[EM] nastier median+L1 counterexample

[Warren Smith]

OK, a nastier median+L1 counterexample is as follows.

2 Candidates:   
X = (8,8)
O = (0,0)

8 Voters:
(-1,-1)
(-2, -2)
(-3, -3)
(4+e, 4)
(4, 4+e)
(19, 19)
(-4-B, 12+e+B)
(12+e+B, -4-B)

where e>0 is a very small number and B>0
is a very big number.

Optionally:
place an additional 9th voter exactly at O.

O is the median among the candidates
in both x-coordinate and y-coordinate, 
but X beats O in a majority vote election where voters
prefer candidates closer in L1 distance (or in Lp distance for
any p>=1).

If the axes are rotated by a random angle,
then O is still the median in both axial directions,
with probability 0.9999 (can be made arbitrarily near 1 but not 1 itself).

Also, I think X always beats O in the election no matter what angle
you rotate the coords.

(Also: There is no "2D median" of this point set.)


Warren D Smith
http://rangevoting.org