[EM] Strong FBC, Majority Rule, and Condorcet

[Elisabeth Varin/Stephane Rouillon]

Alex, I agree with your analysis.

I would even continue. Thus it would seem
that no method can satisfy the FBC criteria
(I consider majority rule mandatory).
If anyone thinks he knows one, please
send it to me, I will search for a counter-example.

No method is strategy-free, there is always a ballot
distribution that can encourage insincere preferences.

What we should aim to is to minimize
the occurence of distributions which such a possibility.
Actually, this would be a good way to compare
winning-votes and relative margins criterias for
ranked-pairs.

The main problem I have now is to justify the relative
probability of A>B>C compared to A>B?C.
Anyone has an idea?

Steph.

Alex Small a écrit :

> Suppose we demand that a method satisfy two criteria:
>
> 1)  Strong FBC:  No voter ever has an incentive to insincerely rank
> another candidate ahead of his favorite.
> 2)  Majority Rule:  If a candidate is the first choice of the majority he
> always wins.
>
> Let's look at some method elects the first choice of the majority, and
> otherwise uses an (unspecified for now) auxillary procedure.
>
> Consider this electorate:
>
> 49 B>A,C (Their relative rankings of A, C don't matter for now)
> 1  A>B>C
> 49 the other factions
>
> No first-place majority.  If the auxillary method picks C then the A>B>C
> faction is kicking itself.  So, the auxillary method must have two steps
> to satisfy strong FBC:
>
> Step 1:  If a candidate is a single vote away from having a first-place
> majority, and at least one voter prefers him to the candidate who would
> win according to step 2, elect the "close but no cigar" candidate.
> Step 2:  Some other procedure
>
> Now, say the electorate is
>
> 48 B>A,C
> 2  A>B>C
> 49 The other factions
>
> No first-place majority, and B is two votes away from a first-place
> majority, so we go to step 2.  If step 2 still picks C then the A>B>C
> voters are kicking themselves:  If just one of them had said B>A>C then B
> would be one vote away, and step 1 would elect B instead of their last
> choice.  So, to satisfy strong FBC we modify step 1, and allow for
> candidates who are 2 votes away from the threshold, not just 1.
>
> We could go on and on, but as we lower the threshold we're basically
> saying that if the winner of step 2 loses a pairwise contest to somebody
> just a single vote away from the threshold then the method has failed
> strong FBC.  So we continue to lower the threshold.
>
> The problem is that sometimes all candidates might lose at least one
> pairwise contest.  This isn't rigorous, but it seems to suggest that
> strong FBC is incompatible with a majoritarian criterion.  There will
> always be incentives to insincerely defect and create an insincere
> first-place majority.  Setting lower thresholds to spare voters the
> indignity of betraying their favorite brings in the concept of pairwise
> contests, and we find ourselves confronted with the Condorcet Paradox.
>
> OK, back to my experiments.
>
> Alex
>
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