[EM] Single-candidate DMTBR idea

[Kevin Venzke]

Hi Kristofer,

> > Tricky, to reduce a scenario to this state without breaking mono-raise.
> 
> We could of course just stitch something together, e.g. if there're
> exactly two candidates above 1/3 fpp, elect the candidate who pairwise
> beats the other, otherwise just elect the Plurality winner. This should
> be monotone because raising A doesn't harm A when A has >1/3 fpp, and
> raising B to >1/3 fpp gives him a second chance against A (if B beats A
> pairwise).

Yes, I realized after posting that that would probably work, and was racing to
prove it. So you can have any number of candidates, but you can only have a
runoff if two candidates each have 1/3+ FPs.

This means we have a 4+ candidate method that satisfies Mono-raise and both LNH,
and is not just FPP. So this should be an answer for Craig Carey here. I wonder
if he was aware of it, or what he would have thought about it.

> Actually, now that I think about it, I think the rule I provided
> combined with Condorcet *is* fpA-fpC (disregarding ties for now).

I think so, yes.

I tried to compare C//(this IFPP expansion) with fpA-max(fpC). Note that the way
I define the latter doesn't block a Condorcet loser from winning. But it seems
like this method is better than using the IFPP expansion, as the overall burial
incentive is less.

Both methods appear to satisfy Mono-raise, Plurality, and DMTCBR.

> It's not very elegant: the seams are very obvious. But perhaps elegance
> can come later... or perhaps it will be induced by turning DMT candidate
> BR into full DMTBR.

Incidentally, do you know if there is a DMTBR (not just DMTCBR) satisfaction
proof known for any method? Some form of C//IRV maybe, but it doesn't seem like
an easy thing to show.

> >> Does it also apply to the generalization where you just take the two
> >> candidates with the most first preferences? I'm not sure.
> >
> > In the three-candidate case, electing the pairwise winner between the top two
> > candidates is basically IRV.
> >
> > Without the 1/3 limit it could happen that the FPW gets more votes and changes
> > who the second place candidate is. He might not beat the new one.
> >
> > This is interesting though. The "obvious" way to expand IFPP to many candidates
> > is to eliminate candidates with a below-average vote count. But it seems like
> > the 1/3 rule was the important thing, as it's what enforces that always either
> > one or two candidates are eligible to win, and these candidates can't be harmed
> > by getting more votes.
> 
> Yes, that also explains where the "third" in dominant mutual third comes
> from. Like with Droop proportionatliy, it's the smallest quota so that
> only two candidates can exceed it. And that would also suggest that (at
> least by this approach), third is the best we can do; there's no, say,
> dominant mutual quarter for Condorcet.

Yes. I've been sitting here trying to get a 25% rule to "work" (defining that
very generously), but there is a lot of trouble regulating who is allowed to
benefit from various vote changes when three candidates are eligible.

Kevin