Voting system

Voting system

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

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

In a simple plurality ballot, the voter is expected to mark only one selection.

Different voting systems have different forms for allowing the individual to express his or her vote. In ranked ballot or "preference" voting systems, such as Instant-runoff voting, the Borda count, or a Condorcet method, voters order the list of options from most to least preferred. In range voting, voters rate each option separately on a scale. In plurality voting (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.


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

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 to maximize voting power.

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

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 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 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

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

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

A vote with multiple winners, such as the election of a legislature, has different practical effects than a single-winner vote. Often, participants in a multiple winner election are more concerned with the overall composition of the legislature than exactly which candidates get elected. For this reason, many multiple-winner systems aim for proportional representation, which means that if a given party (or any other political grouping) gets X% of the vote, it should also get approximately X% of the seats in the legislature. Not all multiple-winner voting systems are proportional.

Proportional methods

Truly proportional methods make some guarantee of proportionality by making each winning option represent approximately the same number of voters. This number is called a quota. For example, if the quota is 1000 voters, then each elected candidate reflects the opinions of 1000 voters, within a margin of error. This can be measured using the Gallagher Index.

Most proportional systems in use are based on party-list proportional representation, in which voters vote for parties instead of for individual candidates.[3] For each quota of votes a party receives, one of their candidates wins a seat on the legislature. The methods differ in how the quota is determined or, equivalently, how the proportions of votes are rounded off to match the number of seats.

The methods of seat allocation can be grouped overall into highest averages methods and largest remainder methods. Largest remainder methods set a particular quota based on the number of voters, while highest averages methods, such as the Sainte-Laguë method and the d'Hondt method, determine the quota indirectly by dividing the number of votes the parties receive by a sequence of numbers.

Independently of the method used to assign seats, party-list systems can be open list or closed list. In an open list system, voters decide which candidates within a party win the seats. In a closed list system, the seats are assigned to candidates in a fixed order that the party chooses. The Mixed Member Proportional system is a mixed method that only uses a party list for a subset of the winners, filling other seats with the winners of regional elections, thus having features of open list and closed list systems.

In contrast to party-list systems, the Single Transferable Vote 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.

Semiproportional methods

An alternative method called Cumulative voting (CV) is a semiproportional voting system in which each voter has n votes, where n is the number of seats to be elected (or, in some potential variants, a different number, e.g. 6 votes for each voter where there are 3 seats). Voters can distribute portions of their vote between a set of candidates, fully upon one candidate, or a mixture. It is considered a proportional system in allowing a united coalition representing a m/(n+1) fraction of the voters to be guaranteed to elect m seats of an n-seat election. For example in a 3-seat election, 3/4 of the voters (if united on 3 candidates) can guarantee control over all three seats. (In contrast, plurality at large, which allows a united coalition (majority) (50%+1) to control all the seats.)

This ballot design, used in cumulative voting, allows a voter to split his vote among multiple candidates.

Cumulative voting is a common way of holding elections in which the voters have unequal voting power, such as in corporate governance under the "one share, one vote" rule. Cumulative voting is also used as a multiple-winner method, such as in elections for a corporate board.

Cumulative voting is not fully proportional because it suffers from the same spoiler effect of 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.

Nonproportional and semiproportional 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 nonproportional. 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 semiproportional. Other methods that can be seen as semiproportional 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.

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

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

The most prevalent single-winner voting method, by far, is plurality (also called "first-past-the-post", "relative majority", or "winner-take-all"), where each voter votes for one choice, and the choice that receives the most votes wins, even if it receives less than a majority of votes.

Runoff methods hold multiple rounds of plurality voting to ensure that the winner is elected by a majority. Top-two runoff voting, the second most common method used in elections, holds a runoff election between the two highest polling options if there is no absolute majority (50% plus one). In elimination runoff elections, the weakest candidate(s) are eliminated until there is a majority.

A primary election process is also used as a two round runoff voting system. The two candidates or choices with the most votes in the open primary ballot progress to the general election. The difference between a runoff and an open primary is that a winner is never chosen in the primary, while the first round of a runoff can result in a winner if one candidate has over 50% of the vote.

In the Random ballot method, each voter votes for one option and a single ballot is selected at random to determine the winner. This is mostly used as a tiebreaker for other methods.

Ranked voting methods

In a typical ranked ballot, a voter is instructed to place the candidates in order of preference.

Also known as preferential voting methods, these methods allow each voter to rank the candidates in order of preference. Often it is not necessary to rank all the candidates: unranked candidates are usually considered to be tied for last place. Some ranked ballot methods also allow voters to give multiple candidates the same ranking.

