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From Wikipedia, the free encyclopedia

For other uses, see Voting system (disambiguation).A voting systemor electoral systemis a method by which votersmake a choice between options, often in

an election or on a policy referendum.

A voting system enforces rules to ensure valid voting, and how votes are counted and aggrega ted to yield a

final result. Common voting systems are majority rule, proportional representationor plurality votingwith a

number of variations and methods such as first-past-the-postor preferential voting. The study of formally

defined voting systems is called social choice theoryor voting theory, a subfield of political science,

economics, or mathematics.

With majority rule, those who are unfamiliar with voting theory are often surprised that another voting system

exists, or that disagreements may exist over the definition of what it means to be supported by a

majority[citation needed]. Depending on the meaning chosen, the common "majority rule" systems can produceresults 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.

Contents [hide]

1 Aspects

1.1 Ballot

1.2 Candidates1.3 Weight o f votes

1.4 Status quo

1.5 Cons tituencies

2 Multiple-winner methods

2.1 Proportional methods

2.2 Semi-proportional methods

2.3 Majoritarian methods

3 Single-winner methods

3.1 Single or s equential vote methods

Single-winner [show]

Multiple-winner [show]

Proxy voting [show]

Random s election [show]

Social choice theor y [show]

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0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%9A%E0%A4%A8_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A3%E0%A4%BE%E0%A4%B2%E0%A4%BF%E0%A4%AF%E0%A4%BE%E0%A4%81http://en.wikipedia.org/wiki/Main_Page


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Constitutional

Oligarchy

Aristocracy

Military junta

Plutocracy

Stratocracy

Timocracy

Other

Anarchy

AuthoritarianismAutocracy

Anocracy

Despotism

Dictatorship

Kritarchy

Republic

Theocracy

Totalitarianism

List of forms of government

Politics portalVTE

3.2 Ranked voting methods

3.3 Rated voting methods

4 Evaluating voting systems using criteria

4.1 Mathematical criteria

4.1.1 Compliance of selected systems (table)

4.2 Experimental criteria

4.3 "Soft" criteria

5 History

5.1 Early democracy

5.2 Foundations of voting theory5.3 Single-winner revival

5.4 Influence of game theory

5.5 Post-1980 developments

6 See also

7 Criteria table notes

8 References

8.1 Bibliography

8.2 Remission

9 External li nks

Aspects

A voting system specifies the form of the ballot, the set of allowable votes, and the tallying method, an algorithmfor

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 dist ributed 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 notconsidered part of the voting sys tem. For example, though a voting system specifies the ballot

abstractly, it does not specify whether the actual physical ballot takes the form of a piece of paper, a punch card, or a computer display. A voting system

also does not specify whether or how votes are kept secret, how to verify that votes are counted accurately, or who is allowed to vote. These are aspects of

the broader topic of elections and election systems.

The Electoral Reform Societyis 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 sys tem is 'bad for

voters, bad for government and bad for democracy'.

Ballot

Different voting systems have different forms for allowing the individual to express his or her vote. In ranked ballot

or "preference" voting systems, such as Instant-runoff voting, the Borda count, or a Condorcet method, voters

order the list of options from most to least preferred. In range voting, voters rate each option separately on a

scale. In plurality voting(also known as " first-past-the-post"), voters select only one option, while in approval

slenska

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Simple English

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Svenska

Lea faka-Tonga

Trke

Ting Vit
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In a simple pluralityballot, the voter is

expected to mark only one selection.

voting, they can select as many as they want. In voting systems that allow "plumping", like cumulative voting,

voters may vote for the same candidate multiple times.

Some voting systems include additional choices on the ballot, such as write-in candidates, a none of the above

option, or a no confidence in that candidate option.

Candidates

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

Weight of votes

Main article: Weighted voting

Many elections are based on the principle of "one person, one vote", meaning that every voter's votes are

counted with equal weight. This is not true of all elections, however. Corporateelections, 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 partiesin

a parliament), voting power measures the ability of a group to change the outcome of a vote. Groups may form coalitionsto maximize voting power.

In some German states, most notably Prussiaand Sachsen, there was before 1918 a weighted vote system known as the Prussian three-class franchise,

where the electorate would be divided into three categories based on the amount of income taxpaid. Each categorywould have equal voting power inchoosing the electors.[1]

Status quo

Some voting systems are weighted in themselves, for example if a super majorityis required to change the status quo. An extreme case of this is

unanimous consent, where changing the status quo requires the support of every voting member. If the decision is whether to accept a new member into an

organization, failure of this procedure to admit the new member is called blackballing.

