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

[edit] Single or sequential vote methods

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

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

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

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

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

[edit] Ranked voting methods

Main article: Preferential voting

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

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

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

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

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

Condorcet methods, or pairwise methods, are a class of ranked voting methods that meet the Condorcet criterion. These methods compare every option pairwise with every other option, one at a time, and an option that defeats every other option is the winner. An option defeats another option if a majority of voters rank it higher on their ballot than the other option.

These methods are often referred to collectively as Condorcet methods because the Condorcet criterion ensures that they all give the same result in most elections, where there exists a Condorcet winner. The differences between Condorcet methods occur in situations where no option is undefeated, implying that there exists a cycle of options that defeat one another, called a Condorcet paradox or Smith set. Considering a generic Condorcet method to be an abstract method that does not resolve these cycles, specific versions of Condorcet that select winners even when no Condorcet winner exists are called Condorcet completion methods.

A simple version of Condorcet is Minimax: if no option is undefeated, the option that is defeated by the fewest votes in its worst defeat wins. Another simple method is Copeland's method, in which the winner is the option that wins the most pairwise contests, as in many round-robin tournaments. The Schulze method (also known as "Schwartz sequential dropping", "cloneproof Schwartz sequential dropping" or the "beatpath method") and Ranked pairs are two recently designed Condorcet methods that satisfy a large number of voting system criteria.

The Kemeny-Young method is a Condorcet method that fully ranks all the candidates from most popular to least popular.

[edit] Rated voting methods

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

Rated ballots allow even more flexibility than ranked ballots, but few methods are designed to use them. Each voter gives a score to each option; the allowable scores could be numeric (for example, from 0 to 100) or could be "grades" like A/B/C/D/F.

approval

Rated ballots can be used for ranked voting methods, as long as the ranked method allows tied rankings. Some ranked methods assume that all the rankings on a ballot are distinct, but many voters would be likely to give multiple candidates the same rating on a rated ballot.

[edit] Range voting

In range voting, voters give numeric ratings to each option, and the option with the highest total score wins.

[edit] Approval voting

Approval voting, where voters may vote for as many candidates as they like, can be seen as an instance of range voting where the allowable ratings are 0 and 1.

[edit] Cumulative voting

There are variants within cumulative voting. In the points form, each voter has as many votes as there are choices, and can distribute those votes as desired: all on one choice or spread in any other pattern. Cumulative voting is used in a number of communities as well as corporate boards. It was examined and developed perhaps most thoroughly by Lani Guinier (1994), and has recently been studied by, among others Brams 2003. The latter notes that 'The chief reason for its nonadoption in public elections, and by some societies, seems to be a lack of key “insider” support.'

[edit] Criteria in evaluating single winner voting systems

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

It is impossible for one voting system to pass all criteria in common use. Economist Kenneth Arrow proved Arrow's impossibility theorem, which demonstrates that several desirable features of voting systems are mutually contradictory. For this reason, someone implementing a voting system has to decide which criteria are important for the election.

Using criteria to compare systems does not make the comparison completely objective. For example, it is relatively easy to devise a criterion that is met by one's preferred voting method, and by very few other methods. Doing this, one can then construct a biased argument for the criterion, instead of arguing directly for the method. There is no ultimate authority on which criteria should be considered, but the following are some criteria that are accepted and considered to be desirable by many voting theorists:

* Majority criterion—If there exists a majority that ranks (or rates) a single candidate higher than all other candidates, does that candidate always win?

* Monotonicity criterion—Is it impossible to cause a winning candidate to lose by ranking him higher, or to cause a losing candidate to win by ranking him lower?

* Consistency criterion—If the electorate is divided in two and a choice wins in both parts, does it always win overall?

* Participation criterion—Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below.[3])

* Condorcet criterion—If a candidate beats every other candidate in pairwise comparison, does that candidate always win? (This implies the majority criterion, above)

* Condorcet loser criterion—If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?

* Independence of irrelevant alternatives—If a candidate is added or removed, do the relative rankings of the remaining candidates stay the same?

* Independence of clone candidates—Is the outcome the same if candidates identical to existing candidates are added?

* Later-no-harm criterion—If, in any election, a voter gives an additional ranking, vote or positive rating to a less preferred candidate, can that additional ranking, vote or rating cause a more preferred candidate to lose?[4]

* Reversal symmetry—If individual preferences of each voter are inverted, does the original winner never win?

* Polynomial time—Can the winner be calculated in a runtime that is polynomial in the number of candidates and the number of voters?

* Summability—How much information must be transmitted from each polling station to a central location in order to determine the winner? This is expressed as an order function of the number of candidates N. Slower-growing functions such as O(N) or O(N2) make for easier counting, while faster-growing functions such as O(N!) might make it harder to catch fraud by election administrators.

