Ranked Voting
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Ranked Voting

Ranked voting, also known as ranked-choice voting or preferential voting, is any election 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. There are multiple ways in which the rankings can be counted to determine which candidate or candidates (or other outcome(s)) is or are elected (or adopted), and these different methods may produce different results from the same set of ballots. Ranked voting is different from cardinal voting, where candidates are independently rated, rather than ranked.[1]

The term "ranked-choice voting" (RCV) is used by the US organization FairVote to refer to the use of ranked ballots with specific counting methods. 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] 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] 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".


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

This system simulates a series of runoff elections. Voters may rank all candidates as their 1st choice, 2nd choice, 3rd choice, and so on rather than indicating support for only one candidate.[10] A candidate with the majority (more than 50%) of first choice votes (the number one spot on the ballot) wins the election outright.[11] If no candidate is the first choice of more than half of the voters, then all votes cast for the candidate with the fewest 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.[12][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]

Contingent Vote

With the contingent vote, all candidates other than the two receiving the most first-choice votes are eliminated at once, and choices are reallocated to one of those two.

Single transferable vote

Sample ballot of ranked voting using written names

This method 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 above the quota are redistributed to voters' next choice candidate(s). Once this is done, if not all seats 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, and its outcome should be proportional to the electorate.[13] Voters can also vote for members of different political parties on the same ballot, rather than of just one party.[14]

When single transferable vote is used for single-winner elections, it becomes equivalent to instant-runoff voting.[15] Both methods may be known as ranked-choice voting in the US.

Condorcet method

Condorcet and Smith-efficient methods elect the candidate who has a pairwise majority over every other candidate. Smith-efficient methods always elect from the Smith set, which is the Condorcet winner if it contains only one candidate, and otherwise is the smallest set of candidates for which any member of the Smith set would be the Condorcet winner if all other members were removed.

Many Condorcet methods, including Schulze beatpath, ranked pairs, and minimax, can be counted by the sum of pairwise races, rather than counting all ranked ballots. Not all Condorcet methods are Smith-efficient; minimax, for example, only elects the Condorcet winner when one exists, and otherwise can elect any candidate including the Condorcet loser.

Any rule can be made Condorcet-consistent or Smith-efficient by electing the Condorcet winner if one exists, such as in Black's method, or eliminating all non-Smith candidates before applying the rule, such as with Smith/IRV.

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.[16] Plurality, anti-plurality, and Borda count are the three different methods in a positional voting. A candidate will receive a certain number of points based on the voter's ranking.[17]

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.[]

The Borda count does not exhibit independence of irrelevant alternatives[4] or independence of clones meaning the outcome it selects is dependent on the other choices present. In large scale elections, the Borda Count is only weakly manipulated by adding candidates, called clones, whose views are similar to the preferred candidate's, but in a small committee election it can more easily manipulated. An example of this strategy can be seen in Kiribati's 1991 presidential nomination contest.[18]


Tennessee and its four major cities: Memphis in the south-west; Nashville in the centre, Chattanooga in the south, and Knoxville in the east

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities and that everyone wants to live as near to the capital as possible.

The candidates for the capital are:

  • Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
  • Nashville, with 26% of the voters, near the center of the state
  • Knoxville, with 17% of the voters
  • Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

Instant runoff voting

Instant runoff voting tabulates only the first non-eliminated candidate on each ballot as if the election were by first-past-the-post voting. When no candidate has a simple majority, the candidate with the fewest votes is eliminated, and the tabulation is repeated.

Here, Chattanooga is eliminated, transferring votes to Knoxville. This gives Knoxville more votes in total than Nashville, leaving Nashville with the fewest votes. After Nashville is eliminated, the highest remaining candidate on those ballots is Knoxville, and so votes are transferred to Knoxville. Knoxville then wins with a majority.

Votes in round/
City Choice
1st 2nd 3rd
Memphis 42% 42% 42%
Nashville 26% 26%
Knoxville 17% 32% 58%
Chattanooga 15%

IRV always elects from the mutual majority set, demonstrated here: all ballots have either ranked Memphis above the other three, or ranked the other three above Memphis. A majority have ranked the other three above Memphis, and no pair of candidates are ranked in the top two spots on a majority of ballots. Thus the mutual majority set is Knoxville, Nashville, and Chattanooga.

