Immune set

Immune set: The set of candidates such that any pairwise defeat against a candidate within the set is countered by a string of stronger (or equally strong) pairwise defeats leading back to the defeating candidate.

There is an immune set definition for various possible defeat strength definition, e.g. the WV-defined immune set, the margins-defined immune set, the CWP-defined immune set. It is based on the concept of beatpath strength.

If there is a Condorcet winner, he or she is the only member of the immune set. In a three candidate cycle with unequal defeat strengths, the immune set has one member, i.e. the candidate with the weakest defeat against him or her. When defeats in a cycle have equal strength, or when the cycle is between more than three candidates, the immune set can have multiple members.

The immune set was proposed by Jobst Heitzig.

The immune set is a subset of the Smith set, because the candidates in the Smith set have no defeats against them. When there is a 3-candidate Smith set with no pairwise ties between them, the Smith//Minimax winner is the only candidate in the immune set. Because of this, the immune set has a strong connection to defeat-dropping Condorcet methods.

Notes

The immune set is a subset of the CDTT set, because the CDTT set can be phrased as "the set containing each candidate A who has a majority-strength beatpath to every other candidate B who has a majority-strength beatpath to A", and any candidate in the immune set who has a majority-strength loss to someone not in the set by definition has a majority-strength (of equal strength or greater) beatpath back to that other candidate.

The immune set is useful in situations where there is a candidate who is preferred by a plurality or majority of voters, but some in that plurality prefer someone else over that candidate, with nobody outside the plurality having a preference between the plurality's candidates:

26 A>B

25 B

49 C

10 D

B only pairwise loses to A, but has a stronger beatpath to A: B beats C 51 to 49, with C beating A 49 to 26. So B is the only member of the immune set.

The immune set may be useful for Condorcet PR when generalizing the above example for quotas that are "semi-solid coalitions".