Ballot types

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An example of the four main ballot types (choose-one, approval, ranked, and rated).

A ballot is a voter's expression of preference among the candidates. There are four main ways to do this in the context of voting methods:

  • choose-one a.k.a. single-mark ballots (you mark one candidate that you support out of all candidates; these are often considered as "choose up to as many candidates as there are seats to be elected" a.k.a bloc voting ballots because when there are, say, two winners to be chosen, usually voters are allowed to mark up to two candidates, etc. Cumulative voting ballots can be considered a variation.),
  • Approval ballots (mark all the candidates that you support),
  • ranked ballots (rank the candidates in order of preference: 1st, 2nd, 3rd, etc. Some ranked ballots allow a voter to give multiple candidates the same rank to indicate no preference between those candidates.) and
  • rated ballots (you rate the candidates on a scale, generally starting from 0, going up to any value, often 5 or 10. Usually only certain in-between values are allowed i.e. if the scale is from 0 to 10, it usually isn't allowed for someone to give a rating of, say, 9.35).

There are several intersections between the various ballot types. For example, the information contained on an Approval ballot can also be found from a ranked or rated ballot if an approval threshold is utilized, and in fact, an Approval ballot is itself a type of rated ballot where the only allowed ratings are "disapprove" and "approve" (0 and 1). A choose-one ballot is itself an Approval ballot with the restriction of only marking one candidate, and can also be thought of as a ranked or rated ballot where only the candidate(s) ranked 1st/rated highest are treated as supported. A ranked ballot can be (at least partially) reconstructed from any of the other three ballot types i.e. if a voter scored one candidate higher than another (or marked one candidate but not another), then it is known for certain that that voter would also rank that candidate higher than the other.

Relation table
Choose-one Approval Ranked Rated
Choose-one No

(Is equivalent to an approval, or cumulative, ballot when unlimited numbers of candidates can be marked).

No No
Approval No

(Is equivalent to a choose-one ballot when only one candidate can be approved).

No No

(Is equivalent to a rated ballot when only two ratings are allowed.)

Ranked Yes (one of the candidates ranked 1st would be marked on a choose-one ballot) Only if an approval threshold is used (or if the voter ranked every candidate either 1st or last; it must also be assumed the voter would approve anyone they ranked 1st). Only if the voter ranked every candidate 1st or last.
Rated Yes (one of the candidates scored highest by the voter would be marked on a choose-one ballot) Only if an approval threshold is used (or if the voter rated every candidate either at the highest or lowest score).

(Is equivalent to an approval ballot when only two ratings are allowed.)

Yes (though with a slight caveat: only to the extent that enough scores were allowed i.e. if there are 7 candidates, and the voter has a preference between each of them, but was only allowed to give each of them one of 6 scores, then by pigeonhole principle they would've had to lie and indicate an equal preference between at least 2 of them, so the reconstructed ranking won't be perfectly accurate).

(The intersection between ballot types indicates relations. For example, the cell in the third row and fourth column indicates whether approval ballots give the information required to reconstruct a ranked ballot and with how much reliability).

This table shows that generally speaking, a rated ballot without an approval threshold provides the most information of any ballot type without an approval threshold, and that a rated ballot with an approval threshold provides the most information of any ballot type.As an example of all four ballot types, suppose there are five candidates to consider: Alicia, Brandon, Charlie, David, and Eileen.

Choose-one and Approval ballots are often shown as some form of:

Alicia☑ Brandon☐ Charlie☐ David☐ Eileen☑

meaning "Alicia and Eileen are supported by this voter" or

Alicia>Eileen=(approval threshold)>Brandon=Charlie=David

meaning "Alicia is my 1st choice, Eileen is my 2nd choice, I approve everyone who is my 2rd choice or higher, and all the other candidates are worse than my 2nd choice" Ranked ballots are often shown as:

Alicia>Eileen>Brandon=Charlie=David

meaning "Alicia is better than Eileen, Eileen is better than any of (Brandon, Charlie, David) and Alicia is also better than any of (Brandon, Charlie, David)" or

(this column isn't always shown) 1 voter
1st Alicia
2nd Eileen
3rd Brandon, Charlie, David

which means the same thing.

