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The plurality voting system is a single-winner voting system often used to elect executive officers or to elect members of a legislative assembly which is based on single-member constituencies. This voting method is also used in multi-member constituencies in what is referred to as an exhaustive counting system where one member is elected at a time and the process repeated until the number of vacancies is filled.

The most common system, used in Canada, the lower house (Lok Sabha) in India, the United Kingdom, and some United States elections, is simple plurality, first-past-the-post or winner-takes-all. In this voting system the single winner is the person with the most votes (plurality); there is no requirement that the winner gain an absolute majority of votes.^[1]

In some countries such as France (as well as in some jurisdictions of the United States, such as Louisiana and Georgia) the "two-ballot" or "runoff election" plurality system is used. In order to ensure the winner gains a majority of votes, two rounds of voting occur. If any candidate in the first round gains a majority of votes, then there is no second round. Otherwise, the two highest-voted candidates of the first round compete in a two-candidate second round or all candidates above a certain threshold in the first round compete in a two-, three- or four-candidate second round.

In political science, the use of the plurality voting system alongside multiple, single-winner constituencies to elect a multi-member body is often referred to as single-member district plurality or SMDP.^[2] Plurality voting is also variously referred to as winner-takes-all or relative/simple majority voting; however, these terms can also refer to elections for multiple winners in a particular constituency using bloc voting.

Terminology[link]

The terms Highest vote, Majority vote and Plurality voting are often confused.

First past the post[link]

The term first past the post (abbreviated FPTP or FPP) was coined as an analogy to horse racing, where the winner of the race is the first to pass a particular point (the "post") on the track (in this case a plurality of votes), after which all other runners automatically and completely lose (that is, the payoff is "winner-takes-all"). There is, however, no "post" that the winning candidate must pass in order to win, as the winning candidate is required only to have received the highest number of votes in his or her favour. This results in the alternative name sometimes being "farthest past the post".

In a multiple member first-past-the-post ballot, the first candidates in order of highest vote to cross the "line" (that is, those who have the successivley greater number of votes) are elected. In some cases, the system involves an iteration of the counting of Plurality votes.

Historically, FPTP has been a contentious electoral system, giving rise to the concept of electoral reform and a multiplicity of different voting systems intended to address perceived weaknesses of plurality voting.

Voting[link]

Plurality voting is used for local and/or national elections in 43 of the 191 countries of the United Nations, as well as in the Republic of China (Taiwan). Plurality voting is particularly prevalent in the United Kingdom and former British colonies, including the United States, Canada and India.^[3]

In single winner plurality voting, each voter is allowed to vote for only one candidate, and the winner of the election is whichever candidate represents a plurality of voters, that is, whoever received the largest number of votes. This makes the plurality voting system among the simplest of all voting systems for voters and vote counting officials (it is however very contentious to draw district boundary lines in this system).

In an election for a legislative body, each voter in a given geographically-defined electoral district votes for one candidate from a list of candidates competing to represent that district. Under the plurality system, the winner of the election acts as representative of the entire electoral district, and serves with representatives of other electoral districts.

In an election for a single seat, such as president in a presidential system, the same style of ballot is used and the candidate who receives the largest number of votes represents the entire population.

The two-round voting system uses first-past-the-post voting method in the first round of voting. In this case the two highest polling candidates that cross the line progress to the second round Run-off ballot.

In a multiple member Plurality election the counting of the ballot uses an exhaustive iteration process using the same ballot papers to elect one person each iteration for each vacant position.

Ballot types[link]

Generally plurality ballots (single-mark ballots) can be categorized into two forms. The simplest form is a blank ballot where the name of a candidate is written in by hand. A more structured ballot will list all the candidates and allow a mark to be made by a single candidate, however a structured ballot can also include space for a write-in candidate.

Examples of plurality voting[link]

General elections in the United Kingdom[link]

The United Kingdom, like the United States and Canada, uses single-member districts as the base for national elections: Each electoral district or constituency chooses one member of parliament, namely the candidate that gets the most votes, whether he reaches a majority of votes or not ("first past the post"). In 1992, for example, a Liberal Democrat in Scotland won with just 26 percent of the votes. This system of single-member districts with plurality victors tends to produce two large political parties. (In countries with proportional representation there is not such a great incentive on forming two large parties, and that contributes to multi-party systems.)

Scotland, Wales and Northern Ireland use the first past the post system for general elections in the UK, but use versions of proportional representation for local elections and European elections.

The countries that inherited the British majoritarian system tend toward two large parties, one left, the other right, such as the U.S. Democrats and Republicans. Canada is an exception to this pattern, because its third parties are territorially concentrated, especially the separatist Bloc Québécois. New Zealand used the British system, and it too yielded two large parties. It also left many New Zealanders discontent, because other viewpoints got ignored, so its parliament in 1993 adopted a new electoral law, modeled on Germany's system of proportional representation (PR) with a partial selection by constituencies. New Zealand soon developed a more complex party system.^[4]

Example[link]

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

The candidates for the capital are:

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

The preferences of the voters would be divided like this:

Voting is accomplished whereby each voter in each city selects one city on the ballot (Memphis voters select Memphis, Nashville voters select Nashville, etc.) Votes are tabulated; Memphis is selected with the most votes (42%). Note that this system does not require that the winner have a majority, but only a plurality. Memphis wins because it has the most votes, even though 58% of the voters in this example preferred Memphis least. Notice that this problem does not hold anymore in the two-round system.

