[EM] MMPO and Symmetric Completion

[fsimmons at pcc.edu]

MinMax Pairwise Opposition satisfies the FBC but not the Condorcet Criterion.

MinMax(margins) satisfies the Condorcet Criterion, but not the FBC.

MMPO combined with symmetric completion of all equal rankings and truncations is exactly equivalent 
to MinMax(margins), so symmetric completion of MMPO trades in the FBC for the CC.

But if we exempt the equal top rank from symmetric completion MMPO retains the FBC.  As before the 
CC is sacrificed, but the only thing standing between this version of MMPO and the CC is symmetric 
completion of the top ranked candidates.  Since we value the FBC more than the CC, we refrain from 
symmetric completion of the top rank.  

How far does this take us from the CC?  I believe that the resulting method (MMPO based on symmetric 
completion of all equal ranks except top) still has great Condorcet efficiency whether the voting is zero 
info or perfect info. 

Perhaps it will even satisfy Juho, since it is as close as you can get to MinMax(margins) while satisfying 
the FBC.

Personally, I prefer to go a little further and refrain from symmetric completion of any ranks except equal 
bottom.  This move further towards classical MMPO drastically reduces any incentive for insincere order 
reversals at any level (as in MMPO).  Why not go all the way to MMPO?  Because the symmetric 
completion at the bottom is necessary (and sufficient) for resolving the Approval Bad Example and 
Kevin's MMPO bad example.

Unfortunately, symmetric completion at the equal bottom level destroys Later No Harm compliance, but 
the method still satisfies the following (nameless?) criterion:

If candidate X is advanced on a ballot by moving from a position above the winner to a higher position or 
by moving to a position below (or equal) to the winner from a lower position, then either the winner is 
unchanged or the new winner is the candidate X.

Put this property together with mono-raise and mono-add-equal-top and you have a nice set of 
complementary monotonicity properties.

Now let's come up with a good name for this MMPO with partial symmetric completion.  Actually we 
need a good technical name as well as a catchy name for public proposal.

Forest