Voting system
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/* Criteria in evaluating single winner voting systems */
{{Otheruses}}{{electoral systems}}
A '''voting system''' allows voters to choose between options, often in an [[election]] where candidates are selected for [[public administration|public office]]. Voting can be also used to award prizes, to select between different plans of action, or by a [[computer program]] to find a solution to a problem. Voting can be contrasted with [[consensus decision making]].
A voting system contains rules for valid voting, and how votes are aggregated to yield a final result. The study of formally defined voting systems is called ''voting theory'', a subfield of [[political science]], [[economics]] or [[mathematics]]. Voting theory began formally in the 18th century and many proposals for voting systems have been made.
Voting systems are either [[majority rule]], [[proportional representation]] or [[plurality voting]]. Given the simplicity of majority rule, those who are unfamiliar with voting theory are often surprised that another voting system exists, or that "majority rule" systems can produce results not supported by a majority. If every election had only two choices, the winner would be determined using majority rule alone. However, when there are three or more options, there may not be a single option that is preferred by a majority. Different voting systems may give very different results, particularly in cases where there is no clear majority preference. {{see|effects of different voting systems under similar circumstances}}
==Aspects of voting systems==
A voting system specifies the form of the ''[[ballot]]'', the set of allowable votes, and the ''tallying method'', an [[algorithm]] for determining the outcome. This outcome may be a single winner, or may involve multiple winners such as in the election of a [[legislative body]]. The voting system may also specify how voting power is distributed among the voters, and how voters are divided into subgroups ([[constituency|constituencies]]) whose votes are counted independently.
The real-world implementation of an election is generally ''not'' considered part of the voting system. For example, though a voting system specifies the ballot abstractly, it does not specify whether the actual physical ballot takes the form of a piece of paper, a punch card, or a [[electronic voting|computer display]]. A voting system also does not specify whether or how votes are kept secret, how to verify that votes are counted accurately, or who is [[enfranchisement|allowed to vote]]. These are aspects of the broader topic of elections and [[election system]]s.
===The ballot===
[[Image:Plurality ballot.svg|thumb|left|In a simple [[Plurality voting system|plurality]] ballot, the voter is expected to mark only one selection.]]
Different voting systems have different forms for allowing the individual to express his or her vote. In [[ranked ballot]] or "preference" voting systems, such as [[Instant-runoff voting]], the [[Borda count]], or a [[Condorcet method]], voters order the list of options from most to least preferred. In [[range voting]], voters rate each option separately on a scale. In [[Plurality voting system|plurality voting]] (also known as "first-past-the-post"), voters select only one option, while in [[approval voting]], they can select as many as they want. In voting systems that allow "plumping", like [[cumulative voting]], voters may vote for the same candidate multiple times.
Some voting systems include additional choices on the ballot, such as [[write-in candidate]]s, a [[none of the above]] option, or a no confidence in that candidate option.
===Weight of votes===
Many elections are held to the ideal of "one person, one vote," meaning that every voter's votes should be counted with equal weight. This is not true of all elections, however. [[Corporation|Corporate]] elections, for instance, usually weight votes according to the amount of stock each voter holds in the company, changing the mechanism to "one share, one vote". Votes can also be weighted unequally for other reasons, such as increasing the voting weight of higher-ranked members of an organization.
Voting weight is not the same thing as [[voting power]]. In situations where certain groups of voters will all cast the same vote (for example, [[political party|political parties]] in a [[parliament]]), voting power measures the ability of a group to change the outcome of a vote. Groups may form [[coalition]]s in order to maximize their voting power.
===Status quo===
Some voting systems are weighted in themselves, for example if a [[supermajority]] is required to change the status quo. An extreme case of this is [[unanimous consent]], where changing the status quo requires the support of every voting member. If the decision is whether to accept a new member into an organization, failure of this procedure to admit the new member is called [[blackball (blacklist)|blackballing]].
A different mechanism that favors the status quo is the requirement for a [[quorum]], which ensures that the status quo remains if not enough voters participate in the vote. Quorum requirements often depend only on the total number of votes rather than the number of actual votes cast for the winning option; however, this can sometimes encourage dissenting voters to refrain from voting entirely in order to prevent a quorum.
===Constituencies===
{{main|Constituency}}
Often the purpose of an election is to choose a [[legislative body]] made of multiple winners. This can be done by running a single election and choosing the winners from the same pool of votes, or by dividing up the voters into constituencies that have different options and elect different winners.
Some countries, like [[Israel]], fill their entire parliament using a single multiple-winner district ([[constituency]]) or [[multiple member electorate]], while others, like the [[Republic of Ireland]] or [[Belgium]], break up their national elections into smaller multiple-winner districts, and yet others, like the [[United States]] or the [[United Kingdom]], hold only single-winner elections. Some systems, like the [[Additional member system]], embed smaller districts within larger ones.
The way in which constituencies are created and assigned seats can dramatically affect the results. [[Apportionment (politics)|Apportionment]] is the process by which states, regions, or larger districts are awarded seats, usually according to population changes as a result of a [[census]]. [[Redistricting]] is the process by which the borders of constituencies are redrawn once apportioned. Both procedures can become highly politically contentious due to the possibility of both [[malapportionment]], where there are unequal representative to population ratios across districts, and [[gerrymandering]], where electoral districts are manipulated for political gain.
