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{{Short description|Family of electoral systems}}
{{Short description|Voting systems that use ranked ballots}}
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{{Electoral systems}}'''Ranked voting''' is any [[voting system]] that uses voters' [[Ordinal utility|orderings (rankings)]] of candidates to choose a single winner or multiple winners. More formally, a ranked rule is one that depends only on which of two candidates is preferred by a voter, and as such does not incorporate any information about [[Rated voting|intensity of preferences]]. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them [[Comparison of electoral systems|very different properties]].
{{Electoral systems}}


For example, in the [[Borda count|Borda method]], the 1st, 2nd, 3rd... candidates on each ballot receive {{Math|1, 2, 3…}} points, and the candidate with the smallest number of points is elected. In [[Instant-runoff voting|instant-runoff (ranked-choice) voting]], lower preferences are used as contingencies (back-up preferences), and are only applied when all higher-ranked preferences on a ballot have been eliminated.
The term '''ranked voting''', also known as '''preferential voting''' or '''ranked-choice voting''', pertains to any [[voting system]] where voters indicate a [[Ranking|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.


Ranked voting systems are usually contrasted with [[rated voting]] methods, which allow voters to indicate ''how strongly'' they support different candidates (e.g. on a scale from 0-10).<ref>{{Cite book |last=Riker |first=William Harrison |title=Liberalism against populism: a confrontation between the theory of democracy and the theory of social choice |date=1982 |publisher=Waveland Pr |isbn=0881333670 |pages=29–30 |oclc=316034736 |quote=''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.}}</ref> Ranked vote systems (ordinal systems) produce more information than X voting systems such as [[first-past-the-post voting]]. Rated voting systems use more information than ordinal ballots; as a result, they are not subject to many of the problems with ranked voting (including results like [[Arrow's impossibility theorem|Arrow's theorem]]).<ref>{{Cite web |last=Hamlin |first=Aaron |date=2012-10-06 |title=Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow |url=https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |url-status=dead |archive-url=https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |archive-date=2023-06-05 |accessdate= |work=The Center for Election Science}}<poem>'''Dr. Arrow:''' Well, 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.</poem></ref><ref>{{Cite web |last=Hamlin |first=Aaron |date=2012-10-06 |title=Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow |url=https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |url-status=dead |archive-url=https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |archive-date=2023-06-05 |accessdate= |work=The Center for Election Science}}<poem>'''Dr. Arrow:''' Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.</poem></ref><ref>{{Cite web |last=Hamlin |first=Aaron |date=2012-10-06 |title=Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow |url=https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |url-status=dead |archive-url=https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |archive-date=2023-06-05 |accessdate= |work=The Center for Election Science}} {{pbl|'''CES:''' Now, you mention that your theorem applies to preferential systems or ranking systems.
Ranked voting systems are opposed to [[cardinal voting]] methods, which allow voters to indicate ''how strongly'' they support different candidates (e.g. on a scale from 0-10). Cardinal ballots provide more information than ordinal ballots and as a result, they are not subject to many of the problems in ranked-choice voting (such as [[Arrow's impossibility theorem]]).
'''Dr. Arrow:''' Yes.
'''CES:''' But the system that you're just referring to, [[approval voting]], falls within a class called [[cardinal voting|cardinal systems]]. So not within [[ranked voting|ranking systems]].
'''Dr. Arrow:''' And as I said, that in effect implies more information.}}</ref>


The most commonly-used example of a ranked-choice system is the familiar [[Plurality voting|plurality voting rule]], which gives one "point" (vote) to the candidate ranked first, and zero points to all others (making additional marks unnecessary). This is an example of a [[Positional voting|positional system]], a system that assigns points to candidates based on their ranking in the ordering. Another example ([[Dowdall system|Dowdall's method]]) assigns {{Math|1, {{frac|1|2}}, {{frac|1|3}}}}... points to the 1st, 2nd, 3rd... candidates on each ballot. In addition, some countries elect policymakers by [[instant-runoff voting]], a staged variant of the plurality system.
Although not typically described as such, the most common ranked voting system is the well-known [[Plurality voting|plurality rule]], where each voter gives a single point to the candidate ranked first and zero points to all others. The most common non-[[Degeneracy (mathematics)|degenerate]] ranked voting rule is the closely-related [[Instant-runoff voting|instant-runoff (ranked-choice) voting]], a staged variant of the plurality system that repeatedly eliminates last-place plurality winners.<ref>{{Cite web |title=Bill Status H.424: An act relating to town, city, and village elections for single-seat offices using ranked-choice voting |url=https://legislature.vermont.gov/bill/status/2024/H.424 |access-date=2024-03-23 |website=legislature.vermont.gov |quote=Condorcet winner. If a candidate is the winning candidate in every paired comparison, the candidate shall be declared the winner of the election.}}</ref>


