j.electstud.2003.08.001.
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15. ^ Reilly, Benjamin (2002). "Social Choice in the South Seas: Electoral Innovation and the Borda Count in the Pacific Island Countries". International Political Science Review, Vol 23, No. 4, 355–72
16. ^ Election database February 1, 2004
17. ^
18. ^ Dublin County Returning Officer complete table of votes cast Dublin North (zip file)
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28. ^ The fact that Hong Kong began using preferential voting in 1998 can be seen from two sources:
• Minutes from a 1997 LegCo meeting include a proposal to use "preferential elimination voting" for the three smallest functional constituencies. See, "Legislative Council Bill (Minutes) 11 Sept 97". The Legislative Council Commission. Retrieved July 2, 2012.
• 1998 is the first year "preferential elimination voting" can be found in the Hong Kong yearbook. See, "The Electoral System: b. Functional Constituency". Hong Kong Yearbook 1998. Government Information Centre of Hong Kong. Retrieved July 2, 2012.
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44. ^
45. ^ "London, Ont., votes to become 1st Canadian city to use ranked ballots". CBC News Windsor, May 2, 2017.
46. ^
47. ^ "Instant Runoff Voting (IRV): History of Use in Ann Arbor". Green Party of Michigan. 1998. Archived from the original on February 8, 2012. Retrieved June 29, 2012.
48. ^ Urquhart, Janet (June 28, 2012). "Marks prevails in lawsuit over Aspen election ballots". The Aspen Times. Retrieved June 29, 2012.
49. ^ a b c "Ranked-Choice Voting". Alameda County Registrar of Voters. Retrieved June 29, 2012.
50. ^ McCrea, Lynne (March 3, 2010). "Burlington Voters Repeal Instant Runoff Voting". Vermont Public Radio. Retrieved November 7, 2019.
51. ^ "Choice Voting in Cambridge". FairVote.
52. ^ "New Voting Method for November 6, 2007: Hendersonville Pilots Instant Runoff Voting" (PDF). Henderson County Board of Elections. 2007. Retrieved June 29, 2012.
53. ^ Harbin, John (April 8, 2011). "Hendersonville votes to keep instant runoff ballots". BlueRidgeNow.com. Times-News. Retrieved June 29, 2012.
54. ^ Joyce, Robert (November 2013). "Instant Runoff Voting". Coates' Canons. Retrieved 2018-07-11. The 2013 General Assembly repealed all legislation authorizing instant runoff elections in North Carolina.
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58. ^ Gilbert, Curtis (November 2, 2009). "Instant runoff voting FAQ". Minnesota Public Radio. Retrieved June 29, 2012.
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60. ^ Baran, Madeleine (November 7, 2011). "Election Day in St. Paul Tuesday". Minnesota Public Radio. Retrieved June 29, 2012.
61. ^ "Ranked Voting Information". Ramsey County. Retrieved June 29, 2012.
62. ^ "City of Takoma Park Election 2011". City Of Takoma Park. 2011. Retrieved June 29, 2012.
63. ^ "Instant Runoff Voting Brochure". Town of Telluride. 2011. Retrieved June 29, 2012.
64. ^ "Country Profile: European Union: Learn More". ElectionGuide. International Foundation for Electoral Systems. Retrieved November 7, 2019.
65. ^ "Eurovision Song Contest voting system". European Broadcasting Union. Retrieved February 10, 2020.