[EM] city block distance, Condorcet winners, and social-utility maximizers

[Warren Smith]

>Ossipoff [slightly edited]:
The CW [Condorcet winner] has everything to do with SU, because, when voting is spatial, based
on distance in issue-space, the [honest voter] CW is the SU maximizer every time if that
distance is measured as city-block distance.
If we instead use Pythagorean distance,
also called  Euclidean distance, then the CW is always the SU maximizer under such
commonly-assumed conditions as multidimensional normally distributed voters,
or uniformly distributed voters.

Intuitively satisfying?

--WDS: this is a fascinating claim by Ossipoff, together with his claim (which
I find plausible) that basing things on city block (or "L1") distance is just better.

But is it true?   It is certainly true (and with a trivial proof)
that any candidate located at the median voter (median in
all dimensions simultaneously, that is) automatically will be a Condorcet winner
(assuming no ties).  However, as far as I can see, Ossipoff never proved either
part (i) or (ii) of this

CONJECTURE:
with utilities that are decreasing functions of L1 candidate-voter distances
for a set of N candidates distributed arbitrarily somehow in the space,
(i) there always exists a Condorcet winner, and further
(ii) it is always the max-summed-utility candidate.

Is either true?
I believe the following 2D picture constitutes a disproof of (i) and hence also of (ii).
The voters are * and the candidates are ABC
and the vertical scale is to be chosen so that the 3 candidates
are at the corners of a square and hence the votes are
C>B>A
B>A>C
A>C>B:

................*...
........A........B..
....................
....................
.......*..........*.
.................C..
....................

However, Ossipoff is correct that with L2 distance, the CW
always (i) exists and (ii) is the SU maximizer under such
commonly-assumed conditions as multidimensional normally distributed voters.

Some of those voting academics (I believe the ones with "heads up their asses"?)
proved that and indeed more powerful things in 1972, and you can find the cite at
   http://rangevoting.org/IEVS/Pictures.html

Voting system academics may have their heads up their asses.  But they have the advantage
that when they publish theorems, they are usually correct as opposed to false.

Warren D. Smith
http://rangevoting.org