In the US, the term ''ranked-choice voting'' is most commonly used by organizations like [[FairVote]] and [[RepresentUs]] to refer to [[instant-runoff voting]] or [[single transferable vote]]. However, it has also been used for other ranked voting systems.<ref>{{Cite web |title=Bill Status H.424: An act relating to town, city, and village elections for single-seat offices using ranked-choice voting |url=https://legislature.vermont.gov/bill/status/2024/H.424 |access-date=2024-03-23 |website=legislature.vermont.gov |quote=Condorcet winner. If a candidate is the winning candidate in every paired comparison, the candidate shall be declared the winner of the election.}}</ref>
In the United States and Australia, the terms ''ranked-choice voting'' and ''preferential voting'' are usually used to refer to the [[Instant-runoff voting|alternative]] or [[single transferable vote]], a [[misnomer]] arising by way of [[conflation]]. However, these terms have also been used to mean ranked voting systems in general, leading most [[Social choice theory|social choice theorists]] to recommend the use of more precise terms like [[Instant-runoff voting|instant-runoff voting (IRV)]].


==History of ranked voting==
==History of ranked voting==
[[File:IRV counting flowchart.svg|thumb|Typical counting process of a single-seat ranked choice voting election, developed by William Robert Ware]]
{{main|Electoral system|Social choice theory}}
{{main|Electoral system|Social choice theory}}
The earliest known proposals for a ranked voting system other than [[Plurality voting|plurality]] can be traced to the works of [[Ramon Llull]] in the late 13th century, who developed what would later be known as [[Copeland's method]]. Copeland's method was devised by [[Ramon Llull]] in his 1299 treatise ''Ars Electionis,'' which was discussed by [[Nicholas of Cusa]] in the fifteenth century.<ref name=":0">George G. Szpiro, "Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present" (2010).</ref><ref>{{Cite journal |last=Colomer |first=Josep M. |date=2013-02-01 |title=Ramon Llull: from ‘Ars electionis’ to social choice theory |url=https://doi.org/10.1007/s00355-011-0598-2 |journal=Social Choice and Welfare |language=en |volume=40 |issue=2 |pages=317–328 |doi=10.1007/s00355-011-0598-2 |issn=1432-217X}}</ref>


A second wave of analysis began when [[Jean-Charles de Borda]] published a paper in 1781, advocating for the [[Borda count]], which he called the "order of merit". This methodology drew criticism from the [[Marquis de Condorcet]], who developed [[Condorcet method|his own methods]] after arguing Borda's approach did not accurately reflect group preferences, because it was vulnerable to [[Spoiler effect|spoiler effects]] and did not always elect the [[majority-preferred candidate]].<ref name=":0" />
The earliest known proposals for a ranked voting system other than [[Plurality voting|plurality]] can be traced to the works [[Ramon Llull]] in the late 12th century, who developed what would later be known as [[Copeland's method]].


Interest in ranked voting continued throughout the 19th century. Danish pioneer [[Carl Christoffer Georg Andræ|Carl Andræ]] formulated the [[single transferable vote]] (STV), which was adopted by his native [[Denmark]] in 1855. Condorcet had previously considered the single-winner version of it, the [[Instant-runoff voting|instant-runoff]] system, but immediately rejected it as [[Pathological (mathematics)#In voting|pathological]].<ref>{{Cite journal |last=Nanson |first=E. J. |date=1882 |title=Methods of election: Ware's Method |url=https://archive.org/details/transactionsproc1719roya/page/206 |journal=Transactions and Proceedings of the Royal Society of Victoria |volume=19 |pages=206 |quote=The method was, however, mentioned by Condorcet, but only to be condemned.}}</ref><ref>{{Cite book |last=Condorcet |first=Jean-Antoine-Nicolas de Caritat |url=https://books.google.com/books?id=d0cwAAAAYAAJ&q=Oeuvres+de+Condorcet%2C+Volume+13&pg=PA243 |title=Complete Works of Condorcet |date=1788 |volume=13 |publication-date=1804 |pages=243 |language=fr |section=On the Constitution and the Functions of Provincial Assemblies |quote=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.}}</ref>
The modern analysis of ranked voting began when [[Jean-Charles de Borda]] published a paper in 1781, advocating for the [[Borda count]] method that now bears his name. This methodology drew criticism from the [[Marquis de Condorcet]], who developed [[Condorcet method|his own methods]] after arguing Borda's approach did not accurately reflect group preferences.<ref>George G. Szpiro, "Numbers Rule" (2010).</ref>