Disadvantages[link]

Tactical voting[link]

To a much greater extent than many other electoral methods, plurality electoral systems encourage tactical voting techniques, like "compromising". Voters are pressured to vote for one of the two candidates they predict are most likely to win, even if their true preference is neither, because a vote for any other candidate will likely be wasted and have no impact on the final result.

In the Tennessee example, if all the voters for Chattanooga and Knoxville had instead voted for Nashville, then Nashville would have won (with 58% of the vote); this would only have been the 3rd choice for those voters, but voting for their respective 1st choices (their own cities) actually results in their 4th choice (Memphis) being elected.

The difficulty is sometimes summed up, in an extreme form, as "All votes for anyone other than the second place are votes for the winner", because by voting for other candidates, they have denied those votes to the second place candidate who could have won had they received them. It is often claimed by United States Democrats that Democrat Al Gore lost the 2000 Presidential Election to Republican George W. Bush because some voters on the left voted for Ralph Nader of the Green Party, who exit polls indicated would have preferred Gore at 45% to Bush at 27%, with the rest not voting in Nader's absence.^[5]

Such a mentality is reflected by elections in Puerto Rico and its three principal voter groups: the Independentistas (pro-independence), the Populares (pro-commonwealth), and the Estadistas (pro-statehood). Historically, there has been a tendency for Independentista voters to elect Popular candidates and policies. This phenomenon is responsible for some Popular victories, even though the Estadistas have the most voters on the island. It is so widely recognised that the Puertoricans sometimes call the Independentistas who vote for the Populares "melons", because the fruit is green on the outside but red on the inside (in reference to the party colors).

Because voters have to predict in advance who the top two candidates will be, this can cause significant perturbation to the system:

• Substantial power is given to the media. Some voters will tend to believe the media's assertions as to who the leading contenders are likely to be in the election. Even voters who distrust the media will know that other voters do believe the media, and therefore those candidates who receive the most media attention will nonetheless be the most popular and thus most likely to be in one of the top two.

• A newly appointed candidate, who is in fact supported by the majority of voters, may be considered (due to the lack of a track record) to not be likely to become one of the top two candidates; thus, they will receive a reduced number of votes, which will then give them a reputation as a low poller in future elections, compounding the problem.

• The system may promote votes against more so than votes for. In the UK, entire campaigns have been organised with the aim of voting against the Conservative party by voting either Labour or Liberal Democrat. For example, in a constituency held by the Conservatives, with the Liberal Democrats as the second-place party and the Labour Party in third, Labour supporters might be urged to vote for the Liberal Democrat candidate (who has a smaller majority to close and more support in the constituency) than their own candidate on the basis that Labour supporters would prefer an MP from a competing left/liberal party than a Conservative one.

• If enough voters use this tactic, the first-past-the-post system becomes, effectively, runoff voting - a completely different system - where the first round is held in the court of public opinion, A good example of this is the Winchester by-election, 1997.

Proponents of other single-winner voting systems argue that their proposals would reduce the need for tactical voting and reduce the spoiler effect. Examples include the commonly used two-round system of runoffs and instant runoff voting, along with less tested systems such as approval voting and Condorcet methods.

Fewer political parties[link]

Duverger's law is an idea in political science which says that constituencies that use first-past-the-post systems will become two-party systems, given enough time.

First-past-the-post tends to reduce the number of political parties to a greater extent than most other methods, thus making it more likely that a single party will hold a majority of legislative seats. (In the United Kingdom, 21 out of 24 General Elections since 1922 have produced a single party majority government.)

FPTP's tendency toward fewer parties and more frequent one-party rule can also produce a government that may not consider as wide a range of perspectives and concerns. It is entirely possible that a voter will find that all major parties agree on a particular issue. In this case, the voter will not have any meaningful way of expressing a dissenting opinion through his or her vote.

As fewer choices are offered to the voters, voters may vote for a candidate with whom they largely disagree so as to oppose a candidate with whom they disagree even more. The downside of this is that candidates will less closely reflect the viewpoints of those who vote for them.

It may also be argued that one-party rule is more likely to lead to radical changes in government policy that are only favoured by a plurality or bare majority of the voters, whereas multi-party systems usually require greater consensus in order to make dramatic changes.

Wasted votes[link]

This unreferenced section requires citations to ensure verifiability.

Wasted votes are votes cast for losing candidates or votes cast for winning candidates in excess of the number required for victory. For example, in the UK General Election of 2005, 52% of votes were cast for losing candidates and 18% were excess votes - a total of 70% wasted votes. This is perhaps the most fundamental criticism of FPTP, that a large majority of votes may play no part in determining the outcome. Alternative electoral systems attempt to ensure that almost all votes are effective in influencing the result and the number of wasted votes is consequently minimised.

Gerrymandering[link]

Because FPTP permits a high level of wasted vote, an election under FPTP is easily gerrymandered. Through gerrymandering, constituencies are deliberately designed to unfairly increase the number of seats won by one party at the expense of another.

In brief, suppose that governing party G wishes to reduce the seats that will be won by opposition party O in the next election. It creates a number of constituencies in each of which O has an overwhelming majority of votes. O will win these seats, but a large number of its voters will waste their votes. Then the rest of the constituencies are designed with small majorities for G. Few G votes are wasted, and G will win a large number of seats by small margins. As a result of the gerrymander, O's seats have cost it more votes than G's seats.

Manipulation charges[link]

The presence of spoilers often gives rise to suspicions that manipulation of the slate has taken place. The spoiler may have received incentives to run. A spoiler may also drop out at the last moment, inducing charges that such an act was intended from the beginning.

