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Burial resistance thoughts

KM
Kristofer Munsterhjelm
Thu, Jul 20, 2023 7:42 PM

I've been thinking a bit about burial resistance, and here's a pattern
that might inform the design of burial-resistant methods. It should at
least be sufficient in many cases (but caveats about Condorcet below).

To determine whether A or B should be elected, the only information we
can use are the first preferences after the elimination of candidates
who are not A or B.

Pairwise information like A>B is a special case where every candidate
but A and B have been eliminated; so A>B could be used to determine if A
should be elected rather than B, but not if A should be elected rather
than C.

So, for instance, Plurality judges whether A should be elected rather
than B based on the first preferences of the candidates with nobody
eliminated. It has no burial incentive.

As for IRV: if A is elected rather than B, either B was eliminated
before A -- and that decision is made by the first preferences
restricted to a candidate set containing both A and B -- or A beats B
pairwise in the final round, which is only based on A>B.

The only non-Condorcet method I can think of that doesn't meet this is
the max A>B method, where A's score is A>B with B chosen to maximize the
score. Suppose A's maximum pairwise victory is A>C and B's maximum
victory is B>D. Then the edge A has got over B is A>C - B>D, which
involves two contests that don't include both A and B. Yet for some
reason it still works.

Within the cycle region, it kinda works? Consider fpA-fpC in an ABCA
cycle. Then since A and B's scores depend on first preferences counted
with nobody eliminated, A's edge over B, f(A) - f(B) also depend on
first preferences in an election in a set containing A and B.

But Smith,Plurality has plenty of burial incentive, while the heuristic
suggests otherwise.

I think that is because Plurality doesn't pass DMT, so that stepping
from the CW region to the cycle region causes a discontinuity that can
be used for burial. (More strictly speaking "A is a CW, B is not, so A
should be elected, not B" depends on A>C thus breaking the pattern.)

I'm not sure why DMT/DMTCBR in particular is the way to get burial
resistance with Condorcet, but I suspect it's got to do with the
following reasoning:

The reason we can't just use "elect A instead of B if A beats B
pairwise" is because of a Condorcet cycle. But consider candidates who
have more than 1/3 first preferences. There can only be two of these.
Therefore, for a relation between candidates with more than 1/3 of first
preferences, there can't be a cycle. And thus we can safely use pairwise
relations between them.

But DMTCBR alone isn't robust. I just implemented the "IFPP-like
generalization" of fpA-fpC mentioned at
https://electowiki.org/wiki/FpA-fpC. It's okay with three candidates,
then its burial resistance collapses with four or more. (Of course, this
method doesn't pass plain DMT with any more than three candidates.)

So some kind of stronger criterion is needed, but I'm not sure what
variant of DMTBR will do the job. Two possible guesses for burial
resistance under Condorcet are DMT + DMTCBR and Electowiki's DMTBR.

Perhaps I'll get more information when I've implemented the other
fpA-fpC generalizations. Until then, the heuristic could be useful for
trying to create burial-resistant methods.

-km

I've been thinking a bit about burial resistance, and here's a pattern that might inform the design of burial-resistant methods. It should at least be sufficient in many cases (but caveats about Condorcet below). To determine whether A or B should be elected, the only information we can use are the first preferences after the elimination of candidates who are not A or B. Pairwise information like A>B is a special case where every candidate but A and B have been eliminated; so A>B could be used to determine if A should be elected rather than B, but not if A should be elected rather than C. So, for instance, Plurality judges whether A should be elected rather than B based on the first preferences of the candidates with nobody eliminated. It has no burial incentive. As for IRV: if A is elected rather than B, either B was eliminated before A -- and that decision is made by the first preferences restricted to a candidate set containing both A and B -- or A beats B pairwise in the final round, which is only based on A>B. The only non-Condorcet method I can think of that doesn't meet this is the max A>B method, where A's score is A>B with B chosen to maximize the score. Suppose A's maximum pairwise victory is A>C and B's maximum victory is B>D. Then the edge A has got over B is A>C - B>D, which involves two contests that don't include both A and B. Yet for some reason it still works. Within the cycle region, it *kinda* works? Consider fpA-fpC in an ABCA cycle. Then since A and B's scores depend on first preferences counted with nobody eliminated, A's edge over B, f(A) - f(B) also depend on first preferences in an election in a set containing A and B. But Smith,Plurality has plenty of burial incentive, while the heuristic suggests otherwise. I think that is because Plurality doesn't pass DMT, so that stepping from the CW region to the cycle region causes a discontinuity that can be used for burial. (More strictly speaking "A is a CW, B is not, so A should be elected, not B" depends on A>C thus breaking the pattern.) I'm not sure why DMT/DMTCBR in particular is the way to get burial resistance with Condorcet, but I *suspect* it's got to do with the following reasoning: The reason we can't just use "elect A instead of B if A beats B pairwise" is because of a Condorcet cycle. But consider candidates who have more than 1/3 first preferences. There can only be two of these. Therefore, for a relation between candidates with more than 1/3 of first preferences, there can't be a cycle. And thus we can safely use pairwise relations between them. But DMTCBR alone isn't robust. I just implemented the "IFPP-like generalization" of fpA-fpC mentioned at https://electowiki.org/wiki/FpA-fpC. It's okay with three candidates, then its burial resistance collapses with four or more. (Of course, this method doesn't pass plain DMT with any more than three candidates.) So some kind of stronger criterion is needed, but I'm not sure what variant of DMTBR will do the job. Two possible guesses for burial resistance under Condorcet are DMT + DMTCBR and Electowiki's DMTBR. Perhaps I'll get more information when I've implemented the other fpA-fpC generalizations. Until then, the heuristic could be useful for trying to create burial-resistant methods. -km