[EM] Smith Sets with >3 members

[Forest Simmons]

On Wed, 27 Feb 2002, Markus Schulze wrote:

> Dear Forest,
>
> you wrote (27 Feb 2002):
> > Markus wrote (27 Feb 2002):
> > > Forest wrote (26 Feb 2002):
> > > > It seems to me that another problem of Copeland methods is that they
> > > > encourage favorite betrayal in the same way IRV does: if your compromise
> > > > has a better chance of winning the election than your favorite, but your
> > > > favorite has a good chance of beating your compromise, then you (and like
> > > > minded voters) vote your compromise above your favorite to maximize your
> > > > compromise's chance of getting one more win.
> > >
> > > In so far as Copeland is monotonic, you cannot increase your favorite's
> > > chance of winning the elections by voting another candidate above your
> > > favorite.
> >
> > But you can increase your compromise's chance by voting you compromise
> > above your favorite.
>
> Yes, but this is true for every Condorcet method.
>
> Blake Cretney calls this strategy "compromising". Every Condorcet method
> is vulnerable to "compromising".

I suspected as much. At first I hoped that Copeland would be immune to
compromising, because a narrow victory is as good as a decisive victory in
Copeland methods (except perhaps for tie breakers).  This has the effect
of solving the minor spoiler problem (as IRV does) delaying the necessity
of compromise to the stage where favorite is big enough to threaten
compromise.

So, if we want a method that is not vulnerable to compromising but always
picks a member of the Smith set, we have to use some hybrid method that
requires information outside of the pairwise comparisons, perhaps
information not available in preference ballots.

Demorep's ACMA seems to be invulnerable to compromising. It makes use of
Approval information in addition to the pairwise win matrix.  It satisfies
the Condorcet Criterion, but may not always pick a member of the Smith
set.

What if we used the Demorep style ballots (candidates ranked and approved
or disapproved) to carry out Approval Seeded Bubble Sort?

Bubble Sort (like Single Elimination) always puts a member of the Smith
set at the top.

Would this method be vulnerable to compromising?  Not much, if at all.

Case 1a.  Compromise is seeded above favorite and is never challenged by
favorite. No problem.

Case 1b.  Compromise is seeded above favorite and is already defeated by
at least one other candidate before being challenged by favorite. No
problem.

Case 1c.  Compromise is at the top when challenged but not defeated by
favorite. No problem.

Case 1d.  Compromise is at the top when challenged and defeated by
favorite. Then favorite will remain at the top until defeated by a
candidate that also defeated compromise.

Case 2a.  Favorite is seeded above Compromise and is never challenged by
Compromise. Disapproving Favorite wouldn't help, because Compromise's
order among the remaining candidates would be unchanged.

Case 2b.  Favorite is seeded above Compromise and is challenged and
defeated by Compromise. No problem.

Case 2c.  Favorite is at the top when challenged (but not defeated) by
Compromise.  Favorite has dashed the hopes of Compromise. Anybody who
deposes favorite will have to beat Compromise first. No problem.

Case 2d. Favorite has already been defeated before being challenged (but
not defeated) by Compromise. At this point we would have some regret that
Compromise was blocked from further chance of proceeding to the top.
However, given that Compromise had less approval than Favorite and that
Favorite beat Compromise, we shouldn't expect Compromise to have great
chances of beating all the guys ahead of Favorite, notwithstanding
pre-race polls to the contrary.

This last case (2d) would be a pretty flimsy excuse for voting Compromise
over Favorite. I suppose it could happen if your preference of Favorite
over Compromise were extremely weak, or if you had extremely precise,
detailed information about voter preferences that gave Compromise a good
chance of beating every candidate that could beat Favorite despite
Compromise having less approval than Favorite and being unable to beat
Favorite.

Note that this Approval Seeded Bubble Sort has less vulnerability to
compromising than Random Candidate Single Elimination, because in this
latter method, Favorite can get into a position to block Compromise's
progress by random (as opposed to merit based) means.

Forest