Schwartz set

In elections, the Schwartz set is the union of all minimum undominierten quantities. A minimal amount undominierte is a non-empty set S of candidates, which applies to:

A Schwartz Set provides a way for an optimal election results. Choice processes in which a candidate always wins from the Schwartz Set, meet the Schwartz criterion. The set is named after the political scientist Thomas Schwartz.

Properties

  • The Schwartz set is never empty - there is always a minimum amount undominierte
  • Two different minimum undominierte sets are disjoint.
  • If there is a Condorcet winner, he is the only member of the Schwartz set. If the Schwartz Set contains only one candidate, there is at least a weak Condorcet winner.
  • Contains a minimal amount undominierte only one candidate, he is a weak Condorcet winner. Contains a minimum quantity undominierte several candidates, they are all in a Beatpath cycle together, a top cycle.
  • Two candidates from different minimal amounts beat undominierten not ( undecided ).

Compare with the Smith quantity

The Schwartz set is always a subset of the Smith set. The Smith quantity is only greater when a candidate in the Schwartz Set draw stacks up pairwise comparison with a candidate from outside the Schwartz set. An example:

  • 3 voters prefer candidates A before B before C
  • 1 voters preferred candidate B before C before A
  • 1 voters preferred candidate C before A before B
  • 1 voters preferred candidate C before B before A

A beats B, B beats C, and A is drawn with C in pairwise comparison. A is thus the only member of the Schwartz Set, while all applicants are element of the Smith set.

Algorithms

The Schwartz Set can be calculated with the algorithm of Floyd and Warshall the complexity, or with a version of the algorithm of Kosaraju same complexity.

Schwartz criterion

A selection mode satisfies the Schwartz criterion, if it always selects an item from the respective Schwartz Set. This is the case for example for the Schulze method.

Credentials

  • Benjamin Ward: Majority Rule and Allocation. In: Journal of Conflict Resolution. 5, No. 4, 1961, pp. 379-389. doi: 10.1177/002200276100500405. In an analysis of serial decision-making based on majority rule, describes the set of Smith and Schwartz set, but do not seem to realize that the Schwartz set can have several components.
  • Thomas Schwartz: On the Possibility of Rational Policy Evaluation. In: Theory and Decision. 1, 1970, pp. 89-106. doi: 10.1007/BF00132454. Introduces the concept of the Schwartz set at the end of the paper as a possible alternative to Maximization, in the presence of cyclic settings as default rational choice.
  • Thomas Schwartz: Rationality and the Myth of the maximum. In: Nous, Vol 6, No. 2 (ed.): Nous. 6, No. 2, 1972, pp. 97-117. doi: 10.2307/2216143. Does an axiomatic characterization and justification of the Schwartz set as a possible standard for optimal, rational collective choice.
  • Deb, Rajat: On Schwart 's Rule. In: Journal of Economic Theory. 16, 1977, p 103-110. doi: 10.1016/0022-0531 (77 ) 90125-9. Prove that the Schwartz set is the set of undominated elements of the transitive closure of pairwise relationship.
  • Thomas Schwartz: The Logic of Collective Choice. Columbia University Press, New York, 1986, ISBN 0-231-05896-9. Discusses the Smith set (named GETCHA ) and the Schwartz set (named GOTCHA ) as standards for optimal, rational collective choice.
Disjoint sets Floyd–Warshall algorithm Digital Object Identifier
717350
de