The most common ranked voting method is instant-runoff voting (IRV), also known as the "alternative vote" or simply preferential voting, which uses voters' preferences to simulate an elimination runoff election without multiple voting events. As the votes are tallied, the option with the fewest first-choice votes is eliminated. In successive rounds of counting, the next preferred choice still available from each eliminated ballot is transferred to candidates not yet eliminated. The least preferred option is eliminated in each round of counting until there is a majority winner, with all ballots being considered in every round of counting.

The Borda count is a simple ranked voting method in which the options receive points based on their position on each ballot. A class of similar methods is called positional voting systems.

Other ranked methods include Coombs' method, Supplementary voting, Bucklin voting, and Condorcet method.

Condorcet methods, or pairwise methods, are a class of ranked voting methods that meet the Condorcet criterion. These methods compare every option pairwise with every other option, one at a time, and an option that defeats every other option is the 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, the Schulze method, and the ranked pairs method are Condorcet methods that fully rank all the candidates from most popular to least popular.

Rated voting methods

On a rated ballot, the voter may rate each choice independently.

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.

An approval voting ballot does not require ranking or exclusivity.

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 give numeric ratings to each option, and the option with the highest total or average score wins. In majority judgment, similar ballots are used, but the winner is the candidate with the highest median score.

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

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

Evaluating voting systems using criteria

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.

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 at the top, higher than all other candidates, does that candidate always win?
  • Mutual majority criterion (MMC)—If there exists a majority that ranks (or rates) a group of candidates higher than all others, does one of those candidates always win? 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.
  • Monotonicity criterion (Monotone)—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.[4])
  • Condorcet criterion—If a candidate beats every other candidate in pairwise comparison, does that candidate always win? (This implies the majority criterion, above)
  • Condorcet loser criterion (Cond. loser)—If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?
  • Independence of irrelevant alternatives (IIA)—If a candidate is added or removed, do the relative rankings of the remaining candidates stay the same?
  • Independence of clones criterion (Cloneproof)—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?
  • Polynomial time (Polytime)—Can the winner be calculated in a runtime that is polynomial in the number of candidates and the number of voters?
  • Summability (Summable)—How much information must be transmitted from each polling station to a central location in order to determine the winner? This 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.
  • Allows equal rankings—Can a voter choose whether to rank any two candidates equally at any position on the ballot? This can reduce the prevalence of spoiled ballots due to overvotes, and can give a less-dishonest alternative to some tactical voting strategies.
  • Allows later preferences (later prefs)—Can a voter indicate different levels of support through ranking or rating candidates?
  • Later-no-harm criterion and Later-no-help criterion—Can adding a later preference to a ballot 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.[5]

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.



Equal rankings
can exist
Approval[nb 1] 3Ambig­uous 1Yes 2Yes[nb 2] 4No[nb 2] 5No 3Ambig­uous 3Ambig.­[nb 3] 1Yes 1Yes 1O(N) 1Yes No[nb 4]
Borda count 5No 1Yes 1Yes 5No 1Yes 5No 5No (teaming) 1Yes 1Yes 1O(N) 5No 1Yes 5No
IRV (AV) 1Yes 5No 5No 5No 1Yes 5No 1Yes 5No 1Yes 5O(N!)­[nb 5] 5No 1Yes 1Yes
Kemeny-Young 1Yes 1Yes 5No 1Yes 1Yes 4No (but ISDA) 5No (teaming) 1Yes 5No 2O(N2[nb 6] 1Yes 1Yes 5No
Majority Judg­ment[nb 7] 1Yes[nb 8] 1Yes 4No[nb 9] 4No[nb 2] 4No[nb 10] 2Yes 1Yes 4No[nb 11] 1Yes 1O(N)­[nb 12] 1Yes 1Yes 3Yes/No
Minimax 2Yes/No 1Yes 5No 2Yes[nb 13] 5No 5No 5No (spoilers) 5No 1Yes 2O(N2) 3Some variants 1Yes 4No[nb 13]
Plurality 2Yes/No 1Yes 1Yes 5No 5No 5No 5No (spoilers) 5No 1Yes 1O(N) 5No No[nb 4]
Range voting[nb 1] 5No 1Yes 2Yes[nb 2] 4No[nb 2] 5No 2Yes[nb 14] 3Ambig.­[nb 3] 1Yes 1Yes 1O(N) 1Yes 1Yes 5No
Ranked pairs 1Yes 1Yes 5No 1Yes 1Yes 4No (but ISDA) 1Yes 1Yes 1Yes 2O(N2) 1Yes 1Yes 5No
Runoff voting 2Yes/No 5No 5No 5No 1Yes 5No 5No (spoilers) 5No 1Yes 2O(N)­[nb 15] 5No 4No[nb 16] 2Yes[nb 17]
Schulze 1Yes 1Yes 5No 1Yes 1Yes 4No (but ISDA) 1Yes 1Yes 1Yes 2O(N2) 1Yes 1Yes 5No
Random winner/
arbitrary winner[nb 18]
5No 5NA 1Yes 5No 5No 1Yes 1No 5NA 1Yes 0O(1) 5No 5No
Random ballot[nb 19] 5No 1Yes 1Yes 5No 5No 1Yes 1Yes 1Yes 1Yes 1O(N) 5No 5No