Theoretical exploration of electoral processes was revived by a 1948 paper from [[Duncan Black]]<ref name="black2">Duncan Black, "On the Rationale of Group Decision-making" (1948).</ref> and [[Kenneth Arrow]]'s investigations into [[social choice theory]], a branch of [[welfare economics]] that extends [[Decision theory|rational choice]] to include community decision-making processes.<ref name="Arrow 1963232">{{Cite book |last=Arrow |first=Kenneth Joseph Arrow |url=http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf |title=Social Choice and Individual Values |date=1963 |publisher=Yale University Press |isbn=978-0300013641 |archive-url=https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf |archive-date=2022-10-09 |url-status=live}}</ref>
Interest in ranked voting continued throughout the 19th century. Danish pioneer [[Carl Christoffer Georg Andræ|Carl Andræ]] formulated the [[single transferable vote]] (STV) system, which was adopted by his native [[Denmark]] in 1855. Condorcet had previously considered the similar [[Instant-runoff voting|instant-runoff]] system before rejecting it as paradoxical.<ref>{{Cite journal |last=Nanson |first=E. J. |date=1882 |title=Methods of election: Ware's Method |url=https://archive.org/details/transactionsproc1719roya/page/206 |journal=Transactions and Proceedings of the Royal Society of Victoria |volume=19 |pages=206 |quote=The method was, however, mentioned by Condorcet, but only to be condemned.}}</ref><ref>{{Cite book |last=Condorcet |first=Jean-Antoine-Nicolas de Caritat |url=https://books.google.com/books?id=d0cwAAAAYAAJ&q=Oeuvres+de+Condorcet%2C+Volume+13&pg=PA243 |section=On the Constitution and the Functions of Provincial Assemblies |date=1788 |title=Complete Works of Condorcet |volume=13 |publication-date=1804 |pages=243 |language=fr |quote=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.}}</ref>

Theoretical exploration of electoral processes was initiated by a 1948 paper from [[Duncan Black]],<ref name="black">Duncan Black, "On the Rationale of Group Decision-making" (1948).</ref> 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]], a branch of [[welfare economics]].


=== Adoption ===
=== Adoption ===
[[Plurality voting]] is the most common voting system. It has been in widespread use since the earliest [[Democracy|democracies]].
[[Plurality voting]] is the most common voting system, and has been in widespread use since the earliest [[Democracy|democracies]].{{Cn|date=July 2024}}


The [[single transferable vote]] (STV) system was first invented by [[Carl Christoffer Georg Andræ|Carl Andræ]] in [[Denmark]], where it was used briefly before being abandoned.{{Cn|date=July 2024}} It was later rediscovered by British lawyer [[Thomas Hare (political reformer)|Thomas Hare]], whose writings soon spread the method throughout the [[British Empire]]. [[Tasmania]] adopted the method in the 1890s, with broader adoption throughout [[Australia]] beginning in the 1910s and 1920s.<ref>Farrell and McAllister, The Australian Electoral System, p. 17</ref> It has been adopted in [[Ireland]], [[South Africa]], [[Malta]], and approximately 20 cities each [[Ranked-choice voting in the United States|in the United States]] and [[Canada]].{{Cn|date=July 2024}}
The [[single transferable vote]] (STV) system was adopted by [[Denmark]] in 1855.


In more recent years, STV has seen a comeback in the United States. In November 2016, the voters of [[Maine]] narrowly passed Question 5, approving [[Instant-runoff voting|ranked-choice voting]] 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.<ref>{{cite web |author=<!--Not stated--> |date=2022-08-23 |title=Ranked Choice Voting in Maine |url=https://www.legislature.maine.gov/lawlibrary/ranked-choice-voting-in-maine/9509 |access-date=2022-11-20 |website=legislature.maine.gov |publisher=State of Maine |quote=}}</ref><ref>{{cite web |last=Piper |first=Kelsey |date=2020-11-19 |title=Alaska voters adopt ranked-choice voting in ballot initiative |url=https://www.vox.com/2020/11/19/21537126/alaska-measure-2-ranked-choice-voting-results |access-date=2022-11-20 |website=vox.com |publisher=Vox Media |quote=}}</ref> However, as before, the system has faced strong opposition. After a series of [[Pathological (mathematics)|electoral pathologies]] in Alaska's [[2022 Alaska's at-large congressional district special election|2022 congressional special election]], a poll found 54% of Alaskans supported a repeal of the system; this included a third of the voters who had supported [[Mary Peltola|Peltola]], the ultimate winner in the election.<ref name="North">{{cite web |title=North to the Future: Alaska’s Ranked Choice Voting System is Praised and Criticized Nationally |url=https://alaskapublic.org/2023/09/19/north-to-the-future-alaskas-ranked-choice-voting-system-is-praised-and-criticized-nationally/ |publisher=Alaska Public Media}}</ref>
Ranked voting first saw governmental use in the 1890s in Tasmania, deploying the STV system. Its broader adoption in Australia began in the 1910s.<ref>Farrell and McAllister, The Australian Electoral System, p. 17</ref> 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. STV was adopted for governmental elections in Ireland, Malta, and Canada (specifically Alberta and Manitoba) beginning in the 1920s.


