Approval cutoff

The approval cutoff, approval threshold or neutral preference level is an option that may be used with a ranked ballot. It allows the collection of the information necessary to create an Approval ballot. Note that the concept of an approval threshold can also be used in a theoretical sense i.e. if a voter is deciding who to vote for in Approval voting, they may do so by thinking about their ranked preferences and mentally deciding who their least-preferred candidate is that they consider acceptable.

Specifying a neutral preference level indicates that candidates above a certain rank have positive preference, while candidates below that rank have negative preference. Positive preference is equivalent to an approval vote, while negative preference receives no approval vote.

Sometimes candidates at the approval threshold are also considered approved, instead of being treated neutrally. There may be a loose terminological tendency to use "approval cutoff" to indicate the neutral preference level option, and "approval threshold" to indicate this option.

When a method uses a ranked ballot with approval cutoff, the usual procedure is that the neutral preference level is set below all ranked candidates, and above unranked candidates.

A voter can set the approval cutoff above the default level by ranking an extra "Neutral Preference" candidate.

A voter may decide to set the neutral preference level between some ranked candidates in order to indicate that

• among the non-approved candidates, some are more preferred than others.
• the ranking of candidates above and below the line is not as important as having a candidate above the line defeat all candidates below the line.

It is also possible to consider fractional approval thresholds i.e. one threshold which gives candidates 100% approval, a second threshold that gives them 90%, etc. Thesr thresholds would be consecutive, so for example, a voter giving their 2nd choice 100% approval but their 4th choice 90% approval would also give their 3rd choice 90% approval. This would allow the collection of rated ballot information.

The approval threshold is usually indicated with a |, which can be visually highlighted by putting space after it. Example:

30: A>B| >C>D>E

This means 30 voters approve both A and B, but not C, D, and E.

Another way to implement the approval threshold in practice would be to let voters mark approval of the candidates directly (similar to Approval voting) alongside their rankings, and then for whichever of the approved candidates the voter ranked lowest, all candidates ranked equal to or higher than that candidate are also considered approved.

Preference-approval

A preference-approval is a preference order that combines preference with approval. It can contain either weak or strong preferences. A complete preference-approval is a total preference order.

Rationality Restrictions

Here are some rationality restrictions on preference-approvals. Suppose there exists two alternatives, x and y:

1) If a given voter prefers x over y, and approves y, then she must approve x.

2) If a given voter prefers x over y, and does not approve x, then she must not approve y.

3) If a given voter is indifferent between x and y, and approves x, then she must approve y.

4) If a given voter is indifferent between x and y, and does not approve x, then she must not approve y.

Note that these same restrictions can be applied to rated ballots or fractional/weighted approval thresholds i.e. a voter who prefers one candidate more than or equally to another must give the first candidate at least as high a score or degree of support as the second. The KP transform can be used to model this in terms of Approval ballots.

He are some expressions of preference-approvals and translations into natural language:

|x>y: "The voter prefers x over y, but approves neither." |x=y: "The voter is indifferent between x and y, but approves neither."

x|y: "The voter prefers x over y, but only approves x."

x>y|: "The voter prefers x over y, but approves both."

Steven Brams and Peter Fishburn used preference-approvals in their book "Approval Voting" in 1983, though it probably was used before then.

There are 2, 8, 44, 308, ... different preference-approvals for 1, 2, 3, 4, ... candidates (Sloan's A005649).

Total preference order

A total preference order is a complete preference-approval. In other words, it is a preference-approval that contains all alternatives competing in a given election.

Notes

Sometimes an approval threshold is also indicated using an ">=", for example, A>B| can be written as A>=B.

The logic of the approval threshold is that if a voter approves a candidate, and they prefer some other candidate(s) over the candidate they approved, then it's very likely that voter would also approve of those more-preferred candidates if voting with an Approval ballot, since otherwise they'd be giving more support to a candidate they prefer less than the candidates they'd be disapproving.

An approval threshold can be used in the context of rated methods as well. This can be useful for cardinal PR, since it could be possible to allow, for example, a Green Party voter to approve both Green Party candidates and Democratic candidates, using the approvals to ensure one of those preferred candidates wins, and then score the candidates in such a way as to maximize the odds that of the preferred candidates, one of the Green Party candidates wins.

It is generally assumed that a voter who doesn't indicate an approval threshold at least approves all of their 1st choice candidates. So if they had voted A=B=C>D>E, then A, B, and C would be considered approved.

See Smith//Approval for an example of the use of the approval threshold.

Sources

Brams, Steven J. & Fishburn, Peter C. Approval Voting. Cambridge, MA: Birkhäuser, Boston, 1983.