Voting system
Overview

A voting system or electoral system is a method by which voters
Voting
Voting is a method for a group such as a meeting or an electorate to make a decision or express an opinion—often following discussions, debates, or election campaigns. It is often found in democracies and republics.- Reasons for voting :...

make a choice between options, often in an election or on a policy referendum
Referendum
A referendum is a direct vote in which an entire electorate is asked to either accept or reject a particular proposal. This may result in the adoption of a new constitution, a constitutional amendment, a law, the recall of an elected official or simply a specific government policy. It is a form of...

.

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 algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

ic in nature, and, since it involves polling the sentiments of a person, this represents affective data.
Encyclopedia
A voting system or electoral system is a method by which voters
Voting
Voting is a method for a group such as a meeting or an electorate to make a decision or express an opinion—often following discussions, debates, or election campaigns. It is often found in democracies and republics.- Reasons for voting :...

make a choice between options, often in an election or on a policy referendum
Referendum
A referendum is a direct vote in which an entire electorate is asked to either accept or reject a particular proposal. This may result in the adoption of a new constitution, a constitutional amendment, a law, the recall of an elected official or simply a specific government policy. It is a form of...

.

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 algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

ic in nature, and, since it involves polling the sentiments of a person, this represents affective data. Together, with the exception of proxy voting
Proxy voting
Proxy voting has two forms: delegable voting and delegated voting, which are procedures for the delegation to another member of a voting body of that member's power to vote in his absence, and/or for the selection of additional representatives, as in the case with transitive proxies...

, this corresponds to in-degree centrality in graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

and social network analysis, with votes as directed edges, and voters and candidates as nodes. Common voting systems are majority rule
Majority rule
Majority rule is a decision rule that selects alternatives which have a majority, that is, more than half the votes. It is the binary decision rule used most often in influential decision-making bodies, including the legislatures of democratic nations...

, proportional representation
Proportional representation
Proportional representation is a concept in voting systems used to elect an assembly or council. PR means that the number of seats won by a party or group of candidates is proportionate to the number of votes received. For example, under a PR voting system if 30% of voters support a particular...

or plurality voting with a number of variations and methods such as first-past-the-post
First-past-the-post
First-past-the-post voting refers to an election won by the candidate with the most votes. The winning potato candidate does not necessarily receive an absolute majority of all votes cast.-Overview:...

or preferential voting
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...

. The study of formally defined voting systems is called voting theory, a subfield of political science
Political science
Political Science is a social science discipline concerned with the study of the state, government and politics. Aristotle defined it as the study of the state. It deals extensively with the theory and practice of politics, and the analysis of political systems and political behavior...

, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, or mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

.

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
Ballot
A ballot is a device used to record choices made by voters. Each voter uses one ballot, and ballots are not shared. In the simplest elections, a ballot may be a simple scrap of paper on which each voter writes in the name of a candidate, but governmental elections use pre-printed to protect the...

, the set of allowable votes, and the tallying method, an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

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
Electronic voting
Electronic voting is a term encompassing several different types of voting, embracing both electronic means of casting a vote and electronic means of counting votes....

. 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
Electoral Reform Society
The Electoral Reform Society is a political pressure group based in the United Kingdom which promotes electoral reform. It is believed to be the oldest organisation concerned with electoral systems in the world.-Aims:...

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'.

### The ballot

Different voting systems have different forms for allowing the individual to express his or her vote. In ranked ballot or "preference" voting systems, such as Instant-runoff voting
Instant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

, the Borda count
Borda count
The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...

, or a Condorcet method
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...

, voters order the list of options from most to least preferred. In range voting
Range voting
Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...

, voters rate each option separately on a scale. In plurality voting
Plurality voting system
The plurality voting system is a single-winner voting system often used to elect executive officers or to elect members of a legislative assembly which is based on single-member constituencies...

(also known as "first-past-the-post"), voters select only one option, while in approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

, they can select as many as they want. In voting systems that allow "plumping", like cumulative voting
Cumulative voting
Cumulative voting is a multiple-winner voting system intended to promote more proportional representation than winner-take-all elections.- History :...

, voters may vote for the same candidate multiple times.

Some voting systems include additional choices on the ballot, such as write-in candidate
Write-in candidate
A write-in candidate is a candidate in an election whose name does not appear on the ballot, but for whom voters may vote nonetheless by writing in the person's name. Some states and local jurisdictions allow a voter to affix a sticker with a write-in candidate's name on it to the ballot in lieu...

s, a none of the above
None of the above
None of the Above or against all is a ballot option in some jurisdictions or organizations, designed to allow the voter to indicate disapproval of all of the candidates in a voting system...

option, or a no confidence in that candidate option.

### Candidates

Some methods call for a primary election
Primary election
A primary election is an election in which party members or voters select candidates for a subsequent election. Primary elections are one means by which a political party nominates candidates for the next general election....

first to determine which candidates will be on the ballot.

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
Corporation
A corporation is created under the laws of a state as a separate legal entity that has privileges and liabilities that are distinct from those of its members. There are many different forms of corporations, most of which are used to conduct business. Early corporations were established by charter...

