Schwartz set
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The Schwartz set is a term used in regard to voting systems. Given a particular election using preferential votes, the Schwartz set is the union of all possible sets of candidates such that for every member set:
- Every candidate inside the set is pairwise unbeatable by any other candidate outside the set, which includes the case of a tie.
- No proper (smaller) subset of the set fulfills the first property.
If there are any weak Condorcet winners in the election, then the Schwartz set consists of exactly all of them.
The Schwartz set is always a subset of the Smith set and almost always equal to it. If they do differ, then it is because one or more certain key pairwise comparisons ended in a tie. 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 weak Condorcet winner and thus the only member of the Schwartz set, while the Smith set on the other hand consists of all the candidates.