"Yes/No", in a column which covers two related criteria, signifies that the given system passes the first criterion and not the second one.

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.[6] 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.[7][8] It found that Range voting consistently scored as either the best system or among the best across the various conditions studied.[9]

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).[10]

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 extremist candidates should have a chance to win, to prevent forcing candidates into taking either position. Their data showed that 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).[11]

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.[12]

"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.[13]

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.[14] 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.

Jean-Charles de Borda, an early voting theorist

Foundations of voting theory

Voting theory became an object of academic study around the time of the French Revolution.[13] 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.[15]

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.[16]

The Marquis de Condorcet, another early voting theorist

Later in the 18th century, the related topic of apportionment began to be studied. The impetus for research into fair apportionment methods came, in fact, from the United States Constitution, which mandated that seats in the United States House of Representatives had to be allocated among the states proportionally to their population, but did not specify how to do so.[17] 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.[17]

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.[18]

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.[19] 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.[20]

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.[13][Need quotation to verify] The use of mathematical criteria to evaluate voting systems was introduced when Kenneth Arrow showed in Arrow's impossibility theorem that certain intuitively desirable criteria were actually mutually contradictory, demonstrating the inherent limitations of voting theorems. These circumscribe nonetheless to ordinal voting systems, and do not apply to cardinal ones like range voting, as John Harsanyi pointed out. 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.[21]

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,[22][23][24] and on political stability.[25][26] A few scholars also studied what effects caused a nation to change for a particular voting system.[27][28][29][30][31][32] 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 other 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 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 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.

It has been argued and shown that more in-depth and fine-tuned voting systems lie at the core of the development of e-democracy,[33] which consists of the digitization of democratic processes, including voting.[34]

