[EM] Example of Condorcet Missing the Highest Utility Center Candidate

Forest Simmons fsimmons at pcc.edu
Thu Jun 4 19:11:00 PDT 2015

Suppose that a town with 100 voting citizens has 33 voters residing at each
of the three vertices of an equilateral triangle (with two mile sides,
say), and one voter residing within two hundred yards of the the center of
the triangle.

Proposed sites for the new community center are M1, M2, and M3, at the
respective midpoints of the three sides of the triangle, as well as site C
at the center of the triangle’ a couple of hundred yards from the lone
voter that we just mentioned.

Assuming that voters prefer closer sites over more distant sites the
preferences are

33 M1=M2>C

33 M1=M3>C

33 M2=M3>C

01 C

Note that C is the Condorcet Loser, since each of the M’s beats C pairwise
by almost a two-thirds majority, 66 to 34.

On the other hand, C is the IA winner with 100 percent implicit approval.

Candidate C is also the IA-MPO winner with a score of 100-66, compared with
66-33 for the other alternatives.

How about average distance of voters from each of the alternatives?

The average distance to alternative C is 0.66 times the square root of
three miles, or about 1.14 miles.

The average distance to any of the M’s is about 1.24 miles.

If the M’s were moved directly away from their midpoint positions to a
position nearly twice as far from the center as the midpoint position, the
preference schedule based on distances would not change, but the average
distance from voter to any of the M’s would go up from about 1.24 to about
1.52 miles, more than 33 percent farther than the average distance to C.

Think of it: the center location is about 33 percent better on average.  It
cuts the distance in half for the faction that ends up furthest from the
winning M, and doesn’t give a lopsided solution where 33 voters have to go
twice as far (forever after) as the other 66 voters on the vertices of the
triangle.  It is the geometrical center solution and the approval solution,
but it is the Condorcet Loser.
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