[EM] Inclusion/Exclusion Counts

[Forest Simmons]

That is a great insight!

My most recent message in this thread adds a term to the defeat strength
that achieves reverse symmetry, but it comes at the cost of increasing
burial incentive.

The term is Bad(B) defined as ...

Sum bottom(Z) | B does not defeat Z

in the context of defeat strength, where B is the pairwise loser of the
defeat in question. Bottom(Z) is the percentage of bottom ballot positions
occupied by Z.

So if B is the Condorcet Loser, then Bad(B) is 100%.

Obviously if this term is given equal weight with its reverse symmetry
counterpart, it will contribute an appreciable burial incentive.

The reverse symmetry counterpart is Good(A) defined by

Sum Top(X) | X does not defeat A

in the context of defeat strength of a defeat where A is the pairwise
winner.
Top(X) is the percentage of Too ballot positions occupied by X.

So if A is the CW, then Good(A) is 100percent.

So jt's probably not a good idea to use Bad(B) to create reverse symmetry.
However, perhaps it could be given non-symmetrical,  infinitesimal weight
for tie breaking purposes only.

Also, perhaps their is a milder version of Good(A), whose reverse symmetry
counterpart Bad(B), would have a tolerable burial incentive.

-Forest

On Mon, Oct 24, 2022, 2:43 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 10/23/22 04:55, Forest Simmons wrote:
>
> > Critics have always maintained that this idea shows a lack of awareness
> > of clone dependence. But that judgment assumes that just because there
> > is a bad way of using those ballots, there can be no good way.
>
> Here's a thought that occurred to me, that would explain why the
> (seemingly out of nowhere) implication that we can't have both reversal
> symmetry and DMTCBR.
>
> First preferences are unaffected by burial, and last preferences are
> unaffected by compromising. Suppose we had a method that were DMTCBR and
> reversal symmetric. Then we could freely translate between a method
> that's very strong against burial and very strong against compromise by
> just reversing the ballots, since the reversed ballots' first
> preferences would be last preferences.
>
> Thus a method that passes both DMTCBR and rev. sym. would be extremely
> resistant to both burial and to compromise. But since the favorite
> betrayal criterion is so hard to pass, we have reason to believe that
> this is impossible. So no such method can be rev. sym -- which is what
> we at least see with Condorcet methods!
>
> It's thus quite that the implication is stronger: that we can't have all
> of DMTBR, majority, and reversal symmetry. But the proof is probably a
> lot harder to find, too.
>
> So all of the above implies that when creating a resistant ranked
> method, we can't both have extreme resistance to burial and compromising
> - we have to pick one. Fortunately (as James Green-Armytage originally
> showed), we already get a great deal of compromising resistance from the
> Condorcet criterion itself (since, for instance, it does the right thing
> under center squeeze). Thus it's more sensible to choose further burial
> resistance over further compromise resistance if we can only have one.
>
> (Unless we consider maximum compromise resistance absolutely
> non-negotiable, e.g. Mike O's insistence on the FBC.)
>
> ...
>
> Finally, it might be useful to see just what the analog of the DMTCBR is
> for a reversed DMTCBR-compliant method. It's something like...
>
> Suppose that more than 1/3 of the ranks some Condorcet loser last. Then
> nobody who prefers this loser to the current winner can make the loser
> win by upranking him.
>
> -km
>
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