Accounting for only ballots in the mutual majority, a majority rank these rank both Chattanooga and Knoxville in the top two positions. Because of this, only Chattanooga or Knoxville can win. Among the ballots ranking Chattanooga And Knoxville in the first two positions, a majority ranked Knoxville first, so only Knoxville can win.

Ranked Pairs

Ranked pairs counts the pairwise results by giving each candidate on a ballot a vote against each lower-ranked candidate and each unranked candidate in their respective two-candidate elections. Unranked and tied candidates receive no votes against one another. The pairwise elections are sorted from largest to smallest strength of victory--either by the number of votes the winning candidate receives or the percent of votes they receive; which measure is chosen generally doesn't affect the outcome.

Pair Winner
Chattanooga (83%) vs. Knoxville (17%) Chattanooga 83%
Nashville (68%) vs. Knoxville (32%) Nashville 68%
Nashville (68%) vs. Chattanooga (32%) Nashville 68%
Memphis (42%) vs. Nashville (58%) Nashville 58%
Memphis (42%) vs. Chattanooga (58%) Chattanooga 58%
Memphis (42%) vs. Knoxville (58%) Knoxville 58%

Above, Nashville defeats all opponents and is the Condorcet winner. This gives a clear outcome.

Hypothetically, if Knoxville had received a majority over Nashville, this would produce Chattanooga defeating Knoxville, which then defeats Nashville, which then defeats Chattanooga. Such a situation is ambiguous, as none of these three are defeated by any other option, but they are also all reachable by following a path of defeats.

Ranked pairs handles this by accepting each pairwise election one at a time, from strongest to weakest, except that if an election would cause a cycle--if a candidate can be found to reach itself by following a path of wins--that pairwise election is discarded. In this hypothetical, the last pair added--Nashville defeating Chattanooga--is discarded, and Chattanooga wins.

Pros and cons of RCV


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.[19] 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.[20]

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.[21][22][23] In 2021, party supporters called for the Labor Party to fully embrace negative campaigning as a necessary tactic to win elections.[24]

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.[25]

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.[26]


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.[27]

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.

Uniqueness of votes

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.[28] 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.[29] For example, in the 2002 Irish general election, the electronic votes were published for the Dublin North constituency.[30] 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


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


Use by politics

Countries and regions

Country Years in use Type Notes
Australia 1918-present[33][34] Single transferable vote, instant-runoff voting From 1949, the single transferable vote method has been used for upper house legislative elections.[35] Instant-runoff voting is used for lower house elections.[36]
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,[37] 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.[35] This is no longer the case.[38]
Fiji 1998-present Instant-runoff voting [39]
Hong Kong 1998-present[40] Instant-runoff voting[41] Instant-runoff voting is only used in the 4 smallest of Hong Kong's 29 functional constituencies.[42] Officially called preferential elimination voting, the system is identical to the instant-runoff voting.[41]
Ireland 1922-present Single transferable vote[43] Single transferable vote is prescribed by Constitution or statute for all public elections.[43] 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.[43]
Malta 1921-present Single transferable vote [35]
Nauru 1968-present[35] Borda count Nauru uses the Dowdall system, a variant of the Borda count that behaves more like FPTP.[44][45]
New Zealand 2004-present[46] Single transferable vote[47] Instant-runoff voting is used in only some single-seat elections, such as district health boards as well as some city and district councils.[47]
Northern Ireland 1973-present[35] Single transferable vote[48] Used for local government, European Parliament and the regional legislature, but not elections to Westminster.
Papua New Guinea 2007-present[49] Instant-runoff voting[9] Between 1964 and 1975, Papua New Guinea used a system that allowed voters the option of ranking candidates.[35] Currently, voters can rank only their top three choices.[50]
Slovenia 2000-present[51] Borda count Only two seats, which are reserved for Hungarian and Italian minorities, are decided using a Borda count.[52]
Sri Lanka 1978-present Contingent vote and open list Contingent vote is used for presidential elections,[35] and open list for legislative elections.[53]
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[54] 1979-1985 Instant-runoff voting Was only used for white candidates