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
Political party
A political party is a political organization that typically seeks to influence government policy, usually by nominating their own candidates and trying to seat them in political office. Parties participate in electoral campaigns, educational outreach or protest actions...

in a parliament), voting power measures the ability of a group to change the outcome of a vote. Groups may form coalition
Coalition
A coalition is a pact or treaty among individuals or groups, during which they cooperate in joint action, each in their own self-interest, joining forces together for a common cause. This alliance may be temporary or a matter of convenience. A coalition thus differs from a more formal covenant...

s to maximize voting power.

In some German states, most notably Prussia
Kingdom of Prussia
The Kingdom of Prussia was a German kingdom from 1701 to 1918. Until the defeat of Germany in World War I, it comprised almost two-thirds of the area of the German Empire...

and Sachsen
Kingdom of Saxony
The Kingdom of Saxony , lasting between 1806 and 1918, was an independent member of a number of historical confederacies in Napoleonic through post-Napoleonic Germany. From 1871 it was part of the German Empire. It became a Free state in the era of Weimar Republic in 1918 after the end of World War...

, there was before 1918 a weighted vote system known as the Prussian three-class franchise
Prussian three-class franchise
After the 1848 revolutions in the German states, the Prussian three-class franchise system was introduced in 1849 by the Prussian king Friedrich Wilhelm IV for the election of the Lower House of the Prussian state parliament. It was completely abolished only in 1918...

, where the electorate would be divided into three categories based on the amount of income tax
Income tax
An income tax is a tax levied on the income of individuals or businesses . Various income tax systems exist, with varying degrees of tax incidence. Income taxation can be progressive, proportional, or regressive. When the tax is levied on the income of companies, it is often called a corporate...

paid. Each category would have equal voting power in choosing the electors. they are known as candidates

### Status quo

Some voting systems are weighted in themselves, for example if a supermajority
Supermajority
A supermajority or a qualified majority is a requirement for a proposal to gain a specified level or type of support which exceeds a simple majority . In some jurisdictions, for example, parliamentary procedure requires that any action that may alter the rights of the minority has a supermajority...

is required to change the status quo. An extreme case of this is unanimous consent
Unanimous consent
In parliamentary procedure, unanimous consent, also known as general consent, or in the case of the parliaments under the Westminster system, leave of the house, is a situation in which no one present objects to a proposal. The chair may state, for instance: "If there is no objection, the motion...

, 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
Blackball (blacklist)
Blackballing is a rejection in a traditional form of secret ballot, where a white ball or ballot constitutes a vote in support and a black ball signifies opposition. This system is typically used where a club's rules provide that, rather than a majority of the votes, one or two objections are...

.

A different mechanism that favors the status quo is the requirement for a quorum
Quorum
A quorum is the minimum number of members of a deliberative assembly necessary to conduct the business of that group...

, 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.

### 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
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...

, fill their entire parliament using a single multiple-winner district (constituency), while others, like the Republic of Ireland
Republic of Ireland
Ireland , described as the Republic of Ireland , is a sovereign state in Europe occupying approximately five-sixths of the island of the same name. Its capital is Dublin. Ireland, which had a population of 4.58 million in 2011, is a constitutional republic governed as a parliamentary democracy,...

or Belgium
Belgium
Belgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...

, 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
Lower house
A lower house is one of two chambers of a bicameral legislature, the other chamber being the upper house.Despite its official position "below" the upper house, in many legislatures worldwide the lower house has come to wield more power...

) and multi-member electorates for its Senate
Senate
A senate is a deliberative assembly, often the upper house or chamber of a legislature or parliament. There have been many such bodies in history, since senate means the assembly of the eldest and wiser members of the society and ruling class...

(upper house). Some systems, like the Additional member system
The Additional Member System is the term used in the United Kingdom for the mixed member proportional representation voting system used in Scotland, Wales and the London Assembly....

, 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
Redistricting
Redistricting is the process of drawing United States electoral district boundaries, often in response to population changes determined by the results of the decennial census. In 36 states, the state legislature has primary responsibility for creating a redistricting plan, in many cases subject to...

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
Gerrymandering
In the process of setting electoral districts, gerrymandering is a practice that attempts to establish a political advantage for a particular party or group by manipulating geographic boundaries to create partisan, incumbent-protected districts...

, 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
Gallagher Index
The Gallagher Index is used to measure the disproportionality of an electoral outcome, that is the difference between the percentage of votes received and the percentage of seats a party gets in the resulting legislature. This is especially useful for comparing proportionality across electoral...

.

Most proportional systems in use are based on party-list proportional representation
Party-list proportional representation
Party-list proportional representation systems are a family of voting systems emphasizing proportional representation in elections in which multiple candidates are elected...