[[Ranked-choice voting in the United States|In the United States]], single-winner ranked voting is used to elect politicians in [[Maine]]<ref>{{cite web |title=Ranked Choice Voting in Maine |url=https://legislature.maine.gov/lawlibrary/ranked-choice-voting-in-maine/9509 |access-date=21 October 2021 |website=Maine State Legislature}}</ref> and [[Alaska]].<ref>{{cite web |title=Alaska Better Elections Implementation |url=https://www.elections.alaska.gov/Core/RCV.php |access-date=21 October 2021 |website=Alaska Division of Elections}}</ref> [[Nauru]] uses a [[Positional voting|positional method]] called the [[Dowdall system]]. Some [[Local elections in New Zealand|local elections]] in New Zealand use the [[single transferable vote]].<ref>{{Cite web |date=19 September 2017 |title=New Zealand Cities Voting to Implement Ranked Choice Voting |url=https://www.fairvote.org/new_zealand_cities_voting_to_implement_ranked_choice_voting}}</ref>
In more recent years, ranked choice voting has seen greater adoption in the United States. In November 2016, the voters of [[Maine]] narrowly passed Question 5, approving [[instant-runoff voting]] 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.<ref>{{cite web |url=https://www.legislature.maine.gov/lawlibrary/ranked-choice-voting-in-maine/9509 |title=Ranked Choice Voting in Maine |author=<!--Not stated--> |date=2022-08-23 |website=legislature.maine.gov |publisher=State of Maine |access-date=2022-11-20 |quote=}}</ref><ref>{{cite web |url=https://www.vox.com/2020/11/19/21537126/alaska-measure-2-ranked-choice-voting-results |title=Alaska voters adopt ranked-choice voting in ballot initiative|last=Piper |first=Kelsey |date=2020-11-19 |website=vox.com |publisher=Vox Media |access-date=2022-11-20 |quote=}}</ref>


== {{Anchor|Equal ranks|Tied ranks}}Equal-ranked ballots ==
Single-winner ranked voting is used in the election of national-level politicians from the states of [[Maine]]<ref>{{cite web |title=Ranked Choice Voting in Maine |url=https://legislature.maine.gov/lawlibrary/ranked-choice-voting-in-maine/9509 |access-date=21 October 2021 |website=Maine State Legislature}}</ref> and [[Alaska]]<ref>{{cite web |title=Alaska Better Elections Implementation |url=https://www.elections.alaska.gov/Core/RCV.php |access-date=21 October 2021 |website=Alaska Division of Elections}}</ref> within the United States. [[2022 Nauruan parliamentary election|Nauru uses a version of ranked voting]] called the [[Dowdall system]]. For certain [[Local elections in New Zealand|local elections]] in New Zealand, ranked voting is used.<ref>{{Cite web |date=19 September 2017 |title=New Zealand Cities Voting to Implement Ranked Choice Voting |url=https://www.fairvote.org/new_zealand_cities_voting_to_implement_ranked_choice_voting}}</ref> Moreover, [[Ranked-choice voting in the United States|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.
In [[voting]] with [[Ranked voting|ranked ballots]], a '''tied''' or '''equal-rank''' ballot is one where multiple candidates receive the same rank or rating.


In [[Ranked choice voting|ranked-choice runoff]] and [[first-preference plurality]], such ballots are generally discarded in practice. However, in [[social choice theory]] these methods are generally modeled by assuming equal-ranked ballots are "split" evenly between all equal-ranked candidates (e.g. in a two-way tie, each candidate receives half a vote).
A form of ranked voting was also used in [[Slovenia]] for electing the two minority representatives; it was replaced by the first-past-the-post system in 2021.<ref>{{cite web |date=20 February 2021 |title=Narodni skupnosti v prihodnje z enostavnejšim načinom izvolitve predstavnikov |url=https://www.rtvslo.si/radio-koper/prispevki/novice/narodni-skupnosti-v-prihodnje-z-enostavnejsim-nacinom-izvolitve-predstavnikov/570202 |website=Radio Koper - RTV Slovenija}}</ref>

By contrast, the [[Borda count]] and [[Condorcet method|Condorcet methods]] can use different rules for handling equal-ranks. Such rules turn out to have extremely different mathematical properties and behaviors, particularly under [[strategic voting]].