Federal provinces or states

Province/state Country Years in use Type Notes
Alaska[55] United States 2022 Instant-runoff voting Approved by Alaska voters in 2020 via ballot measure.
Alberta[35] Canada 1952-1954 Instant-runoff voting
Australian Capital Territory[35] Australia 1993-present Single transferable vote
British Columbia[35] Canada 1926-1955 Instant-runoff voting
Maine[56] 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.
Manitoba[35] Canada 1927-1936 Instant-runoff voting
New South Wales[35] 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[57] 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[35] 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.[58] In the 2018 elections, the first ones conducted under the new legislation, the city of London used ranked balloting,[59] while the cities of Kingston and Cambridge held referendums on whether to adopt ranked ballots for the next municipal elections in 2022.[60]
Queensland[35] 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[35] 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[35] 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[35] 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[35] 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.


City/town Years in use Type Notes
Ann Arbor, MI[61] 1975 only Instant-runoff voting
Aspen, CO[62] 2009 only Instant-runoff voting
Berkeley, CA[63] 2010-present Instant-runoff voting
Burlington, VT[64][65][66] 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[67] 1941-present Single transferable vote
Hendersonville, NC[68] 2007-2013 Instant-runoff voting part of a statewide pilot program,[69] deauthorized in 2013[70]
London 2000-present[71] Supplementary vote[72]
London, Ontario[73] 2018 - present Instant-runoff voting
Memphis, TN[12] 2011-present Instant-runoff voting
Minneapolis, MN[74] 2009-present Instant-runoff voting
New York City, NY[75][76] 2021-present Instant-runoff voting Only applies to primaries and special elections for municipal offices
Oakland, CA[63] 2010-present Instant-runoff voting
Portland, ME[12] 2011-present Instant-runoff voting
San Francisco, CA 2004-present[77] Instant-runoff voting[12]
San Leandro, CA[63] 2010-present Instant-runoff voting
Santa Fe, NM 2018-present Instant-runoff voting
St. Paul, MN 2011-present[78] Instant-runoff voting[79]
Takoma Park, MD[80] 2006-present Instant-runoff voting
Telluride, CO[81] 2011-present Instant-runoff voting

International organizations

Organization Years in use Type Notes
European Union[82] option to use single transferable vote Member countries can use either party-list proportional representation (not a type of preferential voting)[] 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.[83]