, 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 method
Highest averages method
The highest averages method is the name for a variety of ways to allocate seats proportionally for representative assemblies with party list voting systems....

s and largest remainder method
Largest remainder method
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems...

s. Largest remainder methods set a particular quota based on the number of voters, while highest averages methods, such as the Sainte-Laguë method
Sainte-Laguë method
The Sainte-Laguë method is one way of allocating seats approximately proportional to the number of votes of a party to a party list used in many voting systems. It is named after the French mathematician André Sainte-Laguë. The Sainte-Laguë method is quite similar to the D'Hondt method, but uses...

and the d'Hondt method
D'Hondt method
The d'Hondt method is a highest averages method for allocating seats in party-list proportional representation. The method described is named after Belgian mathematician Victor D'Hondt who described it in 1878...

, 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
Open list
Open list describes any variant of party-list proportional representation where voters have at least some influence on the order in which a party's candidates are elected...

system, voters decide which candidates within a party win the seats. In a closed list
Closed list
Closed list describes the variant of party-list proportional representation where voters can only vote for political parties as a whole and thus have no influence on the party-supplied order in which party candidates are elected...

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
Single transferable vote
The single transferable vote is a voting system designed to achieve proportional representation through preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate, and then, after candidates have been either elected or eliminated, any surplus or...

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.)

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
Spoiler effect
The spoiler effect describes the effect a minor party candidate with little chance of winning has in a close election, when that candidate's presence in the election draws votes from a major candidate similar to them, thereby causing a candidate dissimilar to them to win the election...

of the plurality voting system
Plurality voting system
The plurality voting system is a single-winner voting system often used to elect executive officers or to elect members of a legislative assembly which is based on single-member constituencies...

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
Limited Voting
Limited voting is a voting system in which electors have fewer votes than there are positions available. The positions are awarded to the candidates who receive the most votes absolutely...

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
Single non-transferable vote
The single non-transferable vote or SNTV is an electoral system used in multi-member constituency elections.- Voting :In any election, each voter casts one vote for one candidate in a multi-candidate race for multiple offices. Posts are filled by the candidates with the most votes...

(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
Landslide victory
In politics, a landslide victory is the victory of a candidate or political party by an overwhelming margin in an election...

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
Single non-transferable vote
The single non-transferable vote or SNTV is an electoral system used in multi-member constituency elections.- Voting :In any election, each voter casts one vote for one candidate in a multi-candidate race for multiple offices. Posts are filled by the candidates with the most votes...

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
Tactical voting
In voting systems, tactical voting occurs, in elections with more than two viable candidates, when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.It has been shown by the Gibbard-Satterthwaite theorem that any voting method which is...

or strategic nomination
Strategic nomination
Strategic nomination is the manipulation of an election through its candidate set...

.

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
Parallel voting
Parallel voting describes a mixed voting system where voters in effect participate in two separate elections for a single chamber using different systems, and where the results in one election have little or no impact on the results of the other...

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

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
Two-round system
The two-round system is a voting system used to elect a single winner where the voter casts a single vote for their chosen candidate...

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
Random ballot
The random ballot is a hypothetical voting method; in an election or referendum, the ballot of a single voter is selected at random, and that ballot decides the result of the election...

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
Instant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

(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 system
Positional voting system
A positional voting system is a ranked voting method in which the options receive points based on their position on each ballot, and the option with the most points wins....

s.

Other ranked methods include Coombs' method
Coombs' method
The Coombs' method is a voting system created by Clyde Coombs used for single-winner elections in which each voter rank the candidates in order of preference. It is very similar to instant-runoff voting , a more common preferential voting system.-Procedures:Each voter rank-orders all of the...

, Supplementary voting, Bucklin voting
Bucklin voting
Bucklin voting is a class of voting systems that can be used for single-member and multi-member districts. It is named after its original promoter, James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system...

, 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
Smith set
In voting systems, the Smith set, named after John H. Smith, is the smallest non-empty set of candidates in a particular election such that each member beats every other candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome...

. 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
Copeland's method
Copeland's method or Copeland's pairwise aggregation method is a Condorcet method in which candidates are ordered by the number of pairwise victories, minus the number of pairwise defeats....

, in which the winner is the option that wins the most pairwise contests, as in many round-robin tournament
Round-robin tournament
A round-robin tournament is a competition "in which each contestant meets all other contestants in turn".-Terminology:...

s. The Schulze method
Schulze method
The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners...

(also known as "Schwartz sequential dropping", "cloneproof Schwartz sequential dropping" or the "beatpath method") and Ranked pairs
Ranked Pairs
Ranked pairs or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners....

are two recently designed Condorcet methods that satisfy a large number of voting system criteria.

The Kemeny-Young method
Kemeny-Young method
The Kemeny–Young method is a voting system that uses preferential ballots and pairwise comparison counts to identify the most popular choices in an election...

, the Schulze method
Schulze method
The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners...

, and the ranked pairs method
Ranked Pairs
Ranked pairs or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners....

are Condorcet methods that fully rank all the candidates from most popular to least popular.