A different mechanism that favors the s tatus quo is the requirement for a quorum, which ensures that the status quo remains if not enough voters

participate in the vote. Quorum requirements often depend only on the total number of votes cast, rather than the number of votes cast for the winning

option. This can sometimes encourage dissenting voters to refrain from voting, in order to prevent a quorum.

Constituencies

Main article: Constituency

Often the purpose of an election is to choose a legislative body made of multiple winners. This can be done by running a single election and choosing the

winners from the same pool of votes, or by dividing up the voters into constituencies that have different options and elect different winners.

Some countries, like Israel, fill their entire parliament using a single multiple-winner district (constituency), while others, like the Republic of Irelandor

Belgium, break up their national elections into smaller multiple-winner districts , and yet others, like the United States or the United Kingdom, hold only

single-winner elections. The Australian bicameralParliament has single-member electorates for the legislative body ( lower house) and multi-member

electorates for its Senate(upper house). Some systems, like the Additional member system, embed smaller districts (constituencies) within larger ones.
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[edit]Seats w on by each party in the 2005 German

federal election, an example of a proportional

voting sys tem.

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. Redistrictingis the process by which the borders of

constituencies are redrawn once apportioned. Both procedures can become highly politically contentious due to the possibility of both malapportionment ,

where there are unequal representative to population ratios across districts, and gerrymandering, where electoral distric ts 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 sys tem, 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

Main article: Proportional representation

Truly proportional methods make some guarantee of proportionality by making each winning option

represent approximately the same number of voters. This number is called a quota. For example, if the quota is 1000 voters, then each elected candidate

reflects the opinions of 1000 voters, within a margin of error. This can be measured using the Gallagher Index.

Most proportional systems in use are based on party-list proportional representation, in which voters vote for parties instead of for individual candidates. [2]

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 methodsand largest remainder methods. Largest remainder methods set a

particular quota based on the number of voters, while highest averages methods, such as the Sainte-Lagu methodand the d'Hondt method, determine the

quota indirectly by dividing the number of votes the parties receive by a sequence of numbers.

Independently of the method used to assign seats, party-list systems can be open listor closed list. In an open listsystem, voters decide which

candidates within a party win the seats. In a closed listsystem, the seats are assigned to candidates in a fixed order that the party chooses. The Mixed

Member Proportionalsystem is a mixed method that only uses a party list for a subset of the winners, filling other seats with the winners of regional

elections, thus having features of open list and closed list systems.

In contrast to party-list systems, the Single Transferable Vote(STV) is a proportional representation system in which voters rank individual candidates in

order of preference. Unlike party-list systems, STV does not depend on the candidates being grouped into political parties. Votes are transferred between

candidates in a manner similar to instant runoff voting, but in addition to t ransferring votes from candidates who are eliminated, excess votes are also

transferred from candidates who already have a quota.
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This ballot design, used in cumulative

voting, allow s a v oter to split his vote amongmultiple candidates.

Different proportional representation systems use different geographic divisions. In some party-lis t or STV systems, all representatives are elected at large,

with votes that may come from anywhere in the electorate. In others, the larger area is divided up into multimember districts, causing a trade-off between

greater proportional accuracy for larger districts and greater geographical spec ificity for smaller districts. The mixed member proportional system,

mentioned above, has some distric t-based winners and some at-large winners. And biproportional apportionmentsystems can achieve proportionality with

districts as small as one member each, because each dis trict result is adjusted by effectively transferring votes between same-party candidates in different

districts.

Semi-proportional methods

An alternative method called Cumulative voting(CV) is a semi-proportional voting sys tem in which each voter has nvotes, where nis 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 votebetween a set of candidates, fully upon one candidate, or a mixture. It is considered aproportionalsystem in allowing a united coalition representing a

m/(n+1)fraction of the voters to be guaranteed to elect mseats of an n-seat election. For example in a 3-seat election, 3/4of the voters (if united on 3

candidates) can guarantee control over all three seats. (In contrast, plurality at largeallows 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 effectof the plurality voting

systemwithout 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 deserveby their size. The level of proportionality depends on how

well-coordinated the voters are.

Limited votingis a multi-winner system that gives voters fewer votes than the number of seats to be decided. The

simplest and most common form of limited voting is Single Non-Transferable Vote(SNTV). It can be considered

a special variation of cumulative voting where a full vote cannot be divided among more than one candidate. It

depends on a statis tical distributions of voters to smooth out preferences that CV can do by individual voters.

For example, in a 4-seat elect ion a candidate needs 20% to guarantee election. In this case a coalition of 40%

of voters can obtain 2 of the 4 seats by splitting their votes as individuals equally between 2 candidates. In

comparison, SNTV tends towards collectively dividing 20% between each candidate by assuming every coalition

voter flipped a coin to dec ide which candidate to support with their single vote. This limitation simplifies voting

and counting, at the cost of more uncertainty of results.

