[EM] An ABE solution

[fsimmons at pcc.edu]

From: "C.Benham" 

> 
> Forest Simmons, responding to questions from Mike Ossipff, wrote 
> (19 Nov 
> 2011):
> 
> > > 4. How does it do by FBC? And by the criteria that bother some
> > > people here about MMPO (Kevin's MMPO bad-example) and MDDTR 
> > (Mono-Add-Plump)?
> >
> > I think it satisfies the FBC.
> 
> Forest's definition of the method being asked about:
> 
> > Here’s my current favorite deterministic proposal: Ballots are 
> Range 
> > Style, say three slot for simplicity.
> >
> > When the ballots are collected, the pairwise win/loss/tie 
> relations are
> > determined among the candidates.
> >
> > The covering relations are also determined. Candidate X covers 
> > candidate Y if X
> > beats Y as well as every candidate that Y beats. In other 
> words row X 
> > of the
> > win/loss/tie matrix dominates row Y.
> >
> > Then starting with the candidates with the lowest Range 
> scores, they are
> > disqualified one by one until one of the remaining candidates 
> X covers 
> > any other
> > candidates that might remain. Elect X.
> 
> 
> Forest,
> 
> Doesn't this method meet the Condorcet criterion? Compliance 
> with 
> Condorcet is incompatible with FBC, so
> why do you think it satisfies FBC?

When the range ballots have only two slots, the method is simply Approval, which does satisfy the 
FBC.  Does Approval satisfy the Condorcet Criterion?  I would say no, but it does satisfy the "votes only 
Condorcet Criterion." which means that the Approval winner X pairwise beats every other candidate Y 
according to the ballots, i.e. X is rated above Y on more ballots than Y is rated above X.

Now consider the case of range ballots with three slots: and suppose that optimal strategy requires the 
voters to avoid the middle slot.  Then the method reduces to Approval, which does satisfy the FBC.


> 
> 
> http://lists.electorama.com/pipermail/election-methods-
> electorama.com/2005-June/016410.html
> 
> > Hello,
> >
> > This is an attempt to demonstrate that Condorcet and FBC are 
> incompatible.> I modified Woodall's proof that Condorcet and 
> LNHarm are incompatible.
> > (Douglas R. Woodall, "Monotonicity of single-seat preferential 
> > election rules",
> > Discrete Applied Mathematics 77 (1997), pages 86 and 87.)
> >
> > I've suggested before that in order to satisfy FBC, it must be 
> the case
> > that increasing the votes for A over B in the pairwise matrix 
> can never
> > increase the probability that the winner comes from {a,b}; 
> that is, it 
> > must
> > not move the win from some other candidate C to A. This is 
> necessary 
> > because
> > if sometimes it were possible to move the win from C to A by 
> increasing> v[a,b], the voter with the preference order B>A>C 
> would have incentive to
> > reverse B and A in his ranking (and equal ranking would be 
> inadequate).>
> > I won't presently try to argue that this requirement can't be 
> avoided 
> > somehow.
> > I'm sure it can't be avoided when the method's result is 
> determined solely
> > from the pairwise matrix.

Note that in our method the Cardinal Ratings order (i.e. Range Order) is needed in addition to the 
pairwise matrix; the covering information comes from the pairwise matrix, but candidates are dropped 
from the bottom of the range order.

In the two slot case can the approval order be determined from the pairwise matrix?  If so, then this is a 
counterexample to the last quoted sentence above in the attempted proof of the incompatibility of the CC 
and the FBC.

Forest