On Jul 9, 2010, at 9:09 PM, [email protected] wrote:
Given the following preference strengths for the A and B supporters,
40 A>>C>B
30 B>C>>A,
which of the following preference schedules is more credible or
likely for the C supporters?
(1) 30 C>A>B ? or
(2) 30 C>B>A ?
You might think that there is no way of knowing that one is more
likely than the other. But try mapping
them out in an issue space. It is very easy to make preference
order (2) graphically consistent with the
A and B supporter preference strengths, but impossible to do the
same for preference order (1), unless
we allow extremely non-symmetric metrics, where the distance from x
to y is very different from the
distance from y to x.
I believe that Jameson Quinn is right when he says that most
Condorcet cycles are probably artificial,
i.e. they are caused by strategic truncation or strategic burial.
There may be also other more common cases.
1) Random(ish) variation in votes when there are three almost tied
candidates. Note that in two-party elections the result is very often
very close to a tie. The political campaigns tend to seek ties when
the candidates try to optimize their messages. Near ties may be common
also when there are more than two serious candidates. The probability
of a cycle in a situation where the candidates are so close to
pairwise ties that the result might as well be X>Y or Y>X in all
pairwise comparisons (there are thus no meaningful dependencies
between different pairwise results; transitive opinions are no more
probable than cyclic ones) approaches 25%.
2) True opinion cycles in situations where the opinion space is not as
geographical distance based as in the classical one dimensional and
two dimensional examples. The classical example is one where there are
three voter groups, three hot topics and three main candidates. Group
A wants X. Group B wants Y. Group C wants Z. Candidate M supports X
and lightly also Y. Candidate N supports Y and lightly also Z.
Candidate O supports Z and lightly also X. In this set-up a cycle is
probable. This set-up is quite possible (not unnatural). This means
that although (nearly) geographic one and two-dimensional opinion
spaces may be common, there are also opinions spaces that are not as
tightly based on geographical distances. As in the example that I gave
above, having multiple hot topics (and voters that give different
weights to different hot topics) may create natural cyclic opinion
spaces.
Condorcet efficient methods that discourage these two strategies
will almost always find a Condorcet
Winner.
The only exception should be in the case of a “low utility CW,” and
since ordinal ballots cannot discern
between high and low utility CW’s, there either has to be an
approval cutoff for the purpose of detecting
low utility alternatives or else there has to be a strategic option
for defeating low utility CW’s.
Low utility Condorcet winners may not be good winners. One must
however note that if elected, and if the opinions stay as in your
example below, C can still be a good leader since her opinions may
always have majority support (when there are three proposals on the
table, one from A, one from B and one from C). C could still be the
best leader in a majority based political system. A C led government
might last longer than an A or B led government.
Another interesting point that has not been discussed that much on
this list is question on what kind of candidates will be nominated as
candidates in an election. If we take a two-dimensional opinion space
an allocate the candidates of your example below in it the candidates
could form a triangle (C slightly closer to the centre than the
others). The point is that in this set-up there is plenty of space in
the middle of the opinions space. One could expect that some new
fourth candidate might emerge close to the centre point of the
triangle. Or alternatively the three candidates would change their
policy / opinions and marketing / campaign strategy so that they would
move closer to the centre. It is anyway a very typical political
phenomenon (almost guaranteed) that candidates tend to move close to
other major players in the hope of getting support from some voters
that would otherwise support the other candidates. Typically this
means moving closer to the centre. The point thus is that if this kind
of movements typically happen in elections then the weak Condorcet
winner will actually disappear. It could be be replaced (won) by a
stronger Condorcet winner, or the candidates (maybe the weak Condorcet
winner) could change their policies so that they are more popular
among the voters. My scenario above is not a complete proof, just a
demonstration on how this kind of dynamics might work in real life.
But I believe this phenomenon is quite common in elections with
reasonably free candidate nomination policy or environment that allows
candidates to slowly change the balance of their opinions, and where
the voter distribution in the opinion space is sufficiently continuous.
Consider the following electorate profile, for example:
40 A>>C>B
30 B>>C>A
30 C
Alternative C is a low utility CW. Any Condorcet efficient method
gives C the victory. But alternative A
is the obvious approval winner, so A should have at least some
probability of winning. The natural way to
accomplish this is for the A supporters to bury C.
Careful with the burial recommendations. At the election day the
opinions might have changed. Or maybe the A supporters only claim that
C is a weak Condorcet winner although she is not. Widespread burial
may easily make the results also worse. I'm not sure if the dynamics
of Condorcet methods can be kept sensible if speculation and use of
burial (different scenarios, not only one, with changing poll results,
strategic claims/analysis of the preferences etc.) will be widespread.
Deterministic Condorcet efficient methods may or may not give the
victory to A as a result of such a
burial. But a method that uses a certain amount of randomness to
resolve cycles would give alternative
A some share of the probability, because such a method would not
discourage the burial of a low utility
alternative.
In particular, when the method reduces to random ballot applied to
the Smith set in the case of three
alternatives, in that case it punishes the burial of a high utility
CW but does not punish the burial of a low
utility CW.
Any Condorcet efficient method that doesn’t use at least a modicum
of randomness to resolve cycles
should allow for approval cutoffs and incorporate the approval
information in a way that does not punish
the burial of low utility CW’s.
This is a bit too strong statement for me. I think Condorcet methods
may work fine also without these tricks (I think the Condorcet
elections that have been held so far point in this direction (although
they have mostly not been highly competitive political elections)).
Approval information might add something useful in some cases
(although maybe not required in typical political elections to make
the system work). Maybe approval might be more useful as additional
information than for strategic reasons (e.g. to point out when the
winning alternative of the election does not have sufficient support
to justify changing the leader (or other state of affairs) => new
elections, or the old leader / state of affairs can be kept for the
time being).
Juho
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