[EM] Sincere Condorcet Cycles

[Kristofer Munsterhjelm]

robert bristow-johnson wrote:

>>> I think it would be possible to replace the points with Gaussians and 
>>> still have a cyclical outcome.
>>
>> Yes. This means that the strange looking scenario where all the voters 
>> are exactly at the same opinion space points as the candidates is not 
>> totally unrealistic but rather a rough simplification that may indeed 
>> (at some accuracy level) model typical real life situations.
>>
>>> Then, the interpretation would be: each candidate has some core 
>>> support (center of Gaussian placed at candidate's spot in opinion 
>>> space). As you travel further away from a candidate, the voters at 
>>> that point become more scarce.
> 
> somebody worked out that if the 40 with A were randomly distributed 
> according to a 2-dim gaussian distribution centered at (0,2) that all 40 
> of them would prefer A>B>C>D?  i am certain that such is not the case 
> assuming sigma_x is the same as sigma_y (dunno what else to call it in 
> this context) otherwise there would be an additional assumption about 
> the "slant" of these 40 A voters..  if the sigma_y is big enough there 
> will be some of these 40 "A voters" that will be south of B.  they will 
> be voting B>A and even B>C>A.

I haven't worked out whether the particular case given would work with 
Gaussians as well, though you could make a mathematical argument that 
when everybody's at exactly the candidate position, that's like a 
Gaussian centered on each with an extremely low std deviation. While 
that trick works mathematically, it doesn't resolve your objection which 
is that the distribution is unrealistic. However, it doesn't rule out 
that Gaussians with more plausible sigmas could exhibit cycles.

> now i do not know how the sums of voters will add up, but with 4 
> candidates, there are 4! different ways to order them.  there would be 
> 24 different ways of marking a ballot and a vote count for each of 
> them.  and then i would be curious in how to mathematically set up the 
> conditions on those 24 counts that would suffice to meet the definition 
> of a cycle.  i'm a novice here, but without setting the problem up like 
> that, and making some kind of assumption on the voter distribution on 
> that plane, i do not know how to begin drawing conclusions about how 
> many votes there were in any condorcet pair, then to draw a conclusion 
> on whether there is a cycle or no.  there is still something very 
> fundamental that i am missing.

The easiest way would probably to generate n Gaussians in issue space, 
of varying sigma, then draw many voters from the distributions. The 
position of the voters in issue space, along with the fixed positions of 
the candidates, would be sufficient to assign a ranked ballot to each 
voter according to distance (closer is better).

Sum up the ballots and check for a Condorcet winner. If there is none, 
you've just found a "multiple Gaussians" example of a cycle. Checking 
whether there is a Condorcet winner can be done in linear time: list the 
candidates in random order, then compare the first two in the list, 
removing them from the list and inserting whoever wins the one-on-one. 
Once you're left with a single candidate, check if he beats all the others.

> i s'pose what you could do is draw equidistant lines between the points 
> represented by two candidates (like between A and B, the line would be 
> y=1), then for a given distribution that would be the mixing of 4 
> distributions centered at A, B, C, D (where the mixing coefficients 
> would be 0.4 for A and 0.2 for the rest), then you could integrate both 
> half planes and get a count for the A vs. B race, right?

When you have few candidates, that would be possible. Just determine the 
size of each area where, if a voter is located there, he votes the same 
preference. Then integrate the part of the Gaussians that reside in that 
area, and you have the number of voters who vote a particular way.

I'm not good enough at multidimensional integration to do that, so when 
I do simulations, I just pick the candidates randomly from the 
distribution (as mentioned above) - a sort of Monte Carlo approach.

> what is the formal process that these simulations are built on to run 
> experiments?  sorry, this is the electrical engineer in me.  i hope that 
> might explain why i am a loss about the methodology here.

I think Simmons's example was found through reasoning (like a 
mathematical puzzle), not by simulation, although I could be wrong.

>>> In other words, that's a factionalized society where the great 
>>> majority of the voters have a single favorite they really like, and 
>>> where they don't much like the rest.
>>>
>>> If two candidates are close, they may share each other's voters, but 
>>> there will still be more voters at the candidate points than in the 
>>> middle between the two.
> 
> well, that can be modeled into distribution functions, but i still am 
> dubious that most voters are solidly behind a particular candidate.  and 
> certainly not the case for preferences lower down.  and those 
> preferences count when deciding if there is a cycle or not.  i would 
> guess that, unless the particular voter is a teeny-bit activist that, 
> particularly with 2nd or 3rd choices, she might be using her dartboard 
> to grade the candidates.

