Schwartz set

In voting systems, the Schwartz set is the union of all Schwartz set components. A Schwartz set component is any non-empty set S of candidates such that Every candidate inside the set S is pairwise unbeaten by every candidate outside S; and No non-empty proper subset of S fulfills the first property. A set of candidates that meets the first requirement is also known as an undominated set. The Schwartz set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Schwartz set pass the Schwartz criterion. The Schwartz set is named for political scientist Thomas Schwartz. The Schwartz set is always non-empty—there is always at least one Schwartz set component. Any two distinct Schwartz set components are disjoint. If there is a Condorcet winner, it is the only member of the Schwartz set. If there is only one member in the Schwartz set, it is at least a weak Condorcet winner. If a Schwartz set component contains multiple candidates, they are all in a beatpath cycle with each other, a top cycle. Any two candidates that are in different Schwartz set components are pairwise tied with each other. The Schwartz set is closely related to and is always a subset of the Smith set. The Smith set is larger if and only if a candidate in the Schwartz set has a pairwise tie with a candidate that is not in the Schwartz set. For example, if 3 voters prefer candidate A to B to C, 1 voter prefers candidate B to C to A, 1 voter prefers candidate C to A to B, 1 voter prefers candidate C to B to A, then we have A pairwise beating B, B pairwise beating C, and A tying with C in their pairwise comparison, making A the only member of the Schwartz set, while the Smith set on the other hand consists of all the candidates. The Schwartz set can be calculated with the Floyd–Warshall algorithm in time Θ(n3) or with a version of Kosaraju's algorithm in time Θ(n2). The Schulze method always chooses a winner from the Schwartz set.