[EM] Thoughts on Burial

[Juho]

On Jul 9, 2010, at 9:09 PM, fsimmons at pcc.edu 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