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Voting system
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A voting system (also referred to as an electoral system) allows voters to choose between options, often in an election where candidates are selected for 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 and hierarchical or authoritarian systems.
A voting system contains rules for valid voting, and how votes are aggregated to yield a final result.
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Encyclopedia
A voting system (also referred to as an electoral system) allows voters to choose between options, often in an election where candidates are selected for 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 and hierarchical or authoritarian systems.
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.
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 (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 ballot
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.
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. 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 in order to maximize their 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.
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 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
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 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 Boroughs, abandoned towns who still were represented in parliament.
Single-winner methods
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
The most prevalent single-winner voting method, by far, is 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.
Ranked voting methods
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 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: 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 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 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
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.
Range voting
In range voting, voters give numeric ratings to each option, and the option with the highest total score wins.
Approval voting
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.
Multiple-winner methods
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 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
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 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 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.)
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. 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. There is no 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.)
- Condorcet criterion—If a candidate beats every other candidate in 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?
- 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.
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.
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.
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.
Bayesian regret is the sum of voter utilities for the candidate who maximizes the sum of voter utilities, minus the sum of utilities for the candidate who is elected. Finding the average Bayesian regret gives a numerical measure of the quality of a voting system that is sensitive to subjective assumptions about the candidates and electorate.
History
Early democracy
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.
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. The 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. 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.
Foundations of voting theory
Voting theory became an object of academic study around the time of the French Revolution. 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.
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.
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. 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.
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.
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. 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.
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. Bucklin is no longer used in any government elections, and has even been declared unconstitutional in Minnesota.
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. 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 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 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, and on political stability. A few scholars also studied what effects caused a nation to change for a particular voting system. 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 and 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 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.
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 Parliamentary elections in 1993 and Single Transferable Vote for some local elections in 2004 (see Electoral reform in New Zealand). The Canadian province of British Columbia will hold a second referendum on adopting STV in conjunction with its election on May 12, 2009. The Province of Ontario held a Referendum on October 10, 2007, on whether to adopt a Mixed Member Proportional system. It was defeated. An even wider range of voting systems is now seen in non-governmental organizations.
See also
General references
- Arrow, Kenneth J. (1951, 2nd ed., 1963) Social Choice and Individual Values. New Haven: Yale University Press. ISBN 0-300-01364-7
- Boix, Charles (1999). . American Political Science Review 93, 609–624.
- Dummett, Michael (1997). Principles of Electoral Reform. New York: Oxford University Press. ISBN 0-19-829246-5.
- 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.
- 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
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- 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
- from
- : electoral and legislative voting rules
- A mailing list for technical discussions about election methods.
- A wiki that focuses on voting theory.
- . Concise definitions of various methods.
- Canadian site.
- Paper by Warren D. Smith.
- by Matt Corks
- category on voting systems
- Software for computing a variety of voting systems including IRV, STV, and Condorcet.
- : tutorial, evaluation, and calculator
- by Alex Bogomolny. Illustrates various concepts of choice using Java applets.
- by Marcus Pivato.
- : election calculator and other resources
- by Paul E. Johnson. A textbook-style overview of voting methods and their mathematical properties.
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- Freely available on-line referendum engine, competence vs. opinion scatter plots, competence-weighted voting, analysis tools, and fraud reduction methods.
- by Blake Cretney
- by Rob LeGrand
- : Tutorial and essays by James Green-Armytage
- , report by Minority Rights Group, 2006
- A new decision making system based on ant society.
Advocacy
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- Advocates using IRV in the United States.
- Advocates proportional elections in local elections.
- Advocates Condorcet voting and provides links to vote-tallying software.
- A Northern Ireland-based organisation promoting inclusive voting procedures
- A Discover article on Approval voting and the Borda Count, by Dana Mackenzie.
Research papers
- Proceedings of a seminar at the Mathematical Research Institute at Oberwolfach, Germany.
- An article by Roger B. Myerson that analyzes voting systems economically.
- PhD seminar on by Robert Nau.
- by Vincent Conitzer and Tuomas Sandholm.
- by Markus Schulze (, ). Introduces the Schulze method and its use in the Debian project.
- by Edith Elkind and Helger Lipmaa.
- by Vincent Merlin and Fabrice Valognes.
- by Martin van Hees and Keith Dowding. Examines strategic voting from an ethical point of view.
- by Vincent Conitzer and Tuomas Sandholm.
- by Scott Moser.
- 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 utilities for various systems.
- by Rohit Parikh and Eric Pacuit.
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