Resistant set

The resistant set or inner burial set is a set defined for strict rank-order preferences, where electing from the set confers burial resistance. It was devised by Kristofer Munsterhjelm in 2023.

Definition

A sub-election of an election is the resulting election after some candidates have been eliminated and preferences transferred. Exhausted ballots are automatically removed from a sub-election and are thus not counted.

A candidate X disqualifies another candidate Y if: in every sub-election where X and Y are both present, X has more than 1/k of the first preferences, where k is the number of non-eliminated candidates in that sub-election.

The resistant set consists of every candidate who is not disqualified by someone else.

Properties

The set itself is monotone, as raising candidate X can't decrease his first preference count in any sub-election, nor can it raise someone else above the 1/k threshold who wasn't already above it. The disqualification relation is also cycle-free.

Methods that elect from this set automatically pass dominant mutual third candidate resistance. In addition, if a method always elects from the resistant set, then voters who prefer a candidate outside the set to the current winner have no incentive to bury the winner.

As a consequence, if there's only one candidate in the set, then that candidate is completely immune to burial. If that candidate is also an absolute majority Condorcet winner, the candidate is immune to both burial and compromising.

However, methods of the type Resistant//M fail monotonicity if M passes the majority criterion. The resistant set is not clone-proof as a set: cloning a candidate in it can introduce another candidate to the set by undoing the cloned candidate's disqualification of another candidate.

The resistant set always contains at least one member of the Smith set. Thus, the Condorcet winner, when one exists, is always a member of the resistant set.

Although Yee diagrams and numerical simulations suggest that there exist methods of the form Resistant,M that are monotone, this has not been proven.

Complying methods

The following methods elect from the resistant set:

• Benham's method
• Improved First Past the Post
• Instant-runoff voting
• Smith//IRV

If a method M elects from the resistant set, so do Smith,M and Smith//M.

Failing methods

If a method only uses first preference counts, the Condorcet matrix, and/or the positional matrix to determine the winner, that method must sometimes fail to elect from the resistant set. The result rules out the most common ways of achieving summability if a method is to elect from the resistant set. Whether summability is compatible with electing from the resistant set is not known.

Generalizations

The resistant set may be generalized to elections that allow equal-rank and truncation, but in this case, two properties must be true:

• The total number of first preferences in a sub-election does not include voters who rank every non-eliminated candidate equal.
• No voter has more than unit voting weight in total when counting first preferences. In particular A=B=C>D can't be counted as giving one vote to all of A, B, and C.

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