[EM] Burial resistance thoughts

[Kristofer Munsterhjelm]

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