[EM] Strong FBC, Majority Rule, and Condorcet

[Alex Small]

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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