A particularly ill case was the English [[Rotten Borough]]s, abandoned towns who still were represented in parliament.
==Single-winner methods==
{{main article|Single-winner voting systems}}
Single-winner systems can be classified based on their ballot type. '''One vote''' systems are those in which a voter picks one choice at a time. In '''ranked''' voting systems, each voter ranks the candidates in order of preference. In '''rated''' voting systems, voters give a score to each candidate.
===Single or sequential vote methods===
[[Image:TRS ballot papers.svg|thumb|250px|right|An example of [[Two round system|runoff voting]]. Runoff voting involves two rounds of voting. Only two candidates survive to the second round.]]
The most prevalent single-winner voting method, by far, is [[Plurality voting system|plurality]] (also called "first-past-the-post", "relative majority", or "winner-take-all"), in which each voter votes for one choice, and the choice that receives the most votes wins, even if it receives less than a majority of votes.
Runoff methods hold multiple rounds of plurality voting to ensure that the winner is elected by a majority. [[Top-two runoff]] voting, the second most common method used in elections, holds a runoff election between the top two options if there is no majority. In [[elimination runoff]] elections, the weakest candidate is eliminated until there is a majority. In an [[exhaustive runoff]] election, no candidates are eliminated, so voting is simply repeated until there is a majority.
A nonpartisan [[primary election]] is also used as a two round runoff process. The two candidates with the most votes in the open primary pass to the general election. The difference between a runoff and an open primary is that a winner is never chosen in the primary, while the first round of a runoff can result in a winner if one candidate has over 50% of the vote.
[[Random ballot]] is a method in which each voter votes for one option, and a single ballot is selected at random to determine the winner. This is mostly used as a tiebreaker for other methods.
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===Ranked voting methods===
{{main|Preferential voting}}
[[Image:Preferential ballot.svg|thumb|right|In a typical ranked ballot, a voter is instructed to place the candidates in order of preference.]]
Also known as ''preferential voting methods'', these methods allow each voter to rank the candidates in order of preference. Often it is not necessary to rank all the candidates: unranked candidates are usually considered to be tied for last place. Some ranked ballot methods also allow voters to give multiple candidates the same ranking.
The most common ranked voting method is [[instant-runoff voting]] (IRV), also known as the "alternative vote" or simply "preferential voting", which uses voters' preferences to simulate an elimination runoff election without multiple voting events. As the votes are tallied, the option with the fewest first-choice votes is eliminated. In successive rounds of counting, the next preferred choice still available from each eliminated ballot is transferred to candidates not yet eliminated. The least preferred option is eliminated in each round of counting until there is a majority winner, with all ballots being considered in every round of counting.
The [[Borda count]] is a simple ranked voting method in which the options receive points based on their position on each ballot. A class of similar methods is called [[positional voting system]]s.
Other ranked methods include [[Coombs' method]], [[Supplementary Vote|Supplementary voting]], [[Bucklin voting]], and [[Condorcet method]].
Condorcet methods, or ''pairwise methods'', are a class of ranked voting methods that meet the [[Condorcet criterion]]. These methods compare every option pairwise with every other option, one at a time, and an option that ''defeats'' every other option is the winner. An option defeats another option if a majority of voters rank it higher on their ballot than the other option.
These methods are often referred to collectively as ''Condorcet methods'' because the Condorcet criterion ensures that they all give the same result in most elections, where there exists a ''Condorcet winner''. The differences between Condorcet methods occur in situations where no option is undefeated, implying that there exists a cycle of options that defeat one another, called a [[Condorcet paradox]] or [[Smith set]]. Considering a generic Condorcet method to be an abstract method that does not resolve these cycles, specific versions of Condorcet that select winners even when no Condorcet winner exists are called ''Condorcet completion methods''.
A simple version of Condorcet is [[Minimax Condorcet|Minimax]]: if no option is undefeated, the option that is defeated by the fewest votes in its worst defeat wins. Another simple method is [[Copeland's method]], in which the winner is the option that wins the most pairwise contests, as in many [[round-robin tournament]]s. The [[Schulze method]] (also known as "[[Schwartz sequential dropping]]", "[[cloneproof Schwartz sequential dropping]]" or the "[[beatpath method]]") and [[Ranked Pairs]] are two recently designed Condorcet methods that satisfy a large number of [[#Criteria in evaluating single winner voting systems|voting system criteria]].
The [[Kemeny-Young method]] is a Condorcet method that fully ranks all the candidates from most popular to least popular.
===Rated voting methods===
[[Image:Approval ballot.svg|thumb|left|On an [[Approval voting|approval]] ballot, the voter may vote for any number of options.]]
Rated ballots allow even more flexibility than ranked ballots, but few methods are designed to use them. Each voter gives a score to each option; the allowable scores could be numeric (for example, from 0 to 100) or could be "[[grade]]s" like A/B/C/D/F.
In [[range voting]], voters give numeric ratings to each option, and the option with the highest total score wins.
