[EM] The resistant set

[Kristofer Munsterhjelm]

On 8/16/23 13:38, Kristofer Munsterhjelm wrote:

> So point three, the "no-expansion property": a candidate B could be 
> admitted if some C who was previously barring B stops either having 
>  >1/|S| support in every restricted election involving both of them, or 
> stops beating B pairwise. The latter is obviously impossible. But the 
> former... that's what has to be shown, and why I'm not certain.
> 
> You would think that set growing would be possible by a variant of 
> Kevin's counterexample to method X. Let A bar C from the set, then let 
> the base method outcome be C>B>A so that B wins. Then raising B on an 
> ABC ballot could reduce A below the threshold, after which C wins. But 
> somehow I can't get that to work...
> 
> It gives an idea of where you could start if you have a monotonicity 
> checker, though.

The reason I can't prove that is because it isn't true. Here's a 6 
candidate Yee diagram of Resistant,max A>B. (Sorry about the 
near-identical shades of green on the lower left.)

If you can draw a line from a region of one color to the origin of that 
color, and that line passes over a region of a different color, then the 
method is nonmonotone (if I understand correctly). And that's clearly 
the case here.

I kind of feel like I'm advancing up Forest's homotopy method sequence, 
with method X being f_0, and the resistant set compositions being f_1. 
Can we ever reach high enough to get monotonicity? :-)

(Note that it's still possible that *some* composition is monotone. So 
I'd still be interested if you were to check.)

-km
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