|Part of the Politics series|
An electoral system or electoral method consists of the set of rules which must be followed for a vote to be considered valid, and which set out how votes are cast, counted and aggregated to yield a final result of an election or a referendum. Common electoral systems are majority rule, proportional representation and plurality voting, with a number of variations and methods such as first-past-the-post and preferential voting. The study of formally defined voting methods is called social choice theory or voting theory, a subfield of political science, economics, or mathematics.
Those who are unfamiliar with voting theory are often surprised to learn that electoral systems other than majority rule exist, or that disagreements exist over what it means to be supported by a majority. 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 more than two 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.
- 1 Aspects
- 2 Multiple-winner systems
- 3 Single-winner systems
- 4 Comparison of electoral systems
- 5 History
- 6 See also
- 7 References
- 8 External links
An electoral 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, as in the case of a presidential election, or may result in multiple winners, such as in the election of a legislative body. The electoral 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 electoral system. For example, though an electoral 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. An electoral 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.
Different electoral systems have different ways of allowing individuals to express their votes. In ranked ballot or "preference" electoral systems, such as Instant-runoff voting, the Borda count, or a Condorcet method, each voter orders 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 electoral systems that allow "plumping", like cumulative voting, voters may vote for the same candidate multiple times.
Some systems call for a primary election first to determine which candidates will be on the ballot.
Weight of votes
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 the kingdoms of Prussia (German: Preußen) and Saxony (German: Sachsen), before 1918 there was 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.
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 voter participation does not reach the specified threshold. 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.
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 and Belgium, break up their national elections into smaller multiple-winner districts, and yet others, like the United States and the United Kingdom, hold only single-winner elections. The Australian bicameral Parliament has single-member electorates for the 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 counted in 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 possibilities of both malapportionment, i.e. unequal ratios of representatives to population ratios between districts, and gerrymandering, where electoral districts are manipulated for political gain. An extreme example of malapportionment was 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, particularly as many people were not entitled to vote at all. This was a feature of the unreformed House of Commons before the Great Reform Act of 1832.
Most modern democracies use some form of multiple-winner electoral system, with the United States, Canada, the United Kingdom, and India being notable exceptions.
The results of a vote with multiple winners, such as the election of a legislature, are usually different from a set of single-winner votes. Often, voters in a multiple-winner election are more concerned with the overall composition of the legislature than with exactly which candidates get elected. For this reason, many multiple-winner systems aim for proportional representation, which means that if a given party (or other political grouping) gets X% of the total vote, it should also get approximately X% of the seats in the legislature. However, not all multiple-winner voting systems are proportional.
Truly proportional systems make some guarantee of proportionality by making each winning option represent about 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 error 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 in 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.
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 eliminated 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. If no excess votes are transferred, but instead the count stops when there are only as many candidates remaining as seats, this system can be considered to be a multi-member version of instant runoff voting; such a system has apparently been used in South Australia, where it was known as Bottoms-Up preferential voting.
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 more geographically-specific representatives for smaller districts. The mixed member proportional method, mentioned above, has some district-based winners and some at-large winners. And biproportional apportionment methods 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.
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 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 distribution 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.
Many multiple-winner electoral systems are simple extensions of single-winner systems, 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 systems are called "majoritarian".
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 electoral systems
The most prevalent single-winner electoral system, 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 systems 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 (above 50%). 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 electoral 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 system, 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.
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 or "Condorcet extensions".
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), ranked pairs, and maximal lotteries are recently designed Condorcet methods that satisfy a large number of voting system criteria. These four Condorcet methods either fully rank, or can be used to fully rank, all the candidates from most popular to least popular.
Rated voting methods
Rated ballots allow even more flexibility than ranked ballots. 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 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' from 'influential members.'
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
Comparison of electoral systems
Electoral systems can be compared by different means. Attitudes towards systems are highly influenced by the systems' impact on groups that one supports or opposes, which 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 method 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 method passes perfectly, and then see how often or how close to passing various methods 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 method according to these criteria. This approach can look at aspects of voting methods which the other two approaches miss, but both the definitions of these criteria and the evaluations of the methods are still inevitably subjective.
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 approval voting to elect their Great Council.
The Venetians' method 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 method 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, 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.
Development of new systems
Jean-Charles de Borda proposed the Borda count in 1770 as a method for electing members to the French Academy of Sciences. His method 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. However, recent research has shown that the philosopher Ramon Llull devised 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, apportionment methods came to prominence due to 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 devised in the United States were in a sense rediscovered in Europe in the 19th century, as seat allocation methods for the newly proposed method of party-list proportional representation. The result is that many apportionment methods have two names; 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 (STV) method was devised by Carl Andræ in Denmark in 1855 and in the United Kingdom by Thomas Hare in 1857. 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 began to be used to elect European legislatures in the early 20th century, with Belgium the first to implement it for its 1900 general elections. Since then, proportional and semi-proportional methods have come to be used in almost all democratic countries, with most exceptions being former British colonies.