### Rated voting methods

Rated ballots allow even more flexibility than ranked ballots, but few methods are designed to use them. Each voter gives a score to each option; the allowable scores could be numeric (for example, from 0 to 100) or could be "grade
Grades are standardized measurements of varying levels of comprehension within a subject area. Grades can be assigned in letters , as a range , as a number out of a possible total , as descriptors , in percentages, or, as is common in some post-secondary...

s" 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
Range voting
Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...

, voters give numeric ratings to each option, and the option with the highest total or average score wins. In majority judgment
Majority Judgment
Majority Judgment is a single-winner voting system proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner. If more than one candidate has the...

, similar ballots are used, but the winner is the candidate with the highest median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

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
Cumulative voting
Cumulative voting is a multiple-winner voting system intended to promote more proportional representation than winner-take-all elections.- History :...

. 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
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....

proved Arrow's impossibility theorem
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...

, 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
Majority criterion
The majority criterion is a single-winner voting system criterion, used to compare such systems. The criterion states that "if one candidate is preferred by a majority of voters, then that candidate must win"....

—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
Mutual majority criterion
The mutual majority criterion is a criterion used to compare voting systems. It is also known as the majority criterion for solid coalitions and the generalized 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
Majority loser criterion
The majority loser criterion is a criterion to evaluate single-winner voting systems. The criterion states that if a majority of voters prefers every other candidate over a given candidate, then that candidate must not win....

—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
Monotonicity criterion
The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting systems. A voting system is monotonic if it satisfies one of the definitions of the monotonicity criterion, given below.Douglas R...

(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
Consistency criterion
A voting system is consistent if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same choice being selected, an election of the entire electorate also selects that alternative...

—If the electorate is divided in two and a choice wins in both parts, does it always win overall?
• Participation criterion
Participation criterion
The participation criterion is a voting system criterion. It is also known as the "no show paradox". It has been defined as follows:* In a deterministic framework, the participation criterion says that the addition of a ballot, where candidate A is strictly preferred to candidate B, to an existing...

—Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below.)
• Condorcet criterion
Condorcet criterion
The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates...

—If a candidate beats every other candidate in pairwise comparison, does that candidate always win? (This implies the majority criterion, above)
• Condorcet loser criterion
Condorcet loser criterion
In single-winner voting system theory, the Condorcet loser criterion is a measure for differentiating voting systems. It implies the majority loser criterion....

(Cond. loser)—If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?
• Independence of irrelevant alternatives
Independence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....

(IIA)—If a candidate is added or removed, do the relative rankings of the remaining candidates stay the same?
• Independence of clones criterion
Independence of clones criterion
In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman first formulated the criterion, which states that the addition of a candidate identical to one already present in an election will not cause the winner...

(Cloneproof)—Is the outcome the same if candidates identical to existing candidates are added?
• Reversal symmetry
Reversal symmetry
Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, the Kemeny-Young method, and the Schulze method...

—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
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

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
Overvote
An overvote occurs when one votes for more than the maximum number of selections allowed in a contest. The result is a spoilt vote which is not included in the final tally....

—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
Tactical voting
In voting systems, tactical voting occurs, in elections with more than two viable candidates, when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.It has been shown by the Gibbard-Satterthwaite theorem that any voting method which is...

strategies.
• Allows later preferences
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...

(later prefs)—Can a voter indicate different levels of support through ranking or rating candidates?
• Later-no-harm criterion
Later-no-harm criterion
The later-no-harm criterion is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a less preferred candidate cannot cause a more preferred candidate to lose.- Complying methods :Single...

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.

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.

Major­ity
Majority criterion
The majority criterion is a single-winner voting system criterion, used to compare such systems. The criterion states that "if one candidate is preferred by a majority of voters, then that candidate must win"....

/
MMC
Mutual majority criterion
The mutual majority criterion is a criterion used to compare voting systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion...

Mono­tone
Monotonicity criterion
The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting systems. A voting system is monotonic if it satisfies one of the definitions of the monotonicity criterion, given below.Douglas R...

Consist­ency
Consistency criterion
A voting system is consistent if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same choice being selected, an election of the entire electorate also selects that alternative...

/
Particip­ation
Participation criterion
The participation criterion is a voting system criterion. It is also known as the "no show paradox". It has been defined as follows:* In a deterministic framework, the participation criterion says that the addition of a ballot, where candidate A is strictly preferred to candidate B, to an existing...

Condorcet
Condorcet criterion
The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates...

Cond.
loser
Condorcet loser criterion
In single-winner voting system theory, the Condorcet loser criterion is a measure for differentiating voting systems. It implies the majority loser criterion....

IIA
Independence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....

Cloneproof
Reversal
symmetry
Reversal symmetry
Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, the Kemeny-Young method, and the Schulze method...

Poly­time
Summ­able
Equal rankings
can exist
Later
prefs
Later-no-help/
Later-no-harm
Approval
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

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.
Ambig­uous  Yes Yes NoIn 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
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

. 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.
No Ambig­uous Ambig.­The original independence of clones criterion
Strategic nomination
Strategic nomination is the manipulation of an election through its candidate set...

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.
Yes Yes O(N) Yes NoApproval 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.
Borda count
Borda count
The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...

