[EM] Method X

[Forest Simmons]

On Wed, Aug 2, 2023, 4:15 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 8/2/23 07:07, Forest Simmons wrote:
> > Thanks for opening up new vistas!
> >
> > How about restricting method X to Smith?
> >
> > Or using some other method to pare down the field to manageable size?
>
> It's even more strange if I do so. Here's Smith//X:
>
> Burial, no compromise:  4262    0.0875496
> Compromise, no burial:  9268    0.190382
> Burial and compromise:  1933    0.0397075
> Two-sided:              8103    0.166451
> Other coalition strats: 0       0
> ==========================================
> Manipulable elections:  23566   0.48409
>
> See how there's no "Other"? So it too most likely is monotone! Just how
> is that possible? Schwartz//X is monotone too.
>
> (I don't think the difference between manipulability here and for
> Smith,IRV is significant, as much as it would be fun to say I've not
> just met but beat IRV :-)
>
> I tried to check Landau//X. It had plenty of Other strats, which is not
> unexpected given your proof about independence of covered alternatives
> plus first preference tiebreak implying nonmonotonicity... but I think
> my implementation of Landau is suspect, so I'm not including it here.
>

Let M be the matrix whose (j,k) element is one (else zero) if candidate j
defeats candidate k.

Then candidate j is uncovered iff the j_th row of (I+M)^2 is devoid of
zeros  ... because the (j,k) element of this matrix is the number of
distinct short beatpaths (i.e. of length 0,1, or 2) from candidate j to
candidate k.


> I could try other variants like "whenever someone is eliminated,
> everybody he beats pairwise must also be eliminated". If that were done
> on a global ordering (like IRV), that should give us Landau (and
> Banks?), but I don't know if our method construction guidelines still
> hold when the elimination order is different for different candidates.
> This is rather uncharted territory!
>
> -km
>
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