==Theoretical foundations of ranked voting==
==Theoretical foundations of ranked voting==
===Condorcet criterion===
{{main|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]].


=== Majority-rule ===
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.
{{main|Condorcet efficiency|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]], the candidate preferred over all others by a majority of voters. A voting system that always elects this candidate is called a [[Condorcet methods|Condorcet method]].

However, it is possible for an election to have no Condorcet winner, a situation called a [[Condorcet cycle]]. Suppose an election with 3 candidates ''A'', ''B'', and ''C'' has 3 voters''.'' One votes ''A–C–B'', one votes ''B–A–C'', and one votes ''C–B–A''. In this case, no Condorcet winner exists: ''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 rock-paper-scissors style cycle with no Condorcet winner.

=== Social well-being ===
{{Main|Social utility efficiency|Implicit utilitarian voting}}
Voting systems can also be judged on their ability to deliver results that maximize the overall [[Social utility efficiency|well-being of society]], i.e. to choose the best candidate for society as a whole.<ref name=":1">{{Cite journal |last=Weber |first=Robert J. |date=September 1978 |title=Comparison of Public Choice Systems |url=https://cowles.yale.edu/publications/cfdp/cfdp-498 |journal=Cowles Foundation Discussion Papers |publisher=Cowles Foundation for Research in Economics |volume= |pages=16, 38, 62 |id=No. 498 |via=}}</ref>


===Spatial voting models===
===Spatial voting models===
{{main|Median voter theorem}}
{{main|Median voter theorem}}
{{Image frame|width=365|content=[[File:IRVCopeland.png|355px]]|caption=Spatial model diagram}}

{| class="wikitable floatright" style="text-align:center"
! 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.
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.
[[File:IRVCopeland.png|left|thumb|420x420px|A spatial model of voting]]

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


Empirical research has generally found that spatial voting models give a highly accurate explanation of most voting behavior.<ref>T. N. Tideman and F. Plassman, "Modeling the Outcomes of Vote-Casting in Actual Elections" (2012).</ref>
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===
===Other theorems===
{{main|Arrow's impossibility theorem|Gibbard's theorem}}
{{main|Arrow's impossibility theorem|Gibbard's theorem}}
[[Arrow's impossibility theorem]] is a generalization of Condorcet's result on the impossibility of majority rule. It demonstrates that every ranked voting algorithm is susceptible to the [[spoiler effect]]. [[Gibbard's theorem]] provides a closely-related corollary, that no voting rule can have a single, always-best strategy that does not depend on other voters' ballots.
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.


== Examples ==
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,<ref>T. N. Tideman and F. Plassman, "Modeling the Outcomes of Vote-Casting in Actual Elections" (2012).</ref> 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 ==
=== Borda count ===
{{main|Borda count}}
{{main|Borda count|Positional voting}}
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 the Borda count is simple to administer, it does not meet the Condorcet criterion. It is heavily affected by the entry of candidates who have no real chance of winning.
{| class="wikitable floatright" style="text-align:center"
==== Other positional systems ====
|-
Systems that award points in a similar way but possibly with a different formula are called [[Positional voting|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 [[Borda count#dowdall|Dowdall system]] and (1,&nbsp;0,...&nbsp;0) equates to [[first-past-the-post]].
! 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.


=== Instant-runoff (Ranked-choice) voting ===
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 voting|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 [[Borda count#dowdall|Dowdall system]] and (1,&nbsp;0,...&nbsp;, 0) equates to [[first-past-the-post]].

== Alternative vote (instant-runoff voting) ==
{{Main|Instant-runoff voting}}
{{Main|Instant-runoff voting}}
Instant-runoff voting, often conflated with ranked-choice voting in general, is a voting method that recursively eliminates the [[Plurality voting|plurality]] loser of an election until only one candidate is left.
{| class="wikitable floatright" style="text-align:center"
! rowspan="2"| Candidate
! colspan="3"| 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.<ref>[https://www.landtag.ltsh.de/infothek/wahl18/umdrucke/1900/umdruck-18-1916.pdf Alternative II.: Einführung einer Ersatzstimme (in German)]</ref> This proposed system is referred to as the [[spare vote]].<ref>[http://www.dualvoting.com/ What is Dual Level Voting?]</ref>

== 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.

