[EM] Help me understand some notation

[Kristofer Munsterhjelm]

On 20.01.2022 12:26, Daniel Carrera wrote:

> 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.

Yes. I forgot to mention two things, though:

- Borda is particularly nice in that Copeland//Borda is monotone. This
happens because you can infer the Borda score from the pairwise matrix,
and the remaining pairwise matrix entries don't change when you remove
candidates. So although as a rule X//Y is nonmonotone, some monotone
pairwise-based set restrictions of Borda (in particular Smith and
Copeland) will be monotone. However, Borda is unusually susceptible to
strategy, which is a problem, and neither Borda nor Copeland are cloneproof.

- X//Y methods give up monotonicity, but they get something in return:
if the X set itself is independent of candidates not in it (i.e. you
can't make the X set smaller or larger by introducing candidates not in
it), then X//Y is independent of X-dominated candidates. E.g. in
Smith//Plurality it doesn't matter if there's vote-splitting among the
losers as long as there's no vote-splitting in the Smith set itself. But
in Smith,Plurality, losers outside the Smith set may draw votes away
from the Smith set, making someone else inside it win.

(Note that Copeland fails the "independence of candidates not in it"
criterion. So Copeland//Borda doesn't get you independence of
Copeland-dominated candidates -- but it does get you independence of
Smith-dominated candidates.)

-km