[EM] Three forms of reversal symmetry, and an LIIA implication

Forest Simmons forest.simmons21 at gmail.com
Wed Nov 24 18:20:36 PST 2021

```Very Good!

And see an inline comment on my symmetrization suggestion ... a practical
consideration or two.

El mié., 24 de nov. de 2021 6:55 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> On 23.11.2021 05:58, Forest Simmons wrote:
> > If I am not mistaken, here's a way to modify any type one method to
> > confer type three reverse symmetry:
> >
> > For any ballot set S Let F1 be the finish order for the base method
> > applied to S. Let F2 be the finish order for the base method applied to
> > the set of reversed ballots S'.
> >
> > Now pairwise sort F1, F2, and their reverse orders with the same bubble
> > sort algorithm.

Or, use both bubble and sink sort on both F1 and F2 .... where bubble and
sink, respectively, prioritize out of order swaps for adjacent pairs
nearest the bottom and top, respectively .... one starts near the bottom
and bubbles up, while the other starts near the top and settles down.

One could use weakest margin of defeat  to prioritize the swaps (a sorted
margins variant), and thereby simplify the last step of identifying F and
F'... but that creates a potential problem when two nearby adjacent pairs
have the same margin of defeat.

Of these four beatpaths, let F be the strongest, i.e.
> > the one whose weakest pairwise margin is the greatest in absolute value.
> >
> > Then F and its reversal F' have the same strength.  Whichever of these
> > two orders makes the most sense as a finish order for S is the new
> > finish order ... the other one will then turn out to be the new finish
> > order for S'.
> >
> > Whether for ballot set S or S' the same four beatpaths will result ....
> > so the set {F, F'} will also be the same. It's a simple matter to check
> > which is a beatpath for the "forward" ballots and which for the reverse.
> >
> > Make sense?
>
> I think that would work, but you might be able to weaken the type one
> requirement to an "extra weak reversal symmetry":
>
> Type 0: If the forward election does not produce a tie anywhere in the
> social ordering, then reversing the election should not lead to the same
> ordering.
>
> As long as at least one pair is different, you could (theoretically)
> distinguish between the two and assign F to one and F' to the other.
> Maybe bubble sort will require something between type 0 and 1 reversal
> symmetry to ensure that the beatpath orders are always all different,
> though.
>
> And speaking of weakenings, I think I can discard the requirement for
> monotonicity in my 2+LIIA->3 proof.
>
> Suppose there are two candidates and they don't tie. Then 2 immediately
> requires that a social order of A>B for the forward election implies B>A
> for the reverse, which is all we need to provide the base of the
> induction chain.
>
> -km
>
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