# 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 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.[1]

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".

The immune set is related to the set of winners of the Split Cycle method: replacing the term "stronger" in the definition of the immune set with "at least as strong" produces the set of Split Cycle winners.[2]

Due to the relation to Smith//Minimax, any method that elects from the immune set fails dominant mutual third candidate burial resistance.

## References

1. Heitzig, Jobst (2004-05-01). "Examples with 4 options for immune methods". Election-methods mailing list archives.
2. Holliday, Wesley H.; Pacuit, Eric (2020-04-05). "Split Cycle: A New Condorcet Consistent Voting Method Independent of Clones and Immune to Spoilers". arXiv:2004.02350v8 [cs.GT].