[[Approval voting]] where voters may vote for as many candidates as they like can be seen as an instance of range voting where the allowable ratings are 0 and 1.
Rated ballots can be used for ranked voting methods, as long as the ranked method allows tied rankings. Some ranked methods assume that all the rankings on a ballot are distinct, but many voters would be likely to give multiple candidates the same rating on a rated ballot.
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==Multiple-winner methods==
[[Image:German federal election, 2005 - Final.png|thumb|270px|right|Seats won by each party in the [[German federal election, 2005|2005 German federal election]], an example of a proportional voting system.]]
A vote with multiple winners, such as the election of a legislature, has different practical effects than a single-winner vote. Often, participants in a multiple winner election are more concerned with the overall composition of the legislature than exactly which candidates get elected. For this reason, many multiple-winner systems aim for [[proportional representation]], which means that if a given party (or any other political grouping) gets X% of the vote, it should also get approximately X% of the seats in the legislature. Not all multiple-winner voting systems are proportional.
===Non-proportional and semi-proportional methods===
Many multiple-winner voting methods are simple extensions of single-winner methods, without an explicit goal of producing a proportional result. [[Bloc voting]], or ''plurality-at-large'', has each voter vote for ''N'' options and selects the top ''N'' as the winners. Because of its propensity for [[landslide victory|landslide]] victories won by a single winning slate of candidates, bloc voting is non-proportional. Two similar plurality-based methods with multiple winners are the [[Single Non-Transferable Vote]] or SNTV method, where the voter votes for only one option, and [[cumulative voting]], described above. Unlike bloc voting, elections using the Single Nontransferable Vote or cumulative voting may achieve proportionality if voters use [[tactical voting]] or [[strategic nomination]].
Because they encourage proportional results without guaranteeing them, the Single Nontransferable Vote and cumulative voting methods are classified as ''semi-proportional''. Other methods that can be seen as semi-proportional are ''mixed methods'', which combine the results of a plurality election and a party-list election (described below). [[Parallel voting]] is an example of a mixed method because it is only proportional for a subset of the winners.
===Proportional methods===
{{main|Proportional representation}}
Truly proportional methods make some guarantee of proportionality by making each winning option represent approximately the same number of voters. This number is called a ''quota''. For example, if the quota is 1000 voters, then each elected candidate reflects the opinions of 1000 voters, within a margin of error. This can be measured using the [[Gallagher Index]].
Most proportional systems in use are based on [[party-list proportional representation]], in which voters vote for parties instead of for individual candidates. For each quota of votes a party receives, one of their candidates wins a seat on the legislature. The methods differ in how the quota is determined or, equivalently, how the proportions of votes are rounded off to match the number of seats.
The methods of seat allocation can be grouped overall into [[highest averages method]]s and [[largest remainder method]]s. Largest remainder methods set a particular quota based on the number of voters, while highest averages methods, such as the [[Sainte-Laguë method]] and the [[d'Hondt method]], determine the quota indirectly by dividing the number of votes the parties receive by a sequence of numbers.
Independently of the method used to assign seats, party-list systems can be ''open list'' or ''closed list''. In an [[open list]] system, voters decide which candidates within a party win the seats. In a [[closed list]] system, the seats are assigned to candidates in a fixed order that the party chooses. The [[Mixed Member Proportional]] system is a mixed method that only uses a party list for a subset of the winners, filling other seats with the winners of regional elections, thus having features of open list and closed list systems.
In contrast to party-list systems, the [[Single Transferable Vote]] is a proportional representation system in which voters rank individual candidates in order of preference. Unlike party-list systems, STV does not depend on the candidates being grouped into political parties. Votes are transferred between candidates in a manner similar to [[instant runoff voting]], but in addition to transferring votes from candidates who are eliminated, excess votes are also transferred from candidates who already have a quota.
===Semi-proportional methods===
A simpler method called [[Cumulative voting]] (CV) is a semi-proportional voting system in which each voter has ''n'' votes, where ''n'' is the number of seats to be elected. Voters can distribute portions of their vote between a set of candidates, fully upon one candidate, or a mixture. It is considered a ''proportional'' system in allowing a united coalition representing a ''m/(n+1)'' fraction of the voters to be guaranteed to elect ''m'' seats of an ''n''-seat election. For example in a 3-seat election, 3/4 of the voters (if united on 3 candidates) can guarantee control over all three seats. (In contrast, [[plurality at large]], which allows a united coalition (majority) (50%+1) to control all the seats.)
[[Image:Cumulative ballot.svg|thumb|left|This ballot design, used in [[cumulative voting]], allows a voter to split his vote among multiple candidates.]]
Cumulative voting is a common way of holding elections in which the voters have unequal voting power, such as in corporate governance under the "one share, one vote" rule. Cumulative voting is also used as a multiple-winner method, such as in elections for a corporate board.
Cumulative voting is not fully proportional because it suffers from the same [[spoiler effect]] of [[plurality]] without a run-off process. A group of like-minded voters divided among "too many" candidates may fail to elect any winners, or elect fewer than they ''deserve'' by their size. The level of proportionality depends on how well-coordinated the voters are.
