[EM] Strong FBC, Majority Rule, and Condorcet

[Alex Small]

I think I've made the analysis more rigorous:

Suppose that we elect the first choice of the majority, if one exists. 
Otherwise, we elect a candidate by some other method M.  M need not treat
all voters and candidates equally, it need not rely solely on rankings (it
could also look at approval cutoffs, for instance), in fact, we really
don't need to worry about the details of M.

Suppose that in a 3-way race candidate B is one vote away from a majority
and M would pick a candidate C whom B defeats pairwise.  Voters with the
preference A>B>C have an incentive to betray their favorite (there's a
loophole that I'll remove later).  So, supplement our method to elect any
candidate who's a single vote away from a majority if he pairwise defeats
the choice of M.

Now suppose B is 2 away from a majority, and M selects C.  If B defeats C
pairwise then there are at least two voters with the preference A>B>C.  If
one of those voters betrays his favorite, and votes B>A>C, B is a single
vote away, and the previous clause applies.  So, add another supplement to
our method, for the case when a candidate is 2 votes away from a majority
and defeats the choice of M.

By induction we see that we must also add supplementary rules to cover the
case where B defeats C (the choice of M) pairwise but is 3 votes away from
a majority, 4 votes,...n votes away.

So, as long as there is a candidate who pairwise defeats the choice of M,
that other candidate should be elected and M should be circumvented.

Before I declare strong FBC to be impossible, there are three things to
consider:

1)  Suppose that favorite betrayal by the A>B>C faction causes M to select
A or B.  The majority rule analysis doesn't apply, but M itself fails
strong FBC, so by extension the composite method of M plus supplementary
conditions for majority rule must also satisfy strong FBC.

2)  Suppose that the A>B>C faction need not vote insincerely to defend its
interests, i.e. suppose that changing their vote to A>C>B would cause M to
elect either A or B instead of C.  In that case M is non-monotonic, since
the voter has caused C to lose by ranking C higher without changing the
relative rankings of A and B.  I'm working right now on the case where we
allow non-monotonicity.

So, if we impose monotonicity we are requiring that (in the case of 3
candidates) our method never elect a candidate who is pairwise defeated. 
Hence, in general strong FBC, majority rule, and monotonicity are
incompatible.

Note that imposing monotonicity and strong FBC implies that the A>B>C
faction should never be able to obtain a better result from insincere
voting if C is the winner.  This does not violate the
Gibbard-Satterthwaite Theorem, because we've only said that there are
particular circumstances where a particular group of voters can't obtain a
better result from insincere strategy.  We haven't required that NO voter
ever be able to obtain a better result from some form of insincere voting.
 The "vote for top 2" ranked method that I described a few days ago
satisfies strong FBC and monotonicity and is still non-dictatorial.

3)  Does this analysis apply to more than 3 candidates?  I'm thinking
about that right now.  It's not a priori obvious to me.

So, now I'm thinking about removing the monotonicity requirement and
extending to the case of 4+ candidates.  If I could just do that I'd have
a pretty cool result.



Alex


----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em