[EM] Whoops! Condorcet vis-a-vis L1 distance situation = even worse than I thought

[Warren Smith]

I erroneously stated (following Ossipoff, who had also claimed this)
that a "trivial proof" would show that the location of the median voter (median in
all dimensions simultaneously) would automatically be
(if a candidate happened to be there) a Condorcet winner.

That's false.  Here is a counterexample:

o\xBxx|xxxxxxxxxxxxxxxx
oo\xxx|xxxxxxxxxxxxxxxx
ooo\xx|xxxxxxxxxxxxxxxx
oooo\x|xxxxxxxxAxxxxxxx
ooooo\|xxxxxxxxxxxxxxxx
oooooo|xxXxxxxxxxxxxxxx
oooooo|\xxxxxxxxxxxxxxx
oooooo|o\xxxxxxxxxxxxxx
------O----------------
oooooo|ooo\xxxxxxxxxxxx
oooooo|oooo\xxxxxxxxxxx
oooooo|ooooo\xxxxxxxxxx
ooCooo|oooooo\xxxDxxxxx
oooooo|ooooooo\xxxxxxxx
oooooo|oooooooo\xxxxxxx
oooooo|ooooooooo\xxxxxx
oooooo|oooooooooo\xxxxx

In this picture, there are two candidates "O" and "X."
O is located at the origin which is
the median x-coordinate and median Y-coordinate of all voters.
There are 10 voters in the quadrant labeled "A", all in the "x" region.
There is 1 voter located at B, and one located at D.
Finally, there are 10 voters in the quadrant labeled "C".

The voters support X over O by 12 to 10.

---

It is true, however, that IF a candidate is located at the all-way-median,
and IF utilities are decreasing functions of L1 distance, THEN that
candidate is the utility maximizer (albeit not necessarily the Condorcet winner).

In conclusion, essentially everything Ossipoff has ever said about Condorcet
winner vis-a-vis city block distance, has now been shown to be
false and does not constitute any
reason to prefer Condorcet methods.

Warren D Smith
http://rangevoting.org