{| class="wikitable floatright" style="text-align:center;"
! rowspan="2" | Candidate
! colspan="4" | Count
|-
! 1st round !! 2nd round !! 3rd round !! 4th round
|-
! A (votes marked A-B)
| 26 (elected) || colspan="3"| Elected
|-
! B (votes marked B-E)
| 18 || 26 (elected) || colspan="2"| Elected
|-
! C (votes marked C-D)
| 17 || 17 || 17 (elected) || 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 || colspan="2"| Eliminated
|-
! F (votes marked F-B)
| 8 || colspan="3"| Eliminated
|}

{{Main|Single transferable vote}}

The [[Single transferable vote]] (STV) is a [[multiwinner voting|multi-winner]] method and a [[Proportional representation|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|Minimax Condorcet method}}
{| class="wikitable floatright" style="text-align:center;border:1px solid"
|-
! colspan="2" rowspan="2" style="white-space: normal;" | Candidate preference comparison
! colspan="3" | Second candidate
|-
! A !! B !! C
|-
! rowspan="3" | 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 and Copeland's method ==
{{Main|Copeland's method}}

{| class="wikitable floatright" style="text-align:center;border:1px solid"
! scope="col" | Candidate
! scope="col" | 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.<ref>{{Cite book|title=Liberalism against populism: a confrontation between the theory of democracy and the theory of social choice|last=Riker|first=William Harrison|date=1982|publisher=Waveland Pr|isbn=0881333670|pages=29–30|oclc=316034736|quote=''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.}}</ref>

* 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.<ref name="jga">J.&nbsp;Green-Armytage, T.&nbsp;N.&nbsp;Tideman and R.&nbsp;Cosman, "Statistical Evaluation of Voting Rules" (2015).</ref>

* 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.<ref>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)</ref>

* [[Nanson's method|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.<ref name=darlington>R. B. Darlington, "Are Condorcet and Minimax Voting Systems the Best?" (v8, 2021).</ref>

* [[Bucklin voting|Bucklin's method]]: Exists in several forms, some of which are Condorcet-consistent.


In the given example, Candidate A is declared winner in the third 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.
* Ranked Pairs and Related Methods: The [[ranked pairs]] method{{Snd}}including the [[Schulze method]], [[Schulze STV]], and [[Split cycle (voting method)|split cycle method]]{{Snd}} are Condorcet-consistent methods of medium computational complexity based on analyzing the cycle structure of ballots.<ref>W. H. Holliday and E. Pacuit, [[arxiv:2004.02350|"Split Cycle: A New Condorcet Consistent Voting Method Independent of Clones and Immune to Spoilers"]] (2021).</ref>


IRV is notable in that it does not fulfill the [[Condorcet winner criterion]], and as a result will not always elect [[majority-preferred candidate]].
* [[Dodgson's method]]: Devised by [[Lewis Carroll]]. It is Condorcet-consistent but computationally complex.


=== Defeat-dropping Condorcet ===
* Expanding approvals rule
{{main|Round-robin voting}}
The defeat-dropping Condorcet methods all look for a Condorcet winner, i.e. a candidate who is not defeated by any other candidate in a one-on-one majority vote. If there is no Condorcet winner, they repeatedly drop (set the margin to zero) for the one-on-one matchups that are closest to being tied, until there is a Condorcet winner. How "closest to being tied" is defined depends on the specific rule. For [[minimax]], the elections with the smallest margin of victory are dropped, whereas in [[ranked pairs]] only elections that create a cycle are eligible to be dropped (with defeats being dropped based on the margin of victory).


== See also ==
== See also ==

Latest revision as of 18:41, 27 August 2024

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Various types of ranked voting ballot
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Cardinal voting

Proportional representation
Party-list

Quota-remainder methods

Approval-based committees

Fractional social choice

Mixed systems
By results of combination
By mechanism of combination

By ballot type

Other systems
Semi-proportional representation
Paradoxes and pathologies
Spoiler effects

Pathological response

Strategic voting

Paradoxes of majority rule

Social and collective choice
Impossibility theorems

Positive results

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Ranked voting is any voting system that uses voters' orderings (rankings) of candidates to choose a single winner or multiple winners. More formally, a ranked rule is one that depends only on which of two candidates is preferred by a voter, and as such does not incorporate any information about intensity of preferences. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them very different properties.

For example, in the Borda method, the 1st, 2nd, 3rd... candidates on each ballot receive 1, 2, 3… points, and the candidate with the smallest number of points is elected. In instant-runoff (ranked-choice) voting, lower preferences are used as contingencies (back-up preferences), and are only applied when all higher-ranked preferences on a ballot have been eliminated.

