On Mon, 7 Mar 2005, Jobst Heitzig wrote: ...
Perhaps I should make clear again why I propose randomization in the first place:
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Methods such as Condorcet Lottery, RBCC, and RBACC accomplish this ...
But the Condorcet Lottery picks the CW with certainty when there is one. Wouldn't this encourage the creation of a fake CW?
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You then proposed another method, VCRB: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).
That is perhaps better. By definition, each non-winner is beaten by some winner. But is also every winner (except a CW) beaten by a winner?
Yes (except when the CW is the approval winner), if you count being beaten in approval as being beaten.
Suppose that (according to the ballots) candidate A is both the approval winner and the CW. How likely is it that A is a fake CW?
And is it really monotonic?
Yes. First of all, moving up (in approval or pairwise) a viable candidate relative to some other candidate (viable or not) will not help any non-viable candidate become viable, i.e. the new approval cutoff alpha cannot be lower than the old.
The question remains, did the candidate X that moved up relative to candidate Y, suddenly become non-viable? i.e. did the alpha approval cutoff get moved up past X ?
To see that it did not, consider the following characterization of the viable candidates. (For simplicity we assume no ties either in approval or pairwise.)
A candidate X is viable iff there is a sequence of candidates c1, c2, ... cK, such that X=c1 and cK=(the approval winner A), and such that each candidate in the sequence beats its successor either in approval or pairwise.
[The key to this is the maximality of alpha in the definition of viable.]
So before X moved up relative to Y, there was "beat path" of the indicated type from X to A. Assume without loss in generality that candidate X appeared only at the start of that sequence. Then every step of that sequence is still valid, so there is still a "beat path" of that type from X to A, so X is still viable.
Forest
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