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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 enforces rules to ensure valid voting, and how votes are counted and aggregated to yield a final result. 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. 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
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
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
Some methods 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 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 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
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
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
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 (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
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 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
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
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
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 (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 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
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


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.'

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
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
To compare systems fairly and independently of political ideologies, voting theorists use voting system criteria, which define potentially desirable properties of voting systems mathematically.

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
 * Majority criterion (MC)— Will a candidate always win who is ranked as the unique favorite by a majority of voters? This criterion comes in two versions:
 * 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:)
 * 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 criterion explained below.
 * Mutual majority criterion (MMC)— Will a candidate always win who 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— Will a candidate always win who beats every other candidate in pairwise comparisons? (This implies the majority criterion, above)
 * Condorcet loser criterion (Cond. loser)— Will a candidate always win who is not the candidate who loses to every other candidate in pairwise comparisons?
 * Independence of Smith-dominated alternatives (ISDA)— Does the outcome never change 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)— Does the outcome never change if a non-winning candidate is added or removed (assuming votes regarding the other candidates are unchanged)? 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.
 * Local Independence of Irrelevant Alternatives (LIIA)— Does the outcome never change 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)— Does the outcome never change 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.
 * Monotonicity criterion (Monotone)— If candidate W wins for one set of ballots, will W still always win if those ballots change to rank W higher? (This also implies that you cannot cause a losing candidate to win by ranking him lower.)
 * Consistency criterion (CC)— If candidate W wins for one set of ballots, will W still always win if those ballots change by adding another set of ballots where W also wins?
 * Participation criterion (PC)— Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below. )
 * Reversal symmetry—If individual preferences of each voter are inverted, does the original winner never win?
 * Polynomial time (Polytime)— Can the winner be calculated in a runtime that is polynomial in the number of candidates and linear in the number of voters?
 * Resolvable— Can the winner be calculated 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)— Can the winner be calculated 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.
 * Later-no-harm criterion and Later-no-help criterion— Can voters be sure that 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.

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

NA = not applicable.

Experimental criteria
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. 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. It found that Range voting consistently scored as either the best system or among the best across the various conditions studied.

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).

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). 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.

"Soft" criteria
In addition to the above criteria, voting systems are judged using criteria that are not mathematically precise but are still important, such as simplicity, speed of vote-counting, the potential for fraud or disputed results, the opportunity for tactical voting or strategic nomination, and, for multiple-winner methods, the 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.

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. 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. 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.

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.

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 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.

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. 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. 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.

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
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 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.

It has been argued that more in-depth and fine-tuned voting systems lie at the core of the development of e-democracy, which consists of the digitization of democratic processes, including voting. 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.