I like the TACC option the best, but I would like to suggest the following variation (which I will call TACC+ if you don't mind):
After finding the (deterministic) TACC winner, create a lottery based on random ballot among the set of all candidates that have at least as much approval as the TACC winner.
This method is definitely monotone: If the TACC winner or anyone above him in approval is the only candidate to move up in the rankings on the ballots, then the TACC winner remains the TACC winner (assuming that approval counts don't change), so the set on which the lottery is based remains the same, and only the TACC winner has a chance of increasing his winning probability in the lottery.
If the TACC winner A has an approval count increases, advancing him up the approval ladder, a candidate lower on this ladder (but not below A's original position) now has a chance of taking his place as the TACC winner, but that's OK, since A has greater approval than the new TACC winner, so is still included in the new lottery.
This new lottery is based on a subset of the original lottery members (those that are above the new TACC winner were all above A's original position), and the number of ballots on which A is first is the only one that might have increased, so A's winning probability in this new lottery is at least as good as in the old one.
Can you see any holes in that sketch of an argument?
Now here's another lottery method I favor because, although it tends to spread the probability around more promiscuously, it has nice properties:
First find the highest approval score alpha such that no candidate with approval less than alpha beats (pairwise) any candidate with an approval score of alpha or higher.
Then do random ballot relative to the candidates that have approval scores greater than or equal to alpha.
That's it. Let's call it Viable Candidate Random Ballot (VCRB).
This method is monotone and clone proof.
It also has this nice property:
If the candidates with approval less than alpha are eliminated, and the method is run again, then it will produce the exact same winning probabilities as before (as long as approval scores are not adjusted to reflect the new realities).
What should we call this? Independence from zero probability candidates? This would work if we assume that each candidate is ranked first on at least one ballot.
Or we could call it independence from non-viable candidates.
But the best thing about this method (VCRB) is that it seems to consistently punish insincere order reversals.
It handles Kevin's 49C, 24B(sincere>>A), 27A>B>>C, example quite nicely by giving the same lottery .49C+.24B+.27A for both the sincere and the insincere cases. In neither case does any candidate have any approval below alpha=27.
But I'm afraid that neither TACC nor my modification TACC+ would dissuade the B voters from their truncation.
Still, TACC+, with its greater frugality in doling out probability might be better than VCRB for most applications.
Incidently, it was fun proving VCRB's monotonicity. When I have more time I will write it up, but you might enjoy doing it yourself.
Forest
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