Ranked voting systems are usually contrasted with rated voting methods, which allow voters to indicate how strongly they support different candidates (e.g. on a scale from 0-10).[1] Ranked vote systems (ordinal systems) produce more information than X voting systems such as first-past-the-post voting. Rated voting systems use more information than ordinal ballots; as a result, they are not subject to many of the problems with ranked voting (including results like Arrow's theorem).[2][3][4]

Although not typically described as such, the most common ranked voting system is the well-known plurality rule, where each voter gives a single point to the candidate ranked first and zero points to all others. The most common non-degenerate ranked voting rule is the closely-related instant-runoff (ranked-choice) voting, a staged variant of the plurality system that repeatedly eliminates last-place plurality winners.[5]

In the United States and Australia, the terms ranked-choice voting and preferential voting are usually used to refer to the alternative or single transferable vote, a misnomer arising by way of conflation. However, these terms have also been used to mean ranked voting systems in general, leading most social choice theorists to recommend the use of more precise terms like instant-runoff voting (IRV).

History of ranked voting

[edit]
Main articles: Electoral system and Social choice theory

The earliest known proposals for a ranked voting system other than plurality can be traced to the works of Ramon Llull in the late 13th century, who developed what would later be known as Copeland's method. Copeland's method was devised by Ramon Llull in his 1299 treatise Ars Electionis, which was discussed by Nicholas of Cusa in the fifteenth century.[6][7]

A second wave of analysis began when Jean-Charles de Borda published a paper in 1781, advocating for the Borda count, which he called the "order of merit". This methodology drew criticism from the Marquis de Condorcet, who developed his own methods after arguing Borda's approach did not accurately reflect group preferences, because it was vulnerable to spoiler effects and did not always elect the majority-preferred candidate.[6]

Interest in ranked voting continued throughout the 19th century. Danish pioneer Carl Andræ formulated the single transferable vote (STV), which was adopted by his native Denmark in 1855. Condorcet had previously considered the single-winner version of it, the instant-runoff system, but immediately rejected it as pathological.[8][9]

Theoretical exploration of electoral processes was revived by a 1948 paper from Duncan Black[10] and Kenneth Arrow's investigations into social choice theory, a branch of welfare economics that extends rational choice to include community decision-making processes.[11]

Adoption

[edit]

Plurality voting is the most common voting system, and has been in widespread use since the earliest democracies.[citation needed]

The single transferable vote (STV) system was first invented by Carl Andræ in Denmark, where it was used briefly before being abandoned.[citation needed] It was later rediscovered by British lawyer Thomas Hare, whose writings soon spread the method throughout the British Empire. Tasmania adopted the method in the 1890s, with broader adoption throughout Australia beginning in the 1910s and 1920s.[12] It has been adopted in Ireland, South Africa, Malta, and approximately 20 cities each in the United States and Canada.[citation needed]

In more recent years, STV has seen a comeback in the United States. In November 2016, the voters of Maine narrowly passed Question 5, approving ranked-choice voting 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.[13][14] However, as before, the system has faced strong opposition. After a series of electoral pathologies in Alaska's 2022 congressional special election, a poll found 54% of Alaskans supported a repeal of the system; this included a third of the voters who had supported Peltola, the ultimate winner in the election.[15]

In the United States, single-winner ranked voting is used to elect politicians in Maine[16] and Alaska.[17] Nauru uses a positional method called the Dowdall system. Some local elections in New Zealand use the single transferable vote.[18]

Equal-ranked ballots

[edit]

In voting with ranked ballots, a tied or equal-rank ballot is one where multiple candidates receive the same rank or rating.

In ranked-choice runoff and first-preference plurality, such ballots are generally discarded in practice. However, in social choice theory these methods are generally modeled by assuming equal-ranked ballots are "split" evenly between all equal-ranked candidates (e.g. in a two-way tie, each candidate receives half a vote).

By contrast, the Borda count and Condorcet methods can use different rules for handling equal-ranks. Such rules turn out to have extremely different mathematical properties and behaviors, particularly under strategic voting.

Theoretical foundations of ranked voting

[edit]

Majority-rule

[edit]
Main articles: Condorcet efficiency and 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, the candidate preferred over all others by a majority of voters. A voting system that always elects this candidate is called a Condorcet method.

However, it is possible for an election to have no Condorcet winner, a situation called a Condorcet cycle. Suppose an election with 3 candidates A, B, and C has 3 voters. One votes A–C–B, one votes B–A–C, and one votes C–B–A. In this case, no Condorcet winner exists: 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 rock-paper-scissors style cycle with no Condorcet winner.

Social well-being

[edit]
Main articles: Social utility efficiency and Implicit utilitarian voting

Voting systems can also be judged on their ability to deliver results that maximize the overall well-being of society, i.e. to choose the best candidate for society as a whole.[19]

Spatial voting models

[edit]
Main article: Median voter theorem

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.