See also


Notes on systems comparison table


  1. ^ a b These criteria assume that all voters vote their true preference order. This is problematic for Approval and Range, where various votes are consistent with the same order. See approval voting for compliance under various voter models.
  2. ^ a b c d e In Approval, Range, and Majority Judgment, if all voters have perfect information about each other's true preferences and use rational strategy, any Condorcet or Majority winner will win in the Nash equilibrium. In particular if every voter knows that "A or B are the two most-likely to win" and places their "approval threshold" between the two, then the Condorcet winner, if one exists and is in the set {A,B}, will always win. These systems also satisfy the majority criterion in the weaker sense that any majority can force their candidate to win, if it so desires. (However, as the Condorcet criterion is incompatible with the participation criterion and the consistency criterion, these systems cannot satisfy these criteria in the Nash equilibrium. Laslier, J.-F. (2006) "Strategic approval voting in a large electorate," IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique).) While these arguments would apply to Plurality voting as well, Plurality suffers from such a profusion of Nash equilibria that they are irrelevant, and so this table ignores them in that connection.
  3. ^ a b The original independence of clones criterion applied only to ranked voting methods. (T. Nicolaus Tideman, "Independence of clones as a criterion for voting rules", Social Choice and Welfare Vol. 4, No. 3 (1987), pp. 185–206.) There is some disagreement about how to extend it to unranked methods, and this disagreement affects whether approval and range voting are considered independent of clones. If the definition of "clones" is that "every voter scores them within ±ε in the limit ε→0+", then range voting is immune to clones.
  4. ^ a b Approval and Plurality do not allow later preferences. Technically speaking, this means that they pass the technical definition of the LNH criteria - if later preferences or ratings are impossible, then such preferences can not help or harm. However, from the perspective of a voter, these systems do not pass these criteria. Approval, in particular, encourages the voter to give the same ballot rating to a candidate who, in another voting system, would get a later rating or ranking. Thus, for approval, the practically meaningful criterion would be not "later-no-harm" but "same-no-harm" - something neither approval nor any other system satisfies.
  5. ^ The number of piles that can be summed from various precincts is floor((e-1) N!) - 1.
  6. ^ Each prospective Kemeny-Young ordering has score equal to the sum of the pairwise entries that agree with it, and so the best ordering can be found using the pairwise matrix.
  7. ^ Bucklin voting, with skipped and equal-rankings allowed, meets the same criteria as Majority Judgment; in fact, Majority Judgment may be considered a form of Bucklin voting. Without allowing equal rankings, Bucklin's criteria compliance is worse; in particular, it fails Independence of Irrelevant Alternatives, which for a ranked method like this variant is incompatible with the Majority Criterion.
  8. ^ Majority judgment passes the rated majority criterion (a candidate rated solo-top by a majority must win). It does not pass the ranked majority criterion, which is incompatible with Independence of Irrelevant Alternatives.
  9. ^ Balinski and Laraki, Majority Judgment's inventors, point out that it meets a weaker criterion they call "grade consistency": if two electorates give the same rating for a candidate, then so will the combined electorate. Majority Judgment explicitly requires that ratings be expressed in a "common language", that is, that each rating have an absolute meaning. They claim that this is what makes "grade consistency" significant. MJ. Balinski M. and R. Laraki (2007) «A theory of measuring, electing and ranking». Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725.
  10. ^ Majority judgment passes the "majority condorcet loser" criterion; that is, a candidate who loses to all others by a majority cannot win. However, if some of the losses are not by a majority (including equal-rankings), the Condorcet loser can, theoretically, win in MJ, although such scenarios are rare.
  11. ^ Majority judgment can actually pass or fail reversal symmetry depending on the rounding method used to find the median when there are even numbers of voters. For instance, in a two-candidate, two-voter race, if the ratings are converted to numbers and the two central ratings are averaged, then MJ meets reversal symmetry; but if the lower one is taken, it does not, because a candidate with ["fair","fair"] would beat a candidate with ["good","poor"] with or without reversal. However, for rounding methods which do not meet reversal symmetry, the chances of breaking it are on the order of the inverse of the number of voters; this is comparable with the probability of an exact tie in a two-candidate race, and when there's a tie, any method can break reversal symmetry.
  12. ^ Majority Judgment is summable at order KN, where K, the number of ranking categories, is set beforehand.
  13. ^ a b A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.
  14. ^ Range satisfies the mathematical definition of IIA, that is, if each voter scores each candidate independently of which other candidates are in the race. However, since a given range score has no agreed-upon meaning, it is thought that most voters would either "normalize" or exaggerate their vote such that it votes at least one candidate each at the top and bottom possible ratings. In this case, Range would not be independent of irrelevant alternatives. Balinski M. and R. Laraki (2007) «A theory of measuring, electing and ranking». Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725.
  15. ^ Once for each round.
  16. ^ Later preferences are only possible between the two candidates who make it to the second round.
  17. ^ That is, second-round votes cannot harm candidates already eliminated.
  18. ^ Random winner: Uniformly randomly chosen candidate is winner. Arbitrary winner: some external entity, not a voter, chooses the winner. These systems are not, properly speaking, voting systems at all, but are included to show that even a horrible system can still pass some of the criteria.
  19. ^ Random ballot: Uniformly random-chosen ballot determines winner. This and closely related systems are of mathematical interest because they are the only possible systems which are truly strategy-free, that is, your best vote will never depend on anything about the other voters. They also satisfy both consistency and IIA, which is impossible for a deterministic ranked system. However, this system is not generally considered as a serious proposal for a practical method.

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, Carles (1999). "Setting the Rules of the Game: The Choice of Electoral Systems in Advanced Democracies". American Political Science Review 93 (3): 609–624. doi:10.2307/2585577. JSTOR 2585577. 
  • Colomer, Josep M., ed (2004). Handbook of Electoral System Choice. London and New York: Palgrave Macmillan. ISBN 9781403904546. 
  • Cranor, Lorrie. "Vote Aggregation Methods". Declared-Strategy Voting: An Instrument for Group Decision-Making. Retrieved October 3, 2005. 
  • 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 0416683207. 
  • Hermens, Ferdinand A. (1941). Democracy or Anarchy? A Study of Proportional Representation. Notre Dame, Indiana: University of Notre Dame. 
  • Lijphart, Arend
    • Lijphart (1985). "The Field of Electoral Systems Research: A Critical Survey". Electoral Studies 4. 
    • Lijphart, A. (1992). "Democratization and Constitutional Choices in Czecho-Slovakia, Hungary and Poland, 1989-1991". Journal of Theoretical Politics 4 (2): 207–223. doi:10.1177/0951692892004002005. 
    • Lijphart (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." LGDJ.
  • Rae, Douglas W. (1971). The Political Consequences of Electoral Laws. New Haven: Yale University Press. ISBN 0300015178. 
  • Reynolds, Andrew, Reilly, Benjamin and Ellis, Andrew, The New International IDEA Handbook of Electoral System Design, International IDEA, Stockholm 2005.
  • Rogowski, Ronald (1987). Trade and the Variety of Democratic Institutions. 41. International Organization. pp. 203–224. 
  • 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. 


