[EM] An observation from the Ranked Pairs post

Kristofer Munsterhjelm km_elmet at t-online.de
Wed Sep 13 03:17:50 PDT 2023

I used LIIA to show that ties between no-hopes can be shown to have been 
irrelevant to the relative position of the strong candidates.

This got me thinking that there are other implications to LIIA too. Namely:

If X is a set, the method passes the X criterion (ranking members of X 
ahead of non-X members), then automatically it must pass independence of 
non-X alternatives.

Hence: any method that passes Landau and LIIA is nonmonotone (assuming a 
relatively non-contentious additional property). Just like that.

And it also means I can't make a LIIA Resistant/Inner Burial Set method 
and hope to retain monotonicity.

However, there's an out: The method can still pass the Y criterion, 
where Y is always a subset of X. As long as independence of non-Y is 
monotone, we're saved. E.g. it's possible that independence of 
non-Copeland alternatives is monotone, thus this impossibility 
implication doesn't hold.

But such methods must necessarily spread non-Y X candidates over the 
ranking. For instance, suppose that in a particular election, the Landau 
set is {A, B, C} and the Copeland set is {A, B}. Then to pass Y but not 
X, the outcome could be A>B>D>C.


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