* Allows equal rankings—Allows a voter to choose whether to rank any two candidates equally at any position on the ballot. This can reduce the prevalence of spoiled ballots due to overvotes, and can give a less-dishonest alternative to some tactical voting strategies.

[edit] Summary table

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

Majority Monotone Consistency & Participation Condorcet Condorcet loser IIA Clone independence Later-no-harm Reversal symmetry Polynomial time Summability Allows equal rankings

Approval[nb 1] Ambiguous Yes Yes[nb 2] No[nb 2] No Ambiguous Ambiguous[nb 3] No Yes Yes O(N) No[nb 4]

Borda count No Yes Yes No Yes No No (teaming) No Yes Yes O(N) No

Bucklin voting[nb 5] Yes Yes No No No No[nb 5] No[nb 5] (vote-splitting) No No Yes O(N2)[nb 6] Depends on variant

IRV Yes No No No Yes No Yes Yes No Yes O(N!)[nb 7] No

Kemeny-Young Yes Yes No Yes Yes No

(but ISDA) No (teaming possible) No Yes No O(N2)[nb 8] Yes

Minimax Yes Yes No Yes[nb 9] No No No (vote-splitting) No[nb 9] No Yes O(N2) Depends on variant

Plurality Yes Yes Yes No[nb 2] No No No (vote-splitting) Yes No Yes O(N) No

Range voting[nb 1] No[nb 2] Yes Yes[nb 2] No[nb 2] No Yes Ambiguous[nb 3] No Yes Yes O(N) Yes

Ranked pairs Yes Yes No Yes Yes No

(but ISDA) Yes No Yes Yes O(N2) Yes

Runoff voting Yes No No No Yes No No (vote-splitting) Yes No Yes O(N)[nb 10] No

Schulze Yes Yes No Yes Yes No

(but ISDA) Yes No Yes Yes O(N2) Yes

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

It is also possible to simulate large numbers of virtual elections on a computer and see how various voting systems compare in terms of maximum voter satisfaction, called in this context minimum Bayesian regret. Such simulations are sensitive to their assumptions, particularly with regards to voter strategy, but by varying the assumptions they can give repeatable measures that bracket the best and worst cases for a voting system. To date, the only such simulation to compare a wide variety of voting systems was run by a range-voting advocate and has not been peer-reviewed.[7] Simulated elections in a two-dimensional issue space can also be graphed to visually compare election methods; this makes apparent issues like nonmonotonicity and clone- independence.[8]

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

[edit] Single or sequential vote methods

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

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

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

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

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

[edit] Ranked voting methods

Main article: Preferential voting

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

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

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

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

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

Condorcet methods, or pairwise methods, are a class of ranked voting methods that meet the Condorcet criterion. These methods compare every option pairwise with every other option, one at a time, and an option that defeats every other option is the winner. An option defeats another option if a majority of voters rank it higher on their ballot than the other option.

These methods are often referred to collectively as Condorcet methods because the Condorcet criterion ensures that they all give the same result in most elections, where there exists a Condorcet winner. The differences between Condorcet methods occur in situations where no option is undefeated, implying that there exists a cycle of options that defeat one another, called a Condorcet paradox or Smith set. Considering a generic Condorcet method to be an abstract method that does not resolve these cycles, specific versions of Condorcet that select winners even when no Condorcet winner exists are called Condorcet completion methods.

A simple version of Condorcet is Minimax: if no option is undefeated, the option that is defeated by the fewest votes in its worst defeat wins. Another simple method is Copeland's method, in which the winner is the option that wins the most pairwise contests, as in many round-robin tournaments. The Schulze method (also known as "Schwartz sequential dropping", "cloneproof Schwartz sequential dropping" or the "beatpath method") and Ranked pairs are two recently designed Condorcet methods that satisfy a large number of voting system criteria.

The Kemeny-Young method is a Condorcet method that fully ranks all the candidates from most popular to least popular.

[edit] Rated voting methods

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

Rated ballots allow even more flexibility than ranked ballots, but few methods are designed to use them. Each voter gives a score to each option; the allowable scores could be numeric (for example, from 0 to 100) or could be "grades" like A/B/C/D/F.

approval

Rated ballots can be used for ranked voting methods, as long as the ranked method allows tied rankings. Some ranked methods assume that all the rankings on a ballot are distinct, but many voters would be likely to give multiple candidates the same rating on a rated ballot.

[edit] Range voting

In range voting, voters give numeric ratings to each option, and the option with the highest total score wins.

[edit] Approval voting

Approval voting, where voters may vote for as many candidates as they like, can be seen as an instance of range voting where the allowable ratings are 0 and 1.

