[EM] Remember Toby

[fsimmons at pcc.edu]

----- Original Message -----
From: Jameson Quinn

> 2011/6/5 Dave Ketchum
>
> > I see this as Approval with a complication - that Jameson
> calls SODA. It
> > gets a lot of thought here, including claimed Condorcet
> compliance. I offer
> > what I claim is a true summary of what I would call smart
> Approval. What I
> > see:
> > . Candidates each offer draft Approval votes which voters
> can know in
> > making their decisions.
> >
>
> You are close, but apparently Forest and I haven't explained the
> system well
> enough. Candidates offer full or truncated rankings of other
> candidates.
>
> > . Vote by Approval rules.
> > . If there is no winner, then each candidate gets to vote
> above draft
> > once for each ballot that bullet voted for that candidate.
> >
>
> Candidates may vote any approval ballot consistent with the
> ranking above
> once for each ballot. They do so simultaneously, once, after the full
> results and all candidate's rankings have been published.
> "Consistent with"
> means that they simply set an approval cutoff - a lowest
> approved candidate
> - and all candidates above that in their ranking are approved.
>
>
> > . If a voter is thinking bullet voting, but wants to avoid
> the above -
> > voting also for an unreal write-in will avoid giving the
> candidate a draft
> > vote.

Instead of an "unreal write-in" it could be a virtual candidate whose name is
"No proxy for me" meaning "I do not delegate my approvals to any candidate."


>
> Yes.
>
> You've left out one extra check on this system, wherein the top
> two approval
> candidates are recounted in a virtual runoff without any "delegated
> approvals" between those two.
>
>
> >
> > I do not see the claimed compliance, for voters do not get to
> do ranking.
> > I see a couple uses of thoughts that imply ranking - they are
> so rare that
> > they look like typos to me.
> >
>
> I'll give a formal proof showing in what sense and in what
> circumstances this system is more compliant than Condorcet
> systems later this week, when I
> have time to write it out. You are right that individual voters
> cannot do
> ranking, and so if there's a significant constituency with a
> shared ranking
> which is neither represented by a candidate nor balanced out by random
> noise, then that constituency is faced with the strategic
> choices typical of
> approval, and the system as a whole does not guarantee
> compliance. However,
> if that is not true - that is, if the electorate can be
> characterized as a
> set of known coherent candidate-led constituencies plus a
> leftover which is
> exactly 50/50 on any candidate pair - then this system, unlike actual
> Condorcet systems, is compliant, not just for honest votes, but
> always for
> any rational strategic votes.
>

It is also possible to consider the (non-proxy) approval ballots as ordinal
ballots with the approved candidates equal ranked first and the unapproved
candidates truncated.  Then putting these "rankings" together with the candidate
rankings gives a basis for defining a "ballot CW."   Then we can argue that this
ballot CW is very likely to be the same as the actual CW when there is one.  

In fact, it is well known that when there is a real CW, the CW will be a strong
equilibrium Approval winner, assuming near perfect information.  Couple this
fact with the fact that the ballot CW for a set of approval ballots (interpreted
as ranked ballots with lots of equal rankings and truncations) is always the
same as the approval winner, and you are well on your way to showing that the
ballot CW is the same as the actual CW.

These considerations suggest a modification of SODA: for each voter submitted
approval ballot fill out a pairwise matrix.  Add these matrices to the pairwise
matrices of the candidate rankings (weighted according to their respective
numbers of bullet voters).  If, according to the total pairwise matrix, there is
a CW, then elect that candidate.  Else have the candidates indicate their
approval cutoffs, and elect the resulting approval winner.