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#REDIRECT [[Approval cutoff#Preference-approval]]<br />
A [[preference-approval]] is a [[preference order]] that combines preference with approval. It can contain either weak or strong preferences.
[[Category:Voting theory]]
==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 y, then she must not approve x.
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.
Because of these rationality restrictions, some preference-approvals can be expressed as follows for alternatives x and y:
|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 ''Italic text''Approval Voting''Italic text'' in 1983, though it probably was used before then.
==Sources==
Brams, Steven J. & Fishburn, Peter C. ''Approval Voting''. Cambridge, MA: Birkhäuser, Boston, 1983.
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