[edit] Cumulative voting

There are variants within cumulative voting. In the points form, each voter has as many votes as there are choices, and can distribute those votes as desired: all on one choice or spread in any other pattern. Cumulative voting is used in a number of communities as well as corporate boards. It was examined and developed perhaps most thoroughly by Lani Guinier (1994), and has recently been studied by, among others Brams 2003. The latter notes that 'The chief reason for its nonadoption in public elections, and by some societies, seems to be a lack of key “insider” support.'

[edit] Criteria in evaluating single winner voting systems

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

It is impossible for one voting system to pass all criteria in common use. Economist Kenneth Arrow proved Arrow's impossibility theorem, which demonstrates that several desirable features of voting systems are mutually contradictory. For this reason, someone implementing a voting system has to decide which criteria are important for the election.

Using criteria to compare systems does not make the comparison completely objective. For example, it is relatively easy to devise a criterion that is met by one's preferred voting method, and by very few other methods. Doing this, one can then construct a biased argument for the criterion, instead of arguing directly for the method. There is no ultimate authority on which criteria should be considered, but the following are some criteria that are accepted and considered to be desirable by many voting theorists:

* Majority criterion—If there exists a majority that ranks (or rates) a single candidate higher than all other candidates, does that candidate always win?

* Monotonicity criterion—Is it impossible to cause a winning candidate to lose by ranking him higher, or to cause a losing candidate to win by ranking him lower?

* Consistency criterion—If the electorate is divided in two and a choice wins in both parts, does it always win overall?

* Participation criterion—Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below.[3])

* Condorcet criterion—If a candidate beats every other candidate in pairwise comparison, does that candidate always win? (This implies the majority criterion, above)

* Condorcet loser criterion—If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?

* Independence of irrelevant alternatives—If a candidate is added or removed, do the relative rankings of the remaining candidates stay the same?

* Independence of clone candidates—Is the outcome the same if candidates identical to existing candidates are added?

* Later-no-harm criterion—If, in any election, a voter gives an additional ranking, vote or positive rating to a less preferred candidate, can that additional ranking, vote or rating cause a more preferred candidate to lose?[4]

* Reversal symmetry—If individual preferences of each voter are inverted, does the original winner never win?

* Polynomial time—Can the winner be calculated in a runtime that is polynomial in the number of candidates and the number of voters?

* Summability—How much information must be transmitted from each polling station to a central location in order to determine the winner? This is expressed as an order function of the number of candidates N. Slower-growing functions such as O(N) or O(N2) make for easier counting, while faster-growing functions such as O(N!) might make it harder to catch fraud by election administrators.

* Allows equal rankings—Allows a voter to choose whether to rank any two candidates equally at any position on the ballot. This can reduce the prevalence of spoiled ballots due to overvotes, and can give a less-dishonest alternative to some tactical voting strategies.

[edit] Summary table

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

Majority Monotone Consistency & Participation Condorcet Condorcet loser IIA Clone independence Later-no-harm Reversal symmetry Polynomial time Summability Allows equal rankings

Approval[nb 1] Ambiguous Yes Yes[nb 2] No[nb 2] No Ambiguous Ambiguous[nb 3] No Yes Yes O(N) No[nb 4]

Borda count No Yes Yes No Yes No No (teaming) No Yes Yes O(N) No

Bucklin voting[nb 5] Yes Yes No No No No[nb 5] No[nb 5] (vote-splitting) No No Yes O(N2)[nb 6] Depends on variant

IRV Yes No No No Yes No Yes Yes No Yes O(N!)[nb 7] No

Kemeny-Young Yes Yes No Yes Yes No

(but ISDA) No (teaming possible) No Yes No O(N2)[nb 8] Yes

Minimax Yes Yes No Yes[nb 9] No No No (vote-splitting) No[nb 9] No Yes O(N2) Depends on variant

Plurality Yes Yes Yes No[nb 2] No No No (vote-splitting) Yes No Yes O(N) No

Range voting[nb 1] No[nb 2] Yes Yes[nb 2] No[nb 2] No Yes Ambiguous[nb 3] No Yes Yes O(N) Yes

Ranked pairs Yes Yes No Yes Yes No

(but ISDA) Yes No Yes Yes O(N2) Yes

Runoff voting Yes No No No Yes No No (vote-splitting) Yes No Yes O(N)[nb 10] No

Schulze Yes Yes No Yes Yes No

(but ISDA) Yes No Yes Yes O(N2) Yes

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

It is also possible to simulate large numbers of virtual elections on a computer and see how various voting systems compare in terms of maximum voter satisfaction, called in this context minimum Bayesian regret. Such simulations are sensitive to their assumptions, particularly with regards to voter strategy, but by varying the assumptions they can give repeatable measures that bracket the best and worst cases for a voting system. To date, the only such simulation to compare a wide variety of voting systems was run by a range-voting advocate and has not been peer-reviewed.[7] Simulated elections in a two-dimensional issue space can also be graphed to visually compare election methods; this makes apparent issues like nonmonotonicity and clone- independence.[8]

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

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