Approval cutoff: Difference between revisions

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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 approve y. {{Clarify|reason=This seems like it should say "...then she must '''not''' approve y."?|date=April 2020}}

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 [http://www.research.att.com/~njas/sequences/A005649 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 ==

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.

The [[preference-approval]] concept is highly related.

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.

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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.

[[Category:Ballot type]]

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