Schwartz set

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The Schwartz set is a term used in regard to voting systems and is named after Thomas Schwartz. 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:

1. Every candidate inside the set is pairwise unbeatable by any other candidate outside the set, which includes the case of a tie.
2. No proper (smaller) subset of the set fulfills the first property.

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 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 only member of the Schwartz set, while the Smith set on the other hand consists of all the candidates.