No  Yes Yes No  Yes No No (teaming
Strategic nomination
Strategic nomination is the manipulation of an election through its candidate set...

)
Yes Yes O(N) No Yes No
IRV
Instant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

(AV)
Yes  No  No No  Yes No Yes No Yes O(N!)­The number of piles that can be summed from various precincts is floor((e-1) N!) - 1. No Yes Yes
Kemeny-Young
Kemeny-Young method
The Kemeny–Young method is a voting system that uses preferential ballots and pairwise comparison counts to identify the most popular choices in an election...

Yes  Yes No Yes  Yes No (but ISDA) No (teaming
Strategic nomination
Strategic nomination is the manipulation of an election through its candidate set...

)
Yes No O(N2Each 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. Yes Yes No
Majority Judg­ment
Majority Judgment
Majority Judgment is a single-winner voting system proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner. If more than one candidate has the...

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.
YesMajority 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. Yes NoBalinski 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. No NoMajority 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. Yes Yes NoMajority 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. Yes O(N)­Majority Judgment is summable at order KN, where K, the number of ranking categories, is set beforehand. Yes Yes Yes/No
Minimax
Minimax Condorcet
In voting systems, the Minimax method is one of several Condorcet methods used for tabulating votes and determining a winner when using preferential voting in a single-winner election...

Yes/No Yes No YesA variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm. No No No (spoilers) No Yes O(N2) Some variants Yes No
Plurality
Plurality voting system
The plurality voting system is a single-winner voting system often used to elect executive officers or to elect members of a legislative assembly which is based on single-member constituencies...

Yes/No Yes Yes No  No No No (spoilers) No Yes O(N) No No
Range voting
Range voting
Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...

No  Yes Yes No No YesRange 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. Ambig.­ Yes Yes O(N) Yes Yes No
Ranked pairs
Ranked Pairs
Ranked pairs or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners....

Yes  Yes No Yes  Yes No (but ISDA) Yes Yes Yes O(N2) Yes Yes No
Runoff voting
Two-round system
The two-round system is a voting system used to elect a single winner where the voter casts a single vote for their chosen candidate...

Yes/No No No No  Yes No No (spoilers) No Yes O(N)­Once for each round. No NoLater preferences are only possible between the two candidates who make it to the second round. YesThat is, second-round votes cannot harm candidates already eliminated.
Schulze
Schulze method
The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners...

Yes  Yes No Yes  Yes No (but ISDA) Yes Yes Yes O(N2) Yes Yes No
Random winner
Sortition
In politics, sortition is the selection of decision makers by lottery. The decision-makers are chosen as a random sample from a larger pool of candidates....

/
arbitrary winnerRandom 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.
No NA Yes No No Yes No NA Yes O(1) No No
Random ballot
Random ballot
The random ballot is a hypothetical voting method; in an election or referendum, the ballot of a single voter is selected at random, and that ballot decides the result of the election...

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.
No Yes Yes No No Yes Yes Yes Yes O(N) No No

"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. 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. 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
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

is strategic vulnerability. According to the Gibbard-Satterthwaite theorem
Gibbard-Satterthwaite theorem
The Gibbard–Satterthwaite theorem, named after Allan Gibbard and Mark Satterthwaite, is a result about the deterministic voting systems that choose a single winner using only the preferences of the voters, where each voter ranks all candidates in order of preference...

, no voting system can be immune to strategic manipulation
Tactical voting
In voting systems, tactical voting occurs, in elections with more than two viable candidates, when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.It has been shown by the Gibbard-Satterthwaite theorem that any voting method which is...

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
Range voting
Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...

, Borda count
Borda count
The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...

, plurality voting, approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

with two different absolute approval thresholds, Condorcet voting, and majority judgment
Majority Judgment
Majority Judgment is a single-winner voting system proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner. If more than one candidate has the...

, 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
Two-round system
The two-round system is a voting system used to elect a single winner where the voter casts a single vote for their chosen candidate...

, 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).

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
Tactical voting
In voting systems, tactical voting occurs, in elections with more than two viable candidates, when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.It has been shown by the Gibbard-Satterthwaite theorem that any voting method which is...

or strategic nomination
Strategic nomination
Strategic nomination is the manipulation of an election through its candidate set...

, and, for multiple-winner methods, the degree of proportionality produced.

The New Zealand Royal Commission on the Electoral System
Royal Commission on the Electoral System
The Royal Commission on the Electoral System was formed in New Zealand in 1985, and reported in 1986. The decision to form the Royal Commission was taken by the Fourth Labour government, after the Labour party had received more votes, yet won fewer seats than the National Party in both the 1978 and...

listed ten criteria for their evaluation of possible new electoral systems for New Zealand
Electoral reform in New Zealand
Electoral reform in New Zealand has, in recent years, become a political issue as major changes have been made to both Parliamentary and local government electoral systems.- Parliamentary Electoral Reform :...

