[EM] WMA (It's not monotonic or participation compliant, after all)

[fsimmons at pcc.edu]

Kevin,

I'm sure that you are right, but it makes me think that the only reason ordinary
Approval complies with Monotonicity is that the information from polls is not an
official part of the election:  showing (true or alleged) support for a
candidate in the polls can change her from a winner to a loser in the actual
election.

For me there are two reasons for using some kind of DSV: (1) to relieve voters
from the burden of converting their sincere rankings or ratings into strategic
ballots, and (2) incorporating the information necessary for good strategy into
the official method, instead of relying on the unofficial polls that have so
much potential for corruption.

>From this point of view Conditional Approval, WMA, DYN, and Rob LeGrand's
sequential DSV based on Approval strategy A are all better than ordinary
Approval based on disinformation from the unofficial polls.

Here's a way (in the context of WMA) to lower the probability of inadvertently
changing the winner when increasing her ballot support: instead of using full
random ballot probabilities select a small subset of the ballots at random, and
use the first place proportions from that subset in place of the full random
ballot probabilities.  For example in an election with a million voters, take a
random sample of an hundred ballots to approximate the percentage of first place
support for each candidate.  Then in the vast majority of cases (99.99 percent
of the time), raising a candidate (even to first place) would not change the
probabilities on which the DSV strategy is based, so we could say the method
would be at least 99.99 percent monotonic.

Even if we only used a sample size of ten random ballots, I think that the
method would be better than ordinary Range or Approval based on informal
disinformation.  It would be well worth the tiny sacrifice in monotonicity.

Here's a related thought.  Although Rob LeGrand's sequential approval DSV method
(based on strategy A) fails Participation, if we take the following formulation
of Participation too literally, his sequential method satisfies it to the max:

If one more ballot B is counted after the election winner W has already been
determined, the winner can only change to somebody ranked higher than W on that
new ballot B.

In fact, since Rob's sequential method puts the approval cutoff adjacent to the
current winner, the only candidates that can possibly benefit more than W from
the new ballot B are the ones ranked strictly ahead of W on B.

[Here we assumed that Rob takes the ballots in the sequence that they are
submitted, etc., which is not really true, so the method doesn't really satisfy
Participation.]

>
> Hi Forest,
>
> --- En date de?: Lun 3.5.10, fsimmons at pcc.edu
> a ?crit?:
> > De: fsimmons at pcc.edu
> > Objet: [EM] WMA (It's not monotonic or participation
> compliant, after all)
> > ?: election-methods at lists.electorama.com
> > Date: Lundi 3 mai 2010, 18h06
> > Kevin and Chris,
> >
> > You?re both right, my ?proof? only showed that
> > raising the winner W could not hurt her prospects on
> > ballots where she was already approved or on ballots where
> > she was rated at the highest currently
> > unapproved level.? But , as Kevin pointed out, if
> > there were candidates at two or more levels above W,
> > and not all of these levels were above the current cutoff,
> > the cut off could lower without lowering all of the
> > way down to W.
> >
> > Back to the drawing board.? I?m starting to think
> > that it may be impossible to have any kind of
> > monotonic, non-trivial DSV method for automatically
> > choosing approval cutoffs.
>
> I definitely think it is impossible. If you let voters choose their
> cutoff based on "X," it's impossible to guarantee anything about
> how you
> are using "X." And "X" is probably affected by raising candidates.
>
> I think you will have to greatly constrain how much "freedom"
> the voters
> have to place the cutoff. It will have to be based on as little
> information as possible.
>
> For instance you can view the situation in Bucklin as that
> voters lower
> their threshold continually until the method ends. If you raise
> a winner
> the method may just end sooner, but with the same outcome.
> Though for
> a DSV method this leaves voters relatively quite "blind." They can't
> "react to threats," they just gradually become impatient and
> start to
> compromise.
>
> I like my old (non-monotonic) "Conditional Approval" method
> where (if
> we assume a three-slot ballot) voters repeatedly add in their
> second-slot
> approval whenever the current leader is a disapproved candidate. No
> one can retract second-slot approval once granted. The method
> ends with
> the round where nothing else changes.
>
> The justification for failing LNHarm feels more tangible than is often
> the case: If we didn't count your lower preference (and those
> like it)
> when we did, the winner would've been somebody that you said you
> didn'tlike.
>
> Kevin