DSV (Designated Voting Strategy) Approval of any stripe has never turned out to be monotonic, as far as I know ... but it's the striving that we learn from...
Ballots are ranked preference style with optional approval marks allowed.
1. Compute the pairwise win, lose, tie matrix WLT:
WLT(i, j) is 1, -1, or zero respectively in the repective cases candidate i pairwise beats, is beaten by, or ties candidate j .
2. On each ballot find the lowest positioned candidate C that pairwise beats every candidate ranked ahead of it.
3. Mark as approved ...
.... every candidate ranked above C,
.... every candidate ranked equal top,
.....any candidate ranked above the lowest ranked candidate optionally marked approved by the voter
4. Cross out the name of any candidate explicitly disapproved by the voter ... nullifying (by so doing) any approval from step 3 above in the process.
5. If candidate C is no longer ranked, or never was to begin with, count it as not approved.
6. Otherwise, if candidate C is ranked equal top or otherwise approved by step three, count it as fully approved.
7. Otherwise, count it as half approved.
Note that if all ballots are complete rankings of the candidates, and if no optional approval/disapproval marks are made, then the Condorcet Candidate C (if there is one) will get the most approval.
To see this, first notice that in this case the CC will be the cutoff candidate C on every ballot, so she will get exactly half of the total possible approval N plus an additional half point for every top position occupied.
No other candidate can get more approval than N/2 ... without being ranked ahead of C on more than half of the ballots... not possible if C is truly the CC.
It seems to me that this method is monotonic. Am I overlooking something?
It also seems to be at least marginally clone independent ... marginal in the sense that Approval is clone free on the assumption that precise clones will always end up on the same side of an approval cutoff.
It satisfies Plurality and is not vulnerable to "chicken" attacks.
And it is hard to imagine how it could fail the FBC.