# 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.[1]

## 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.[2]

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.[2] 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.[3] 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.[4]

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:[5]

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.[6] 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.