[[Limited voting]] is a multi-winner system that gives voters fewer votes than the number of seats to be decided. The simplest and most common form of limited voting is [[Single Non-Transferable Vote]] (SNTV). It can be considered a special variation of cumulative voting where a full vote cannot be divided among more than one candidate. It depends on a statistical distributions of voters to smooth out preferences that CV can do by individual voters.
For example, in a 4-seat election a candidate needs 20% to guarantee election. A coalition of 40% can guarantee 2-seats in CV by perfectly splitting their votes as individuals between 2 candidates. In comparison, SNTV tends towards collectively dividing 20% between each candidate by assuming every coalition voter flipped a coin to decide which candidate to support with their single vote. This limitation simplifies voting and counting, at the cost of more uncertainty of results.
==Criteria in evaluating single winner voting systems==
In the real world, attitudes toward voting systems are highly influenced by the systems' impact on groups that one supports or opposes. This can make the objective comparison of voting systems difficult. In order to compare systems fairly and independently of political ideologies, voting theorists use ''voting system criteria'', which define potentially desirable properties of voting systems mathematically.
It is impossible for one voting system to pass all criteria in common use. Economist [[Kenneth Arrow]] proved [[Arrow's impossibility theorem]], which demonstrates that several desirable features of voting systems are mutually contradictory. For this reason, someone implementing a voting system has to decide which criteria are important for the election.
Using criteria to compare systems does not make the comparison completely objective. For example, it is relatively easy to devise a criterion that is met by one's preferred voting method, and by very few other methods. Doing this, one can then construct a biased argument for the criterion, instead of arguing directly for the method. No one can be the ultimate authority on which criteria should be considered, but the following are some criteria that are accepted and considered to be desirable by many voting theorists:
* [[Majority criterion]] — If there exists a majority that ranks (or rates) a single candidate higher than all other candidates, does that candidate always win?
* [[Monotonicity criterion]] — Is it impossible to cause a winning candidate to lose by ranking him higher, or to cause a losing candidate to win by ranking him lower?
* [[Consistency criterion]] — If the electorate is divided in two and a choice wins in both parts, does it always win overall?
* [[Participation criterion]] — Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below.<ref>One criterion does not necessarily imply the other. For example, range voting complies with participation and consistency, but median ratings satisfies participation and fails consistency.</ref>)
* [[Condorcet criterion]] — If a candidate beats every other candidate in [[Condorcet method|pairwise comparison]], does that candidate always win?
* [[Condorcet loser criterion]] — If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?
* [[Independence of irrelevant alternatives]] — Is the outcome the same after adding or removing non-winning candidates?
* [[Strategic nomination|Independence of clone candidates]] — Is the outcome the same if candidates identical to existing candidates are added?
* [[Reversal symmetry]] — If individual preferences of each voter are inverted, does the original winner never win?
The following table shows which of the above criteria are met by several single-winner systems.
{{Voting system criteria}}
In addition to the above criteria, voting systems are judged using criteria that are not mathematically precise but are still important, such as simplicity, speed of vote-counting, the potential for fraud or disputed results, the opportunity for [[tactical voting]] or [[strategic nomination]], and, for multiple-winner methods, the [[Gallagher Index|degree of proportionality]] produced.
It is also possible to simulate large numbers of virtual elections on a computer and see how various voting systems compare in terms of voter satisfaction. Such simulations are sensitive to their assumptions, particularly with regards to voter strategy, but by varying the assumptions they can give repeatable measures that bracket the best and worst cases for a voting system. To date, the only such simulation to compare a wide variety of voting systems was run by a range-voting advocate and has not been peer-reviewed.<ref>Results are available in an appendix of his unpublished paper here: [http://math.temple.edu/~wds/homepage/rangevote.pdf].</ref>
The [[New Zealand]] [[Royal Commission on the Electoral System]] listed ten criteria for their evaluation of possible [[Electoral reform in New Zealand|new electoral systems for New Zealand]]. These included fairness between political parties, effective representation of minority or special interest groups, political integration, effective voter participation and legitimacy.
[[Bayesian regret]] is an objective numerical measure of quality of a voting system that is sensitive to subjective assumptions about the candidates and electorate. Its least contentious use is to give best-case and worst-case values for a given system.