. 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
Athenian democracy
Athenian democracy developed in the Greek city-state of Athens, comprising the central city-state of Athens and the surrounding territory of Attica, around 508 BC. Athens is one of the first known democracies. Other Greek cities set up democracies, and even though most followed an Athenian model,...

. 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
Sortition
In politics, sortition is the selection of decision makers by lottery. The decision-makers are chosen as a random sample from a larger pool of candidates....

), as well as rotation of office. One of the earliest recorded elections in Athens was a plurality vote
Plurality voting system
The plurality voting system is a single-winner voting system often used to elect executive officers or to elect members of a legislative assembly which is based on single-member constituencies...

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
History of democracy
The history of democracy traces back to Athens to its re-emergence and rise from the 17th century to the present day. According to one definition, democracy is a political system in which all the members of the society have an equal share of formal political power...

were held using plurality voting or some variant, but as an exception, the state of Venice
Venice
Venice is a city in northern Italy which is renowned for the beauty of its setting, its architecture and its artworks. It is the capital of the Veneto region...

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
Electoral college
An electoral college is a set of electors who are selected to elect a candidate to a particular office. Often these represent different organizations or entities, with each organization or entity represented by a particular number of electors or with votes weighted in a particular way...

of nine members elected 40 people by approval voting; those 40 were reduced to form a second electoral college of 12 members by drawing lots again. The second electoral college elected 25 people by approval voting, which were reduced to form a third electoral college of nine members by drawing lots. The third electoral college elected 45 people, which were reduced to form a fourth electoral college of 11 by drawing lots. They in turn elected a final electoral body of 41 members, who ultimately elected the Doge. Despite its complexity, the system had certain desirable properties such as being hard to game and ensuring that the winner reflected the opinions of both majority and minority factions. This process was used with little modification from 1268 until the end of the Republic of Venice
Republic of Venice
The Republic of Venice or Venetian Republic was a state originating from the city of Venice in Northeastern Italy. It existed for over a millennium, from the late 7th century until 1797. It was formally known as the Most Serene Republic of Venice and is often referred to as La Serenissima, in...

in 1797, and was one of the factors contributing to the durability of the republic.

### Foundations of voting theory

Voting theory became an object of academic study around the time of the French Revolution
French Revolution
The French Revolution , sometimes distinguished as the 'Great French Revolution' , was a period of radical social and political upheaval in France and Europe. The absolute monarchy that had ruled France for centuries collapsed in three years...

. Jean-Charles de Borda proposed the Borda count in 1770 as a method for electing members to the French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

. His system was opposed by the Marquis de Condorcet
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...

, 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
Ramon Llull
Ramon Llull was a Majorcan writer and philosopher, logician and tertiary Franciscan. He wrote the first major work of Catalan literature. Recently-surfaced manuscripts show him to have anticipated by several centuries prominent work on elections theory...

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
Apportionment (politics)
Apportionment is the process of allocating political power among a set of principles . In most representative governments, political power has most recently been apportioned among constituencies based on population, but there is a long history of different approaches.The United States Constitution,...

began to be studied. The impetus for research into fair apportionment methods came, in fact, from the United States Constitution
United States Constitution
The Constitution of the United States is the supreme law of the United States of America. It is the framework for the organization of the United States government and for the relationship of the federal government with the states, citizens, and all people within the United States.The first three...

, which mandated that seats in the United States House of Representatives
United States House of Representatives
The United States House of Representatives is one of the two Houses of the United States Congress, the bicameral legislature which also includes the Senate.The composition and powers of the House are established in Article One of the Constitution...

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
Alexander Hamilton
Alexander Hamilton was a Founding Father, soldier, economist, political philosopher, one of America's first constitutional lawyers and the first United States Secretary of the Treasury...

, Thomas Jefferson
Thomas Jefferson
Thomas Jefferson was the principal author of the United States Declaration of Independence and the Statute of Virginia for Religious Freedom , the third President of the United States and founder of the University of Virginia...

, and Daniel Webster
Daniel Webster
Daniel Webster was a leading American statesman and senator from Massachusetts during the period leading up to the Civil War. He first rose to regional prominence through his defense of New England shipping interests...

. 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
Sainte-Laguë method
The Sainte-Laguë method is one way of allocating seats approximately proportional to the number of votes of a party to a party list used in many voting systems. It is named after the French mathematician André Sainte-Laguë. The Sainte-Laguë method is quite similar to the D'Hondt method, but uses...

, while Hamilton's method is identical to the Hare largest remainder method.

The Single Transferable Vote
Single transferable vote
The single transferable vote is a voting system designed to achieve proportional representation through preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate, and then, after candidates have been either elected or eliminated, any surplus or...

system was devised by Carl Andrae in Denmark
Denmark
Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. 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
Tasmania
Tasmania is an Australian island and state. It is south of the continent, separated by Bass Strait. The state includes the island of Tasmania—the 26th largest island in the world—and the surrounding islands. The state has a population of 507,626 , of whom almost half reside in the greater Hobart...

in 1896 after its use was promoted by Andrew Inglis Clark
Andrew Inglis Clark
Andrew Inglis Clark was an Australian barrister, politician, electoral reformer and jurist. He initially qualified engineer, however he re-trained as a barrister in order to effectively fight for social causes which deeply concerned him...

