Voting system

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For other uses, see Voting system (disambiguation).
Queueing to vote at a polling station for the Australian 2016 federal election

A voting method or electoral method consists of the set of rules which must be followed for a vote to be considered valid, and which set out how votes are cast, counted and aggregated to yield a final result of an election or a referendum. Common voting methods are majority rule, proportional representation and plurality voting, with a number of variations and methods such as first-past-the-post and preferential voting. The study of formally defined voting methods is called social choice theory or voting theory, a subfield of political science, economics, or mathematics.

Those who are unfamiliar with voting theory are often surprised to learn that voting methods other than majority rule exist, or that disagreements exist over what it means to be supported by a majority. Depending on the meaning chosen, the common "majority rule" methods can produce results that the majority does not support. If every election had only two choices, the winner would be determined using majority rule alone. However, when there are more than two options, there may not be a single option that is most liked or most disliked by a majority. A simple choice does not allow voters to express the ordering or the intensity of their feeling. Different voting methods may give very different results, particularly in cases where there is no clear majority preference.


An electoral method specifies the form of the ballot, the set of allowable votes; and the tallying method, an algorithm for determining the outcome. This outcome may be a single winner, as in the case of a presidential election, or may result in multiple winners, such as in the election of a legislative body. The electoral method 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 method, and is instead referred to as a voting system. For example, though a voting method 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 method also does not specify whether or how votes are kept secret, how to verify that votes are counted accurately, or who is allowed to vote. These are aspects of the broader topic of elections and election systems.



Different voting methods have different ways of allowing individuals to express their votes. In ranked ballot or "preference" voting methods, such as Instant-runoff voting, the Borda count, or a Condorcet method, each voter orders the list of options from most to least preferred. In range voting, voters rate each option separately on a scale. In plurality voting (also known as "first-past-the-post"), voters select only one option, while in approval voting, they can select as many as they want. In voting methods that allow "plumping", like cumulative voting, voters may vote for the same candidate multiple times.




Weight of votes[edit]

Main article: Weighted voting



In some German states, most notably the kingdoms of Prussia (German: Preußen) and Saxony (German: Sachsen), before 1918 there was a weighted vote method known as the Prussian three-class franchise, where the electorate would be divided into three categories based on the amount of income tax paid. Each category would have equal voting power in choosing the electors.[1]

Status quo[edit]

Some voting methods are weighted in themselves, for example if a super majority is required to change the status quo. An extreme case of this is unanimous consent, where changing the status quo requires the support of every voting member. If the decision is whether to accept a new member into an organization, failure of this procedure to admit the new member is called blackballing.

A different mechanism that favors the status quo is the requirement for a quorum, which ensures that the status quo remains if voter participation does not reach the specified threshold. Quorum requirements often depend only on the total number of votes cast, rather than the number of votes cast for the winning option. This can sometimes encourage dissenting voters to refrain from voting, in order to prevent a quorum.


Main article: Constituency


Some countries, like Israel, fill their entire parliament using a single multiple-winner district (constituency), while others, like the Republic of Ireland and Belgium, break up their national elections into smaller multiple-winner districts, and yet others, like the United States and the United Kingdom, hold only single-winner elections. The Australian bicameral Parliament has single-member electorates for the lower house and multi-member electorates for its Senate (upper house). Some methods, like the additional member system, embed smaller districts (constituencies) within larger ones.

The way constituencies are created and assigned seats can dramatically affect the results. Apportionment is the process by which states, regions, or larger districts are awarded seats, usually according to population counted in a census. Redistricting is the process by which the borders of constituencies are redrawn once apportioned. Both procedures can become highly politically contentious, due to the possibilities of both malapportionment, i.e. unequal ratios of representatives to population ratios between districts, and gerrymandering, where electoral districts are manipulated for political gain. An extreme example of malapportionment was the UK rotten and pocket boroughs, parliamentary constituencies that had a very small electorate (e.g. an abandoned town) and could thus be used by a patron to gain undue and unrepresentative influence within parliament, particularly as many people were not entitled to vote at all. This was a feature of the unreformed House of Commons before the Great Reform Act of 1832.

Multiple-winner methods[edit]


Most modern democracies use some form of multiple-winner voting method, with the United States, Canada, the United Kingdom, and India being notable exceptions.