Majoritarian methods

Main article: Election by list

Many multiple-winner voting methods are simple extensions of s ingle-winner methods, without an explic it goal of producing a proportional result. Bloc

voting, orplurality-at-large, has each voter vote for Noptions and selects the top Nas the winners. Because of their propensity for landslide victorieswon by

a single winning slate of candidates, bloc voting and similar nonproportional methods are called "majoritarian".
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An example of runoff voting. Runoff voting

involves tw o rounds of voting. Only two

candidates continue to the second round.

Single-winner methods

Main article: Single-winner voting sys tems

Single-winner syst ems can be c lassified based on their ballot type. In one votesystems, a voter picks one choice at a time. In rankedvoting systems,

each voter ranks the candidates in order of preference. In ratedvoting 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 runoffvoting, the second most common method used in elections, holds a runoff election between

the two highest polling options if there is no absolute majority (above 50%). In elimination runoffelections,

the weakest candidate(s) are eliminated until there is a majority.

A primary election process is also used as a two round runoff voting sys tem. 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 ballotmethod, each voter votes for one option and a single ballot is selected at random todetermine the winner. This is mostly used as a tiebreaker for other methods.

Ranked voting methods

Main article: Preferential voting
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In a typical ranked ballot, a voter is

instructed to place the c andidates in order of

preference.

,

preference. Often it is not necessary to rank all the candidates: unranked candidates are usually considered to

be tied for last place. Some ranked ballot methods also allow voters to give multiple candidates the same

ranking.

The most common ranked voting method is instant-runoff voting(IRV), also known as the " alternative vote" or

simply preferential voting, which uses voters' preferences to simulate an elimination runoff election without

multiple voting events. As the votes are tallied, the option with the fewest first-choice votes is eliminated. In

successive rounds of counting, the next preferred choice still available from each eliminated ballot is t ransferred

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 sys tems.

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

Condorcet methods, orpairwise 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 defeatsevery other option is the Condorcet winnersometimes called the pairwise champion. An option

defeats another option if more voters rank the first option higher on their ballot than the number of voters who

rank the second option higher. This is called apairwise defeat.

These methods are often referred to collectively as Condorcet methodsbecause the Condorcet criterion ensures that they all give the same result in most

elections, where there exists a Condorcet winner. The differences between Condorcet methods occur in situations where no option is undefeated, implying

that there exists a cycle of options that defeat one another, called a Condorcet paradox or Smith set. Considering a generic Condorcet method to be an

abstract method that does not resolve these cycles, specific versions of Condorcet that select winners even when no Condorcet winner exists are called

Condorcet completion methods.

A simple version of Condorcet is Minimax: if no option is undefeated, the option that is defeated by the fewest votes in its worst defeat wins. Another simple

method is Copeland's method, in which the winner is the option that wins the most pairwise contests, as in many round-robin tournaments.

The Kemeny-Young method, the Schulze method(also known as " Schwartz sequential dropping", "cloneproof Schwartz sequential dropping" or the

"beatpath method") and Ranked pairsare recently designed Condorcet methods that satis fy a large number of voting system criteria. These three

Condorcet methods either fully rank, or can be used to fully rank, all the candidates from most popular to least popular.

Rated voting methods

Main article: Cardinal voting systems

Rated ballots allow even more flexibility than ranked ballots, but few

methods are designed to use them. Each voter gives a score to each

option; the allowable scores could be numeric (for example, from 0 to

100) or could be "grades" like A/B/C/D/F.

Rated ballots can be used for ranked voting methods, as long as the

ranked method allows tied rankings. Some ranked methods assume
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On a rated ballot, the voter may rate eachchoice independently.

An approval voting ballot does not requireranking or exclusivity.

that all the rankings on a ballot are distinct, but many voters would be

likely to give multiple candidates the same rating on a rated ballot.

In range voting, voters score or rate each option on a range, and the

option with the highest t otal or average score wins. In majority

judgment, similar ballots are used, but the winner is the candidate

with the highest medianscore.

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 [3]who notes that 'The chief

reason for its nonadoption in public elections, and by some societies,

seems to be a lack of key "insider" support.'

There are variants within cumulative voting. In the points form, each voter has as many votes as there are choices, and can dis tribute 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 s till be arguably biased. A final approach is to c reate 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 c riteria 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 Arrowproved Arrow's imposs ibility theorem, which

demonstrates that several desirable features of voting systems are mutually contradictory. For this reason, someone implementing a voting system has to

decide which criteria are important for the elect ion.

