[EM] Strong FBC, Majority, non-monotonicity, and stability

[Alex Small]

Suppose we have an election method M which is non-monotonic, elects the
first choice of the majority (when such a candidate exists), and satisfies
strong FBC.

Consider this situation with 2m-1 voters, 3 candidates, and 1 winner:

m-1  B>A,C
p    A>B>C
q    A>C>B
x    C>B>A
y    C>A>B  (Assume p,q,x,y>0, x+y<m-1, p+q<m-1)

Say that M elects candidate A.  A voter with the preference C>B>A has an
incentive to betray his favorite by ranking B first on his ballot (giving
B a majority of first place votes), unless B or C wins in the following
situation (where a C>B>A voter has defended his interests by using an
insincere but favorite-loyal strategy):

m-1  B>A,C
p    A>B>C
q    A>C>B
x-1  C>B>A
y+1  C>A>B

If B wins in this situation then the majority criterion provides no
incentive for favorite betrayal by any voter, since no act of favorite
betrayal by any voter can give A or C a majority.  It is possible that an
act of favorite betrayal could cause M to select a candidate whom that
"disloyal" voter prefers, but then reversing his 2nd and 3rd preferences
must also give that result.  We will stick to the consequences of
combining majority rule and strong FBC here, however.

Now, suppose that C won in the above situation.  The majoritarian
criterion gives a voter who prefers A>B>C an incentive to betray his
favorite unless reporting A>C>B elects A or B.  If that strategy elects A,
then a voter who prefers C>B>A has an incentive to betray his favorite
unless switching to C>A>B elects B or C.  And so forth.....

We could go on, but the point is that if a method is non-monotonic,
majoritarian, and compliant with strong FBC then when a candidate is a
single vote away from a majority the method either selects that candidate
or it's highly unstable.  Now, granted, all majoritarian methods will be
relatively unstable when a candidate is a single vote away from a
majority.  Strong FBC will make the method even more unstable, however.

Also, if candidate B is 2 votes away from a majority, and somebody other
than B wins, and somebody other than B also wins when B is 1 vote away
from a majority, then arguments similar to those above suggests serious
instabilities.

Now, go back to the geometric picture I was toying with a few weeks ago: 
Work in a 6-dimensional space where each dimension corresponds to the
number of voters with a given preference order.  Look at a 2-D slice of
this space, where we see the transition from B being a vote short of a
majority to having a majority.  The bars, lines, and dashes are supposed
to be boundaries, and inside each "bounded" region is the letter
indicating who wins.  The geometry looks something like this.


___________________________
A|C|A|C|A|                |
--------|                 |
C|A|C|A|                  |
------|      B            |
A|C|A|                    |
-----|                    |
C|A|C|                    |
_____|____________________|


As a physicist, this reminds me of pictures from books on nonlinear
dynamics.  I am not even suggesting a rigorous proof of anything in this
message (Note to Craig:  See, I'm being honest!).  Still, the arguments
above make it plausible to suspect that majoritarian methods satisfying
strong FBC will be quite unstable, if they are even possible.

(Craig:  I know that these arguments are not proofs, and I'm not even
suggesting it.  However, when pondering a scientific question, people
generally start from hunches based on plausibility arguments and then work
to find rigorous evidence and arguments which confirm or reject the
hunches.  If you find this whole matter to be beneath you then you can
ignore it and stop lurking here.)




Alex


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