[EM] Sincere Condorcet Cycles

[Kristofer Munsterhjelm]

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. 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.

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 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.