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
- 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.
- Rated ballots
- 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).
With all ballot types, it is generally assumed that unmarked/unranked/unrated candidates are considered worse than all marked candidates, and that the voter has no preference between any of them.
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.
(Is equivalent to an approval, or cumulative, ballot when unlimited numbers of candidates can be marked).
(Is equivalent to a choose-one ballot when only one candidate can be approved).
(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[clarification needed]
Alicia>Eileen=(approval threshold)>Brandon=Charlie=Davidmeaning "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=Davidmeaning "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|
|3rd||Brandon, Charlie, David|
which means the same thing.Rated ballots are often shown as:
Alicia:5 Eileen:4 Brandon:3 Charlie:3 David:3meaning "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>COne can simply write
3: A>B>C 1: D>EWhen 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|
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. This is the reason that, in the two-candidate case, one can either allow a voter to "vote" for one candidate or the other, or allow them to rank their preference between both candidates, without it reducing the voter's expressiveness.
With a rated ballot, again, either the score can be written next to each candidate, or the voter can bubble it in like so:
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).
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.
See Declaration of Election-Method Reform Advocates: Ban Single-Mark Ballots for a voting reform impassioned plea to ban single-mark ballots.
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.
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).
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.
The general idea of rating is that a voter's pairwise preferences are connected i.e. if a voter indicates A is maximally better than B (by giving A the max score and B the min score), then they must indicate B is no better than C.
See Preferential voting. A ranked ballot involves ranking candidates i.e. A>B>C means A is better than B, and B is better than C, with A being implied to be better than C as well. Some ranked ballot implementations allow you to skip rankings i.e. A>skipped ranking>B, and also allow you to rank candidates equally i.e. A>B=C>D=E=F means A is better than everyone else, B and C are equal but better than everyone except A, and D, E, and F are worse than the other ranked candidates, but the voter has no preference between them.
A ranked ballot can be thought of as either imposing transitivity on voters' preferences in every possible runoff (based on pairwise counting), or as asking voters who they would elect if it was up to them, then asking them who'd they elect if that candidate was ineligible to win, etc. Note that a ranked ballot can be reconstructed from a voter's pairwise preferences using a Copeland ranking (i.e. the candidate(s) whom the voter prefers against the most other candidates are their 1st choices, etc.), but that a rated ballot can't be, indicating that ranked ballots collect less information.
The fundamental idea of ranking is generally that voters are treated as having maximal preferences between every pair of candidates they indicate a preference between; this explains why practically all ranked methods pass the majority criterion in the two-candidate case.
A slot or preference-level is a tier in which candidates may be placed, such that they are treated as preferred more than any candidate in a lower tier, less than any candidate in a higher tier, and equally to any candidate in the same level. Approval and choose-one ballots offer two slots (supported or not supported), while rated and ranked ballots tend to offer more. Note that the ballot paper need only allow voters to mark one less slot than the number of slots the voter is meant to be able to mark, if unmarked candidates are assumed to be in the lowest slot. For example, an Approval ballot doesn't usually let voters indicate who they don't approve of, since this is implicit when observing the candidates the voter didn't mark as approved.
A ballot can allow a voter to support candidates that aren't actually named on that ballot by letting them "write in" those candidates' names on the ballot. Generally, voters are only allowed to write in one candidate.
Supporting write-ins vary in difficulty from method to method. Voting methods where only one data value is recorded per candidate, such as FPTP, Approval voting, and Score voting, can easily support write-ins, while voting methods based on pairwise counting (see Category:Pairwise counting-based voting methods) can have a more difficult time. There are various ways to handle this; see Pairwise counting#Notes.
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.Note that ranked, rated, and Approval ballots can be generalized into one ballot type: allowing the voters to express their rated preferences in every head-to-head matchup. With some simplification, this can be visualized as (example for a single voter, with 6 candidates A through F):
A>B=C>DSo this voter expressed a ranked preference, and also expressed, in the head-to-head matchup table, their strength of preference in every head-to-head matchup between each of the candidates in each rank. "1st" here refers to "1st choice", and "20%" here can be read as "20% of a vote" or "20% support", equivalent to 0.2 votes (or a 2 out of 10 on a rated ballot). This can be read as, for example, "1st>3rd" referring to the voter's support for A>D, and "2nd>last" referring to the voter's support for either B or C over all candidates they prefer less than D. This table captures the margin in strength of preference; it is instead possible to capture the strength of preference in a way that captures both margins and "winning votes"-relevant information (i.e. the voter's rated preference for both candidates in the matchup) by, instead of writing 20% for the more-preferred candidate and 0% for the less-preferred candidate, writing, say, 80% and 60% respectively, if that's what the voter's actual preference was. Certain minimum requirements for transitivity are apparent simply from looking at this table; for example, since the voter expressed a 50% difference in support for their 2nd choice>3rd choice, it wouldn't have made sense for them to express less than 50% support for their 1st choice>3rd choice. Another example is that, because they expressed 20% support for 1st>2nd, they must have had at least 20% support for 1st>3rd as well.
Margins-based rated matchups table 1st 2nd 3rd Last 1st --- 20% 60% 75% 2nd 0% --- 50% 60% 3rd 0% 0% --- 40% Last 0% 0% 0% ---
This approach is a generalization of the above 3 ballot types in the sense that if every voter expresses the same margins-based or winning votes-based preference for each candidate in each head-to-head matchup as they would if they were rating them on a scale with all other candidates (i.e. a voter who would give a candidate 80% support on a rated ballot's scale would give that candidate a 30% margin in a head-to-head matchup against a candidate they'd rate a 50% on the same scale), then it reduces to a rated ballot (with the same logic following for an Approval ballot, since an Approval ballot is a restricted form of a rated ballot), and if every voter expresses a maximal preference for their preferred candidate in each matchup, then it reduces to a ranked ballot. See Pairwise counting#Cardinal methods and Order theory#Strength of preference for more information on this ballot type.