Voting Methods

Various Voting Methods Used or Proposed Today

How can a large number of voters can express their opinions about multiple candidates on a single ballot, and how then can one of those candidates be chosen for office?

Different ballots can ask for different information; and even ballots that collect the same information can elect different candidates, depending on which rule is used to process that information. 

Even with identical ballots and the same general rule, different implementations can differ in fine details, such as how to resolve rare ties, or how to interpret incomplete, or partially illogical, ballots.

Following are summaries of the various voting methods used or proposed today. After most is a reference [ ] to an end note where material may be found examining that method more deeply

METHODS WITH BALLOTS WHERE VOTERS MARK ONLY A SINGLE CANDIDATE

  • Usable only if but two candidates run. The ballot allows the voter to mark which of the two candidates he prefers take office. Each ballot awards 1 vote to the candidate marked as preferred; a blank ballot awards 0 votes; and a ballot where both candidates are marked as preferred is deemed to be spoiled and awards neither candidate a vote. The votes awarded each candidate are totaled, and the candidate with the highest total wins.

    Resolving rare ties: The two candidates can have their vote totals tie. Before the election the elections official shall have done a random draw, and published the result of that draw, that sets which candidate will win in case of a tie.

  • Usable if any number of candidates run. The ballot allows a voter to mark the name of a single candidate whom that voter will support for the office. Each ballot awards 1 vote to the candidate marked as supported; a blank ballot awards 0 votes; a ballot where two or more candidates are marked as supported is deemed to be spoiled and award no candidate a vote. The votes awarded each candidate are totaled, and the candidate with the highest total wins.

    Resolving rare ties: Two or more candidates could tie for having the highest total. Before the election the elections official shall have chosen and published a list of the candidates in a random order; of the candidates involved in the tie, the candidate closest to the head of that list wins.

METHODS WITH BALLOTS WHERE VOTERS MARK A LIST OF CANDIDATES

These methods use a ballot where a voter can submit a  list of candidates in the order of his most preferred to his least preferred, with no candidates permitted to share the same place on the list. A voter need not include on his list all the candidates running; candidates left off his list are deemed to rank below all the candidates on the list, and are ranked “equally” last, in a sense that varies from method to method. [1]

This ballot is generally referred to as a Ranked Choice Ballot. Typically such a ballot is implemented by presenting the voter with a list of the candidates, which he marks to indicate his first, or most preferred choice; then his second choice; and so on down to the last candidate he chooses to include on his list. 

There are a number of ways in which votes can be counted when using this type of ballot.

1 Delicate questions of interpretation arise when a voter ranks four candidates not as 1, 2, 3, and 4, but marks just two candidates as his 2nd and 3rd choices; or marks one candidate as his 1st choice but two candidates equally as his 2nd. We do not deal with these questions here, but assume that voters mark the candidates in order as their 1st, 2nd, and so in order to the last candidate they choose to mark; or that a badly marked ballot has already been interpreted into that form.

  • A runoff is used to remove candidates from contention one by one until only one is left, who is the winner. In each round of the runoff, each ballot awards 1 vote to the candidate still in contention who is placed highest on that ballot; a ballot none of whose marked candidates are still in contention awards no vote. In each round the votes are totaled, and the candidate with the lowest total is removed from contention.

    Resolving rare ties: Two or more candidates could tie for having the lowest total. Before the election the elections official shall have chosen and published a list of the candidates in a random order; of the candidates involved in the tie, the candidate lowest on that list is the one removed from contention.

  • Condorcet methods rely on comparisons of pairs of candidates. In the comparison of candidate A and candidate B, a ballot awards a vote to A if candidate A is placed higher on that ballot than B, and awards a vote to B if candidate B is placed higher on that ballot than A. If one of candidates A and B is left unmarked on a ballot, the vote is awarded to his opponent; if both are left unmarked, that ballot ‘abstains’ from that comparison and awards no vote. The candidate with the most votes wins that comparison. 

    The defining characteristic of a Condorcet method is that if a candidate wins his comparison with each of his rivals, that candidate is elected; such a candidate is therefore called a Condorcet winner.  

    Condorcet methods differ only on how to choose whom to elect if there is no Condorcet winner.  Almost every election will have such a winner, so disagreement among Condorcet methods is rare; but still their common choice of whom to elect can disagree with the candidate chosen by Instant Runoff.