A spatial model of voting

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 the candidate closest to the median of the voter distribution.

Empirical research has generally found that spatial voting models give a highly accurate explanation of most voting behavior.[20]

Other theorems

[edit]
Main articles: Arrow's impossibility theorem and Gibbard's theorem

Arrow's impossibility theorem is a generalization of Condorcet's result on the impossibility of majority rule. It demonstrates that every ranked voting algorithm is susceptible to the spoiler effect. Gibbard's theorem provides a closely-related corollary, that no voting rule can have a single, always-best strategy that does not depend on other voters' ballots.

Examples

[edit]

Borda count

[edit]
Main articles: Borda count and Positional voting

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 the Borda count is simple to administer, it does not meet the Condorcet criterion. It is heavily affected by the entry of candidates who have no real chance of winning.

Other positional systems

[edit]

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.

Instant-runoff (Ranked-choice) voting

[edit]
Main article: Instant-runoff voting

Instant-runoff voting, often conflated with ranked-choice voting in general, is a voting method that recursively eliminates the plurality loser of an election until only one candidate is left.

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

IRV is notable in that it does not fulfill the Condorcet winner criterion, and as a result will not always elect majority-preferred candidate.

Defeat-dropping Condorcet

[edit]
Main article: Round-robin voting

The defeat-dropping Condorcet methods all look for a Condorcet winner, i.e. a candidate who is not defeated by any other candidate in a one-on-one majority vote. If there is no Condorcet winner, they repeatedly drop (set the margin to zero) for the one-on-one matchups that are closest to being tied, until there is a Condorcet winner. How "closest to being tied" is defined depends on the specific rule. For minimax, the elections with the smallest margin of victory are dropped, whereas in ranked pairs only elections that create a cycle are eligible to be dropped (with defeats being dropped based on the margin of victory).

See also

[edit]

References

[edit]
  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. ^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.

    Dr. Arrow: Well, 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.

  3. ^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.

    Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.

  4. ^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.
    CES: Now, you mention that your theorem applies to preferential systems or ranking systems.
    Dr. Arrow: Yes.
    CES: But the system that you're just referring to, approval voting, falls within a class called cardinal systems. So not within ranking systems.
    Dr. Arrow: And as I said, that in effect implies more information.
  5. ^ "Bill Status H.424: An act relating to town, city, and village elections for single-seat offices using ranked-choice voting". legislature.vermont.gov. Retrieved 2024-03-23. Condorcet winner. If a candidate is the winning candidate in every paired comparison, the candidate shall be declared the winner of the election.
  6. ^ a b George G. Szpiro, "Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present" (2010).
  7. ^ Colomer, Josep M. (2013-02-01). "Ramon Llull: from 'Ars electionis' to social choice theory". Social Choice and Welfare. 40 (2): 317–328. doi:10.1007/s00355-011-0598-2. ISSN 1432-217X.
  8. ^ 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.
  9. ^ Condorcet, Jean-Antoine-Nicolas de Caritat (1788). "On the Constitution and the Functions of Provincial Assemblies". Complete Works of Condorcet (in French). Vol. 13 (published 1804). 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.
  10. ^ Duncan Black, "On the Rationale of Group Decision-making" (1948).
  11. ^ Arrow, Kenneth Joseph Arrow (1963). Social Choice and Individual Values (PDF). Yale University Press. ISBN 978-0300013641. Archived (PDF) from the original on 2022-10-09.
  12. ^ Farrell and McAllister, The Australian Electoral System, p. 17
  13. ^ "Ranked Choice Voting in Maine". legislature.maine.gov. State of Maine. 2022-08-23. Retrieved 2022-11-20.
  14. ^ Piper, Kelsey (2020-11-19). "Alaska voters adopt ranked-choice voting in ballot initiative". vox.com. Vox Media. Retrieved 2022-11-20.
  15. ^ "North to the Future: Alaska's Ranked Choice Voting System is Praised and Criticized Nationally". Alaska Public Media.
  16. ^ "Ranked Choice Voting in Maine". Maine State Legislature. Retrieved 21 October 2021.
  17. ^ "Alaska Better Elections Implementation". Alaska Division of Elections. Retrieved 21 October 2021.
  18. ^ "New Zealand Cities Voting to Implement Ranked Choice Voting". 19 September 2017.
  19. ^ Weber, Robert J. (September 1978). "Comparison of Public Choice Systems". Cowles Foundation Discussion Papers. Cowles Foundation for Research in Economics: 16, 38, 62. No. 498.
  20. ^ T. N. Tideman and F. Plassman, "Modeling the Outcomes of Vote-Casting in Actual Elections" (2012).
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