Using criteria to compare systems does not make the comparison completely objective. For example, it is relatively easy to devise a criterion that is met

by one's preferred voting method, and by very few other methods. Doing this, one can then construct a biased argument for the criterion, instead of arguing

directly for the method. There is no ultimate authority on which criteria should be considered, but the following are some criteria that usefully distinguish

between various sys tems and are considered to be desirable by many voting theorists:

Result criteria
http://en.wikipedia.org/wiki/File:Approval_ballot.svghttp://en.wikipedia.org/wiki/File:Rated_voting.pnghttp://en.wikipedia.org/wiki/File:Rated_voting.pnghttp://en.wikipedia.org/wiki/File:Rated_voting.pnghttp://pdfcrowd.com/http://pdfcrowd.com/redirect/?url=http%3a%2f%2fen.wikipedia.org%2fwiki%2fVoting_system&id=ma-121107115951-9a9f81abhttp://pdfcrowd.com/customize/http://pdfcrowd.com/html-to-pdf-api/?ref=pdfhttp://en.wikipedia.org/w/index.php?title=Voting_system&action=edit&section=15http://en.wikipedia.org/w/index.php?title=Voting_system&action=edit&section=16http://en.wikipedia.org/wiki/File:Rated_voting.pnghttp://en.wikipedia.org/wiki/File:Rated_voting.pnghttp://en.wikipedia.org/wiki/File:Approval_ballot.svghttp://en.wikipedia.org/wiki/File:Approval_ballot.svghttp://en.wikipedia.org/wiki/Range_votinghttp://en.wikipedia.org/wiki/Majority_judgmenthttp://en.wikipedia.org/wiki/Medianhttp://en.wikipedia.org/wiki/Cumulative_votinghttp://books.google.com/books?id=aRF7XdcCLq0C&q=inauthor:%22Lani+Guinier%22&dq=inauthor:%22Lani+Guinier%22&hl=en&ei=9i4FTbTjOISisAOZ_tWBDQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC4Q6AEwAQhttp://en.wikipedia.org/wiki/Kenneth_Arrowhttp://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem


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Absolute: Will a candidate always win who...

Majority criterion ... is ranked as the unique favorite by a majority of voters? This criterion comes in two versions:

1. Rankedmajority criterion, in which an option which is merely preferred over the others by a majority must win. (Passing the ranked MC is

denoted by "yes", because implies also passing the following:)

2. Ratedmajority criterion, in which only an option which is uniquely given a perfectrating by a majority must win. The ranked and rated MC

are synonymous for ranked voting systems, but not for rated or graded ones. The ranked MC, but not the rated MC, is incompatible with

the IIA and FBC criteria explained below.

Mutual majority criterion(MMC) ...is among a group of candidates ranked above all others by a majority of voters? This also implies the

Majority loser criterionif a majority of voters prefers every other candidate over a given candidate, then does that candidate not win?

Therefore, of the sys tems lis ted, all pass neither or both criteria, except for Borda, which passes Majority Loser while failing Mutual Majority.

Condorcet criterion ...beats every other candidate in pairwise comparisons? (This implies the majority criterion, above)

Strategic, Ma jority Condorcet criterion ...beats every other candidate in pairwise comparison by a majority of all voters , and voters are

aware of each other's preferences and vote strategically? (Any method that passes the Condorcet criterion above passes this criterion easily.

However, if some voters were indifferent between two candidates, one candidate could be the Condorcet winner but not the Majority Condorcet

winner.)

Condorcet loser criterion(Cond. loser) ...is not the candidate who loses to every other candidate in pairwise comparisons?

Relative a s nominees change: Does the outcome never change...

Independence of Smith-dominated alternatives(ISDA) ...if a Smith-dominated candidate is added or removed (assuming votes regarding the

other candidates are unchanged)? Candidate C is Smith-dominated if there is some other candidate A such that C is beaten by A and everycandidate B that is not beaten by A etc. Note that although this criterion is classed here as nominee-relative, it has a strong absolute component

in exc luding Smith-dominated candidates from winning. In fact, it implies all of the absolute criteria above.

Independence of Irrelevant Alternatives(IIA) ...if a non-winning candidate is added or removed (assuming votes regarding the other candidates

are unchanged)?[4]For instance, plurality rule fails IIA; adding a candidate X can cause the winner to change from W to Y even though Y receives

no more votes than before.

Local Independence of Irrelevant Alternatives(LIIA) ...if the alternative that would finish last is removed? (And could the alternative that

finishes second fail to become the winner if the winner were removed?)

Independence of Clone Alternatives(Cloneproof) ...if non-winning candidates similar to an existing candidate are added? There are three

different phenomena which could cause a system to fail this c riterion:

Spoilersare candidates which decrease the chance of any of the similar or clone candidates winning, also known as a spoiler effect.Teamsare sets of similar candidates whose mere presence helps the chances of any of them winning.

