Uncovered set

The uncovered set is defined for a set of rank-order preferences. Usually, the uncovered set is defined only for situations without pairwise ties. When there are no pairwise ties, then the uncovered set is identical to the set called Fishburn winners:

Select the candidate or candidates that are not Fishburn losers. A candidate i is a Fishburn loser if there is some other candidate j such that every candidate that pairwise beats j also pairwise beats i and there is at least one candidate that pairwise beats i but does not pairwise beat j.

When there are pairwise ties, a likely equivalent definition is:

In voting systems, the Landau set (or uncovered set, or Fishburn set) is the set of candidates x such that for every other candidate z, there is some candidate y (possibly the same as x or z) such that y is not preferred to x and z is not preferred to y.

The uncovered set is a nonempty subset of the Smith set. The reason is that every candidate in the Smith set is preferred to every candidate not in the Smith set, therefore each candidate in the Smith set can be considered a candidate x and be their own candidate y; since a candidate can't be preferred to themselves (y is not preferred to x), and since candidates in the Smith set being preferred to every candidate not in the Smith set implies that candidates not in the Smith set are not preferred to candidates in the Smith set (z is not preferred to y), the uncovered set must be a subset of the Smith set.

Independence of covered alternatives says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the uncovered set. Independence of covered alternatives implies Smith and thus Condorcet. If a method is independent of covered alternatives, then the method fails monotonicity if perfect ties can always be broken in favor of a choice W by using ballots ranking W first.