[EM] Enhanced DMC

[fsimmons at pcc.edu]

One of the main approaches to Democratic Majority Choice was through the idea that if X beats Y and 
also has greater approval than Y, then Y should not win.

If we disqualify all that are beaten pairwise by someone with greater approval, then the remaining set P 
is totally ordered by approval in one direction, and by pairwise defeats in the other direction.  DMC 
solves this quandry by giving pairwise defeat precedence over approval score; the member of P that 
beats all of the others pairwise is the DMC winner.  

The trouble with this solution is that the DMC winner is always the member off P with the least approval 
score.  Is there some reasonable way of choosing from P that could potentially elect any of its members?

My idea is based on the following observation: 

There is always at least one member of P, namely the DMC winner, i.e. the lowest approval member of 
P, that is not covered by any member of P.

So why not elect the highest approval member of P that is not covered by any member of P?

By this rule, if the approval winner is uncovered it will win.  If there are five members of P and the upper 
two are covered by members of the lower three, but the third one is covered only by candidates outside 
of P (if any), then this middle member of P is elected.

What if this middle member X is covered by some candidate Y outside of P?  How would X respond to 
the complaint of Y, when Y says, "I beat you pairwise, as well as everybody that you beat pairwise, so 
how come you win instead of me?"

Candidate X can answer, "That's all well and good, but I had greater approval than you, and one of my 
buddies Z from P beat you both pairwise and in approval.  If Z beat me in approval, then I beat Z pairwise, 
and somebody in P covers Z.  If you were elected, both Z and the member of P that covers Z would have 
a much greater case against you than you have against me."