The results of a vote with multiple winners, such as the election of a legislature, are usually different from a set of single-winner votes. Often, voters in a multiple-winner election are more concerned with the overall composition of the legislature than with exactly which candidates get elected. For this reason, many multiple-winner methods aim for proportional representation, which means that if a given party (or other political grouping) gets X% of the total vote, it should also get approximately X% of the seats in the legislature. However not all multiple-winner voting methods are proportional.[2]

Proportional methods[edit]

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

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


Independently of the method used to assign seats, party-list methods can be open list or closed list. In an open list method, voters decide which candidates within a party win the seats. In a closed list method, the seats are assigned to candidates in a fixed order that the party chooses.

In contrast to party-list methods, the single transferable vote (STV) is a proportional representation method in which voters rank individual candidates in order of preference. Unlike party-list methods, STV does not depend on the candidates being grouped into political parties. Votes are transferred between eliminated candidates in a manner similar to instant runoff voting, but in addition to transferring votes from candidates who are eliminated, excess votes are also transferred from candidates who already have a quota. If no excess votes are transferred, but instead the count stops when there are only as many candidates remaining as seats, this method can be considered to be a multi-member version of instant runoff voting; such a method has apparently been used in South Australia, where it was known as Bottoms-Up preferential voting.[4]

Different proportional representation methods use different geographic divisions. In some party-list or STV methods, all representatives are elected at large, with votes that may come from anywhere in the electorate. In others, the larger area is divided up into multimember districts, causing a trade-off between greater proportional accuracy for larger districts and more geographically-specific representatives for smaller districts. The mixed member proportional method, mentioned above, has some district-based winners and some at-large winners. And biproportional apportionment methods can achieve proportionality with districts as small as one member each, because each district result is adjusted by effectively transferring votes between same-party candidates in different districts.

Semi-proportional methods[edit]

An alternative method called cumulative voting (CV) is a semi-proportional voting method 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 method 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, 34 of the voters (if united on 3 candidates) can guarantee control over all three seats. (In contrast, plurality at large allows a united coalition (majority) (50%+1) to control all the seats.)



Cumulative voting is not fully proportional because it suffers from the same spoiler effect of the plurality voting method without a run-off process. A group of like-minded voters divided among "too many" candidates may fail to elect any winners, or elect fewer than they deserve by their size. The level of proportionality depends on how well-coordinated the voters are.

Limited voting is a multi-winner method that gives voters fewer votes than the number of seats to be decided. The simplest and most common form of limited voting is single non-transferable vote (SNTV). It can be considered a special variation of cumulative voting where a full vote cannot be divided among more than one candidate. It depends on a statistical distribution of voters to smooth out preferences that CV can do by individual voters.


Majoritarian methods[edit]

Main article: Election by list


Single-winner methods[edit]

A political cartoon about the Greek referendum, 1920, depicting a voter at the ballot box being watched by the spirit of War. Shows a 2-compartment ballot box of the kind used in 19th-early 20th century Greece for approval voting.

Single-winner methods can be classified based on their ballot type. In one vote methods, a voter picks one choice at a time. In ranked voting methods, each voter ranks the candidates in order of preference. In rated voting methods, voters give a score to each candidate.

Single or sequential vote methods[edit]




A primary election process is also used as a two-round runoff voting method. 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.


Ranked voting methods[edit]

Main article: Ranked voting method




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




The Kemeny–Young method, the Schulze method (also known as Schwartz sequential dropping, cloneproof Schwartz sequential dropping, or the beatpath method), ranked pairs, and maximal lotteries are recently designed Condorcet methods that satisfy a large number of voting method 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[edit]

An approval voting ballot does not require ranking or exclusivity.




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[5] who notes that 'The chief reason for its nonadoption in public elections, and by some societies, seems to be a lack of key "insider" support' from 'influential members.'


Evaluating voting methods using criteria[edit]

In the real world, attitudes toward voting methods are highly influenced by the methods' impact on groups that one supports or opposes. This can make the objective comparison of voting methods difficult.

There are several ways to address this problem. Criteria can be defined mathematically, such that any voting method either passes or fails. This gives perfectly objective results, but their practical relevance is still arguable. Another approach is to define ideal criteria that no voting method passes perfectly, and then see how often or how close to passing various methods are over a large sample of simulated elections. This gives results which are practically relevant, but the method of generating the sample of simulated elections can still be arguably biased. A final approach is to create imprecisely defined criteria, and then assign a neutral body to evaluate each method according to these criteria. This approach can look at aspects of voting methods which the other two approaches miss, but both the definitions of these criteria and the evaluations of the methods are still inevitably subjective.