Disproportionate influence of smaller parties[link]

Smaller parties can disproportionately change the outcome of an FPTP election by swinging what is called the 50-50% balance of two party systems, by creating a faction within one or both ends of the political spectrum which shifts the winner of the election from an absolute majority outcome to a simple majority outcome favouring the previously less favoured party. In comparison, for electoral systems using proportional representation small groups win only their proportional share of representation. However in PR systems, small parties can become decisive in Parliament so gaining a power of blackmail against the Government, a problem which is generally reduced by the FPTP system.^[6][7]

Issues specific to particular countries[link]

Solomon Islands[link]

In August 2008, Sir Peter Kenilorea commented on what he perceived as the flaws of a first-past-the-post electoral system in the Solomon Islands:

An[...] underlying cause of political instability and poor governance, in my opinion, is our electoral system and its related problems. It has been identified by a number of academics and practitioners that the First Past the Post system is such that a Member elected to Parliament is sometimes elected by a small percentage of voters where there are many candidates in a particular constituency. I believe that this system is part of the reason why voters ignore political parties and why candidates try an appeal to voters' material desires and relationships instead of political parties. [...] Moreover, this system creates a political environment where a Member is elected by a relatively small number of voters with the effect that this Member is then expected to ignore his party’s philosophy and instead look after that core base of voters in terms of their material needs. Another relevant factor that I see in relation to the electoral system is the proven fact that it is rather conducive, and thus has not prevented, corrupt elections practices such as ballot buying.

—^[8]

Advantages[link]

Obvious preservation of "one person, one vote" principle[link]

The arguments for a plurality voting system rely on the preservation of the "one person, one vote" principle (often shortened to OMOV for "one man, one vote" or more recently "one member, one vote"), as cited by the Supreme Court of the United States, wherein each voter is only able to cast one vote in a given election, where that vote can only go to one candidate. Plurality voting systems elect the candidate who is preferred first by the largest number of voters, although this need not be an absolute majority. Other voting systems, such as instant-runoff voting, party-list proportional representation or single transferable vote also preserve OMOV, but it is not as obvious that they do so, because they rely on lower voter preference to enable a candidate to earn either an absolute majority (single member district) or a quota (multi-member district), respectively.

Moderation[link]

Some other voting systems can end up giving a higher chance of victory to a candidate perceived as having extreme views. Under a first-past-the-post system, voters are often afraid of "wasting" their vote on a candidate unlikely to win, so they cast their vote towards their most preferable choice possible of victory. Advocates of plurality voting suggest that this results in most candidates having to field a fairly moderate or centrist position. This is debated by advocates of other systems, who argue that preferential voting or range voting systems, by getting more information from voters, allow a more rigorous definition of the word "moderate" and can be designed to explicitly favor candidates fitting that description.

Fewer minority parties[link]

Plurality voting tends to promote two-party systems (even more so than other single-winner systems). Supporters^[who?] view this as beneficial, as parliamentary governments, or other coalitions, are typically more stable in two-party systems, and thus small minorities are not given undue voice. First-past-the-post minimizes the influence of third parties and thus arguably keeps out extremists. However, this can also deny fair representation to positive third parties, racial minorities, women, and others.

Advantages compared to proportional representation[link]

Plurality is often conflated with Single-winner voting systems in general, in order to contrast it with Proportional representation. In this context it shares advantages, such as local accountability, with other single-winner systems.

Current events[link]

The United Kingdom continues to use the first-past-the-post electoral system for general elections, and for local government elections in England and Wales. Changes to the UK system have been proposed, and alternatives were examined by the Jenkins Commission in the late 1990s. After the formation of a new coalition government in 2010, it was announced as part of the coalition agreement that a referendum would be held on 5 May 2011 on switching to the alternative vote system. The Parliamentary Voting System and Constituencies Act 2011 received Royal Assent in February 2011 to allow the referendum to occur. It was rejected by British voters on May 6.

Canada also uses this system for national and provincial elections. In May 2005 the Canadian province of British Columbia had a referendum on abolishing single-member district plurality in favour of multi-member districts with the Single Transferable Vote system after the Citizens' Assembly on Electoral Reform made a recommendation for the reform. The referendum obtained 57% of the vote, but failed to meet the 60% requirement for passing. An October 2007 referendum in the Canadian province of Ontario on adopting a Mixed Member Proportional system, also requiring 60% approval, failed with only 36.9% voting in favour.

Northern Ireland, Scotland, Wales, the Republic of Ireland, Australia and New Zealand are notable examples of countries within the UK, or with previous links to it, that use non-FPTP electoral systems (Northern Ireland, Scotland and Wales use First Past the Post in United Kingdom general elections, however).

Recent examples of nations which have undergone democratic reforms but have not adopted the FPTP system include South Africa, almost all of the former Eastern bloc nations, Russia, Afghanistan and Iraq.

Use[link]

Countries that use this system to elect the lower or only house of their legislature include:

The plurality election system is used in the Republic of China on Taiwan for executive offices such as county magistrates, mayors, and the president, but not for legislative seats which used the single non-transferable vote system. This has produced an interesting party structure in which there are two broad coalitions of parties which cooperate in executive elections but which compete internally in legislative elections.^[9]

See also[link]

• 2006 Texas Governor Election (a real-life example where incumbent Rick Perry won re-election despite gaining less than 40 percent of the vote)
• Cube rule
• Plurality-at-large voting
• List of democracy and elections-related topics
• Proportional representation
• Single non-transferable vote
• Single transferable vote
• Runoff voting
• Voting system

Notes[link]

External links[link]