See also


  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 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.
  10. ^ FairVote.org. "Ranked Choice Voting / Instant Runoff". FairVote. Retrieved .
  11. ^ a b "Ranked-choice voting (RCV)". Ballotpedia. Retrieved .
  12. ^ a b c d Bialik, Carl (May 14, 2011). "Latest Issue on the Ballot: How to Hold a Vote". Wall Street Journal. Retrieved 2012.
  13. ^ "Glossary". ElectionGuide. International Foundation for Electoral Systems. Archived from the original on July 17, 2012.
  14. ^ Affairs, The Department of Internal. "STV Information". www.stv.govt.nz. Retrieved .
  15. ^ "Q&A: Electoral reform and proportional representation". BBC. 2010-05-11. Retrieved 2010.
  16. ^ Saari, Donald G. (1995). Basic Geometry of Voting. Springer-Verlag. pp. 101-103. ISBN 3-540-60064-7.
  17. ^ Regenwetter, Michel; Tsetlin, Ilia (2004). "Approval voting and positional voting methods: Inference, relationship, examples". Social Choice and Welfare. 22 (3): 539-566. doi:10.1007/s00355-003-0232-z. ISSN 0176-1714. JSTOR 41106611. S2CID 16226738.
  18. ^ 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
  19. ^ "Is ranked-choice voting a better way to decide elections?". news.yahoo.com. Retrieved .
  20. ^ Bol, Damien; Verthé, Tom (2019-09-30). "Strategic Voting Versus Sincere Voting". Oxford Research Encyclopedia of Politics. doi:10.1093/acrefore/9780190228637.013.932. ISBN 9780190228637. Retrieved .
  21. ^ Carson, Andrea (2014-04-03). "Negative campaigning, issue salience and vote choice: assessing the effects of the Australian Labor party's 2016 "Mediscare" campaign". Journal of Elections, Public Opinion and Parties. 30 (1): 83-104. doi:10.1080/17457289.2018.1563093.
  22. ^ "Federal election 2016: negative tactics cast shadow over poll". The Australian. June 24, 2016.
  23. ^ "Here come the negative ads". August 20, 2013.
  24. ^ Kevin Rudd. "Winning at all costs: Rudd's case for courage and conflict".
  25. ^ "Pros and Cons of Instant Runoff (Ranked Choice) Voting". MyLO. 2017-12-22. Retrieved .
  26. ^ "Ranked Choice Voting Can Make Government Look More Like Us". www.wbur.org. Retrieved .
  27. ^ "In Our View: Ranked-choice voting cons outweigh the pros". The Columbian. Retrieved .
  28. ^ Election database February 1, 2004
  29. ^ First Report of the Commission on Electronic Voting, Part 5: Accuracy and Secrecy of the System, First Report of the Commission on Electronic Voting - December 2004
  30. ^ Dublin County Returning Officer complete table of votes cast Dublin North (zip file)
  31. ^ Wilf, Herbert S. (January 1994) [1990]. "Chapter 5: Analytic and asymptotic methods". generatingfunctionology (Second ed.). Academic Press. pp. 175-76. ISBN 0127519564. Retrieved .
  32. ^ Sloane, N. J. A. (ed.). "Sequence A007526". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  33. ^ "Our electoral system". About Australia. Australian Department of Foreign Affairs and Trade. May 2008. Retrieved 2012.
  34. ^ Brent, Peter. Short History of Preferential Voting. Mumble Blog, The Australian. April 17, 2011.
  35. ^ a b c d e f g h i j k l m n o p q r Sawer, Marian (2001). Elections: Full, Free & Fair. Federation Press. pp. 93-95. ISBN 978-1862873957.
  36. ^ "Country Profile: Australia: Learn More". ElectionGuide. International Foundation for Electoral Systems. Retrieved 2019.
  37. ^ "Liberals vote overwhelmingly in favour of one-member, one-vote". Liberal.ca. 2 May 2009. Archived from the original on 4 May 2011. Retrieved 2011.
  38. ^ "Country Profile: Estonia: Learn More". ElectionGuide. International Foundation for Electoral Systems. Retrieved 2019.
  39. ^ "Section 54: Voting and other matters". Constitution of Fiji. International Constitutional Law Project. Retrieved 2012.
  40. ^ 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 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 2012.
  41. ^ a b "Chapter. 3, Functional Constituencies: The Preferential Elimination System of the 4 SFCs" (PDF). Guidelines on Election-related Activities in respect of the Legislative Council Election. Hong Kong Electoral Affairs Commission. Retrieved 2012.
  42. ^ "Functional Constituency Elections". 2000 Legislative Council Elections. Hong Kong Electoral Affairs Commission. 2000. Retrieved 2012.
  43. ^ a b c Farrell, David; Synott, Richard (2017). "The electoral system". In Coakley, John; Gallagher, Michael (eds.). Politics in the Republic of Ireland. Routledge. ISBN 978-1-317-31269-7.
  44. ^ "Country Profile: Nauru: Learn More". ElectionGuide. International Foundation for Electoral Systems. Retrieved 2019.
  45. ^ Fraenkel, Jon; Grofman, Bernard (2014-04-03). "The Borda Count and its real-world alternatives: Comparing scoring rules in Nauru and Slovenia". Australian Journal of Political Science. 49 (2): 186-205. doi:10.1080/10361146.2014.900530. S2CID 153325225.
  46. ^ "STV legislation, background and further information". New Zealand Department of Internal Affairs. Retrieved 2015.
  47. ^ a b "STV - It's Simple To Vote". New Zealand Department of Internal Affairs. 2010. Retrieved 2012.
  48. ^ "Voting systems in Northern Ireland". Electoral Office for Northern Ireland. Retrieved 2020.
  49. ^ Blackwell, Eoin (June 20, 2012). "Observers urge peaceful PNG election". The Sydney Morning Herald. Retrieved 2012.
  50. ^ "Voting". Electoral Commission of Papua New Guinea. 2011. Retrieved 2012.
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