On Jul 24, 2010, at 3:46 AM, [email protected] wrote:
You guys have come up with some interesting ideas about the
likelihood of sincere cycles, but my idea
is not that complicated:
Usually in the high stakes elections that I have witnessed there are
just a few issues that most voters
feel strongly about, and opinions on these issues are highly
correlated (or anti-correlated) so that the
voter distribution in issue space is basically cigar shaped.
Perpendicular to the long axis of that cigar find a plane that
divides the voters into two equal subsets
(plus or minus one). The candidate closest to that plane is very
likely a Condorcet candidate.
But this Condorcet candidate can be buried as easily as a Condorcet
candidate can be buried in a
precisely one dimensional issue space.
I like Condorcet methods that discourage burial in one dimensional
cases. I don't care so much about
the case where the candidates are distributed on the vertices of an
acute triangle, i.e. the triangle is
close to equilateral. In that case burial may serve a useful
purpose of decreasing the probability of
winning for a low utility Condorcet candidate.
In particular, the sincere profile
40 A>C>>B
30 B>C>A
30 C>A>>B
could easily come from a one dim or cigar shaped issue space. Any
condorcet method that doesn't
make burial of C risky for the A faction in this context is going to
end up with more artificial cycles than
real ones.
Thanks, this is at least a well defined case where strategic cycles
might occur. (I guess "B>C>A" should be read "either B>C>A" or
"B>>C>A", and in addition to "C>A>>B" votes there could be also some
"C>A>B" and even few "C>B>A" votes.)
I'm not sure if this case would lead to artificial cycles very easily.
75% of the A supporters should vote strategically to make the strategy
work. A smaller number of strategic voters (50%) is sufficient to
create an artificial cycle. There are many possible ways this strategy
can fail. For example the B supporters prefer C to A. If they know
that A supporters will try a strategy and win, then the B supporters
might vote directly for C and thereby guarantee that the strategy of
the A supporters will not work. The preferences may also change before
the election, and part of the C supporters may sincerely rank A lower
because of the attempted strategy that tries to steal the victory from
their favourite. In short, if some society is so strategic that they
would try this strategy then there could be also other strategic
moves, and the whole election (and future elections) might become a
chaos. It is possible that in some "very strategically oriented"
societies with very stable opinions (e.g. assuming that B can not win
even if A supporters would rank B higher, and C supporters will not
stop liking A because of the strategy) we would get strategic cycles
this way, but it seems probable to me that in most societies this kind
of chaos would not emerge (assuming that some percentage of voters
want to vote sincerely rather than steal the victory etc.) (sorry, no
clear proof available).
Note that random ballot on Smith is adequate for preventing the
burial without any defensive strategy on
the part of the C supporters.
On the other hand the profile
40 A>>C>B
30 B>>C>A
30 C>>A>B
could not arise from a one dimensional or cigar shaped issue space.
And candidate C has such low
uility, it wouldn't be bad if A got a share of the probability
through a burial of C.
One could also consider low utility Condorcet winners to be worth
being elected with 100% probability. The difference in philosophy is
if one tries to find a winner that would offer best sum of utility to
the voters (=> sum of ratings like philosophy) or if one wants to find
a winner that can rule the society thanks to having majority support.
Juho
Random Ballot Smith doesn't discourage burial in this case, in which
C retains only 30% of the
probability. Without more detailed information it would be
impossible to prove that C deserved more than
that amount.
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