• A handbook of Electoral System Design from International IDEA
• Atlas of Electoral Systems of the World Colour-coded world maps showing the electoral systems used by every democratic country in the world, also available with more details as tables of text. Part of the information on Electoral Systems offered by the World Policy Institute (WPI)'s Project for Global Democracy & Human Rights
• ACE Project: What is the electoral system for Chamber 1 of the national legislature?
• ACE Project: First Past The Post - Detailed explanation of first-past-the-post voting
• ACE Project: Experiments with moving away from FPTP in the UK
• ACE Project: Electing a President using FPTP
• ACE Project: FPTP on a grand scale in India
• The Citizens' Assembly on Electoral Reform says the new proportional electoral system it proposes for British Columbia will improve the practice of democracy in the province.
• ASSEMBLY AUDIO AND VIDEO Below, you'll find audio and video recordings of six Learning Phase weekends, a meeting held in Prince George and the six Deliberation Phase weekends. The Learning Phase and Deliberation Phase recordings were broadcast on Hansard TV during 2004. - week 5 gives a detailed description by David Farrell, of the University of Manchester (England), Elizabeth McLeay of Victoria University of Wellington, New Zealand.
• Districting in FPTP systems, The unusual "earmuff" shape of the 4th Congressional District of Illinois connects two Hispanic neighborhoods while remaining contiguous by narrowly tracing Interstate 294.
• The Problem With First-Past-The-Post Electing (data from UK general election 2005)

A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum.

A voting system contains rules for valid voting, and how votes are counted and aggregated to yield a final result. Since voting involves counting, it is algorithmic in nature, and, since it involves polling the sentiments of a person, this represents affective data. Together, with the exception of proxy voting, this corresponds to in-degree centrality in graph theory and social network analysis, with votes as directed edges, and voters and candidates as nodes.^[1] Common voting systems are majority rule, proportional representation or plurality voting with a number of variations and methods such as first-past-the-post or preferential voting. The study of formally defined voting systems is called social choice theory or voting theory, a subfield of political science, economics, or mathematics.

With majority rule, those who are unfamiliar with voting theory are often surprised that another voting system exists, or that disagreements may exist over the definition of what it means to be supported by a majority^{[citation needed]}. Depending on the meaning chosen, the common "majority rule" systems can produce results that the majority does not support. 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 most liked or most disliked by a majority. A simple choice does not allow voters to express the ordering or the intensity of their feeling. Different voting systems may give very different results, particularly in cases where there is no clear majority preference.

Aspects[link]

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 (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 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 allowed to vote. These are aspects of the broader topic of elections and election systems.

The Electoral Reform Society is a political pressure group based in the United Kingdom, believed to be the oldest organisation concerned with electoral systems in the world. The Society advocates scrapping First Past the Post (FPTP) for all National and local elections arguing that the system is 'bad for voters, bad for government and bad for democracy'.

Ballot[link]

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 (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 candidates, a none of the above option, or a no confidence in that candidate option.

Candidates[link]

Some methods call for a primary election first to determine which candidates will be on the ballot.

Weight of votes[link]

Many elections are based on the principle of "one person, one vote", meaning that every voter's votes are counted with equal weight. This is not true of all elections, however. 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 parties in a parliament), voting power measures the ability of a group to change the outcome of a vote. Groups may form coalitions to maximize voting power.

In some German states, most notably Prussia and Sachsen, there was before 1918 a weighted vote system known as the Prussian three-class franchise, where the electorate would be divided into three categories based on the amount of income tax paid. Each category would have equal voting power in choosing the electors.^[2]

Status quo[link]

Some voting systems are weighted in themselves, for example if a super majority 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 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 cast, rather than the number of votes cast for the winning option. This can sometimes encourage dissenting voters to refrain from voting, in order to prevent a quorum.

Constituencies[link]

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), 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. The Australian bicameral Parliament has single-member electorates for the legislative body (lower house) and multi-member electorates for its Senate (upper house). Some systems, like the Additional member system, embed smaller districts (constituencies) within larger ones.

The way constituencies are created and assigned seats can dramatically affect the results. 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. An example of this were the UK Rotten and pocket boroughs, parliamentary constituencies that had a very small electorate - e.g. an abandoned town - and could thus be used by a patron to gain undue and unrepresentative influence within parliament. This was a feature of the unreformed House of Commons before the Great Reform Act of 1832.

Multiple-winner methods[link]

Seats won by each party in the 2005 German federal election, an example of a proportional voting system.

Most Western democracies use some form of multiple-winner voting system, with the United States and the United Kingdom being notable exceptions.

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.

Proportional methods[link]

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.^[3] 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 methods and largest remainder methods. 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 (STV) 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.

Different proportional representation systems use different geographic divisions. In some party-list or STV systems, all representatives are elected at large, with votes that may come from anywhere in the electorate. In others, the larger area is divided up into multimember districts, causing a trade-off between greater proportional accuracy for larger districts and greater geographical specificity for smaller districts. The mixed member proportional system, mentioned above, has some district-based winners and some at-large winners. And biproportional apportionment systems can achieve proportionality with districts as small as one member each, because each district result is adjusted by effectively transferring votes between same-party candidates in different districts.

Semi-proportional methods[link]

An alternative 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 (or, in some potential variants, a different number, e.g. 6 votes for each voter where there are 3 seats). 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.)

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 the plurality voting system 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. In this case a coalition of 40% of voters can obtain 2 of the 4 seats by splitting their votes as individuals equally 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.

Majoritarian methods[link]

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 their propensity for landslide victories won by a single winning slate of candidates, bloc voting and similar nonproportional methods are called "majoritarian".

Single-winner methods[link]

Single-winner systems can be classified based on their ballot type. In one vote systems, 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[link]

An example of runoff voting. Runoff voting involves two rounds of voting. Only two candidates continue to the second round.

