Definitions

# 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

1. Every candidate inside the set S is pairwise unbeaten by any candidate outside S; and
2. 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.

## Properties

• 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 only a single candidate, that candidate is 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.

## Smith set comparison

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, given:

• 3 voters preferring candidate A to B to C
• 1 voter preferring candidate B to C to A
• 1 voter preferring candidate C to A to B
• 1 voter preferring 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.

## Algorithms

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).

## Complying methods

The Schulze method always chooses from the Schwartz set.

## References

• Ward, Benjamin (1961). "Majority Rule and Allocation". Journal of Conflict Resolution 5 379–389. In an analysis of serial decision making based on majority rule, describes the Smith set and the Schwartz set, but apparently fails to recognize that the Schwartz set can have multiple components.
• Schwartz, Thomas (1970). "On the Possibility of Rational Policy Evaluation". Theory and Decision 1 89–106. Introduces the notion of the Schwartz set at the end of the paper as a possible alternative to maximization, in the presence of cyclic preferences, as a standard of rational choice.
• Schwartz, Thomas (1972). "Rationality and the Myth of the Maximum". Noûs 6 97–117. Gives an axiomatic characterization and justification of the Schwartz set as a possible standard for optimal, rational collective choice.
• Deb, Rajat (1977). "On Schwart's Rule". Journal of Economic Theory 16 103–110. Proves that the Schwartz set is the set of undominated elements of the transitive closure of the pairwise preference relation.
• Schwartz, Thomas (1986). The Logic of Collective Choice. New York: Columbia University Press. Discusses the Smith set (named GETCHA) and the Schwartz set (named GOTCHA) as possible standards for optimal, rational collective choice.