Schwartz set

The Schwartz set is the union of all of the smallest groups of candidates such that any candidate in any group is preferred by at least as many voters as any candidate not in that group (when looking at pairwise matchups).



The Schwartz set represents one approach to identifying a subset of candidates from which an election method should choose a winner.

The Smith set is a closely related superset that can only differ when there are pairwise ties.

Beatpath Order
The Schwartz set is the union of all sets that are Schwartz set components.

A Schwartz set component is any non-empty set S of candidates, such that:


 * 1) Every candidate in S pair-wise beats or ties any other candidate outside of S, and
 * 2) No non-empty proper (smaller) subset of S fulfills the first property

The Schwartz set is defined for any election in which every candidate is evaluated against every other candidate in pair-wise contests that each result either in a tie or with one of the candidates beating the other.

The Schwartz set is the union of all of the cycle equivalence classes that are maximal in the  beatpath order. A set is a Schwartz set component if and only if it is a cycle equivalence class that is maximal in the beatpath order.

This characterization of the Schwartz set follows from two characteristics of any set S that has the first property of a Schwartz set component:
 * No beatpath starts outside and finishes inside S.
 * If X is in S, then the set S \ {Y : Y is in S and there is no beatpath from Y to X} also has the first property of a Schwartz set component.

Smith Set
The Schwartz set is a subset of the Smith set. The Schwartz set is equal to the Smith set if and only if every candidate in the Schwartz set pair-wise beats every candidate outside the Schwartz set, that is, no candidate in Schwartz set pair-wise ties with any candidate outside of the Schwartz set.

The Smith set can be constructed from the Schwartz set by repeatedly adding two types of candidates until no more such candidates exist outside the set:
 * candidates that have pair-wise ties with candidates in the set,
 * candidates that start a beatpath that ends in the set.

Note that candidates of the second type can only exist after candidates of the first type have be added.

Other Characteristics
Other characteristics of the Schwartz set include:


 * If a Condorcet winner exists, it is the one and only candidate in the Schwartz set.


 * All weak Condorcet winners are in the Schwartz set. The weak Condorcet winners are the candidates in the Schwartz set that are not in a cycle, so they are each in their own Schwartz set component.


 * If the Schwartz set contains only one candidate, it is the weak Condorcet winner. If that candidate has no pair-wise ties, it is the Condorcet winner.


 * The Schwartz set is never empty. At least one Schwartz set component exists.  (Assuming the number of candidates is finite.)


 * There is no beatpath that starts outside and ends inside a Schwartz set component.


 * Schwartz set components are disjoint. No candidate is in more than one Schwartz set component.


 * Any two candidates from different Schwartz set components are pair-wise tied with each other.


 * Any two distinct candidates X and Y from the same Schwartz set component are in a cycle with each other; there is a beatpath from X to Y and there is a beatpath from Y to X.


 * For any candidate Y not in the Schwartz set, there is at least one candidate X in the Schwartz set that has a beatpath from X to Y. However, it is possible that there exists a candidate outside the Schwartz set that is not beaten by any candidate in the Schwartz set, but instead is pair-wise tied with every candidate in the Schwartz set.


 * A Schwartz set component may contain 1, 3, or more candidates, but never exactly two candidates.


 * If there are no pair-wise ties, as is typical in larger-scale elections, the Schwartz set will consist of either:
 * a single candidate that is the Condorcet winner, or
 * a set of three or more candidates that are all in cycles with each other.

Examples
Examples with 3 candidates are here.

An example with 12 candidates is here.

Algorithms
The Schwartz set can be calculated using versions of either Kosaraju's algorithm or the Floyd-Warshall algorithm. Examples of both algorithms applied to calculating the Smith set and the Schwartz set are available here.

In practice, it is possible to identify the Smith set and then identify all Schwartz set components within the Smith set. If the Smith set has no pairwise ties between its members, then it will be the Schwartz set.