[EM] Sincere Condorcet Cycles

[Kristofer Munsterhjelm]

I'll get to the rest later, but:

robert bristow-johnson wrote:

>>> 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
>>

[snip]

>>> 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.

Okay. What this means is that if the distribution of voters makes up a 
function that decays from a central point (like a Gaussian), and this 
function has only a single peak (at the central point), and the issue 
space is one-dimensional, then there will always be a CW, no matter 
where the central point resides with respect to the candidates.

That does not mean that everybody is for Dear Leader. Consider a shifted 
example of the "centrist" society above:

            ##
          ######
    ##################
  0-------A------------B-1

Now, there will be some people who prefer B to A, but A is still going 
to win. The median voter is also closer to A than to B.

It's possible to generalize this to something that Warren calls the 
"DDH-median" for multidimensional issue space. See 
http://rangevoting.org/BlackSingle.html for a mathematical treatment of 
that.