[EM] Help me understand some notation

[Daniel Carrera]

On Thu, Jan 20, 2022 at 4:22 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 20.01.2022 01:05, Daniel Carrera wrote:
> > I just read this sentence:
> >
> > "And going from Copeland//Borda to Copeland,Borda shouldn't make the
> > method that much harder to understand.">
> > I remember seeing Kristofer say "Smith//IRV" and "Smith,IRV". Evidently
> > "//" and "," have particular meanings. Can someone explain what they
> > are? At least one of those must mean "Restrict to the Smith set then
> > apply IRV".
>
> Smith,IRV is the method where you first do IRV and then you pick the
> highest ranked candidate in the resulting social ordering that's in the
> Smith set. (In IRV's case this means: the last eliminated candidate
> who's in the Smith set.)
>
> Smith//IRV is the method where you first eliminate everybody not in the
> Smith set and *then* do IRV. This is your "restrict to the Smith set
> then apply IRV".
>
> Smith//X methods are often nonmonotone because raising A might insert
> someone else into (or kick someone else out of) the Smith set. If this
> candidate is say, ranked in the middle in the social ordering of X, then
> the whole social ordering can change and turn the winner from A to
> someone else. Smith,X methods don't have that (unless X itself already
> fails).
>


Ha! So I want to look for methods similar to Smith,X. Hmm... let me try:

1) Every candidate has a 1-1 pairwise match against every other candidate.

2) The candidates with the most won matches are "finalists".

3) The finalist with the greatest margin of victory against any other
candidate is elected.

So step (2) basically gives the Copeland set. The whole method should be
"Copeland,X" where "X" is the method "elect the candidate with the largest
victory". Since X must be a monotonic method, would it follow that
Copeland,X is monotonic too? Conversely, if step (3) had said "... against
any other *finalist*" that would have created "Copeland//X" and it would
probably be non-monotonic. Does that sound right?

This is interesting because something like Smith,X or Copeland,X as in my
example allows you to consider methods X that you would normally have
considered too simple to be interesting. This might be a good way to design
good methods (e.g. Smith-efficient and monotone) without them being very
complicated.

Cheers,
-- 
Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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