    Voting using a Condorcet method is sometimes referred to as consensus voting or tournament voting.  `Consensus’ in that while a Condorcet winner may not be the candidate most-preferred by a majority of voters, for every alternative there is some majority of the voters who prefer him to that alternative, though the voters who make up that majority change depending on the alternative being considered. `Tournament’ in that a Condorcet winner can be found just from knowing the outcomes of the comparisons of pairs of candidates, just as one can tell that in a round-robin tournament a player who wins all his matches is the overall winner; and in that when there is no Condorcet winner, the winner can be found knowing only the margins by which the comparisons are won, just as in a round-robin tournament if no player is undefeated the overall winner can be found from the margins by which each player won or lost his matches.

  • This is the simplest method that always elects a Condorcet winner and that uses the method of Instant Runoff to decide elections that have no Condorcet winner.

    The elections official shall determine the results of all the comparisons between the candidates. If after resolving all tied comparisons, one candidate wins his comparison against each of his rivals, he is elected. If not, the winner shall be chosen as by Instant Runoff, above.

    Resolving rare ties: Before the election, the elections official shall prepare and make public a randomized list of all the candidates. If a comparison is tied, the winner of the comparison shall be chosen to be the candidate higher on this list.

  • This method is more formally called Bottom-Two-Runoff Instant Runoff Voting, or BTR-IRV.

    Should there be no Condorcet winner, candidates are removed from contention one by one in a runoff until only one candidate is left, who is elected. In the runoff each ballot awards 1 vote to the candidate still in contention who is placed highest on that ballot; a ballot none of whose marked candidates are still in contention awards no vote. In each round the votes are totaled, and the two candidates with the lowest totals are identified; of these, the candidate who loses his comparison with the other is the one removed from contention.

    Resolving rare ties: Two or more candidates could tie for having the lowest total. Before the election the elections official shall have chosen and published a list of the candidates in a random order. If there is one candidate among them who wins his comparisons with all the rest, he is retained, and the candidate among the rest who is lowest on the list is the one removed from contention. If no candidate among them wins his comparisons with all the rest, then the candidate who is lowest on the list is the one removed from contention.

  • When there is no Condorcet winner, this method uses a variation of Instant Runoff to remove candidates from contention one by one in a runoff until only one is left, who is elected.

    Before each round of the runoff, the candidates in contention are checked to see if among them is a candidate who wins his comparisons with each of his rivals likewise still in contention; if so, he is elected. If no candidate is thus elected, then each ballot awards 1 vote to the candidate still in contention who is placed highest on that ballot; a ballot none of whose marked candidates are still in contention awards no vote. The candidate with the lowest total is removed from contention.

    Resolving rare ties: Two or more candidates could tie for having the lowest total. Before the election, the elections official shall have chosen and published a list of the candidates in a random order; of the candidates involved in the tie, the candidate lowest on this list is removed from contention.

  • When there is a Condorcet winner, he is elected. If not, the method conceptually begins with a collection of all the possible lists in which an election could possibly rank the candidates from first to last, and removes lists from this collection step by step so that only one list remains. The candidate at the head of that list is elected.

    The guiding principle is that if we have three or more comparisons that are collectively inconsistent with such a list---for example, that a majority of voters in a three-candidate race prefer candidate A to candidate B, and another majority prefer candidate B to C, and yet another majority prefer C to A---then the relative ranking suggested by the larger of the majorities must prevail over the ranking suggested by the smallest. Thus in this example if the voters prefer A to B by a margin of 50 votes, and B to C by a margin of 30 votes, and C to A by a margin of 40 votes, then it is the suggestion that B precede C that must yield. Here the ranking of the candidates would be CAB, so that candidate C would be elected.

    Accordingly, the election official shall prepare a list of all the possible two-candidate comparisons, sorted in order from largest majority to smallest. He shall proceed through the comparisons one by one starting with the top one on this list. If all the lists remaining in the collection are inconsistent with the suggestion offered by a comparison, then that suggestion shall be ignored (as inconsistent with suggestions made by previous comparisons each of which has a greater margin); otherwise the suggestion shall be adopted and so all the lists inconsistent with the suggestion shall be discarded from the collection. At the end only one list will remain, and the candidate at the head of it shall be elected.

    In practice an algorithm that accomplishes this same task but without having to build the voluminous initial collection of lists would be used; but the reasoning that justifies that algorithm is as just described.

    Resolving rare ties: Before the election, the elections official shall prepare and make public a list of all the candidates in a random order. If a comparison is tied, the winner shall be chosen to be the candidate higher on this list, and the suggestion shall be that the winner precede the loser. If two or more comparisons have the same margins, then these comparisons on the list of winning comparisons shall be sorted from high to low first by whether the winner is higher on the randomized list, and second by whether the loser is higher.

  • Further information on the Beatpath method can be found in the footnote below.