Mathematical criteria[edit]

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

Using criteria to compare methods 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 methods and are considered to be desirable by many voting theorists:

Result criteria (absolute)
Majority criterion (MC)
  2. Rated majority criterion, in which only an option which is uniquely given a perfect rating by a majority must win. The ranked and rated MC are synonymous for ranked voting methods, but not for rated or graded ones. The ranked MC, but not the rated MC, is incompatible with the IIA criterion explained below.
Mutual majority criterion (MMC)
Will a candidate always win who is among a group of candidates ranked above all others by a majority of voters? This also implies the majority loser criterion – if a majority of voters prefers every other candidate over a given candidate, then does that candidate not win? Therefore, of the methods listed, all pass neither or both criteria, except for Borda, which passes Majority Loser while failing Mutual Majority.
Condorcet criterion
Condorcet loser criterion (cond. loser)
Result criteria (relative)
Independence of Smith-dominated alternatives (ISDA)
Independence of irrelevant alternatives (IIA)
Does the outcome never change if a non-winning candidate is added or removed (assuming votes regarding the other candidates are unchanged)?[7] 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)
Independence of clone alternatives (cloneproof)
Does the outcome never change if non-winning candidates similar to an existing candidate are added? There are three different phenomena which could cause a method to fail this criterion:
Sets of similar candidates whose mere presence helps the chances of any of them winning.
Monotonicity criterion (monotone)
Consistency criterion (CC)
Participation criterion (PC)
Reversal symmetry (reversal)
Ballot-counting criteria
Polynomial time (polytime)
Summability (summable)
Strategy criteria
These are criteria that relate to a voter's incentive to use certain forms of strategy. They could also be considered as relative result criteria; however, unlike the criteria in that section, these criteria are directly relevant to voters; the fact that a method passes these criteria can simplify the process of figuring out one's optimal strategic vote.
Later-no-harm criterion, and later-no-help criterion
Favorite betrayal criterion (FBC)
Ballot format
These are issues relating to the expressivity or information content of a valid ballot.
Ballot type
What information is the voter give on the ballot?
Equal ranks
Can a valid ballot express equal support for more than one candidate (and not just equal opposition to more than one)?
Over 2 ranks
Can a ballot express more than two levels of support/opposition for different candidates?

Note on terminology: A criterion is said to be "weaker" than another when it is passed by more voting methods. 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 method 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 methods (table)[edit]

The following table shows which of the above criteria are met by several single-winner methods. Another table can be found here.

Maj­ority Maj.
Cond­orcet Cond.
LIIA IIA Clone­proof Mono­tone Consis­tency Partic­ipation Rever­sal
Summ­able Later-no- Favorite
betrayal [clarification needed]
Harm Help = >2
Appr­oval Rated[a] No No No[b][c] No No[b] Yes Yes[d] Yes[e] Yes Yes Yes Yes O(N) Yes O(N) No Yes[f] Yes Appr­ovals Yes No
Borda count No Yes No No[b] Yes No No No Teams Yes Yes Yes Yes O(N) Yes O(N) No Yes No Ran­king No Yes
Cope­land Yes Yes Yes Yes Yes Yes No No[b] Teams,
Yes No[b] No[b] Yes O(N2) No O(N2) No[b] No No[b] Ran­king Yes Yes
IRV (AV) Yes Yes Yes No[b] Yes No[b] No No Yes No No No No O(N2) Yes[g] O(N!)[h] Yes Yes No Ran­king No Yes
Kemeny–Young Yes Yes Yes Yes Yes Yes Yes No[b] Spoil­ers Yes No[b]
No[b] Yes O(N!) Yes O(N2)[j] No[b] No No[b] Ran­king Yes Yes
Rated[l] Yes[m] No[n] No[b][c] No No[b] Yes Yes[d] Yes Yes No[o] No[p] Dep­ends[q] O(N) Yes O(N)[r] No[s] Yes Yes Scores[t] Yes Yes
Mini­max Yes No No Yes[u] No No No No[b] Spoil­ers Yes No[b] No[b] No O(N2) Yes O(N2) No[b][u] No No[b] Ran­king Yes Yes
Plura­lity Yes No No No[b] No No[b] No No Spoil­ers Yes Yes Yes No O(N) Yes O(N) N/A[v] N/A[v] No Single mark N/A No
Range voting No No No No[b][c] No No[b] Yes Yes[d] Yes Yes Yes Yes Yes O(N) Yes O(N) No Yes Yes Scores Yes Yes
Ranked pairs Yes Yes Yes Yes Yes Yes Yes No[b] Yes Yes No[b] No[p][b] Yes O(N4) Yes O(N2) No[b] No No[p][b] Ran­king Yes Yes
Runoff voting Yes Yes No No[b] Yes No[b] No No Spoil­ers No No No No O(N)[w] Yes O(N)[w] Yes Yes[x] No Single mark N/A No[y]
Schulze Yes Yes Yes Yes Yes Yes No No[b] Yes Yes No[b] No[p][b] Yes O(N3) Yes O(N2) No[b] No No[p][b] Ran­king Yes Yes
Sortition, arbitrary winner[z] No No No No[b] No No[b] Yes Yes No Yes Yes Yes Yes O(1) No O(1) Yes Yes Yes None N/A N/A
Random ballot[aa] No No No No[b] No No[b] Yes Yes Yes Yes Yes Yes Yes O(N) No O(N) Yes Yes Yes Single mark N/A No

