[EM] covering in the context of range style ballots

[fsimmons at pcc.edu]

It recently struck me that in range we can strengthen the covering relation if we include the range levels 
as virtual candidates:

An alternative beats level L pairwise iff it is rated above L on more ballots than it is rated below L.

Then for an alternative to cover Y, it has to beat Y pairwise, as well as all of the alternatives that Y beats 
pairwise, including each virtual candidate (i.e. level) that Y beats pairwise.

This makes the relation stronger, so that we should have fewer qualms about using covering to over-ride 
other kinds of defeats,

In particular, if X covers Y in this strong sense, then Y should not win, even if Y is the approval winner 
(unless the method is just plain vanilla Approval).

In particular, I'm thinking of the method that goes like this:

Initialize variable X as the highest approval candidate.

Then while X is covered (in the strong sense)
 replace X with the highest (non-virtual) approval candidate that covers X (in the strong sense)
EndWhile

Elect the final value of X.

It would be interesting to see how this changes things, in particular when the approval winner is covered 
in the old weak sense, but uncovered in this new strong sense.

In most realistic cases, the old version is just Smith//Approval.  But what should we mean by "Smith" 
when we have all of these virtual candidates?  If it is included, then the top level virtual candidate 
automatically pairwise beats (and covers) all of the other candidates, both real and virtual. 

Also notice that an ordinary CW covers (in the strong sense) the Bucklin winner, iff the CW has its 
median rating at the same level as the Bucklin winner.