The most prevalent single-winner voting method, by far, is plurality (also called "first-past-the-post", "relative majority", or "winner-take-all"), where 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 two highest polling options if there is no absolute majority (50% plus one). In elimination runoff elections, the weakest candidate(s) are eliminated until there is a majority.

A primary election process is also used as a two round runoff voting system. The two candidates or choices with the most votes in the open primary ballot progress 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.

In the Random ballot method, 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.

Ranked voting methods[link]

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 systems.

Other ranked methods include Coombs' method, 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 Condorcet winner sometimes called the pairwise champion. An option defeats another option if more voters rank the first option higher on their ballot than the number of voters who rank the second option higher. This is called a pairwise defeat.

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: 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 tournaments.

The Kemeny-Young method, the Schulze method (also known as "Schwartz sequential dropping", "cloneproof Schwartz sequential dropping" or the "beatpath method") and Ranked pairs are recently designed Condorcet methods that satisfy a large number of voting system criteria. These three Condorcet methods either fully rank, or can be used to fully rank, all the candidates from most popular to least popular.

Rated voting methods[link]

An approval voting ballot does not require ranking or exclusivity.

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 "grades" like A/B/C/D/F.

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.

In range voting, voters score or rate each option on a range, and the option with the highest total or average score wins. In majority judgment, similar ballots are used, but the winner is the candidate with the highest median score.

Approval voting, where voters may vote for as many candidates as they like, can be seen as an instance of range voting (or majority judgment) where the allowable ratings are 0 and 1. It has recently been studied by, among others Brams 2003 who notes that 'The chief reason for its nonadoption in public elections, and by some societies, seems to be a lack of key "insider" support.'

There are variants within cumulative voting. In the points form, each voter has as many votes as there are choices, and can distribute those votes as desired: all on one choice or spread in any other pattern. Cumulative voting is used in a number of communities as well as corporate boards. It was examined and developed perhaps most thoroughly by Lani Guinier (1994).

Evaluating voting systems using criteria[link]

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.

There are several ways to address this problem. Criteria can be defined mathematically, such that any voting system either passes or fails. This gives perfectly objective results, but their practical relevance is still arguable. Another approach is to define ideal criteria that no voting system passes perfectly, and then see how often or how close to passing various systems are over a large sample of simulated elections. This gives results which are practically relevant, but the method of generating the sample of simulated elections can still be arguably biased. A final approach is to create imprecisely defined criteria, and then assign a neutral body to evaluate each system according to these criteria. This approach can look at aspects of voting systems which the other two approaches miss, but both the definitions of these criteria and the evaluations of the methods are still inevitably subjective.

Mathematical criteria[link]

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. There is no ultimate authority on which criteria should be considered, but the following are some criteria that usefully distinguish between various systems and are considered to be desirable by many voting theorists:

Result criteria

Absolute: Will a candidate always win who...

• Majority criterion—... is ranked as the unique favorite by a majority of voters? This criterion comes in two versions:

1. Ranked majority criterion, in which an option which is merely preferred over the others by a majority must win. (Passing the ranked MC is denoted by "yes", because implies also passing the following:)
2. Rated majority criterion, in which only an option which is uniquely given a perfect rating by a majority must win. The ranked and rated MC are synonymous for ranked voting systems, but not for rated or graded ones. The ranked MC, but not the rated MC, is incompatible with the IIA and FBC criteria explained below.

• • Mutual majority criterion (MMC)—... is among a group of candidates ranked above all others by a majority of voters? This also implies the Majority loser criterion—if a majority of voters prefers every other candidate over a given candidate, then does that candidate not win? Therefore, of the systems listed, all pass neither or both criteria, except for Borda, which passes Majority Loser while failing Mutual Majority.
• Condorcet criterion—... beats every other candidate in pairwise comparisons? (This implies the majority criterion, above)

• Strategic, Majority Condorcet criterion—... beats every other candidate in pairwise comparison by a majority of all voters, and voters are aware of each other's preferences and vote strategically? (Any method that passes the Condorcet criterion above passes this criterion easily. However, if some voters were indifferent between two candidates, one candidate could be the Condorcet winner but not the Majority Condorcet winner.)

• Condorcet loser criterion (Cond. loser)—... is not the candidate who loses to every other candidate in pairwise comparisons?

Relative as nominees change: Does the outcome never change ...

• Independence of Smith-dominated alternatives (ISDA)—... if a Smith-dominated candidate is added or removed (assuming votes regarding the other candidates are unchanged)? Candidate C is Smith-dominated if there is some other candidate A such that C is beaten by A and every candidate B that is not beaten by A etc. Note that although this criterion is classed here as nominee-relative, it has a strong absolute component in excluding Smith-dominated candidates from winning. In fact, it implies all of the absolute criteria above.
• Independence of Irrelevant Alternatives (IIA)—... if a non-winning candidate is added or removed (assuming votes regarding the other candidates are unchanged)?^[4] For instance, plurality rule fails IIA; adding a candidate X can cause the winner to change from W to Y even though Y receives no more votes than before.

Note: IIA and ISDA are actually logically incompatible, but they are grouped in the same column because they have similar practical impact.

• Local Independence of Irrelevant Alternatives (LIIA)—... if the alternative that would finish last is removed? (And could the alternative that finishes second fail to become the winner if the winner were removed?)

• Independence of Clone Alternatives (Cloneproof)—... if non-winning candidates similar to an existing candidate are added? There are three different phenomena which could cause a system to fail this criterion:

• Spoilers are candidates which decrease the chance of any of the similar or clone candidates winning, also known as a spoiler effect.
• Teams are sets of similar candidates whose mere presence helps the chances of any of them winning.
• Crowds are additional candidates who affect the outcome of an election without either helping or harming the chances of their factional group, but instead affecting another group.

