[EM] Sincere Condorcet Cycles

[Juho]

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.
>
> 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.

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.

Juho


>
> 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.
>
> In an opinion space model, voters prefer candidates closer to them  
> in issue space. So if a voter is at 0, it prefers someone at 0.5 to  
> someone at 1.
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