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#REDIRECT [[Approval cutoff]]<br />
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.
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.
==Sources==
Brams, Steven J. & Fishburn, Peter C. ''Approval Voting''. Cambridge, MA: Birkhäuser, Boston, 1983.
[[Category:Voting theory]]
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