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 (IRV). 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, proposed the straightforward Condorcet method known as Dodgson's method as well as a proportional multiwinner method based on proxy voting.
Ranked voting methods 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, some municipalities began to use Bucklin voting, although this is no longer used in any government elections, and has even been declared unconstitutional in Minnesota.
The use of game theory to analyze voting methods led to discoveries about the effects of certain methods; research led Steven Brams and Peter Fishburn to formally define and promote the use of approval voting in 1977. Political scientists of the 20th century published many studies on the effects that the voting methods have on voters' choices and political parties, and on political stability. A few scholars also studied which effects caused a nation to switch to a particular electoral 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.
The study of voting methods 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 representation for the 1993 general elections and STV for some local elections in 2004. After plurality voting was a key factor in the contested results of the 2000 presidential elections in the United States, various municipalities in the United States began to adopt IRV, although some of them subsequently returned to their prior method. However, attempts at introducing more proportional systems were not always successful; in Canada there were two referendums in British Colombia in 2005 and 2009 on adopting an STV method, both of which failed. In the United Kingdom, a 2011 referendum on adopting AV+ saw the proposal rejected.
In other countries there were calls for the restoration of plurality or majoritarian systems; a referendum was held in Ecuador in 1994 on the adoption the two round system, but the idea was rejected. In Romania a proposal to switch to a two-round system for parliamentary elections failed only because voter turnout in the referendum was too low. Attempts to reintroduce single-member constituencies in Poland (2015) and two-round system in Bulgaria (2016) via referendums both also failed due to low turnout.
- Types and classifications:
- DRE voting machine
- Paillier cryptosystem § Electronic voting
- Homomorphic secret sharing § Example: decentralized voting protocol
- ThreeBallot § Method
- Bingo voting
- Security aspects:
- Voter-verified paper audit trail
- Vote rigging
- Blind signature
- Dining cryptographers problem
- Arrow, Kenneth J (1963) , Social Choice and Individual Values (2nd ed.), New Haven: Yale University Press, ISBN 0-300-01364-7.
- Balinski, M; Laraki, R (2007), "Election by Majority Judgement: Experimental Evidence", Cahier du Laboratoire d’Econométrie de l’Ecole Polytechnique (28)
- Balinski, M; Laraki, R (2011) , "Election by Majority Judgement: Experimental Evidence", in Dolez, Bernard; Grofman, Bernard; Laurent, Annie, In Situ and Laboratory Experiments on Electoral Law Reform: French Presidential Elections, Springer.
- Boix, Carles (1999). "Setting the Rules of the Game: The Choice of Electoral Systems in Advanced Democracies". American Political Science Review. 93 (3): 609–24. doi:10.2307/2585577. JSTOR 2585577. SSRN .
- Colomer, Josep M., ed. (2004). Handbook of Electoral System Choice. London and New York: Palgrave Macmillan. ISBN 978-1-4039-0454-6.
- Cranor, Lorrie. "Vote Aggregation Methods". Declared-Strategy Voting: An Instrument for Group Decision-Making. Retrieved October 3, 2005.
- Emerson, Peter (2012). Defining Democracy. Springer. ISBN 978-3-642-20903-1.
- Farrell, David M. (2001). Electoral Systems: A Comparative Introduction. New York: St. Martin's Press. ISBN 0-333-80162-8.
- 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. ISBN 0-416-68320-7.
- Hermens, Ferdinand A. (1941). "Democracy or Anarchy? A Study of Proportional Representation". Notre Dame, Indiana: University of Notre Dame.
- Lijphart, Arend
- Lijphart, Arend (1985). "The Field of Electoral Systems Research: A Critical Survey". Electoral Studies. 4. doi:10.1016/0261-3794(85)90028-9.
- Lijphart, Arend (1992). "Democratization and Constitutional Choices in Czecho-Slovakia, Hungary and Poland, 1989–1991". Journal of Theoretical Politics. 4 (2): 207–23. doi:10.1177/0951692892004002005.
- Lijphart, Arend (1994). Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945–1990. Oxford: Oxford University Press. ISBN 0-19-828054-8.
- Owen, Bernard (2002), Le système électoral et son effet sur la représentation parlementaire des partis: le cas européen (in French), LGDJ, ISBN 9782275023045.
- Poundstone, William (2008), Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It), New York: Hill and Young.