Relative as voters change: If candidate W wins for one set of ballots, will W still always win if those ballots change...

• Monotonicity criterion (Monotone)—... to rank W higher? (This also implies that you cannot cause a losing candidate to win by ranking him lower.)
• Consistency criterion—... by adding another set of ballots where W also wins?
  • 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.^[5])
• Reversal symmetry—If individual preferences of each voter are inverted, does the original winner never win?

Election administration criteria: Can the winner be calculated...

• Polynomial time (Polytime)—... in a runtime that is polynomial in the number of candidates and linear in the number of voters?
  • Resolvable—... in almost all cases, without using any random processes such as flipping coins? That is, are exact ties, in which the winner could be one of two or more candidates, vanishingly rare in large elections?
• Summability (Summable)—... by tallying ballots at each polling station separately and simply adding up the individual tallies? The amount of information necessary for such tallies is expressed as an order function of the number of candidates N. Slower-growing functions such as O(N) or O(N²) make for easier counting, while faster-growing functions such as O(N!) might make it harder to catch fraud by election administrators.

Voter's criteria

Ballot model: A voter states their choices in the form of...

• ... a single mark choosing one single candidate (e.g. "A is my favorite candidate.")
• ... approvals, i.e. a set of candidates the voter approves (e.g. "I approve candidates A, B and E.")
• ... a (strict) ranking, i.e. an list of candidates strictly ordered by the preferences of the voter (e.g. "A > B > E > D > H" or "I prefer A to B, B to E, E to D and D to H.")
• ... ranking with equality (e.g. "A = B > D = E > H" or "I prefer A and B to D and E; D and E to H; but I have no opinion between A and B or between D and E.") This is a generalized form of a ranking, so every voting system that can handle comparisons, also can be used if the ballots are rankings.
• ... scores, the voter places each candidate in one of several numerical or verbal categories, such as numbered categories 0-100 (e.g. "A gets score 100, E gets 0, H 47, B 12 and D also 12")

... which therefore allows voters to choose whether to use...

• ... equal rankings (= ranks)— A voter can rank any two candidates equally at any position on the ballot? This can reduce the prevalence of spoiled ballots due to overvotes, and can give a less-dishonest alternative to some tactical voting strategies.
• ... later preferences (>2 ranks)— A voter can indicate different levels of support (beyond the two levels of voted and unvoted) through ranking or rating candidates.

Strategy guarantees: Can voters be sure that...

• Later-no-harm criterion and Later-no-help criterion—... adding a later preference to a ballot will not harm/help any candidate already listed? Note that these criteria are not applicable to methods which do not allow later preferences; although such methods technically pass, they can be said to fail from a voter's perspective.^[6]
• Favorite Betrayal Criterion—... they do not need to rank any other candidate above their favorite in order to obtain a result they prefer? ^[7]

Note on terminology: A criterion is said to be "weaker" than another when it is passed by more voting systems. Frequently, this means that the conditions for the criterion to apply are stronger. For instance, the majority criterion (MC) is weaker than the multiple majority criterion (MMC), because it requires that a single candidate, rather than a group of any size, should win. That is, any system which passes the MMC also passes the MC, but not vice versa; while any required winner under the MC must win under the MMC, but not vice versa.

Compliance of selected systems (table)[link]

The following table shows which of the above criteria are met by several single-winner systems.

NA = not applicable.

Experimental criteria[link]

It is possible to simulate large numbers of virtual elections on a computer and see how various voting systems compare in practical terms. Since such investigations are more difficult than simply proving that a given system does or does not satisfy a given mathematical criterion, results are not available for all systems. Also, these results are sensitive to the parameters of the model used to generate virtual elections, which can be biased either deliberately or accidentally.

One desirable feature that can be explored in this way is maximum voter satisfaction, called in this context minimum Bayesian regret. Such simulations are sensitive to their assumptions, particularly with regard to voter strategy, but by varying the assumptions they can give repeatable measures that bracket the best and worst cases for a voting system.^[8] To date, the only such simulation to compare a wide variety of voting systems was run by a range-voting advocate and was not published in a peer-reviewed journal.^[9][10] It found that Range voting consistently scored as either the best system or among the best across the various conditions studied.^[11]

Another aspect which can be compared through such Monte Carlo simulations is strategic vulnerability. According to the Gibbard-Satterthwaite theorem, no voting system can be immune to strategic manipulation in all cases, but certainly some systems will have this problem more often than others. M. Balinski and R. Laraki, the inventors of the majority judgment system, performed such an investigation using a set of simulated elections based on the results from a poll of the 2007 French presidential election which they had carried out using rated ballots. Comparing range voting, Borda count, plurality voting, approval voting with two different absolute approval thresholds, Condorcet voting, and majority judgment, they found that range voting had the highest (worst) strategic vulnerability, while their own system majority judgment had the lowest (best).^[12]

Balinski and Laraki also used the same data to investigate how likely it was that each of those systems, as well as runoff voting, would elect a centrist. Opinions differ on whether this is desirable or not. Some argue that systems which favor centrists are better because they are more stable; others argue that electing ideologically purer candidates gives voters more choice and a better chance to retrospectively judge the relative merits of those ideologies; while Balinski and Laraki argue that both centrist and extremist candidates should have a chance to win, to prevent forcing candidates into taking either position. According to their model, plurality, runoff voting, and approval voting with a higher approval threshold tended to elect extremists (100%, 98%, and 94% of the time, respectively); majority judgement elected both centrists and extremists (56% extremists); and range, Borda, and approval voting with a lower approval threshold elected centrists (6%; 0.25%-13% depending on the number of candidates; and 6% extremists; respectively).^[13] However, their model did not take into account voters' strategic reactions to the system used, such as "lesser of two evils" voting under plurality.

