[EM] Improved Condorcet Approval (ICA)

[Simmons, Forest]

Kevin, your ICA method interests me.  In particular, your creative use of "equal ranked top" might be called "power top" analogous to what Mike Ossipoff recently called "power truncation" for equal (non)ranking at the bottom.
 
I suggest that we consider methods that sum two modified pairwise matrices in addition to the basic pairwise matrix:
 
(This description is at the ballot level)
 
In the ordinary pairwise matrix M, the (i,j) entry is a one or a zero depending on whether or not candidate i is ranked ahead of candidate j on the ballot.
 
In the "Pro modification"  PM,  if candidate i is ranked equal first, then row i is filled in with ones.
 
In the "Con modification" CM,  if candidate k is truncated, then column k is filled in with ones. This is Ossipoff's "power truncation" matrix.
 
I'll leave it as an exercise to restate the definition of ICA in terms of  M and CM, assuming that "least approved rank" is treated like a candidate.
 
Here's another possible application that comes to mind suggested by the question, "What happens when an irresistible force comes up against and immoveable object?"
 
We pit the candidate with the strongest offense against the candidate with the strongest defense:
 
The offensive winner is the candidate for whom the minimal row element of the PM matrix is maximal, i.e. the MMPO winner with power truncation.
 
The defensive winner is the candidate for whom the maximal element of the CM matrix is minimal.
 
If these two winners are different, then the ordinary pairwise matrix M decides between them.
 
It seems like this method might satisfy the FBC:  the process of picking the offensive winner must satisfy the FBC for the same reason that MMPO does.  And it seems to me that the process of picking the defensive winner satisfies the FBC for the same reason that ICA does.
 
What do you think?
 
Forest
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