==History==
===Early democracy===
Voting has been used as an essential feature of [[democracy]] since the 6th century BC, when democracy was introduced by the [[Athenian democracy]]. One of the earliest recorded elections in Athens was a [[Plurality voting system|plurality vote]] that it was undesirable to "win": in the process called [[ostracism]], voters chose the citizen they most wanted to exile for ten years. Most elections in the early [[history of democracy]] were held using plurality voting or some variant, but as an exception, the state of [[Venice]] in the 13th century adopted the system we now know as [[approval voting]] to elect their Great Council.<ref name="mactutorhistory">{{cite web | author=J. J. O'Connor and E. F. Robertson | title=The history of voting | work=The MacTutor History of Mathematics Archive | url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Voting.html | accessmonthday=October 12 | accessyear=2005 }}</ref>
The Venetians' system for [[Doge of Venice#Choosing of the Doge|electing the Doge]] was a particularly convoluted process, consisting of five rounds of drawing lots and five rounds of approval voting. By drawing lots, a body of 30 electors was chosen, which was further reduced to 9 electors by drawing lots again. The electoral college of 9 members elected 40 people by approval voting; those 40 were reduced to form a second electoral college of 12 members by drawing lots again. The second electoral college elected 25 people by approval voting, which were reduced to form a third electoral college of 9 members by drawing lots. The third electoral college elected 45 people, which were reduced to form a fourth electoral college of 11 by drawing lots. They in turn elected a final electoral body of 41 members, who ultimately elected the Doge. Despite its complexity, the system had certain desirable properties such as being hard to game and ensuring that the winner reflected the opinions of both majority and minority factions.<ref name="dogeelection">{{cite web | author=Miranda Mowbray and Dieter Gollmann | title=Electing the Doge of Venice: Analysis of a 13th Century Protocol | url=http://www.hpl.hp.com/techreports/2007/HPL-2007-28R1.html | accessmonthday=July 12 | accessyear=2007 }}</ref> This process was used with little modification from 1268 until the end of the [[Republic of Venice]] in 1797, and was one of the factors contributing to the durability of the republic.
[[Image:Jean Charles Borda.jpg|thumb|left|182px|[[Jean-Charles de Borda]], an early voting theorist.]]
[[Image:Condorcet.jpg|thumb|right|182px|The [[Marquis de Condorcet]], another early voting theorist.]]
===Foundations of voting theory===
Voting theory became an object of academic study around the time of the [[French Revolution]].<ref name="mactutorhistory" /> [[Jean-Charles de Borda]] proposed the [[Borda count]] in 1770 as a method for electing members to the [[French Academy of Sciences]]. His system was opposed by the [[Marquis de Condorcet]], who proposed instead the method of pairwise comparison that he had devised. Implementations of this method are known as [[Condorcet method]]s. He also wrote about the [[Condorcet paradox]], which he called the ''intransitivity of majority preferences''.<ref>{{cite web | author=J. J. O'Connor and E. F. Robertson | title=Marie Jean Antoine Nicolas de Caritat Condorcet | work=The MacTutor History of Mathematics Archive | url=http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Condorcet.html | accessmonthday=October 12 | accessyear=2005 }}</ref>
While Condorcet and Borda are usually credited as the founders of voting theory, recent research has shown that the philosopher [[Ramon Llull]] discovered both the Borda count and a pairwise method that satisfied the [[Condorcet criterion]] in the 13th century. The manuscripts in which he described these methods had been lost to history until they were rediscovered in 2001.<ref>{{cite journal | author=G. Hägele and F. Pukelsheim | title=Llull's writings on electoral systems | journal=Studia Lulliana | year=2001 | volume=3 | pages=3–38 | url=http://www.math.uni-augsburg.de/stochastik/pukelsheim/2001a.html }}</ref>
Later in the 18th century, the related topic of [[Apportionment (politics)|apportionment]] began to be studied. The impetus for research into fair apportionment methods came, in fact, from the [[United States Constitution]], which mandated that seats in the [[United States House of Representatives]] had to be allocated among the states proportionally to their population, but did not specify how to do so.<ref>{{cite web | author=Joseph Malkevitch | title=Apportionment | work=AMS Feature Columns | url=http://www.ams.org/featurecolumn/archive/apportion1.html | accessmonthday=October 13 |accessyear=2005 }}</ref> A variety of methods were proposed by statesmen such as [[Alexander Hamilton]], [[Thomas Jefferson]], and [[Daniel Webster]]. Some of the apportionment methods discovered in the [[United States]] were rediscovered in [[Europe]] in the 19th century, as seat allocation methods for the newly proposed system of [[party-list proportional representation]]. The result is that many apportionment methods have two names: for instance, ''Jefferson's method'' is the same as the [[d'Hondt method]], and ''Webster's method'' is the [[Sainte-Laguë method]].<ref>{{cite web | author=Joseph Malkevitch | title=Apportionment II | work=AMS Feature Columns | url=http://www.ams.org/featurecolumn/archive/apportionII1.html | accessmonthday=October 13 | accessyear=2005 }}</ref>
The [[Single Transferable Vote]] system was devised by [[Carl Andrae]] in [[Denmark]] in 1855, and also in [[England]] by [[Thomas Hare]] in 1857. Their discoveries may or may not have been independent. STV elections were first held in Denmark in 1856, and in [[Tasmania]] in 1896 after its use was promoted by [[Andrew Inglis Clark]]. Party-list proportional representation was first implemented to elect European legislatures in the early 20th century, with [[Belgium]] implementing it first in 1900. Since then, proportional and semi-proportional methods have come to be used in almost all democratic countries, with most exceptions being former [[British Empire|British]] colonies.<ref>{{cite web | title=Proportional Voting Around the World | work=FairVote.org | url=http://www.fairvote.org/?page=53 | accessmonthday=October 13 | accessyear=2005 }}</ref>
===The single-winner revival===
Perhaps influenced by the rapid development of multiple-winner voting methods, theorists began to publish new findings about single-winner methods in the late 19th century. This began around 1870, when [[William Robert Ware]] proposed applying STV to single-winner elections, yielding [[instant runoff voting]].<ref>{{cite web | title=The History of IRV | work=FairVote.org | url=http://www.fairvote.org/irv/vt_lite/history.htm | accessmonthday=November 9 |accessyear=2005 }}</ref> Soon, mathematicians began to revisit Condorcet's ideas and invent new methods for Condorcet completion. [[Edward J. Nanson]] combined the newly described [[instant runoff voting]] with the [[Borda count]] to yield a new Condorcet method called [[Nanson's method]]. [[Charles Dodgson]], better known as Lewis Carroll, published pamphlets on voting theory, focusing in particular on Condorcet voting. He introduced the use of [[Matrix (mathematics)|matrices]] to analyze Condorcet elections, though this, too, had already been done in some form in the then-lost manuscripts of [[Ramon Llull]]. He also proposed the straightforward Condorcet method known as [[Dodgson's method]].