Simulated elections in a two-dimensional issue space can also be graphed to visually compare election methods; this illustrates issues like nonmonotonicity, clone-independence, and tendency to elect centrists vs extremists.^[14]

"Soft" criteria[link]

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 degree of proportionality produced.

The New Zealand Royal Commission on the Electoral System listed ten criteria for their evaluation of possible 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.

History[link]

Early democracy[link]

Voting has been used as a feature of democracy since the 6th century BC, when democracy was introduced by the Athenian democracy. However, in Athenian democracy, voting was seen as the least democratic among methods used for selecting public officials, and was little used, because elections were believed to inherently favor the wealthy and well-known over average citizens. Viewed as more democratic were assemblies open to all citizens, and selection by lot (known as sortition), as well as rotation of office. One of the earliest recorded elections in Athens was a 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.^[15]

The Venetians' system for electing the Doge was a particularly convoluted process, consisting of five rounds of drawing lots (sortition) and five rounds of approval voting. By drawing lots, a body of 30 electors was chosen, which was further reduced to nine electors by drawing lots again. An electoral college of nine 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 nine 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.^[16] This process, with slight modifications, was central to the politics of the Republic of Venice throughout its remarkable lifespan of over 500 years, from 1268 to 1797.

Jean-Charles de Borda, an early voting theorist

Foundations of voting theory[link]

Voting theory became an object of academic study around the time of the French Revolution.^[15] 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 methods. He also wrote about the Condorcet paradox, which he called the intransitivity of majority preferences.^[17]

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.^[18]

The Marquis de Condorcet, another early voting theorist

Later in the 18th century, the related topic of 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.^[19] 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 in a sense 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 equivalent to the d'Hondt method, as is Webster's method to the Sainte-Laguë method, while Hamilton's method is identical to the Hare largest remainder method.^[19]

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 1899. Since then, proportional and semi-proportional methods have come to be used in almost all democratic countries, with most exceptions being former British colonies.^[20]

Single-winner revival[link]

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.^[21] 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 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 as well as a proportional multiwinner method based on proxy voting.

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 Progressive era, various municipalities began to use Bucklin voting. Bucklin is no longer used in any government elections, and has even been declared unconstitutional in Minnesota.^[22]

Influence of game theory[link]

Kenneth Arrow, a pioneer of modern voting 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. One such tool was the idea of a strong Nash equilibrium, based on the work of John Nash in the 1950s. This and similar methods led to significant new results that changed the field of voting theory.^{[15][Need quotation to verify]} 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. Among the criteria Arrow considered desirable was one which requires the voting system to use ordinal (ranking) information; thus, as John Harsanyi pointed out, cardinal (rated) voting systems such as approval voting, range voting, and majority judgment can successfully meet all the other criteria. 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, for any deterministic voting system, whether cardinal or ordinal.

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 often leads to a two-party system. Further research into this and other 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 ordinal preference rankings.^[23]

Public choice theory, a field which uses economic modeling tools including game theory to study the behavior of politicians and voters, began to form in the 1950s, and was both influenced by and influential on voting theory.

Post-1980 developments[link]

Voting theory has come to focus on 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 scientists of the 20th century published many studies on the effects that the voting systems have on voters' choices and political parties,^[24][25][26] and on political stability.^[27][28] A few scholars also studied what effects caused a nation to change for a particular voting system.^{[29][30][31][32][33][34]} 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 clones. Also, Donald G. Saari has brought renewed interest to the Borda count with the books he has published since 2001. Saari uses geometric models of positional voting systems to promote the Borda count.

The increased availability of computer processing has increased the practicality of using the Kemeny-Young, ranked pairs, and 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 mathematical fields, voting theory is generally accessible enough to non-experts that new results can be discovered by amateurs, and frequently are.

The study of voting systems has influenced a new push for electoral reform beginning around the 1990s, with proposals being made to replace plurality voting in governmental elections with other methods. New Zealand adopted Mixed Member Proportional for Parliamentary elections in 1993 and Single Transferable Vote for some local elections in 2004 (see Electoral reform in New Zealand). After plurality voting was a key factor in the contested results of the 2000 US presidential election, various municipalities in the United States began to adopt instant-runoff voting, although some of them subsequently returned to their prior system. The Canadian province of British Columbia held two unsuccessful referendums (in 2005 and 2009) to adopt an STV system, and Ontario, another Canadian province, held an unsuccessful referendum on October 10, 2007 on whether to adopt a Mixed Member Proportional system. An even wider range of voting systems is now seen in non-governmental organizations.

Though theorists are divided on which particular voting system is best, many agree that several options are superior to plurality. For instance, the "Declaration of Election-Method Reform Advocates" supports single-winner reforms including approval voting, various Condorcet methods, majority judgment, and range voting. ^[35][36]

It has been argued that more in-depth and fine-tuned voting systems lie at the core of the development of e-democracy,^[37] which consists of the digitization of democratic processes, including voting.^[38] Research is ongoing into the use of homomorphic encryption for making voting systems that are transparent and voter-verifiable without compromising ballot secrecy. However, others are skeptical of the security of computerized voting, and argue that only paper ballots (whether machine-printed and voter-verified, or even exclusively hand-marked) give a reliable audit trail.