Crowdsare additional candidates who affect the outcome of an elect ion without either helping or harming the chances of their factional group,

but instead affecting another group.

Relative as voters change: If candidate W wins for one set of ballots, will W still always win if those ballots change...

Monotonicity criterion(Monotone) ...to rank W higher? (This also implies that you cannot cause a losing candidate to win by ranking him

lower.)

Consistency criterion ...by adding another set of ballots where W also wins?

Participation criterion Is voting honestly always better than not voting at all? (This is grouped with the distinct but s imilar Consistency
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Criterion in the table below.[5])

Reversal symmetryIf individual preferences of each voter are inverted, does the original winner never win?

Election administration criteria: Can the winner be calculated...

Polynomial time(Polytime) .. .in a runtime that is polynomial in the number of candidates and linear in the number of voters?

Resolvable ...in almost all cases, without using any random processes such as flipping coins? That is, are exact ties, in which the winner

could be one of two or more candidates, vanishingly rare in large elections?

Summability(Summable) ...by tallying ballots at each polling station separately and simply adding up the individual tallies? The amount of

information necessary for such tallies is expressed as an order functionof the number of candidates N. Slower-growing functions such as O(N) or

O(N

2

) make for easier counting, while faster-growing functions such as O(N!) might make it harder to catch fraud by election administrators.Voter's criteria

Ballot model: A voter states their choices in the form of...

...a single markchoosing one single candidate (e.g. "A is my favorite candidate.")

...approvals, i.e. a set of candidates the voter approves (e.g. "I approve candidates A, B and E. ")

...a (strict) ranking, i.e. an list of candidates strictly ordered by the preferences of the voter (e.g. "A > B > E > D > H" or "I prefer A to B, B to E,

E to D and D to H.")

...ranking with equality(e.g. "A = B > D = E > H" or "I prefer A and B to D and E; D and E to H; but I have no opinion between A and B or

between D and E.") This is a generalized form of a ranking, so every voting system that can handle comparisons, also can be used if the ballots

are rankings.

...scores, the voter places each candidate in one of several numerical or verbal categories, such as numbered categories 0-100 (e.g. "A getsscore 100, E gets 0, H 47, B 12 and D also 12")

...which therefore allows voters to choose whether to use...

...equal rankings(= ranks) A voter can 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 votingstrategies.

...later preferences(>2 ranks) A voter can indicate different levels of support (beyond the two levels of voted and unvoted) through ranking or

rating candidates.

Strategy guarantees: Can voters be sure that.. .

Later-no-harm criterionand Later-no-help criterion ...adding a later preference to a ballot will not harm/help any candidate already lis ted? 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.[6]

Favorite Betrayal Criterion... they do not need to rank any other candidate above their favorite in order to obtain a result they prefer?[7]

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.
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absolute nominee-

relative voter-relative administrative ballot allows... voter guarantees

sort

by:

Majority/

MMC

Condorcet/

Strategic,

Majority

Condorcet

Cond.

loserISDA LIIA IIA

Clone-

proof

Mono-

tone

Consist-

ency/

Particip-

ation

Rever-

sal

sym-

metry

Polytime/

Resolv-

able

Summ-

able

ballot

type

=

ranks

>2

ranks

Later-no-

harm/

Later-no-

help

FBC:No

favorite

betrayal

Appro-val Rated[nb 1] No No[nb 2] Yes[nb 3] No No[nb 2] Yes Yes Yes[nb 4] Yes Yes Yes Yes O(N) Yes O(N) appro-vals Yes No

[nb 5] Yes

Borda

countNo No No No Yes No No No

No:

teamsYes Yes Yes Yes O(N) Yes O(N) ranking No Yes No Yes No

Cope-

landYes Yes Yes Yes Yes Yes No

No[nb 2]

teams,

crowdsYes

No[nb 2]

No[nb 2] Yes O(N

2) No O(N2) ranking Yes YesNo

[nb 2] No No

IRV(AV) Yes YesNo

[nb 2] No YesNo

[nb 2] No No Yes No No No No O(N2) Yes O(N!)[nb 6] ranking No Yes Yes Yes No

Keme-

ny-

Young

Yes Yes Yes Yes Yes Yes YesNo

[nb 2]

No:

spoil-

ers

Yes

No[nb 2]

[nb 7]

No[nb 2] Yes O(N!) Yes

O(N2)[nb 8] ranking Yes Yes

No[nb 2] No No

Majority

Judg-

ment[nb 9]

RatedNo

[nb 10]No

[nb 2]Yes[nb 3] No

No[nb 2] Yes Yes Yes Yes

No[nb 11] No

Dep-

ends[nb 12]

O(N) Yes O(N)[nb 13]scores[nb 14] Yes Yes

No[nb 15] Yes Yes

Mini-

maxYes No

Yes[nb 16] Yes No No No

No[nb 2]