Ranked voting systems eventually gathered enough support to be adopted for use in government elections. In Australia, IRV was first adopted in 1893, and continues to be used along with STV today. In the [[United States]] in the early 20th century, various municipalities began to use [[Bucklin voting]], but the results were not satisfying to voters. Bucklin is no longer used in any government elections, and has even been declared unconstitutional in [[Minnesota]].<ref>{{cite web | title=Municipal Voting System Reform: Overcoming the Legal Obstacles | author=Tony Anderson Solgård and Paul Landskroener | url=http://www2.mnbar.org/benchandbar/2002/oct02/voting.htm | work=Bench & Bar of Minnesota | accessmonthday=November 16 | accessyear=2005 }}</ref>
===Influence of game theory===
After [[John von Neumann]] and others developed the mathematical field of [[game theory]] in the 1940s, new mathematical tools were available to analyze voting systems and strategic voting. This led to significant new results that changed the field of voting theory.<ref name="mactutorhistory" /> The use of mathematical criteria to evaluate voting systems was introduced when [[Kenneth Arrow]] showed in [[Arrow's impossibility theorem]] that certain intuitively desirable criteria were actually mutually contradictory, demonstrating the inherent limitations of voting theorems. Arrow's theorem is easily the single most cited result in voting theory, and it inspired further significant results such as the [[Gibbard-Satterthwaite theorem]], which showed that strategic voting is unavoidable in certain common circumstances.
The use of game theory to analyze voting systems also led to discoveries about the emergent strategic effects of certain systems. [[Duverger's Law]] is a prominent example of such a result, showing that [[Plurality voting system|plurality voting]] often leads to a [[two-party system]]. Further research into the game theory aspects of voting led [[Steven Brams]] and [[Peter Fishburn]] to formally define and promote the use of [[approval voting]] in 1977. While approval voting had been used before that, it had not been named or considered as an object of academic study, particularly because it violated the assumption made by most research that single-winner methods were based on preference rankings.
===Post-1980 developments===
Voting theory has come to focus on [[Voting system|voting system criteria]] almost as much as it does on particular voting systems. Now, any description of a benefit or weakness in a voting system is expected to be backed up by a mathematically defined criterion. Recent research in voting theory has largely involved devising new criteria and new methods devised to meet certain criteria.
[[Political scientist]]s of the 20th century published many studies on the effects that the voting systems have on voters choices and political parties,<ref>Duverger 1954</ref><ref>Rae 1971</ref><ref>Taagapera and Shugart 1989</ref> and on political stability.<ref>Hermens 1941</ref><ref>Lijphart 1994</ref> A few scholars also studied what effects caused a nation to change for a particular voting system.<ref>Lijphart 1985 and Lijphart 1992</ref><ref>Rokkan 1970</ref><ref>Rogowski 1987</ref><ref>Boix 1999</ref><ref>Cox 1997, pp. 15-16</ref> One prominent current voting theorist is [[Nicolaus Tideman]], who formalized concepts such as [[strategic nomination]] and the [[spoiler effect]] in the [[independence of clones criterion]]. Tideman also devised the [[ranked pairs]] method, a [[Condorcet method]] that is not susceptible to [[clone (voting)|clones]]. Also, [[Donald G. Saari]] has brought renewed interest to the [[Borda count]] with the books he has published since 2001. Saari and uses geometric models of [[positional voting system]]s to promote the Borda count.
The increased availability of computer processing has increased the practicality of using the [[Kemeny-Young method|Kemeny-Young]], [[ranked pairs]], and [[Schulze method|Schulze]] methods that fully rank all the choices from most popular to least popular.
The advent of the Internet has increased the interest in voting systems. Unlike many other mathematical fields, voting theory is generally accessible enough to non-experts that new results can be discovered by amateurs, and frequently are. As such, many recent discoveries in voting theory come not from published papers, but from informal discussions among hobbyists on online forums and mailing lists.{{Fact|date=August 2007}}
The study of voting systems has influenced a new push for [[electoral reform]] that is going on today, with proposals being made to replace plurality voting in governmental elections with other methods. Various municipalities in the [[United States]] have begun to adopt [[instant-runoff voting]] in the 2000s. [[New Zealand]] adopted [[Mixed Member Proportional]] for [[Elections in New Zealand|Parliamentary elections]] in 1993 and [[Single Transferable Vote]] for some local elections in 2004 (see [[Electoral reform in New Zealand]]). The [[Canada|Canadian]] province of [[British Columbia]] will hold a [[BC-STV|second referendum]] on adopting [[Single Transferable Vote|STV]] in 2008. The Province of Ontario will hold a Referendum on [[October 10]], [[2007]], on whether to adopt a Mixed Member Proportional system. Moreover, on September 2007, the [[United New Democratic Party]] of [[South Korea]] started the world's first Mobile Voting System to use their presidential primary.<ref>[http://english.kbs.co.kr/news/newsview_sub.php?menu=2&key=2007092827 UNDP Promote Mobile Phone Voting], KBS Global, Retrieved on September 28, 2007</ref> An even wider range of voting systems is now seen in non-governmental organizations.