Experimental criteria[edit]

It is possible to simulate large numbers of virtual elections on a computer and see how various voting methods compare in practical terms. Since such investigations are more difficult than simply proving that a given method does or does not satisfy a given mathematical criterion, results are not available for all methods. 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 method.[11] To date, the only such simulation to compare a wide variety of voting methods was run by a range-voting advocate and was not published in a peer-reviewed journal.[12][13] It found that Range voting consistently scored as either the best method or among the best across the various conditions studied.[14]

Another aspect which can be compared through such Monte Carlo simulations is strategic vulnerability. According to the Gibbard–Satterthwaite theorem, no voting method can be immune to strategic manipulation in all cases, but certainly some methods will have this problem more often than others. M. Balinski and R. Laraki, the inventors of the majority judgment method, performed such an investigation using a set of simulated elections based on the results from a poll of the 2007 French presidential election which they had carried out using rated ballots. Comparing range voting, Borda count, plurality voting, approval voting with two different absolute approval thresholds, Condorcet voting, and majority judgment, they found that range voting had the highest (worst) strategic vulnerability, while their own method majority judgment had the lowest (best).[15]

Balinski and Laraki also used the same information to investigate how likely it was that each of those methods, as well as runoff voting, would elect a centrist. Opinions differ on whether this is desirable or not. Some argue that methods which favor centrists are better because they are more stable; others argue that electing ideologically purer candidates gives voters more choice and a better chance to retrospectively judge the relative merits of those ideologies; while Balinski and Laraki argue that both centrist and extremist candidates should have a chance to win, to prevent forcing candidates into taking either position. According to their model, plurality, runoff voting, and approval voting with a higher approval threshold tended to elect extremists (100%, 98%, and 94% of the time, respectively); majority judgement elected both centrists and extremists (56% extremists); and range, Borda, and approval voting with a lower approval threshold elected centrists (6%; 0.25–13% depending on the number of candidates; and 6% extremists; respectively).[15] However, their model did not take into account voters' strategic reactions to the method used, such as "lesser of two evils" voting under plurality.


"Soft" criteria[edit]


The New Zealand Royal Commission on the Electoral System listed ten criteria for their evaluation of possible new electoral methods for New Zealand. These included fairness between political parties, effective representation of minority[17] or special interest groups, political integration, effective voter participation and legitimacy.


Early democracy[edit]



Jean-Charles de Borda, an early voting theorist

Foundations of voting theory[edit]



The Marquis de Condorcet, another early voting theorist



Single-winner revival[edit]



Influence of game theory[edit]

Kenneth Arrow, a pioneer of modern voting theory

After John von Neumann and others developed the mathematical field of game theory in the 1940s, new mathematical tools were available to analyze voting methods and strategic voting. One such tool was the idea of a strong Nash equilibrium, based on the work of John Nash in the 1950s. This and similar methods led to significant new results that changed the field of voting theory.[18][need quotation to verify] The use of mathematical criteria to evaluate voting methods was introduced when Kenneth Arrow showed in Arrow's impossibility theorem that certain intuitively desirable criteria were actually mutually contradictory, demonstrating the inherent limitations of voting methods. Among the criteria Arrow considered desirable was one which requires the voting method to use ordinal (ranking) information; thus, as John Harsanyi pointed out, cardinal (rated) voting methods such as approval voting, range voting, and majority judgment can successfully meet all the other criteria. Arrow's theorem is easily the single most cited result in voting theory, and it inspired further significant results such as the Gibbard–Satterthwaite theorem, which showed that strategic voting is unavoidable in certain common circumstances, for any deterministic voting method, whether cardinal or ordinal.