- Rae, Douglas W. (1971). The Political Consequences of Electoral Laws. New Haven: Yale University Press. ISBN 0-300-01517-8.
- Reynolds, Andrew; Reilly, Benjamin; Ellis, Andrew, eds. (2005). The New International IDEA Handbook of Electoral System Design. Stockholm: International IDEA. ISBN 91-85391-18-2.
- Rogowski, Ronald (1987). "Trade and the Variety of Democratic Institutions". 41. International Organization: 203–24. doi:10.1017/s0020818300027442.
- Rokkan, Stein (1970). "Citizens, Elections, Parties: Approaches to the Comparative Study of the Process of Development". Oslo: Universitetsforlaget.
- Taagapera, Rein; Shugart, Matthew S. (1989). Seats and Votes: The Effects and Determinants of Electoral Systems. New Haven: Yale University Press.
- Ludwig Windthorst, Speech in Favor of Reforming the Prussian Suffrage, in the Prussian House of Deputies, 26 November 1873
- Batto, NF., Huang, C., Tan, AC. and Cox, G. (Ed.) (2016) Mixed-Member Electoral Systems in Constitutional Context: Taiwan, Japan and Beyond. Ann Arbor: University of Michigan Press.
- Douglas J. Amy, "How proportional representation elections work", PR Library
- "Bottoms-Up" preferential voting use in Australia". prsa.org.au.
- Brams (2003), Theory to practice (PDF), New York city, NY, USA: NYU
- Lani Guinier (1994).
- J.J. O'Connor & E. F. Robertson The history of voting MacTutor History of Mathematics archive
- Miranda Mowbray & Dieter Gollmann (2007) "http://www.hpl.hp.com/techreports/2007/HPL-2007-28R1.pdf Electing the Doge of Venice: Analysis of a 13th Century Protocol]
- G. Hägele & F. Pukelsheim (2001) "Llull's writings on electoral systems", Studia Lulliana Vol. 3, pp3–38
- Apportionment: Introduction American Mathematical Society
- Proportional Voting Around the World FairVote
- The History of IRV FairVote
- Tony Anderson Solgård & Paul Landskroener (2002) "Municipal Voting System Reform: Overcoming the Legal Obstacles", Bench & Bar of Minnesota, Vol. 59, no. 9
- Poundstone, William (2008) Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It), Hill and Young, p198
- Duverger, Maurice (1954) Political Parties, Wiley ISBN 0-416-68320-7
- Douglas W. Rae (1971) The Political Consequences of Electoral Laws, Yale University Press ISBN 0-300-01517-8
- Rein Taagapera & Matthew S. Shugart (1989) Seats and Votes: The Effects and Determinants of Electoral Systems, Yale University Press
- Ferdinand A. Hermens (1941) Democracy or Anarchy? A Study of Proportional Representation, University of Notre Dame.
- Arend Lijphart (1994) Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945–1990 Oxford University Press ISBN 0-19-828054-8
- Arend Lijphart (1985) "The Field of Electoral Systems Research: A Critical Survey" Electoral Studies, Vol.4
- Arend Lijphart (1992) "Democratization and Constitutional Choices in Czecho-Slovakia, Hungary and Poland, 1989–1991" Journal of Theoretical Politics Vol. 4 (2), pp207–23
- Stein Rokkan (1970) Citizens, Elections, Parties: Approaches to the Comparative Study of the Process of Development, Universitetsforlaget
- Ronald Rogowski (1987) "Trade and the Variety of Democratic Institutions", International Organization Vol. 41, pp203–24
- Carles Boix (1999) "Setting the Rules of the Game: The Choice of Electoral Systems in Advanced Democracies", American Political Science Review Vol. 93 (3), pp609–24
- Existing electoral methods and related practical considerations
- Electoral method reform advocacy
- Proportional Representation Society of Australia Electoral reform NGO
- The Center for Election Science
- Center for Range Voting
- Center for Voting and Democracy Advocates using IRV in the United States.
- Electoral Reform Society – Pushing to reform democracy in the United Kingdom
- Declaration of Election-Method Reform Advocates
- Theoretical resources
- Accurate Democracy: electoral and legislative voting rules
- Electowiki A wiki that focuses on voting theory
- VoteRoll is a vote blog roll method developing statistics for people voting online.
- OpenSTV Software for computing a variety of voting methods including IRV, STV, and Condorcet.
- Student's Social Choice by Alex Bogomolny. Illustrates various concepts of choice using Java applets.
- Voting and Election Reform: election calculator and other resources
- Math in Society chapter: Voting Theory
- Voting Systems by Paul E. Johnson. A textbook-style overview of voting methods and their mathematical properties.
- Practical multi-candidate election method
- Balbis: Voting method democracy oriented