==See also==
*[[E-democracy]]
*[[Electoral reform]]
*[[Poll]]
*[[Social choice theory]]
*[[Table of voting systems by nation]]
*[[Vote counting system]]
*[[Voting machine]]
==References==
<!-- Note to editors: If you'd like to know how to add automatically-formatted references like these, see [[Wikipedia:Template messages/Sources of articles]]. -->
===General references===
*Arrow, Kenneth J. (1951, 2nd ed., 1963), ''[[Social Choice and Individual Values]]''. New Haven: Yale University Press. ISBN 0-300-01364-7
*{{cite book | author=Colomer, Josep M. ed.| title=Handbook of Electoral System Choice | location=London and New York | publisher=Palgrave-Macmillan| year=2004 | id=ISBN 1-4039-0454-5}}
*{{cite web | author=Cretney, Blake | title=Election Methods Resource | work=condorcet.org | url=http://condorcet.org/emr/ | accessmonthday=October 3 | accessyear=2005 }}
*{{cite web | author=Cranor, Lorrie | title=Vote Aggregation Methods | work=Declared-Strategy Voting: An Instrument for Group Decision-Making | url=http://lorrie.cranor.org/pubs/diss/node4.html | accessmonthday=October 3 | accessyear=2005 }}
*{{cite book | author=Farrell, David M. | title=Electoral Systems: A Comparative Introduction | location=New York | publisher=St. Martin's Press | year=2001 | id=ISBN 0-333-80162-8}}
===Notes===
{{reflist}}
===References===
*Boix, Carles 1999. ''[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=159213 Setting the Rules of the Game: The Choice of Electoral Systems in Advanced Democracies]'' American Political Science Review 93, 609-624.</ref>
*Duverger, Maurice. 1954. Political Parties. New York: Wiley.
*Hermens, Ferdinand A. 1941. Democracy or Anarchy? A Study of Proportional Representation. Notre Dame: University of Notre Dame.
*[[Arend Lijphart|Lijphart, Arend]]
**1985 ''The Field of Electoral Systems Research: A Critical Survey,'' Electoral Studies 4:
**1992 ''Democratization and Constitutional Choices in Czecho-Slovakia, Hungary and Poland, 1989-1991,'' Journal of Theoretical Politics 4: 207-223.
**1994 ''Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945-1990''. Oxford: Oxford University Press, 1994. ISBN 0-19-828054-8.
*Rae, Douglas W. 1971. The Political Consequences of Electoral Laws. New Haven: Yale University Press.
*Rogowski, Ronald. 1987. ''Trade and the Variety of Democratic Institutions,'' International Organization
41: 203-224.
*Rokkan, Stein. 1970. ''Citizens, Elections, Parties: Approaches to the Comparative Study of the Process of Development.'' Oslo: Universitetsforlaget.
*Taagapera, Rein and Matthew S. Shugart. 1989. Seats and Votes: The Effects and Determinants of Electoral Systems. New Haven: Yale University Press.
==External links==
===General===
*[http://www.aceproject.org ACE Electoral Knowledge Network] Expert site providing encyclopedia on Electoral Systems and Management, country by country data, a library of electoral materials, latest election news, the opportunity to submit questions to a network of electoral experts, and a forum to discuss all of the above
*[http://www.idea.int/publications/esd/index.cfm A handbook of electoral system Design] from [http://www.idea.int International IDEA]
*[http://www.accuratedemocracy.com/ Accurate Democracy]: electoral and legislative voting rules
*[http://electorama.com/em Election methods list] A mailing list for technical discussions about election methods.
*[http://wiki.electorama.com/wiki/Main_Page Electowiki] A wiki that focuses on voting theory.
*[http://www.nationalbrainstorm.org National Brainstorming project] Canadian site.
*[http://m-schulze.webhop.net/votedesc.pdf Descriptions of Single-Winner Voting Systems] Paper by Warren D. Smith.
*[http://theorem.ca/~mvcorks/code/voting_methods.html Evaluating Voting Methods] by Matt Corks
*[http://dmoz.org/Society/Politics/Campaigns_and_Elections/Voting_Systems/ Open Directory Project] category on voting systems
*[http://stv.sourceforge.net/ OpenSTV] Software for computing a variety of voting systems including IRV, STV, and Condorcet.