  1. ^
  2. ^ Ludwig Windthorst, Speech in Favor of Reforming the Prussian Suffrage, in the Prussian House of Deputies, 26 November 1873
  4. ^ Consistency implies participation, but not vice versa. For example, range voting complies with participation and consistency, but median ratings satisfies participation and fails consistency.
  5. ^ Woodall, Douglas, Properties of Preferential Election Rules, Voting Matters, Issue 3, December 1994
  6. ^ Poundstone, William, Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It), Hill and Young, New York, 2008, p.239
  7. ^ Results are available in an appendix of his unpublished paper here: [1].
  8. ^ Poundstone, William, Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It), Hill and Young, New York, 2008, p.257: "Range voting is still largely a samizdat enterprise on the fringes of social choice theory. The most glaring example must be Smith's pivotal 2000 paper. It has never been published in a journal."
  9. ^ Poundstone, William, Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It), Hill and Young, New York, 2008, p.240
  10. ^ Balinski M. and R. Laraki (2007) «Election by Majority Judgement: Experimental Evidence». Cahier du Laboratoire d’Econométrie de l’Ecole Polytechnique 2007-28. Chapter in the book: «In Situ and Laboratory Experiments on Electoral Law Reform: French Presidential Elections», Edited by Bernard Dolez, Bernard Grofman and Annie Laurent. Springer, to appear in 2011.
  11. ^ Ibid.
  12. ^ These two-dimensional graphs are called Yee diagrams after their inventor, Ka-Ping Yee. His website includes some sample graphs.
  13. ^ a b c J. J. O'Connor and E. F. Robertson. "The history of voting". The MacTutor History of Mathematics Archive. Retrieved October 12, 2005. 
  14. ^ Miranda Mowbray and Dieter Gollmann. "Electing the Doge of Venice: Analysis of a 13th Century Protocol". Retrieved July 12, 2007. 
  15. ^ J. J. O'Connor and E. F. Robertson. "Marie Jean Antoine Nicolas de Caritat Condorcet". The MacTutor History of Mathematics Archive. Retrieved October 12, 2005. 
  16. ^ G. Hägele and F. Pukelsheim (2001). "Llull's writings on electoral systems". Studia Lulliana 3: 3–38. 
  17. ^ a b Joseph Malkevitch. "Apportionment". AMS Feature Columns. Retrieved October 13, 2005. 
  18. ^ "Proportional Voting Around the World". Retrieved October 13, 2005. 
  19. ^ "The History of IRV". Retrieved November 9, 2005. 
  20. ^ Tony Anderson Solgård and Paul Landskroener. "Municipal Voting System Reform: Overcoming the Legal Obstacles". Bench & Bar of Minnesota. Retrieved November 16, 2005. 
  21. ^ Poundstone, William, Gaming the Vote, p.198
  22. ^ Duverger 1954
  23. ^ Rae 1971
  24. ^ Taagapera & Shugart 1989
  25. ^ Hermens 1941
  26. ^ Lijphart 1994
  27. ^ Lijphart 1985
  28. ^ Lijphart 1992
  29. ^ Rokkan 1970
  30. ^ Rogowski 1987
  31. ^ Boix 1999
  32. ^ Cox 1997, pp. 15-16
  33. ^ Martin Hilbert (April, 2009). The Maturing Concept of E-Democracy: From E-Voting and Online Consultations to Democratic Value Out of Jumbled Online Chatter. Journal of Information Technology and Politics. Retrieved February 24, 2010. 
  34. ^ Hilbert, Martin. "DIGITAL PROCESSES AND DEMOCRATIC THEORY: Dynamics, risks and opportunities that arise when democratic institutions meet digital information and communication technologies." open-access online book 2007 <>

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