No:

spoil-

ers

YesNo

[nb 2]No

[nb 2] No O(N2) Yes O(N2) ranking Yes Yes

No[nb 2]

[nb 16]No No

Plurality Yes No

No

[nb 2] No No

No

[nb 2] No No

No:

spoil-ers Yes Yes Yes No O(N) Yes O(N)

single

mark No No

NA

[nb 17]

NA

[nb 17] No

Range

votingNo No

No[nb 2]

Yes[nb 3] No

No[nb 2] Yes Yes Yes Yes Yes Yes Yes O(N) Yes O(N) scores Yes Yes No Yes Yes

Ranked

pairsYes Yes Yes Yes Yes Yes Yes

No[nb 2] Yes Yes

No[nb 2]

No[nb 2] Yes O(N

4) Yes O(N2) ranking Yes YesNo

[nb 2] No No

Runoff

votingYes No

No[nb 2] No Yes

No[nb 2] No No

No:

spoil-

ers

No No No NoO(N)[nb 18] Yes

O(N)[nb 19]

single

markNo

No[nb 20] Yes

[nb 21] No
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ty_votinghttp://en.wikipedia.org/wiki/Condorcet_loser_criterion#Plurality_votinghttp://en.wikipedia.org/wiki/Independence_of_irrelevant_alternatives#Plurality_voting_systemhttp://en.wikipedia.org/wiki/Independence_of_clones_criterion#Plurality_votinghttp://en.wikipedia.org/wiki/Vote-splittinghttp://en.wikipedia.org/wiki/Reversal_symmetry#Plurality_votinghttp://en.wikipedia.org/wiki/Range_votinghttp://en.wikipedia.org/wiki/Majority_criterion#Range_votinghttp://en.wikipedia.org/wiki/Condorcet_criterion#Majority_Judgmenthttp://en.wikipedia.org/wiki/Condorcet_loser_criterion#Range_votinghttp://en.wikipedia.org/wiki/Later-no-harm_criterion#Range_votinghttp://en.wikipedia.org/wiki/Ranked_pairshttp://en.wikipedia.org/wiki/Condorcet_loser_criterion#Ranked_Pairshttp://en.wikipedia.org/wiki/Independence_of_irrelevant_alternatives#Ranked_pairshttp://en.wikipedia.org/wiki/Consistency_criterion#Ranked_pairshttp://en.wikipedia.org/wiki/Participation_criterion#Ranked_pairshttp://en.wikipedia.org/wiki/Later-no-harm_criterion#Ranked_pairshttp://en.wikipedia.org/wiki/Two-round_systemhttp://en.wikipedia.org/wiki/Independence_of_irrelevant_alternatives#Two-round_systemhttp://en.wikipedia.org/wiki/Vote-splittinghttp://en.wikipedia.org/wiki/Monotonicity_criterion#Instant-runoff_voting_and_the_Two-round_system_are_not_monotonichttp://en.wikipedia.org/wiki/Participation_criterion#Two-round_systemhttp://en.wikipedia.org/wiki/Favorite_betrayal_criterion#Two-round_systemhttp://en.wikipedia.org/wiki/Independence_of_irrelevant_alternatives#Schulze_methodhttp://en.wikipedia.org/wiki/Consistency_criterion#Schulze_methodhttp://en.wikipedia.org/wiki/Participation_criterion#Schulze_methodhttp://en.wikipedia.org/wiki/Later-no-harm_criterion#Schulze_method


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Schulze Yes Yes Yes Yes Yes Yes NoNo

[nb 2] Yes YesNo

[nb 2]No

[nb 2] Yes O(N3) Yes O(N2) ranking Yes Yes

No[nb 2] No No

Random

winne r/

arbitrary

winne r[nb 22]

No NoNo

[nb 2] No NoNo

[nb 2] Yes Yes NA NA No NA NA O(1) No O(1) none NA NA Yes

Random

ballot

[nb 23]

No NoNo

[nb 2] No NoNo

[nb 2] Yes Yes Yes Yes No Yes Yes O(N) No O(N)single

mark

No No NA Yes

NA = not applicable.

Experimental criteria

It is poss ible 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 s imply proving that a given sys tem does or does not satisfy a given mathematical criterion, results are not available for

all sys tems. A lso, these results are sensit ive 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 satis faction, 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 bracketthe best and worst cases for a voting system.[8]To date, the only such s imulation to compare a wide variety of voting systems was run by a range-voting

advocate and was not published in a peer-reviewed journal.[9][10]It found that Range voting consistently scored as either the best system or among the best

across the various conditions studied.[11]

Another aspect which can be compared through such Monte Carlo simulationsis strategic vulnerability. According to the Gibbard-Satterthwaite theorem, no

voting system can be immune to strategic manipulationin 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 electionwhich they had carried out using rated ballots. Comparing range voting, Borda count, plurality voting, approval

votingwith two different absolute approval thresholds, Condorcet voting, and majority judgment, they found that range voting had the highest (worst)

strategic vulnerability, while their own sys tem majority judgment had the lowest (best). [12]

Balinski and Laraki also used the same data to investigate how likely it was that each of those systems, as well as runoff voting, would elect a centrist.

