Spread spectrum analysis: IR vs Condorcet

[Lowell Bruce Anderson]

On Oct 26, 12:05pm, Steve Eppley wrote:
> Ignoring the "self-fulfilling" aspect of that prediction caused by 
> the spoiler and LOE dilemmas, perhaps we could attempt some sort of 
> comparison of IR and Condorcet with varying voter distributions and 
> some rough approximations?  The following is a crude first attempt.  
> (Undoubtedly this has already been covered in the academic literature.
> References, anyone?)
>-- End of excerpt from Steve Eppley

I've heard of analyses involving voter distributions being called "spatial 
models," and, in particular, Steven Brams has written a monograph entitled 
"Spatial Models of Election Competition" (1979).  It may not discuss exactly 
what you have in mind; but it, and its references, could be a good place to 
start.  You could also try searching for "spatial models."

My favorite spatial model is 2-dimensional, say the [0,1]x[0,1] unit square.  
The axes represent positions that could be taken on two important issues (or 
aspects of government) in the election being held, where 0 is at one 
extreme, and 1 is at the other extreme, for each issue.  Each candidate and 
each voter is represented by a (separate) point in that square.  The 
location of each candidate's point depends, of course, on the stand that 
candidate takes concerning the two issues.  Likewise, the location of each 
voter's point depends on the opinions that voter holds concerning these 
issues.  Each voter can then be pictured as ranking the candidate(s) closest 
to his or her position as first in his or her preference, then ranking the 
candidate(s) second closest to his or her position as second in his or her 
preference, and so on.  This model is easily extended to n dimensions if the 
voting situation warrants such an extension.

I think that 2 dimensions is quite realistic in many voting situations.  In 
a 2-dimensional situation, suppose that there are 3 candidates, and suppose 
that the positions of these 3 candidates do not lie on a straight line.  
Then the perpendicular bisectors of the three lines connecting the three 
different (unordered) pairs of candidates divide the unit square into 6 
different regions, one each for each of the 6 permutations (i.e., the 6 
different rankings with no ties) of the 3 candidates.  If a voter's position 
happens to lie on one of these perpendicular bisectors, then that voter is 
indifferent between the two candidates whose positions give rise to that 
bisector.  If a voter's position happens to lie at the intersection of the 
three bisectors, then the voter is indifferent to all 3 candidates.

If one constructs such a 2-dimensional model, and the positions of the 
candidates all turn out to lie on a particular straight line, then the 
situation reduces to being a 1-dimensional situation along that line.

Bruce