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