See also[link]

• Issue voting
• Nakamura number
• Opinion poll
• Proxy voting
• Table of voting systems by nation
• Vote counting system
• Voting machine
• Duverger's law
• Micromega rule
• Experimental political science
• Leader election

Criteria table notes[link]

1. ^ Approval only passes the majority criterion if the majority approve of only one candidate. Though this is strategically rational of them if they know each other's preferences, it may not be the obvious strategy if they do not.
2. ^ ^{a b c} In Approval, Range, and Majority Judgment, if all voters have perfect information about each other's true preferences and use rational strategy, any Majority Condorcet or Majority winner will be strategically forced – that is, win in all of one or more strong Nash equilibria. In particular if every voter knows that "A or B are the two most-likely to win" and places their "approval threshold" between the two, then the Condorcet winner, if one exists and is in the set {A,B}, will always win. These systems also satisfy the majority criterion in the weaker sense that any majority can force their candidate to win, if it so desires. (Laslier, J.-F. (2006) "Strategic approval voting in a large electorate," IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique).)
3. ^ The original Independence of clones criterion applied only to ranked voting methods. (T. Nicolaus Tideman, "Independence of clones as a criterion for voting rules", Social Choice and Welfare Vol. 4, No. 3 (1987), pp. 185–206.) Tideman notes that "in the spirit of independence of clones", "if there were two or more candidates who were so similar that every voter would rank them as tied if given the chance to rank them [...], then the number of perfect clones present would have no effect on whether the perfect clones were in the set of winning candidates under approval voting". So, Approval Voting satisfies this mathematical criterion by definition. However, there is some disagreement about whether considerations of the voter in the process of making up his vote could be tactically influenced by clones (in a way that a voter would dispossess a candidate of his approval when a clone of him is introduced) and whether the definition of clones have to be extended to these considerations additionally to the handling of actual votes.
4. ^ ^{a b} Approval and Plurality do not allow later preferences. Technically speaking, this means that they pass the technical definition of the LNH criteria - if later preferences or ratings are impossible, then such preferences can not help or harm. However, from the perspective of a voter, these systems do not pass these criteria. Approval, in particular, encourages the voter to give the same ballot rating to a candidate who, in another voting system, would get a later rating or ranking. Thus, for approval, the practically meaningful criterion would be not "later-no-harm" but "same-no-harm" - something neither approval nor any other system satisfies.
5. ^ The number of piles that can be summed from various precincts is floor((e-1) N!) - 1.
6. ^ ^{a b} Kemeny-Young and Ranked Pairs also pass a weaker version of the Independence of Irrelevant Alternatives criterion called Local IIA.
7. ^ Kemeny-Young does not pass the consistency criterion for winner, but the consistency criterion for full rankings, that is, if the electorate is divided in two parts and in both parts Kemeny-Young chooses the same ranking, Kemeny-Young will also choose that ranking for the combined electorate.
8. ^ Each prospective Kemeny-Young ordering has score equal to the sum of the pairwise entries that agree with it, and so the best ordering can be found using the pairwise matrix.
9. ^ Bucklin voting, with skipped and equal-rankings allowed, meets the same criteria as Majority Judgment; in fact, Majority Judgment may be considered a form of Bucklin voting. Without allowing equal rankings, Bucklin's criteria compliance is worse; in particular, it fails Independence of Irrelevant Alternatives, which for a ranked method like this variant is incompatible with the Majority Criterion.
10. ^ Majority judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade.
11. ^ Balinski and Laraki, Majority Judgment's inventors, point out that it meets a weaker criterion they call "grade consistency": if two electorates give the same rating for a candidate, then so will the combined electorate. Majority Judgment explicitly requires that ratings be expressed in a "common language", that is, that each rating have an absolute meaning. They claim that this is what makes "grade consistency" significant. MJ. Balinski M. and R. Laraki (2007) «A theory of measuring, electing and ranking». Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725.
12. ^ Majority judgment can actually pass or fail reversal symmetry depending on the rounding method used to find the median when there are even numbers of voters. For instance, in a two-candidate, two-voter race, if the ratings are converted to numbers and the two central ratings are averaged, then MJ meets reversal symmetry; but if the lower one is taken, it does not, because a candidate with ["fair","fair"] would beat a candidate with ["good","poor"] with or without reversal. However, for rounding methods which do not meet reversal symmetry, the odds of breaking it are comparable to the odds of an irresolvable (tied) result; that is, vanishingly small for large numbers of voters.
13. ^ Majority Judgment is summable at order KN, where K, the number of ranking categories, is set beforehand.
14. ^ In fact, Majority Judgment ballots use ratings expressed in "common language" rather than numbers, that is, that each rating have an absolute meaning.
15. ^ Majority judgment meets a related, weaker criterion: ranking an additional candidate below the median grade (rather than your own grade) of your favorite candidate, cannot harm your favorite.
16. ^ ^{a b} A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.
17. ^ Once for each round.
18. ^ Later preferences are only possible between the two candidates who make it to the second round.
19. ^ That is, second-round votes cannot help or harm candidates already eliminated.
20. ^ Random winner: Uniformly randomly chosen candidate is winner. Arbitrary winner: some external entity, not a voter, chooses the winner. These systems are not, properly speaking, voting systems at all, but are included to show that even a horrible system can still pass some of the criteria.
21. ^ Random ballot: Uniformly random-chosen ballot determines winner. This and closely related systems are of mathematical interest because they are the only possible systems which are truly strategy-free, that is, your best vote will never depend on anything about the other voters. However, this system is not generally considered as a serious proposal for a practical method.

References[link]

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