Opinions differ on whether this is desirable or not. Some argue that systems which favor centrists are better because they are more stable; others argue

that electing ideologically purer candidates gives voters more choice and a better chance to retrospectively judge the relative merits of those ideologies;

while Balinski and Laraki argue that both centrist and extremist candidates should have a chance to win, to prevent forcing candidates into taking either

position. According to their model, plurality, runoff voting, and approval voting with a higher approval threshold tended to elect extremists (100%, 98%, and

94% of the time, respectively); majority judgement elected both centrists and extremists (56% ext remists); 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).[12]However, their

model did not take into account voters' strategic reactions to the system used, such as "lesser of two evils" voting under plurality.

Simulated elections in a two-dimensional issue space can also be graphed to visually compare election methods; this illustrates issues like

[13]
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nonmonotonicity, clone-independence, and tendency to elect centrists vs extremists. [13]

"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 votingor strategic nomination, and, for multiple-winner

methods, the degree of proportionality produced.

The New Zealand Royal Commission on the Electoral Systemlist ed ten criteria for their evaluation of possible new electoral sys tems for New Zealand.

These inc luded fairness between political parties, effective representation of minority or special interest groups, political integration, effective voter

participation and legitimacy.

History

Early democracy

Voting has been used as a feature of democracy since the 6th century BC, when democracy was introduced by the Athenian democracy. However, in

Athenian democracy, voting was seen as the least democratic among methods used for selecting public officials, and was little used, because elections

were believed to inherently favor the wealthy and well-known over average citizens. Viewed as more democratic were assemblies open to all citizens, and

selection by lot (known as sortition), as well as rotation of office. One of the earliest recorded elections in Athens was a plurality votethat 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

democracywere held using plurality voting or some variant, but as an exception, the s tate of Venicein the 13th century adopted the system we now know

as approval voting to elect their Great Council.[14]

The Venetians' system for electing the Dogewas 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 collegeof

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 sys tem had certain desirable properties such as being hard to game and

ensuring that the winner reflected the opinions of both majority and minority factions.[15]This process, with slight modifications, was central to the politics

of the Republic of Venicethroughout its remarkable lifespan of over 500 years, from 1268 to 1797.

Foundations of voting theoryVoting theory became an object of academic s tudy around the time of the French Revolution.[14]Jean-Charles de Borda

proposed the Borda count in 1770 as a method for electing members to the French Academy of Sciences. His system

was opposed by the Marquis de Condorcet, who proposed instead the method of pairwise comparison that he had

devised. Implementations of this method are known as Condorcet methods. He also wrote about the Condorcet paradox,

which he called the intransitivity of majority preferences.[16]

While Condorcet and Borda are usually credited as the founders of voting theory, recent research has shown that the

philosopher Ramon Llulldiscovered 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

[17]
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Jean-Charles de Borda, an

early voting theorist

The Marquis de Condorcet,

another early voting theorist

rediscovered in 2001.[17]

Later in the 18th century, the related topic of apportionment began to be studied. The

impetus for research into fair apportionment methods came, in fact, from the United

States Constitution, which mandated that seats in the United States House of

Representativeshad to be allocated among the states proportionally to their

population, but did not specify how to do so. [18]A variety of methods were proposed by

statesmen such as Alexander Hamilton, Thomas Jefferson, and Daniel Webster. Some of the apportionment methods

discovered in the United States were in a sense rediscovered in Europe in the 19th century, as seat allocation methods

for the newly proposed system of party-list proportional representation. The result is that many apportionment methods

have two names: for instance,Jefferson's methodis equivalent to the d'Hondt method, as is Webster's methodto theSainte-Lagu method, while Hamilton's methodis identical to the Hare largest remainder method.[18]

The Single Transferable Votesystem was devised by Carl Andraein Denmarkin 1855, and also in England by Thomas

Harein 1857. Their discoveries may or may not have been independent. STV elections were first held in Denmark in

1856, and in Tasmaniain 1896 after its use was promoted by Andrew Inglis Clark. Party-list proportional representation

was first implemented to elect European legislatures in the early 20th century, with Belgium implementing it first in 1899.

