[EM] five monotonic ways to select winners from the uncovered set

[Forest W Simmons]

Candidate X covers candidate Y 
                        if and only if
 X defeats (pairwise) both Y and each candidate that Y defeats.

An uncovered candidate is one that is not covered by any candidate.

Method 1.  If the approval winner A is uncovered, then elect A.  
Otherwise elect the highest approval uncovered candidate that covers A.

Method 2.  Elect approval winner A if A is uncovered, otherwise among 
the uncovered candidates that cover A elect the one against which A 
scores the least pairwise opposition.

Methods 3.  Let A be the candidate that is ranked (as opposed to 
truncated) on the greatest number of ballots.  If A is uncovered, then 
elect A, else among the uncovered candidates that cover A elect the one 
against which A scores the least pairwise opposition.

Method 4.  Pick A by random ballot.  If A is uncovered, then elect A, 
else among the uncovered candidates that cover A elect the one against 
which A scores the least pairwise opposition.

Method 5.  Pick A by random ballot. If A is uncovered, then elect A, 
else among the uncovered candidates that cover A elect the one ranked 
highest on the (same) random ballot.

[These methods are the result of a simplification that Jobst suggested 
when I asked his opinion about some other methods that were inspired by 
the Condorcet Lottery method that he brought to our attention a couple 
of years ago.]

All of these methods satisfy Smith because uncovered candidates have 
(short) beatpaths to the other candidates.

Of these methods, one and five are my favorites.  But some people don't 
like approval info and some don't like randomness, so I include method 
three, which I think is pretty good if you can live without Later No 
Harm.  I haven't found a case where it violates the weak FBC.

A note about method five:  It seems to me that this method is very 
likely to pick the uncovered candidate ranked highest on the randomly 
selected ballot, which tends against favorite betrayal.  

Think of Yee-Bolson diagrams:  All of the uncovered candidates must lie 
relatively near the center of the voter distribution.  The ones that 
cover A must lie roughly between A and the center. The ones that don't 
cover A must lie relatively further from the line of sight to the 
center, or on the other side of the center.

So we have

                            
                B1               D1
                            C2        
                    A   C1               
                            C3        
                 B2              D2
                             

 
where A beats the B's, the C's cover A, and the D's don't.  An  A 
supporter should prefer one of the C's over any of the D's, at least if 
A's preferences are consistent with the geometry.

[I hope the spaces are preserved in the transmission.]

MONOTONICITY: The key to proving monotonicity is that if  C  is an 
uncovered candidate that covers A, and C moves up in rank on one or 
more ballots without any other candidates moving relative to each 
other, then C remains uncovered and the set of candidates that cover A 
does not change.

Comments are welcome, especially positive ones ;)

Forest