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

Kristofer Munsterhjelm km_elmet at t-online.de
Wed Nov 24 06:53:15 PST 2021

```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. 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
```