[EM] Warren reply

[Michael Ossipoff]

I had said (with Warren’s additions in brackets):
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

I reply:

The CW is defined in terms of actual preferences, with no assumption about 
anyone’s honesty.

Warren continues the quote:


…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.


Warren says:

Intuitively satisfying?

However, as far as I can see, Ossipoff never proved either

part (i) or (ii) of this

I reply:

Forive me, but I don’t know what parts 1 & 2 are. In years previous, I told 
why, with city block distance a CW maximizes SU. I’ll tell it again, later 
in this reply.

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.

I reply:

At no time did I say that, under any particular conditions, there is always 
a CW. So hopefully it isn’t necessary for me to prove part 1, which I never 
claimed.

Warren continues:

Is either true?
I believe the following 2D picture constitutes a disproof of (i) and hence 
also of (ii).

I reply:

Wrong. Disproving 1 doesn’t disprove 2. Even if there isn’t always a CW, it 
could still be that the CW, when there is one, always maximizes SU.

If Warren’s demonstration that, with city-block distance, the CW, when one 
exists, doesn’t always maximize SU consists of showing that there isn’t 
always a CW, then he is mistaken about the validity of his justification for 
his claim.

Warren continues:

However, Ossipoff is correct that with L2 distance, the CW
always (i) exists

I reply:

I’m glad that’s correct, but I still didn’t say it.

Now, why did I claim that, with city-block distance, a CW always maximizes 
SU?

Say there’s a candidate at the multidimensional median point. As Warren 
said, s/he is the CW.

Starting from the CW, say we move some distance from the CW, in a direction 
parallel to one of the axes by which we measure city-block distance. By 
moving in that direction from the median candidate, we’re moving away, by 
city-block distance, from the CW, and from everyone on the other side of the 
CW--that entire half of the candidates who are on the other side of the CW, 
in the dimension in which we’re moving. Of course we’re also moving away 
from everyone whom we’ve passed while moving. Now, say, from there, we move 
in direction perpendicular to the one in which we initially moved. Again, by 
city-block distance, we’re moving away from the CW, and from the half of the 
candidates who are on the other side of the CW, and from everyone whom we’ve 
passed, in the dimension that we’re now moving. In both of those moves, 
we’re moving away from more voters than we’re moving toward, by the same 
distance.

Maybe that isn’t rigorously-stated, and maybe, taking more time, I could 
word it better. But it seems convincing.

Moving, away from more voters than we’re moving toward, by the same 
distance, that can only increased summed distance, decreasing the SU of a 
candidate at that new position.

Mike Ossipoff


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