[EM] Method X, bummer

Mon Aug 7 03:13:57 PDT 2023

A note concerning monotonicity testing - even when there is no possible
pushover strategy for a voter/group of voters, it doesn't necessarily mean
that a given method is monotone.
In these examples, changing ABC to BAC makes C the winner, so the change
doesn't make sense from the voter's point of view and will go under the
radar of any strategy detector, I guess.

Another thing is that there are some election scenarios which an
impartial/spatial simulator might never notice. For the purposes of
strategy/critetia testing it might be good to include, for example, a
ballot generator that will produce random-size groups of voters, rather
that drawing voters one-by-one like a standard impartial generator does.

sob., 5 sie 2023, 13:52 użytkownik Kristofer Munsterhjelm <
km_elmet at t-online.de> napisał:

> On 8/5/23 05:49, Kevin Venzke wrote:
> > Hi Kristofer,
> >
> > It wasn't so easy, but regrettably I think I have a monotonicity
> counter-example:
> >
> > 408: B>C>A
> > 329: A>C>B
> > 126: C>A>B
> > 91: C>B>A
> > 43: A>B>C  -->  B>A>C
> > (total 997)
> >
> > For the first round, A and B votes both exceed 1/3rd (332.33) and so
> only C can be
> > eliminated.
> > The match-up A:B gives B a very slight win of 499 vs 498 for A. C can't
> score anything.
> > Scores: B 499, A 498, C 0.
>
> I can verify that the scores are B: 499 > A: 498 > C: 0.
>
> > Now change the 43 to B>A>C, theoretically helping B further.
> > First round totals become 329 A, 451 B, 217 C. So it is now allowed to
> eliminate A.
> > Both A and B fare worse against C than against each other and so prefer
> to score off of
> > eliminating C.
> > B improves its score to 542 while A's score is reduced to 455.
> > However, when A is eliminated, C can score 546 from their matchup with B.
> > New scores: C 546, B 542, A 455.
>
> And I can verify that the scores are C: 546 > B: 542 > A: 455.
>
> Well done. Well, I would rather have wanted it to be monotone, but it's
> better to know the truth! I guess that makes this "very low
> nonmonotonicity" rather than monotone - now I know how the IRVists feel
> when people complain about nonmonotonicity!
>
> Here's a minimal example produced by linear programming:
>
> 1: A>B>C
> 7: A>C>B
> 8: B>A>C
> 3: C>A>B
> 4: C>B>A
>
> the scores are B: 12 > A: 11 > C: 0, then after changing ABC to BAC the
> scores become C: 14 > B: 13 > A: 10.
>
> Interestingly, for your example, fpA-fpC says that the correct ordering
> for the "before" election is C>B>A, whle Carey says B>A>C. My example,
> on the other hand, doesn't have this distinction... but it has a
> Condorcet cycle both before and after, thus showing that Smith//X won't
> solve the problem.
>
> Despite the example showing that X itself isn't monotone, I'm more
> confident now that (properly phrased) DMTBR is compatible with both
> monotonicity and Condorcet. Prior to method X, we only had the fpA-fpC
> generalizations, IFPP, and IRV; the first were only DMTCBR, while the
> latter two were clearly nonmonotone. I was worried that there might be
> an impossibility theorem of some kind proving that monotonicity would be
> forever out of our grasp for burial-resistant Condorcet methods.
>
> I can also use method X to find out just what kind of DMTBR should hold,
> and then build off that. I think I have another idea that could work,
> but it would be so incredibly ugly - basically "IRV with donations".
>
> Or we could try to find out why X comes so close to monotonicity, since
> it's the closest we've got so far. Doing so would require figuring out
> why max A>B ("max votes-for") is monotone, and why Smith//method X also
> seems to be (nearly) monotone, I think.
>
> > One thing I noticed is that modifying the quota rule allows you at one
> extreme to
> > implement IRV (i.e. by saying that only the candidate with the fewest
> votes can be
> > eliminated each round) and at the other extreme to implement "max
> votes-for wins" (by
> > imposing no quota requirement at all). While the latter is monotone, it
> doesn't satisfy
> > majority favorite.
>
> That's right; making the quota more loose (i.e. giving the method more
> candidates to choose eliminations from in a given round) doesn't seem to
> hurt monotonicity until you go past 1/3, but it does hurt strategy
> resistance. Going in the other direction is not strictly possible
> because if you're in an n-way tie, every candidate has exactly 1/n of
> the first preferences. So you would then need to also allow eliminating
> the lowest scorer no matter what. This would make it more like IRV and
> thus compromise its monotonicity (further).
>
> > It's interesting to consider whether any quota rule could at least
> > preserve monotonicity and add majority favorite. I'm thinking no, though.
> The weakest quota I can think of that will preserve majority is 1/2.
> Suppose A is voted first by a majority. Then A can never be eliminated,
> so for any other candidate B, it eventually ends up being A vs B, and
> since A is a majority favorite, A then wins. However, this is not
> strategy resistant; even a constant quota of 1/3 for everything but the
> final round (which is what I tried first) destroys strategy resistance.
>
> Furthermore, as mentioned above, there seems to be a strange
> relationship between the quota and the degree of nonmonotonicity - at
> least if "Other" is a good indicator. For a three-candidate election,
> 1/3 is equivalent to "normal" method X, which we now know is (barely)
> nonmonotone. However, loosening the quota to 1/2 introduces more
> nonmonotonicity; then getting rid of the quota altogether gets us back
> into the monotone domain.
>
> E.g. with fixed quota 1/2, impartial culture, 5 candidates, 97 voters,
> 7500 elections:
>
> Burial, no compromise:  218     0.0305793
> Compromise, no burial:  1138    0.15963
> Burial and compromise:  435     0.0610184
> Two-sided:              5297    0.743021
> Other coalition strats: 41      0.00575116
> ==========================================
> Manipulable elections:  7129    1
>
> and with fixed quota 1/3:
>
> Burial, no compromise:  403     0.0558017
> Compromise, no burial:  1544    0.213791
> Burial and compromise:  86      0.0119081
> Two-sided:              5149    0.71296
> Other coalition strats: 0       0
> ==========================================
> Manipulable elections:  7182    0.994461
>
> -km
> ----
> Election-Methods mailing list - see https://electorama.com/em for list
> info
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20230807/0fa57e2b/attachment.htm>