. 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
British Empire
The British Empire comprised the dominions, colonies, protectorates, mandates and other territories ruled or administered by the United Kingdom. It originated with the overseas colonies and trading posts established by England in the late 16th and early 17th centuries. At its height, it was the...

colonies.

### The single-winner revival

Perhaps influenced by the rapid development of multiple-winner voting methods, theorists began to publish new findings about single-winner methods in the late 19th century. This began around 1870, when William Robert Ware
William Robert Ware
William Robert Ware , born in Cambridge, Massachusetts into a family of the Unitarian clergy, was an American architect, author, and founder of two important American architectural schools....

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
Edward J. Nanson
Edward John Nanson was a mathematician known for devising Nanson's method , a variation of the Borda count using successive elimination down to the winner....

combined the newly described instant runoff voting with the Borda count to yield a new Condorcet method called Nanson's method
Nanson's method
The Borda count can be combined with an Instant Runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method.- Nanson method :The Nanson method is based on the original work of the mathematician Edward J...

. Charles Dodgson, better known as Lewis Carroll, published pamphlets on voting theory, focusing in particular on Condorcet voting. He introduced the use of matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

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
Dodgson's method
Dodgson's Method is a voting system proposed by Charles Dodgson.-Description:In Dodgson's method, each voter submits an ordered list of all candidates according to their own preference . The winner is defined to be the candidate for whom we need to perform the minimum number of pairwise swaps ...

.

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
Minnesota
Minnesota is a U.S. state located in the Midwestern United States. The twelfth largest state of the U.S., it is the twenty-first most populous, with 5.3 million residents. Minnesota was carved out of the eastern half of the Minnesota Territory and admitted to the Union as the thirty-second state...

.

### Influence of game theory

After John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

and others developed the mathematical field of game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

in the 1940s, new mathematical tools were available to analyze voting systems and strategic voting. This led to significant new results that changed the field of voting theory. The use of mathematical criteria to evaluate voting systems was introduced when Kenneth Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....

showed in Arrow's impossibility theorem
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...

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
John Harsanyi
John Charles Harsanyi was a Hungarian-Australian-American economist and Nobel Memorial Prize in Economic Sciences winner....

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
Gibbard-Satterthwaite theorem
The Gibbard–Satterthwaite theorem, named after Allan Gibbard and Mark Satterthwaite, is a result about the deterministic voting systems that choose a single winner using only the preferences of the voters, where each voter ranks all candidates in order of preference...

, 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
Duverger's law
In political science, Duverger's law is a principle which asserts that a plurality rule election system tends to favor a two-party system. This is one of two hypotheses proposed by Duverger, the second stating that “the double ballot majority system and proportional representation tend to...

is a prominent example of such a result, showing that plurality voting often leads to a two-party system
Two-party system
A two-party system is a system where two major political parties dominate voting in nearly all elections at every level of government and, as a result, all or nearly all elected offices are members of one of the two major parties...

. Further research into the game theory aspects of voting led Steven Brams
Steven Brams
Steven J. Brams is a game theorist and political scientist at the New York University Department of Politics. Brams is best known for using the techniques of game theory and public choice to research voting systems and fair division. He is one of the independent discoverers of approval voting...

and Peter Fishburn to formally define and promote the use of approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

in 1977. While approval voting had been used before that, it had not been named or considered as an object of academic study, particularly because it violated the assumption made by most research that single-winner methods were based on preference rankings.

### Post-1980 developments

Voting theory has come to focus on voting system criteria almost as much as it does on particular voting systems. Now, any description of a benefit or weakness in a voting system is expected to be backed up by a mathematically defined criterion. Recent research in voting theory has largely involved devising new criteria and new methods devised to meet certain criteria.

Political scientists of the 20th century published many studies on the effects that the voting systems have on voters' choices and political parties, and on political stability. A few scholars also studied what effects caused a nation to change for a particular voting system. One prominent current voting theorist is Nicolaus Tideman
Nicolaus Tideman
T. Nicolaus Tideman is a Professor of Economics at Virginia Polytechnic Institute and State University. He received his Bachelor of Arts in economics and mathematics from Reed College in 1965 and his PhD in economics from the University of Chicago in 1969...

, who formalized concepts such as strategic nomination
Strategic nomination
Strategic nomination is the manipulation of an election through its candidate set...

and the spoiler effect
Spoiler effect
The spoiler effect describes the effect a minor party candidate with little chance of winning has in a close election, when that candidate's presence in the election draws votes from a major candidate similar to them, thereby causing a candidate dissimilar to them to win the election...

in the independence of clones criterion
Independence of clones criterion
In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman first formulated the criterion, which states that the addition of a candidate identical to one already present in an election will not cause the winner...