POSITIONAL SCORING METHODS

• Positional Scoring Methods in general [9]

These use a ranked-choice ballot. Assume for simplicity that when N candidates run a voter gives all of them a different rank, from first to last. There are different approaches about what to do if a voter does not give a rank to each candidate; we shall forego delving into those complications here.

A positional scoring method uses some sequence of decreasing positive constants1, . . . ,N; the precise sequence depends upon the method. A single ballot awards the candidate placed first 1 votes, and the candidate placed second 2 votes, down to the candidate placed last who is awarded N votes. The votes for each candidate over all the ballots are totaled, and the candidate who received the most votes wins.

The classic among positional scoring methods is the Borda count, which uses the sequence of constants N, N − 1, . . . , 1, 0. This particular sequence can be implemented using the same comparisons as are defined for Condorcet methods, though Borda is not itself Condorcet.

Resolving rare ties: Two or more candidates could tie for having the largest score. Before the election, the elections official shall prepare and make public a list of the candidates in a random order. Among the candidates involved in the tie the candidate who is highest on this list is elected.

  • The score for a candidate shall be the sum of the margins, positive or negative or zero, against each of his rival candidates. The candidate with the largest score shall be elected.

    When a rare tie occurs: Before the election, the elections official shall prepare and make public a randomized list of all the candidates. Should two or more candidates tie for having the largest score, the candidate who is highest on this list shall be elected.

METHODS WITH OTHER BALLOTS

  • This method generalizes plurality; instead of using a ballot to award one vote to a sole candidate of his choice, a voter may mark his ballot to award one vote to each of as many of the candidates he wishes (the voter is said to ‘approve’ of each such candidate); the candidates he leaves unmarked he awards zero votes (the voter is said to ‘disapprove’ of such candidate.) The votes awarded each candidate are totaled, and the candidate with the most votes wins.

    When a rare tie occurs: Two or more candidates could tie for having the most votes. Before the election, the elections official shall prepare and make public a list of all the candidates in a random order. Among the candidates involved in the tie the candidate who is highest on this list is elected.

  • This method generalizes approval voting; instead of using his ballot to award as many candidates as he wishes each one vote, a voter marks his ballot to award each candidate he wishes, each independently without reference to any other, any number of votes within a finite range. A typical range is 0 to 10. As with approval voting, the votes awarded each candidate are totaled, and the candidate with the most votes wins.

    Some variants do an automatic linear adjustment of each voter’s awards so that his lowest award is 0 and his highest award is 10, though this adjustment makes awards be not integers in general but fractions.

    Resolving rare ties: Two or more candidates could tie for having the most votes. Before the election, the elections official shall prepare and make public a list of all the candidates in a random order. Among the candidates involved in the tie the candidate who is highest on this list shall be elected.

  • This is a variant of Range voting, and uses a ballot identical to that for Range Voting; the acronym “STAR” stands for “Score, Then Automatic Runoff. The two candidates with the top Range scores are identified. The candidate of the two who wins their comparison, as it would be defined by Condorcet methods using the STAR ballots, is elected.

    Specifically, in the comparison of final candidate A and final candidate B, a ballot awards a vote to A if candidate A is given a higher range score than B, and awards a vote to B if candidate B is given a higher range score than A. If one of candidates A and B is left unmarked on a ballot, the vote is awarded to his opponent; if both are left unmarked, or if both are given the same range score, that ballot ‘abstains’ from that comparison and awards no vote. The candidate with the most votes wins that comparison.

    Resolving rare ties: Two or more candidates could tie for having the most Range votes. Before the election, the elections official shall prepare and make public a randomized list of all the candidates. Should two or more candidates tie for having the most Range votes, the candidates selected as the final pair shall be the two highest on this list. The comparison between the final pair could also be tied; if so, the candidate in this pair highest on this list is elected.

END NOTES AND REFERENCES

[1] Resources, The right way to read Ranked-choice ballots: not Instant Runoff, but Ranked Pairs, papers A through D.

[2] Regarding plurality see [1], paper A, section II A on p. 6. 

[3] For a quick comparison of the effects of Instant Runoff and Condorcet methods in routine elections see Resources, Condorcet Versus Instant Runoff in Two Figures.  For a detailed analysis of these differences see [1].  For an analysis of the systematic nature of the failures of Instant Runoff to choose representative candidates when 3 or more candidates run see paper B, section II, pp. 1-4.  For why it is a mistake when Instant Runoff avoids electing the candidate at the median of voter opinion see paper B, sections V and VI, pp.  14-17; for why Condorcet methods consistently elect that median candidate (the median lemma, and its generalizations) see paper B, section IV, pp. 5-14.

