[EM] Visualizing Covering

[fsimmons at pcc.edu]

Warren Smith suggested …

‘How about, as a first step, DEFINING "covering"?’

Fair enough:

There are several common variants to the meaning of “cover” depending on how ties are treated.  For my 
purposes, alternative C covers alternative A iff C is not beaten (pairwise) by any candidate that does not 
beat A, AND C beats at least one candidate that A does not beat, or C ties at least one alternative that 
beats A.

In other words C does at least as well as A (with regard to pairwise win/tie/loss) against each alternative, 
and does better on at least one alternative.

Here I assume that A is tied with itself, so if C is tied with A, it can still cover A, as long as it does as 
well as A against the other alternatives and strictly better against at least one of them.

If the voters are distributed symmetrically around some center in some Euclidean space, then C covers 
A iff C is closer to the center than A.  

If C covers A and among those alternatives that cover A it is the one that has the greatest defeat 
strength against A, then C is arguably a natural compromise candidate for the supporters of A.  As 
Condorcet enthusiasts know, there are many ways to measure defeat strength.  James Green-Armytage 
has argued persuasively for using Cardinal Weighted Pairwise (CWP) as the best measure of defeat 
strength in River, Ranked Pairs, and Schulze CSSD/ Beatpath, when cardinal information is available.

I’m not sure that CWP is the best defeat strength measure in the context of finding the most natural 
compromise candidate for alternative A.

In fact, I think that the alternative C (among those that cover A) against which A has the least CWP 
score is a better choice (as opposed to the C that has the greatest CWP score against A) because then 
A supporters can rate compromise C equal to A without hurting C’s chances.

Forest