Ballot

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; a voter could be allowed to:


 * Choose only one candidate.
 * Choose multiple candidates (either a limited number, or as many as they want).
 * Rank the candidates (1st, 2nd, 3rd, etc.)
 * Rate/score the candidates (this one is a 4 out of 5, that one is a 2 out of 10, etc.)

With all ballot types, it is generally assumed that unmarked/unranked/unrated candidates are considered worse by that voter than all marked candidates, and that the voter has no preference between any of them.

For a given ballot type and a certain number of candidates, there is a finite number of possible ballots that may be cast. The type of ballot significantly alters the amount of expressivity available to voters. The set of potential ballots can be used to define a space of possible elections, which correspond to all possible distinct ways an election may take place. The goal of voting system theory is understand the small subset of this space which is relevant in the real-world, optimizing results for that subset according to some metric.

Binary ballot types
These are ballot types where a voter can only indicate "support or no support" for each candidate.

Choose-one ballot
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, which can be considered a generalization.

Example:"Alicia☑ Brandon☐ Charlie☐ David☐  Eileen☐"These are often also generalized as "choose up to as many candidates as there are seats to be elected" a.k.a plurality-at-large ballots because when there are, say, two winners to be chosen, usually voters are allowed to mark up to two candidates, etc.

Many in the voting reform community consider it imperative to move away from single-mark ballots to ballot types that offer more information.

Approval ballot
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. Usually this is indicated by letting the voter mark the candidates they approve, and assuming that unmarked candidates are disapproved. Approval ballots are used by Approval voting, Explicit approval voting, Proportional approval voting, Combined approval voting, etc.

Example:"Alicia☑ Brandon☐ Charlie☐ David☐  Eileen☑"

More expressive ballot types
These ballot types allow voters to give more information than a simple "yes or no" for each candidate. More formally defined, they offer more pairwise preference information.

Rated ballot
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 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). It is sometimes proposed that voters be able to give negative ratings (indicate disapproval/opposition), as in Evaluative voting.

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.

Comparability of scales
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.

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).

The general idea of rating is that a voter's pairwise preference s 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.

Ranked ballot
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 the candidate they just chose 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. Note that ranked ballots generally only look at the relations between ranks, not the absolute numbers i.e. a voter who ranks one candidate 2nd and another 7th, while not ranking anyone else, is treated as ranking those candidates 1st and 2nd respectively.

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.

Other ballot types
There are several ways to do a cumulative voting ballot.

A generalization of the four main ballot types mentioned above is the rated pairwise preference ballot; it allows voters to submit rated preferences for every head-to-head matchup between the candidates while following certain transitivity requirements.

Generalized notation
Generalized ballot notation is often used in the context of analyzing "election/preference profiles".

It is customary to, instead of repeating the same ballot type multiple times, simply write 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

Specific notations
"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"
 * The information on an Approval ballot can alternatively be seen from an approval threshold, like so:

"1: Alicia>Eileen>Brandon=Charlie=David"meaning "1 voter said that 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 which means the same thing.
 * Ranked ballots are often shown as:

Relationships
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 or ranked ballot where only two slots are allowed (i.e. the only allowed ratings are "disapprove" and "approve" (0 and 1), or the only allowed rankings are 1st choice and last choice).

(The intersection between ballot types indicates relations; specifically, whether the ballot type on the left gives the information given by the ballot type on the top. 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 (with enough scores/gradations) without an approval threshold provides the most information of any ballot type (except possibly Approval ballots, since an intermediate score doesn't indicate whether a voter considers a candidate acceptable/approved), and that a rated ballot with an approval threshold provides the most information of any ballot type.

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. This is because ranking is based on pairwise preference s, which is what all ballot types seek to offer at least partially.

Slot
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 explicitly mark/indicate who they don't approve of, since this is implicit when observing the candidates the voter didn't mark as approved. Another example would be that when doing majority rule in the two-candidate case, one can either allow a voter to "vote" for one candidate or the other (i.e. indicate only their 1st choice), or allow them to rank both candidates, without it reducing the voter's expressiveness.

When there are two or fewer preference levels than the number of candidates, voters are forced to compress their preferences, because of the pigeonhole principle. See Compromising-compression for how this can also be done by strategic voters.

Implementation
This section explains how to actually allow voters to write/provide the information contained in each ballot type i.e. how a ranked ballot might look and be like, a rated ballot, etc.

Ranked ballot
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: 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.

Rated ballot
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).

Write-in option
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.

Preference information
The different ballot types collect different types of preference data. See also Number of supportable candidates in various voting methods:

1st choice information is collected by choose-one and ranked ballots, and is used mostly in FPTP, IRV, and Category:FPTP-based voting methods. It is often associated with the majority criterion.

Approval and scored preferences are used in rated method s. Note that because there is no zero-sum nature to these preferences, it is not always possible to tell, for a set of candidates, which of them a voter supported and to what degree, solely from the vote totals i.e. if A gets 30 approvals and B 20, then there can be anywhere from 30 to 50 voters in the election. Generally, this information is indicated using approval rating s. The other way of doing so, which is showing the candidate's point total as a % of all points received by all candidates, can be less informative, since it doesn't as clearly indicate the % of voters who support each candidate (both measures are equivalent when only looking at 1st choices).

Pairwise preferenc es can be used in Category:Pairwise counting-based voting methods, and are often displayed in a matrix. For further clarity, the matrix can be outfitted to also show the result, margin, and % of votes for each candidate in each Head-to-head matchup.

Ballot features
Several things can be modified or added to various ballot types. For example, any ballot type offering ranked information can be outfitted with an approval threshold to also offer Approval-related information.

Also, it's possible to use checkboxes and other features to allow voters to indicate, for example, how they want their ballot to be processed, or how they want the election to be tabulated. A very wonky example would be if voters not only voted on candidates, but also took a vote on whether to use one of two different voting methods to decide how the winner should be chosen, using those very same ballots. Election officials could then tally up the votes, and then once the voting method to use is established, use the same ballots to tabulate the results of that voting method.

Indicating weak preferences as no preference
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.

Pairwise counting can be done on all four ballot types, though ranked and rated ballots offer the most information for this purpose. Indeed, one metric for evaluating the amount of information a ballot provides is to see how much pairwise information can be gleaned from it.