Ballots are ordinal with approval cutoff, equal rankings allowed.
Let U(A) be the set of uncovered candidates that cover the approval winner A. The member of U(A) with the highest approval is the method winner W.
In particular, if the approval winner A is covered only by itself, then A=W is the method winner.
It seems that W has a strong position: W covers A (i.e. beats every candidate that is beaten by the approval winner A), is maximal in this respect (i.e is not covered by any other candidate), and has the maximum approval among such candidates.
What does it mean if W is different from A? It means that W beat A head to head, and that W beat every candidate that A beats head to head, so ultimately it means that given more accurate polling information, the voters (in all likelihood) would have adjusted their approval cutoffs to give more approval to W than to A; i.e. approval winner A getting more approval than W was probably a mistake due to disinformation about the relative strength or weakness of the other candidates.
If these approval cutoffs were adjusted and the new approval winner did indeed turn out to be W, then the method (reapplied) would still yield W as the winner, since W would still be uncovered.
This suggests another method using cardinal ratings ballots:
First pass: use the ordinal information from the ballots to produce the uncovered set U.
Second pass: set the approval cutoff on each ballot at the mean of the ratings of the members of U on that ballot. (i.e. apply the "above mean" approval strategy, based on the assumption that all members of U are equally likely to win.)
Step three: the approval winner (based on the cutoffs determined in step 2) is the method winner.
Does this method always pick a member of U?
Not necessarily, but in all likelihood the winner will be from the set U. We could enforce this by changing step three to "The member of U with the highest approval is the method winner."
I prefer the simpler version. In practice is would always pick from U, but the psychology of still having hope for all candidates, after identification of the set U, would discourage voters from heeding phony experts with advice about how to keep their "only hope" from being eliminated from U.
[Remember how the "Anybody but Bush" crowd wasted activist resources and set back the progressive movement ten years by bashing Nader (and Nader supporters) as a misguided strategy for saving the hopeless "only hope" Kerry?]
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