I guess real world voters would grade the "important" comparisons. For 
instance, if a voter considers left vs right more important than 
centrist vs autarchic, he might rank the well known right-wing candidate 
ahead of the well known left-wing candidate, but not really bother to 
rank the various left-wing (right-wing) candidates apart from that.

Of course, part of the advantage of a good ranked method like Condorcet 
is that those who do want to be activists, or who do support 
independents, can do so without unduly harming their influence on the 
more mainstream candidates.

So, in short, yes, the example of everybody clustered around their 
favorite is unrealistic. The example's function is more to show that it 
is possible to engineer a cycle in issue-space with honest voters than 
to show that it's realistic. It's a possibility proof.

> boy, i'd like that explained quantitatively.  i am not sure what is 
> meant by "an ideal position".  might you mean that all voters agree 
> about one of two ideal positions (literally poles, making for a 
> polarized electorate: "yer either fer W or you be agin' it")?  if all 
> voters agree about a single ideal position, doesn't that make for a 
> trivial election?  like, here in North Korea, we're all for the Dear 
> Leader.

See my other post. To recap: ideal society means that there's a commonly 
held idea of what's the best political position, where the closer you 
get to that idea, the more people agree with it.

It would seem reasonable that a system should elect the candidate 
closest to this position, but some methods fail to do so. Plurality, for 
instance, may split the space evenly ("Democrats" claiming the left half 
of the Gaussian, "Republicans" claiming the right half). To claim 
voters, the different parties have to look as much like the median voter 
as possible, but because their center of mass is some distance away from 
the center, that ends up being deceptive.

>>> In an opinion space model, voters prefer candidates closer to them in 
>>> issue space.
> 
> i agree with that.
> 
>>> So if a voter is at 0, it prefers someone at 0.5 to someone at 1.
> 
> 
> yes, and i've been going with that model from the beginning.  but it's 
> the whole thing about the assumed distributions and counting the votes 
> that i am not clear about, from how the problem was stated.
> 
> by the way, i can see how we can put 3 candidates on the 3 points of an 
> equilateral triangle (let's say it's centered at (0,0) and we don't give 
> a fig about rotation), and then from polling data of voter preference, 
> determine regions on the plane where a voter's position in that region 
> is logically consistent with their ordered preference.  i am not sure 
> how to do polling to put voters or candidates on a 2-dim grid, just from 
> information regarding who they like and who they don't.

I think it can be done using eigenvector analysis, but I'm not sure 
about this.
Also, it's possible to do it with Range-type votes by using 
dimensionality reduction (artificial coordinates). Artificial 
coordinates are used to model latency on the internet, and the objective 
is to, given distances between points, find n-dimensional points so that 
the distances agree as much as possible. To find the distance between 
two candidates, consider each candidate to be placed in v-dimensional 
space, where v is the number of voters. The coordinates are given by 
those voters' ratings. Then the distance between two candidates is 
simply the Euclidean distances between their v-dimensional points.

In any case, it doesn't matter as far as Condorcet cycles go, because we 
have the privilege of choosing the voter points prior to doing the 
analysis. The argument goes somewhat like:

  1. It's reasonable that voters rank candidates according to how much 
they agree with them on certain issues.
  2. This lends itself to issue space models.
  3. It is possible to engineer an issue space instance where the honest 
ranked ballots derived from it results in a Condorcet cycle.
  4. Thus, unless this instance lies in an area where issue space is 
unlike real elections, honest voting in the real world could lead to a 
Condorcet cycle.

When you say Simmons's example is unrealistic, that argues "this model 
lies in an area where issue space is unlike real elections".

> if the axes of the grid were to represent fundamental sociological 
> orientation, like liberal vs. conservative on the x-axis and libertarian 
> vs. communitarian (some might say "authoritarian") on the y-axis based 
> on questions about values and social issues.  and then rate your 
> candidates on the same basis and mark their position.  in doing that, i 
> am not sure that for two candidates positioned diagonally (that would 
> also have their equidistant boundary line at a diagonal), it would not 
> necessarily be the case that some voter that is closer to A than to B 
> would vote A>B.

Yes, observation of the pattern of "wings" or politial spectra is what 
leads to point 1 above. Though I'm not sure why someone who is closer to 
A than to B would not vote A>B. Do you mean that the grid would be 
insufficient to capture all the factors that might lead the voter to 
prefer B to A, or is there another reason?