For a point-by-point refutation of the Fairvote arguments in favor of Instant Runoff over Condorcet methods see Resources, Critique of the Fairvote analysis that compares Instant Runoff and Condorcet methods, and also the related  Benham, a Condorcet method, compared to Instant Runoff, has for 3 candidates strictly fewer opportunities for tactical voting, and also (on a more technical point) “Spoilers” Spoil Nothing.

[4] Regarding Condorcet methods in general see [1].  For why Condorcet methods consistently elect that median candidate (the median lemma, and its generalizations) see paper B, section IV, pp. 5-14.  For electing the candidate at the median of voter opinion is desirable see paper B, sections V and VI, pp.  14-17.

[5] For the Direct Hybrid, also called the Hybrid method see [1].  For a definition, see A, section V E on p. 14, and also C, section III, pp. 18-20.  For an analysis of its abstract properties see Section III A, pp. 20-22; for a comparison of those properties with those of other election methods see V, pp. 26-29, in particular Table VIII on p. 28.

[6] Regarding the Benham method, see [1].  For a definition, see A, section V D on p. 18, and C, section III, p. 20.  For an analysis of its abstract properties see section III B, pp. 23-25; for a comparison of those properties with those of other election methods see V, pp. 26-29, in particular Table VIII on p. 28.  For an analysis of this methods superiority to Instant Runoff regarding the sheer number of opportunities it affords to tactical voting see Resources, Benham, a Condorcet method, compared to Instant Runoff, has for 3 candidates strictly fewer opportunities for tactical voting.

[7] Regarding Ranked Pairs see [1].  For a definition, see A, section V C on pp. 16-17; for its motivation, see section VI on pp. 18-19, and in particular regarding how it balances representing the intent of voters and avoiding elections being manipulated by the running of essentially identical candidates (``clones’’) see section VIII C on p. 24.  For an analysis of how it establishes an ordered ranking of all possible orders of candidates from first to last, which renders a challenge to the correctness of the outcome to be accepted or rejected using only pen and paper and no computer, see paper D, section X, pp. 24-28.

[8] Regarding Beatpath, see [1].  For a definition, and for how it balances representing the intent of voters and avoiding elections being manipulated by the running of essentially identical candidates (``clones’’) see section paper A, section VIII D on pp. 24-26.  For its abstract properties, and how those compare to other election methods, see paper C, section V, pp. 26-29, in particular Table VIII on p. 28.  For a short discussion of why Ranked Pairs is a method superior to Beatpath see Resources,  CONDORCET METHODS VS. INSTANT RUNOFF, The best Condorcet-compatible election method: Ranked Pairs.  Note however an erratum, Correction to: The best Condorcet-compatible election method: Ranked Pairs, submitted to the Journal of Constitutional Political Economy, and note the further analysis in Resources, Technical Aspects of Election Methods, Dropping one candidate under Beatpath and Ranked Pairs.

[9] Regarding positional scoring methods in general see [1], section II pp. 4-5, in particular p. 5; for detailed analysis of the Borda count, which is one example, see [10].

[10] Regarding the Borda count see [1], Paper A, section III B pp. 6-7. For a formal definition, see section V F on p. 18.  For its vulnerability of its election outcomes being manipulated by running multiple candidates with the same point of view see paper B, section II, pp. 3-4; and paper D, section VII, pp. 20-21.  For its vulnerability to voters casting strategic ballots by deliberately down-ranking their favorite candidate’s closest rival see Resources, Critique of the Fairvote analysis that compares Instant Runoff and Condorcet methods, p. 34.

[11] Resources, ELECTION METHODS WITHOUT RANKED-CHOICE BALLOTS, Approval voting is Condorcet-compatible voting under a constraint: A critique of Approval, and both Range and Star, voting.

[12] Regarding Approval voting and its relation to Condorcet methods see [10], sections I to II on pp. 1-3; for sample elections where Approval voting fails to elect representative candidates see sections III pp. 3-7; for a critique of the arguments used to justify Approval voting see section IV on pp. 7-9 and section VI on pp. 12-14 and section VIII on pp. 14-16.

[13] Regarding Range voting and its relation to Approval and STAR voting see [10]; for a critique of Range voting in general see section V, pp. 9-12; for a criticism of its underlying motivation see section VIII on pp. 14-16.

[14] Regarding STAR voting and its relation to Approval and Range voting see [10]; for a critique of STAR voting in general see section VII, p. 14; for a criticism of its underlying motivation see section VIII on pp. 14-16.