The use of game theory to analyze voting methods also led to discoveries about the emergent strategic effects of certain methods. Duverger's law is a prominent example of such a result, showing that plurality voting often leads to a two-party system. Further research into this and other game theory aspects of voting led Steven Brams and Peter Fishburn to formally define and promote the use of approval voting in 1977. While approval voting had been used before that, it had not been named or considered as an object of academic study, particularly because it violated the assumption made by most research that single-winner methods were based on ordinal preference rankings.[26]


Post-1980 developments[edit]

Voting theory has come to focus on voting method criteria almost as much as it does on particular voting methods. Now, any description of a benefit or weakness in a voting method 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. By contrast, the factors driving voters to accept or reject particular methods in elections tend to be based on other arguments.

Political scientists of the 20th century published many studies on the effects that the voting methods have on voters' choices and political parties,[27][28][29] and on political stability.[30][31] A few scholars also studied which effects caused a nation to switch to a particular voting method.[32][33][34][35][36] One prominent current voting theorist is Nicolaus Tideman, who formalized concepts such as strategic nomination and the spoiler effect in the independence of clones criterion. Tideman also devised the ranked pairs method, a Condorcet method that is not susceptible to clones. Also, Donald G. Saari has brought renewed interest to the Borda count with the books he has published since 2001. Saari uses geometric models of positional voting methods to promote the Borda count.


The advent of the Internet has increased the interest in voting methods. 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 methods has influenced a new push for electoral reform beginning around the 1990s, with proposals being made to replace plurality voting in governmental elections with other methods. New Zealand adopted Mixed Member Proportional for Parliamentary elections in 1993 and Single Transferable Vote for some local elections in 2004 (see Electoral reform in New Zealand). After plurality voting was a key factor in the contested results of the 2000 US presidential election, various municipalities in the United States began to adopt instant-runoff voting, although some of them subsequently returned to their prior method. The Canadian province of British Columbia held two unsuccessful referendums (in 2005 and 2009) to adopt an STV method, and Ontario, another Canadian province, held an unsuccessful referendum on October 10, 2007 on whether to adopt a Mixed Member Proportional method. An even wider range of voting methods is now seen in non-governmental organizations.

Though theorists are divided on which particular voting method is best, many agree that several options are superior to plurality. For instance, the "Declaration of Election-Method Reform Advocates" supports single-winner reforms including approval voting, various Condorcet methods, majority judgment, and range voting.[37][38]

It has been argued that more in-depth and fine-tuned voting methods lie at the core of the development of e-democracy,[39] which consists of the digitization of democratic processes, including voting.[40] Research is ongoing into the use of homomorphic encryption for making voting methods that are transparent and voter-verifiable without compromising ballot secrecy. Tough criteria have been published for [41] and some advocates consider them to have been met for instance in Norway[42] or Estonia[43] – the latter even published its source code online, allowing independent verification at least of the voting-specific code.

However, as of November 2014, the majority of legal and political analysts stand firmly against computerized voting,[44][45][46] and argue that only paper ballots (whether machine-printed and voter-verified, or even exclusively hand-marked) give a reliable audit trail. Perhaps more importantly, they argue that important protections of electoral neutrality and perceived fairness can never be provided in an online voting context:

  • Scrutineers from each political party cannot inspect the process as it occurs in every location.
  • Polling place neutrality is impossible to guarantee, especially in the last instants before vote casting – a popup containing false information, for instance, can appear just as a ballot is cast.

Alongside these proposed and mostly unsuccessful reforms, other types of electoral and mass decision reform succeeded. For instance the rise of disapproval voting in the form of recall elections and referendums on matters that previously would have been left to the decision of a general election or legislature. California elected Arnold Schwarzenegger in a widely contested recall election. Many US states held referenda to specifically disapprove of same-sex marriage, though some of these were later overturned by the courts. And the rise of third parties and minority governments in the British Commonwealth led to several minority governments, coalitions and "hung parliaments" and thus an increase in votes of no confidence causing early elections. Despite many mathematical, political and legal arguments against unadvertised coalitions, 130-candidate first-past-the-post elections as in California, or referenda on what some regard as basic human rights, these measures went ahead anyway, often with results that did not reflect the expectations or traditions of the electoral or party system that was challenged:


Also significant since 1980 is actual experience with voting method referenda and change. As noted above, several US cities adopted instant-runoff voting, and a few discarded it after non-monotonic results. Post-facto analysis of six failed electoral referenda in Canada[48][49][50] and nationwide reform in the UK[51] yielded no consensus about voting method criteria that the public actually applied or expected to be applied to evaluate new methods. The only consensus seemed to be that change is difficult against any active "No" campaign that seeks to preserve the status quo, particularly if it can exploit distrust of political parties.

