Schwartz crowd
In elections, the Schwartz set is the union of all minimal undominated sets . A minimal undominated set is a non-empty set S of applicants for which the following applies:
- Each contestant within set S is pairwise unbeaten by each contestant outside S (i.e., an undominated set).
- No non-empty proper subset of S satisfies the first property (i.e. minimally).
A Schwartz set offers an opportunity for an optimal election result. Electoral procedures in which one candidate from the Schwartz crowd always wins, meet the Schwartz criterion . The crowd is named after the political scientist Thomas Schwartz.
properties
- the Schwartz crowd is never empty - there is always a minimal undominated crowd
- two different minimal undominated sets are disjoint .
- if there is a Condorcet winner , he is the only member of the Schwartz crowd. If the Schwartz crowd includes only one applicant, there will be at least one weak Condorcet winner.
- if a minimum undominated number contains only one applicant, he is a poor Condorcet winner. If a minimal undominated number contains several applicants, they are all in a beatpath cycle with one another, a top cycle .
- two candidates from different minimal undominated sets do not beat each other (tie).
Compare with the Smith crowd
The Schwartz set is always a subset of the Smith set. The Smith set is only greater if an applicant in the Schwartz set comes off in a pairwise comparison with an applicant outside the Schwartz set. An example:
- 3 voters prefer applicants A over B over C.
- 1 voter prefers applicant B over C over A.
- 1 voter prefers applicant C over A over B.
- 1 voter prefers applicant C over B over A.
A beats B, B beats C and A is a draw with C in a pairwise comparison. A is therefore the only member of the Schwartz set, while all applicants are members of the Smith set.
Algorithms
The Schwartz set can be calculated using Floyd and Warshall's algorithm of complexity , or a version of Kosaraju's algorithm of the same complexity.
Schwartz criterion
A voting mode fulfills the Schwartz criterion, provided that it always selects an element from the respective Schwartz set. This is the case, for example, with 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, the Smith Theorem and Schwartz describe it, but apparently fail to recognize that the Schwartz set can have multiple 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 Schwartz-Set at the end of the paper as a possible alternative to maxiaturization, in the presence of cyclical settings as a standard rational choice.
- Thomas Schwartz: Rationality and the Myth of the Maximum . In: Noûs, Vol. 6, No. 2 (Ed.): Noûs . 6, No. 2, 1972, pp. 97-117. JSTOR 2216143 . doi : 10.2307 / 2216143 . Gives 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, pp. 103-110. doi : 10.1016 / 0022-0531 (77) 90125-9 . Proves that Schwartz-Set the set of undominated elements is the transitive conclusion of the pairwise favoritism relationship.
- Thomas Schwartz: The Logic of Collective Choice . Columbia University Press, New York 1986, ISBN 0-231-05896-9 . Explains the Smith-Set (named GETCHA) and Schwartz-Set (named GOCHA) as standards for optimal, rational collective choice.