*[http://userfs.cec.wustl.edu/~rhl1/rbvote/ Ranked Ballot Voting Methods]: tutorial, evaluation, and calculator
*[http://www.maa.org/editorial/knot/LiberalArts.html Student's Social Choice] by Alex Bogomolny. Illustrates various concepts of choice using [[Java applet]]s.
*[http://xaravve.trentu.ca/pivato/Teaching/voting.pdf Voting, Arbitration, and Fair Division] by Marcus Pivato.
*[http://bolson.org/voting/ Voting and Election Reform]: election calculator and other resources
*[http://pj.freefaculty.org/Ukraine/PJ3_VotingSystemsEssay.pdf Voting Systems] by Paul E. Johnson. A textbook-style overview of voting methods and their mathematical properties.
*[http://www.votingmachinesprocon.org/votingsystems.htm U.S. Voting System Analysis]
*[http://dca.tufts.edu/features/aas A New Nation Votes: American Elections Returns 1787–1825]
*[http://logicracy.org/ Logicracy] Freely available on-line referendum engine, competence vs. opinion scatter plots, competence-weighted voting, analysis tools, and fraud reduction methods.
*[http://condorcet.org/emr/criteria.shtml Criteria] by Blake Cretney
*[http://userfs.cec.wustl.edu/~rhl1/rbvote/eval.html Evaluation of ranked ballot voting methods] by Rob LeGrand
*[http://fc.antioch.edu/~james_green-armytage/voting.htm Voting Methods]: Tutorial and essays by James Green-Armytage
*[http://www.minorityrights.org/?lid=1075 Electoral systems and the protection and participation of minorities], report by [[Minority Rights Group]], 2006
===Advocacy===
*[http://approvalvoting.org/ Citizens for Approval Voting]
*[http://rangevoting.org/ Center for Range Voting] [http://rangevoting.org/MotPlusMenu.html CRV simplified entry page]
*[http://fairvote.org/ Center for Voting and Democracy] Advocates using IRV in the United States.
*[http://LocalParty.Org/ California LocalParty.Org] Advocates proportional elections in local elections.
*[http://condorcet.org/ condorcet.org] Advocates Condorcet voting and provides links to vote-tallying software.
* [http://www.deborda.org The De Borda Institute] A Northern Ireland-based organisation promoting inclusive voting procedures
*[http://www.discover.com/issues/nov-00/features/featbestman/ May the Best Man Lose] A Discover article on Approval voting and the Borda Count, by Dana Mackenzie.
===Research papers===
*[http://www.mfo.de/programme/schedule/2004/11b/OWR_2004_14.pdf Analysis and Design of Electoral Systems] Proceedings of a seminar at the Mathematical Research Institute at Oberwolfach, Germany.
*[http://home.uchicago.edu/~rmyerson/research/perspec.pdf Analysis of Democratic Institutions: Structure, Conduct and Performance] An article by Roger B. Myerson that analyzes voting systems economically.
* PhD seminar on [http://faculty.fuqua.duke.edu/~rnau/choice/choice09.pdf Choice Theory] by Robert Nau.
*[http://www-2.cs.cmu.edu/~conitzer/MLEvotingUAI05.pdf Common Voting Rules as Maximum Likelihood Estimators] by Vincent Conitzer and Tuomas Sandholm.
*[http://www.citizensassembly.bc.ca/resources/submissions/csharman-10_0409201706-143.pdf A New Monotonic and Clone-Independent Single-Winner Election Method] by Markus Schulze ([http://lists.gnu.org/archive/html/demexp-dev/2003-09/pdflQW7IlpAfC.pdf mirror1], [http://www.mcdougall.org.uk/VM/ISSUE17/I17P3.PDF mirror2]). Introduces the [[Schulze method]] and its use in the [[Debian]] project.
*[http://www.cs.princeton.edu/~elkind/hybrid.pdf Hybrid Voting Protocols and Hardness of Manipulation] by Edith Elkind and Helger Lipmaa.
*[http://www.hss.caltech.edu/Events/SCW/Papers/merlv.pdf On the impact of indifferent voters on the likelihood of some voting paradoxes] by Vincent Merlin and Fabrice Valognes.
*[http://personal.lse.ac.uk/DOWDING/Files/Manipulaiton_web_04_05.pdf In Praise of Manipulation] by Martin van Hees and Keith Dowding. Examines strategic voting from an ethical point of view.
*[http://www-2.cs.cmu.edu/~conitzer/tweakIJCAI03.pdf Universal voting protocol tweaks to make manipulation hard] by Vincent Conitzer and Tuomas Sandholm.
*[http://www.pubchoicesoc.org/papers2005/Moser.pdf Voting by Adaptive Agents in Multi-candidate Elections] by Scott Moser.
*[http://math.temple.edu/~wds/homepage/rangevote.pdf Range Voting] by Warren D. Smith. After the mathematical advocacy of Range Voting, there is a good monte-carlo comparison of voting systems in virtual elections, which, despite a rudimentary approach to strategy and polling, gives interesting best-case (honest) and worst-case (overstrategic) [[Social welfare function|social utilities]] for various systems.
*[http://web.cs.gc.cuny.edu/~kgb/docs/sincvote.pdf Safe Votes, Sincere Votes and Strategizing] by Rohit Parikh and Eric Pacuit.
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