. Tideman also devised the ranked pairs method, a Condorcet method that is not susceptible to clones. Also, Donald G. Saari
Donald G. Saari
Donald Gene Saari is the Distinguished Professor of Mathematics and Economics and director of the Institute for Mathematical Behavioral Sciences at the University of California Irvine...

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
Electoral reform
Electoral reform is change in electoral systems to improve how public desires are expressed in election results. That can include reforms of:...

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
Instant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

in the 2000s. New Zealand
New Zealand
New Zealand is an island country in the south-western Pacific Ocean comprising two main landmasses and numerous smaller islands. The country is situated some east of Australia across the Tasman Sea, and roughly south of the Pacific island nations of New Caledonia, Fiji, and Tonga...

adopted Mixed Member Proportional for Parliamentary elections in 1993 and Single Transferable Vote
Single transferable vote
The single transferable vote is a voting system designed to achieve proportional representation through preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate, and then, after candidates have been either elected or eliminated, any surplus or...

for some local elections in 2004 (see Electoral reform in New Zealand
Electoral reform in New Zealand
Electoral reform in New Zealand has, in recent years, become a political issue as major changes have been made to both Parliamentary and local government electoral systems.- Parliamentary Electoral Reform :...

). The Canadian province of British Columbia
British Columbia
British Columbia is the westernmost of Canada's provinces and is known for its natural beauty, as reflected in its Latin motto, Splendor sine occasu . Its name was chosen by Queen Victoria in 1858...

held two unsuccessful referendums (in 2005
British Columbia electoral reform referendum, 2005
A referendum was held in the Canadian province of British Columbia on May 17, 2005 to determine whether or not to adopt the recommendations of the Citizens' Assembly on Electoral Reform...

and 2009
British Columbia electoral reform referendum, 2009
A second referendum on electoral reform was held in conjunction with the provincial election on May 12, 2009. The BC-single transferrable vote electoral system was again voted on by the BC electorate. It would have required 60 per cent overall approval and 50 per cent approval in at least 60 per...

BC-STV
BC-STV is a proposed voting system recommended by the Citizens' Assembly on Electoral Reform for use in British Columbia, and belongs to the Single Transferable Vote family of voting systems. BC-STV was supported by a majority of the voters in a referendum held in 2005 but the government had...

system, and Ontario
Ontario
Ontario is a province of Canada, located in east-central Canada. It is Canada's most populous province and second largest in total area. It is home to the nation's most populous city, Toronto, and the nation's capital, Ottawa....

, 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
E-democracy
E-democracy refers to the use of information technologies and communication technologies and strategies in political and governance processes...

, which consists of the digitization of democratic processes, including voting.

• Nakamura number
Nakamura number
In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....

• Opinion poll
Opinion poll
An opinion poll, sometimes simply referred to as a poll is a survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions of a population by conducting a series of questions and then extrapolating generalities in ratio or within confidence...

• Proxy voting
Proxy voting
Proxy voting has two forms: delegable voting and delegated voting, which are procedures for the delegation to another member of a voting body of that member's power to vote in his absence, and/or for the selection of additional representatives, as in the case with transitive proxies...

• Table of voting systems by nation
Table of voting systems by nation
This table deals with voting to select candidates for office, not for the passing of legislation.- Voting systems by country :- Key :Seats per district : Most elections are split into a number of electoral districts. In some elections, there is one person elected per district. In others, there are...

• Vote counting system
Vote counting system
There exist various methods through which the ballots cast at an election may be counted, prior to applying a voting system to obtain one or more winners.-Manual counting:Manual counting requires a physical ballot that represents voter intent...

• Voting machine
Voting machine
Voting machines are the total combination of mechanical, electromechanical, or electronic equipment , that is used to define ballots; to cast and count votes; to report or display election results; and to maintain and produce any audit trail information...

• Experimental political science
Experimental political science
Experimental political science is the use in political science of experiments to implement the scientific method.- Usage :Among the areas that it is used in are:...

In distributed computing, leader election is the process of designating a single process as the organizer of some task distributed among several computers . Before the task is begun, all network nodes are unaware which node will serve as the "leader," or coordinator, of the task...

### General references

• Arrow, Kenneth J. (1951, 2nd ed., 1963) Social Choice and Individual Values
Social Choice and Individual Values
Kenneth Arrow's monograph Social Choice and Individual Values and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor...

. New Haven: Yale University Press. ISBN 0-300-01364-7
• Dummett, Michael
Michael Dummett
Sir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...

(1997). Principles of Electoral Reform. New York: Oxford University Press. ISBN 0-19-829246-5.
• Lijphart, Arend
Arend Lijphart
Arend d'Angremond Lijphart is a world renowned political scientist specializing in comparative politics, elections and voting systems, democratic institutions, and ethnicity and politics. He received his PhD in Political Science at Yale University in 1963, after studying at the University of...

• Owen, Bernard, 2002. "Le système électoral et son effet sur la représentation parlementaire des partis: le cas européen." LGDJ.
• Reynolds, Andrew, Reilly, Benjamin and Ellis, Andrew, The New International IDEA Handbook of Electoral System Design, International IDEA, Stockholm 2005.