Rated ballots are often shown as:

Alicia:5 Eileen:4 Brandon:3 Charlie:3 David:3

meaning "I give Alicia a score of 5/I give Alicia 5 points, Eileen 4, Brandon 3, etc."

Note that with rated ballots, one might mention what the scale is by putting, for example, Alicia:5/5 (meaning Alicia is a 5 out of 5).

It is customary to, instead of repeating the same ballot type multiple times, simply type the number of voters who cast a ballot of a certain type. For example, instead of

A>B>C

A>B>C

A>B>C

D>E

One can simply write

3: A>B>C 1: D>E

When filling out a ranked ballot, often voters are either allowed to write the rank numbers next to each candidate, or they are allowed to bubble it in like so:

Candidates 1st choice 2nd choice 3rd choice
A X
B X
C X
D X

Each voter bubbles in which rank they want to put each candidate at.

Note that with a ranked ballot, only (number of candidates - 1) ranks need be provided (if unranked candidates are assumed to be ranked last); this is because even if the voters indicates a preference between their (number of candidates - 1) favorite candidates and uses up all of their available ranks, the last candidate they didn't rank will be assumed to be ranked last.

With a rated ballot, again, either the score can be written next to each candidate, or the voter can bubble it in like so:

Candidates 0 1 2 3 4 5
A X
B X
C X
D X

If, for example, you wish to use a scale from 0 to some value, and unscored candidates are assumed to be given the lowest score, then voters need only be allowed to score from 1 to the chosen value; this is because unscored candidates will automatically be given a 0 (if it's set up that way).

Approval ballot[edit | edit source]

An approval ballot is a ballot type in which the voter is given a list of candidates and must, for each candidate, either approve them, or disapprove them. Approval ballots are used by Approval voting, Explicit approval voting, Proportional approval voting, Combined approval voting, etc.

Rated ballot[edit | edit source]

Wikipedia has an article on:

A ratings ballot, or cardinal ballot, is a ballot in which, for each candidate, a voter is asked to "rate" the candidate on a scale, for example from 0 to 100 or from −5 to +5. Ratings are usually (though not always) restricted to integer values.

An approval ballot can be thought of as a ratings ballot with an integer scale from 0 to 1 (inclusive).

Voting systems that use a ratings ballot include Score voting, instant runoff normalized ratings, and cardinal pairwise.

Two rated ballots that use different scales can be converted to each other. For example, if one voter gave a candidate a 5 out of 10 and another voter gave a candidate a 3 out of 7, the 5 out of 10 can be interpreted as a 3.5 out of 7, and the 3 out of 7 as a 4.2857 out of 10. In general, all rated ballots can be thought of as approximations of (and transformable into) a scale from 0 to 1 (or 0% to 100%), with 0 being no support and 1 being full support.

Choose-one ballot[edit | edit source]

A single-mark ballot or plurality ballot is a type of ballot in which voters can only make a single mark next to one candidate. It is most commonly-used for first-past-the-post elections, but also for Runoff voting, Asset voting, Random ballot, etc. See Category:Single-mark ballot voting methods. Most of these methods can also be modified to work with a cumulative voting ballot.

Voting methods that use single-mark ballots[edit | edit source][edit | edit source]

From Category:Single-mark ballot voting methods:

External links[edit | edit source]

Notes[edit | edit source]

It's important to keep in mind that voters may be incentivized to show or not show weak preferences between candidates based on ballot type. For example, a voter who thinks candidate A is a 10/10 and candidate B is a 9/10 may rank the two candidates equally or approve both of them on ranked or Approval ballots, respectively, in order to avoid exaggerating their preference between the two. So sometimes converting between one ballot type to another involves losing some of this information.

A common assumption with all four ballot types is that anyone who is not given an explicit marking by the voter is considered worse by that voter than any of their marked candidates, and that the voter has no preference between any of the unmarked candidates.

Pairwise counting can be done on all four ballot types, though ranked and rated ballots offer the most information for this purpose.