Since then, proportional and semi-proportional methods have come to be used in almost all democratic countries, with

most exceptions being former Britishcolonies. [19]

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 thelate 19th century. This began around 1870, when William Robert Wareproposed applying STV to s ingle-winner elections, yielding instant runoff voting. [20]

Soon, mathematicians began to revisit Condorcet's ideas and invent new methods for Condorcet completion. Edward J. Nansoncombined the newly

described instant runoff voting with the Borda count to y ield a new Condorcet method called Nanson's method. Charles Dodgson, better known as Lewis

Carroll, published pamphlets on voting theory, focusing in particular on Condorcet voting. He introduced the use of matricesto 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 methodas well as a proportional multiwinner method based on proxy voting.

Ranked voting systems eventually gathered enough support to be adopted for use in government elections. In Australia, IRV was first adopted in 1893, and

continues to be used along with STV today. In the United States in the early-20th-century Progressive era, various municipalities began to use Bucklin

voting. Buck lin is no longer used in any government elections, and has even been declared unconstitutional in Minnesota.[21]

Influence of game theory

After John von Neumannand others developed the mathematical field of game theoryin the 1940s, new mathematical

tools were available to analyze voting systems and strategic voting. One such tool was the idea of a strong Nash

equilibrium, based on the work of John Nashin the 1950s. This and s imilar methods led to significant new results that

changed the field of voting theory.[14][need quotation to verify]The use of mathematical c riteria to evaluate voting systems

was introduced when Kenneth Arrowshowed in Arrow's impossibility theoremthat certain intuitively desirable criteria

were actually mutually contradictory, demonstrating the inherent limitations of voting theorems. Among the criteria Arrow

considered desirable was one which requires the voting system to use ordinal (ranking) information; thus, as John

Harsanyipointed out cardinal (rated) voting sys tems such as approval voting range voting and majority judgment can
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Kenneth Arrow , a pioneer of

modern voting theory.

Electoral sys tems by country in 2012.

Harsanyipointed out, cardinal (rated) voting sys tems such as approval voting, range voting, and majority judgment can

successfully meet all the other criteria. Arrow's theorem is easily the single most cited result in voting theory, and it

inspired further significant results such as the Gibbard-Satterthwaite theorem, which showed that st rategic voting is

unavoidable in certain common c ircumstances, for any deterministic voting sys tem, whether cardinal or ordinal.

The use of game theory to analyze voting sys tems also led to discoveries about the emergent st rategic effects of certain

systems. Duverger's lawis a prominent example of such a result, showing that plurality voting often leads to a two-party

system. Further research into this and other game theory aspects of voting led Steven Bramsand Peter Fishburnto

formally define and promote the use of approval votingin 1977. While approval voting had been used before that, it had not

been named or considered as an object of academic study, particularly because it violated the assumption made by most research that single-winner

methods were based on ordinal preference rankings.[22]

Public choice theory, a field which uses economic modeling tools including game theory to study the behavior of politicians and voters, began to form in the

1950s, and was both influenced by and influential on voting theory.

Post-1980 developments

Voting theory has come to focus on voting system criteria almost as much as it

does on particular voting systems. Now, any description of a benefit or weakness

in a voting system is expected to be backed up by a mathematically defined

criterion. Recent research in voting theory has largely involved devising new

criteria and new methods devised to meet certain criteria.

Political scientistsof the 20th century published many s tudies on the effects thatthe voting systems have on voters' choices and political parties,[23][24][25] and on

political stability.[26][27]A few scholars also studied what effects caused a nation

to change for a particular voting system. [28][29][30][31][32]One prominent current

voting theorist is Nicolaus Tideman, who formalized concepts such as strategic

nominationand the spoiler effectin the independence of clones criterion.

Tideman also devised the ranked pairs method, a Condorcet method that is not

susceptible to clones. Also, Donald G. Saarihas 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 fullyrank all the choices from most popular to least popular.

The advent of the Internet has increased the interest in voting systems. Unlike many mathematical fields, voting theory is generally accessible enough to

non-experts that new results can be discovered by amateurs, and frequently are.

The study of voting systems has influenced a new push for electoral reformbeginning around the 1990s, with proposals being made to replace plurality

voting in governmental elections with other methods. New Zealandadopted Mixed Member Proportionalfor Parliamentary electionsin 1993 and Single

Transferable Votefor some local elections in 2004 (see Electoral reform in New Zealand). After plurality voting was a key factor in the contested results of

the 2000 US presidential election, various municipalities in the United States began to adopt instant-runoff voting, although some of them subsequently

returned to their prior system. The Canadianprovince of British Columbiaheld two unsuccessful referendums (in 2005and 2009) to adopt an STVsystem,

and Ontario another Canadian province held an unsuccessful referendum on October 10 2007 on whether to adopt a Mixed Member Proportional system
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