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).