See also[edit]

Criteria table notes[edit]

  2. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap Condorcet, Smith and Independence of Smith-dominated alternatives criteria are incompatible with Independence of irrelevant alternatives, Consistency, Participation, Later-no-harm, Later-no-help, and Favorite betrayal[clarification needed] criteria.
  3. ^ a b c In Approval, Range, and Majority Judgment, if all voters have perfect information about each other's true preferences and use rational strategy, any Majority Condorcet or Majority winner will be strategically forced – that is, win in all of one or more strong Nash equilibria. 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 methods also satisfy the majority criterion in the weaker sense that any majority can force their candidate to win, if it so desires. Laslier, J-F (2006), "Strategic approval voting in a large electorate" (PDF), IDEP Working Papers, Marseille, France: Institut D'Economie Publique (405) 
  4. ^ a b c Approval voting, range voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates individually and independently of knowing the available alternatives in the election, using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power or even abstain, despite having meaningful preferences among the available alternatives. If this assumption is not made, these methods fail IIA, as they become more ranked than rated methods.
  7. ^ If the number of candidates grows faster than the square root of the number of voters, this may not be the case, as ties at any point in the process, even between two non-viable candidates, could affect the final result. If the rule for resolving such ties involves no randomness, though, the method does pass the criterion.[citation needed]
  12. ^ Majority Judgment does not always elect a candidate preferred over all others by over half of voters; however, it always elects the candidate uniquely top-rated by over half of voters.
  13. ^ Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters.
  15. ^ Balinski and Laraki, Majority Judgment's inventors, point out that it meets a weaker criterion they call "grade consistency": if two electorates give the same rating for a candidate, then so will the combined electorate. Majority Judgment explicitly requires that ratings be expressed in a "common language", that is, that each rating have an absolute meaning. They claim that this is what makes "grade consistency" significant. Balinski M, MJ; Laraki, R (2007), A theory of measuring, electing and ranking, 104 (21), USA: National Academy of Sciences, pp. 8720–25 
  16. ^ a b c d e In Majority Judgment, Ranked Pairs, and Schulze voting, there is always a regret-free semi-honest ballot for any voter, holding other ballots constant. That is, if they know enough about how others will vote (for instance, in the case of Majority Judgment, the winning candidate and their winning median score), there is always at least one way for them to participate without grading any less-preferred candidate above any more-preferred one. However, this can cease to hold if voters have insufficient information.
  23. ^ a b Once for each round.
  26. ^ Sortition, uniformly randomly chosen candidate is winner. Arbitrary winner, some external entity, not a voter, chooses the winner. These methods are not, properly speaking, voting methods at all, but are included to show that even a horrible method can still pass some of the criteria.
  27. ^ Random ballot, uniformly randomly chosen ballot determines winner. This and closely related methods are of mathematical interest because they are the only possible methods which are truly strategy-free, that is, your best vote will never depend on anything about the other voters. However, this method is not generally considered as a serious proposal for a practical method.



  • Arrow, Kenneth J (1963) [1951], Social Choice and Individual Values (2nd ed.), New Haven: Yale University Press, ISBN 0-300-01364-7 .
  • Balinski, M; Laraki, R (2007), "Election by Majority Judgement: Experimental Evidence", Cahier du Laboratoire d’Econométrie de l’Ecole Polytechnique (28) 
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  27. ^ Duverger 1954.
  28. ^ Rae 1971.
  29. ^ Taagapera & Shugart 1989.
  30. ^ Hermens 1941.
  31. ^ Lijphart 1994.
  32. ^ Lijphart 1985.
  33. ^ Lijphart 1992.
  34. ^ Rokkan 1970.
  35. ^ Rogowski 1987.
  36. ^ Boix 1999.
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External links[edit]

Existing electoral methods and related practical considerations
Electoral method reform advocacy
Theoretical resources