[EM] A look at the low-burial Condorcet methods

[Kevin Venzke]

Hello,

I front-load this with a few method definitions:

FPBO stands roughly for "First-preference winner vs. best opposition." You elect
the pairwise winner between the first-preference winner (FPW) and the "best
opposition" candidate (BO). The BO is the candidate who received the highest vote
count (compared to any other candidate) in their pairwise contest with the FPW.

With Condorcet//FPBO, you just check for a CW first. Smith or Schwartz variants are
also possible, but don't make a lot of difference in my simulations.

Condorcet//KOTH (King of the Hill) checks for a CW first, and if there is none,
performs KOTH. That is, identify the candidate with the most first preferences who
is involved in a majority-strength pairwise contest with the FPW. There may be no
such candidate, in which case elect the FPW. Otherwise, elect the pairwise winner
between the FPW and the other candidate.

For 4+ candidates, I use my own expansions for BPW and SV, which haven't been
defined very well for that scenario. For BPW, chain climbing is used, processing
the candidates in descending order of first preference count. For SV, the ratio
defining each candidate's score uses the maximum value possible for each term. (For
example, for the first preference count of the candidate pairwise beaten, use the
highest such count when multiple candidates were beaten.) Also, a candidate with no
pairwise wins is disqualified from being elected.

"dcCopeland" means "decloned Copeland," a Forest suggestion where wins/losses are
added/subtracted (respectively) as a score weighted by the first-preference count
of the candidate beaten or "beaten by."

***

I have had some respect for Eivind Stensholt's BPW and SV methods because I
perceived them to be basically the best in terms of Condorcet methods that minimize
burial incentive. ("Burial incentive" just means a given scenario will reward
burial, not that a voter would realistically know that they could do this.)

This is true (according to my simulations) in the 3-candidate case: BPW, SV, and
also C//FPBO are top tier. C//IRV, C//KOTH, and dcCopeland are second tier.

For DMTBR burial specifically, these are all about equally good (and the best).
fpA-fpC joins this list. C//KOTH is a bit worse than the others.

However, with 4 candidates I see some change. For burial in general, C//IRV and
Benham join the top tier with C//FPBO, while BPW, SV, and C//KOTH become a much
worse second tier. [Benham was not considered with 3 candidates as it is not
distinct from C//IRV there.]

4-candidate DMTBR sees BPW, C//FPBO, and dcCopeland exit the top tier. C//IRV,
Benham, fpA-max(fpC), and SV remain there.

As you can see, none of the "lowest burial" methods are using implicit approval. I
find that a little surprising, since implicit approval can create an obvious risk
for those adding falsified preferences. Perhaps in real elections such rules would
be effective psychologically in a way I can't see here. The "best" (i.e. least
burial) method of this type that I have is probably TACC, especially when DMTBR is
considered.

What interested me most from looking at this was the C//FPBO method. If I plot it,
it's quite close to BPW. Actually with 3 candidates it's hard to design a scenario
where the two give different results.

With 4 candidates the properties are still pretty similar. BPW, SV, and C//FPBO all
have far worse monotonicity than other methods. (dcCopeland also being pretty bad.)

For compromise incentive, BPW and then C//FPBO are the best within this selection
of methods, however all these methods are mediocre in this area as a rule.

For truncation incentive, BPW and SV are about the best here, with C//FPBO being
somewhat worse for some reason.

As mentioned, there was some difference in the burial results between C//FPBO and
BPW. In the 4-candidate case C//FPBO was the single best Condorcet method at
overall burial, being a bit better than C//IRV. I thought that was interesting so I
wrote this post.

Kevin