[EM] Sincere Condorcet Cycles

[robert bristow-johnson]

On Dec 15, 2009, at 3:49 PM, Juho wrote:

> On Dec 14, 2009, at 7:11 PM, Kristofer Munsterhjelm wrote:
>
>> robert bristow-johnson wrote:
>>> On Dec 14, 2009, at 12:06 AM, Dan Bishop wrote:
>>>> robert bristow-johnson wrote:
>>>>>
>>>>> On Dec 13, 2009, at 7:53 PM, fsimmons at pcc.edu wrote:
>>>>>
>>>>>> Here's a natural scenario that yields an exact Condorcet Tie:
>>>>>>
>>>>>> A together with 39 supporters at the point (0,2)
>>>>>> B together with 19 supporters at (0,0)
>>>>>> C together with 19 supporters at (1,0)
>>>>>> D together with 19 supporters at (4,2)
>>>>>>
>>>>>> D is a Condorcet loser.
>>>>>> A beats B beats C beats A, 60 to 40 in every case.
>>>>>
>>>>>
>>>>> i wouldn't mind if someone could decode or translate the  
>>>>> above.  what does "at the point (x,y)" mean in the present  
>>>>> context?
>>>>>
>>>>> much appreciated.
>>>> They're coordinates in a 2-dimensional political spectrum.   
>>>> Assuming Euclidean distances are used, the ballots are:
>>>>
>>>> 40: A>B>C>D
>>>> 20: B>C>A>D
>>>> 20: C>B>A>D
>>>> 20: D>C>A>B
>>> thanks.  where i am still lacking is understanding how the latter  
>>> is derived from the former.  is there some 2-dimensional  
>>> distribution of voters in this plane and the voter's ballot is  
>>> evaluated and preference is a strictly decreasing function of the  
>>> distance?  or are they all at only those 4 points?  i don't  
>>> consider that natural.  i'm pretty much what South Park typecasts  
>>> as "Aging Hippie Liberal Douche" but you might find me an issue  
>>> where i just do not identify with the Democrats (or in Vermont,  
>>> the Progs).  not every voter who is primarily for A is gonna  
>>> consider B to be better than satan.
>>> i think maybe i now understand how the latter is derived from the  
>>> former.  if i do, then i don't consider the scenario to be  
>>> particularly natural.
>>
>> 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.

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.

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?

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.

>> 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.

> One additional way to explain this set-up is via nomination. If  
> there are this kind of groupings of voters in the society then they  
> quite typically each nominate one candidate. They could nominate  
> also two or more candidates but since most currently used election  
> methods do not handle this very well having one candidate per  
> grouping is typical. The groupings are often represented by parties  
> that those people have formed.
>
>>
>> To take an 1D example, the society would be like this:
>>
>>     ###        ###
>>    #####      #####
>> ######################
>> 0-----A----------B-----1
>>
>> whereas a "centrist" society is like this:
>>
>>           ##
>>         ######
>>   ##################
>> 0-----A----------B-----1
>
> If one looks at these examples from the nomination point of view  
> then it is natural that in the first example both groupings will  
> nominate their own candidate. The first distortion to this clean  
> picture is maybe that strategically it may be wise not to nominate  
> a candidate that represents the mean opinion of the party but  
> nominate a candidate that closer to the mean opinion of all the  
> voters. (It may be also enough if the candidate markets herself as  
> being close to the opinions of the other party voters.) In the  
> first example A and B could thus be closer to each other on the  
> horizontal axis.
>
> If the nominated candidates are close to "centrist opinions" as  
> described above then the largest mass of own party voters are at  
> the "more extreme side" of the candidate and therefore this  
> phenomenon may increase the loyalty of the own party voters too  
> (the other party candidate is definitely not closer). This means  
> that the original model where all the party voters were exactly at  
> the same spot with the candidate may be even one step more credible.
>
> Also general loyalty and fighting spirit may make party members  
> quite loyal to their own candidate even if the candidate would not  
> be an exact match with the opinions of all of the party supporters.
>
>>
>> If all voters agree about an ideal position (like in the centrist  
>> society above), then there will always be a Condorcet winner. If  
>> not, there may be cycles.

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.

>> 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.

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.

sorry to be the skeptic.  but i gotta understand the models  
underneath, and i don't yet.

--

r b-j                  rbj at audioimagination.com

"Imagination is more important than knowledge."