2011/7/14 Kristofer Munsterhjelm
> Jameson Quinn wrote:
>
>> I doubt it's monotonic, though it's probably not a practical problem. That
>> is, it would probably be totally impractical to try to use the
>> nonmonotonicity for anything strategic, and it wouldn't even lead to Yee
>> diagram ugliness
Jameson Quinn wrote:
I doubt it's monotonic, though it's probably not a practical problem.
That is, it would probably be totally impractical to try to use the
nonmonotonicity for anything strategic, and it wouldn't even lead to Yee
diagram ugliness.
Nonmonotonicity could be considered an erro
fsimm...@pcc.edu wrote:
If we abandon the Euclidean metric, then we also abandon Voronoi
Polygons; the corresponding idea for more general metrics is that of
a Dirichlet region.
That's strange. The Wikipedia article on Voronoi diagrams mention
diagrams based on L_1 and Mahalanobis distance. Is
fsimm...@pcc.edu wrote:
Here's a simpler version that is basically the same:
Make use of cardinal ratings so that the rating of candidate X on
ballot b is given by b(X).
Define the closeness of candidate X to candidate Y as the dot product
Sum b(X)*b(Y)
where the sum is taken over all b in t
I doubt it's monotonic, though it's probably not a practical problem. That
is, it would probably be totally impractical to try to use the
nonmonotonicity for anything strategic, and it wouldn't even lead to Yee
diagram ugliness.
2011/7/13
> Here's a simpler version that is basically the same:
>
Here's how to prove it with a non-euclidean metric.
1. Assume that we are dealing with a metric (like all of the L_p metrics) whose
"balls" are symmetric
with respect to the origin.
2. Suppose that AB and DE are line segments with the same midpoint C, and
define the set
pi={x| d(x,A)=s(x,B)}
If we abandon the Euclidean metric, then we also abandon Voronoi Polygons; the
corresponding idea for
more general metrics is that of a Dirichlet region.
It would be amusing to see Yee diagrams based on L_1 and L_infinity metrics
Of course, Yee uses the L_2 metric to make his pictures rotation
Here's a simpler version that is basically the same:
Make use of cardinal ratings so that the rating of candidate X on ballot b is
given by b(X).
Define the closeness of candidate X to candidate Y as the dot product
Sum b(X)*b(Y)
where the sum is taken over all b in the set beta of ballots.
That proof assumes a euclidean distance metric. With a non-Euclidean one,
the "planes" could have kinks in them. I believe I have heard that the
result still holds with, for instance, a city-block metric, but I cannot
intuitively demonstrate it to myself by imagining volumes and planes as in
this p
2011/7/13
>
>
> - Original Message -
> From: Kristofer Munsterhjelm
> > fsimm...@pcc.edu wrote:
> ...
> > There may also be another scenario where Majority Judgement (or
> > median
> > ratings, for that matter) would do better than ranked methods.
> > If it's
> > possible for the voters t
Actually, any centrally symmetric distribution will do, no matter how many
dimensions.
The property that we need about central symmetry is this: any plane (or
hyper-plane in higher
dimensions) that contains the center of symmetry C will have equal numbers of
voters on each side of
the plane..
- Original Message -
From: Kristofer Munsterhjelm
> fsimm...@pcc.edu wrote:
...
> There may also be another scenario where Majority Judgement (or
> median
> ratings, for that matter) would do better than ranked methods.
> If it's
> possible for the voters to agree on what, say, "Good
After looking up some old email threads, it now seems to me that I made
a significant mistake in the post below. It is true that the model
underlying Yee diagrams guarantees that there will always be a Condorcet
winner. But apparently that has nothing to do with the two dimensions
being orthogo
fsimm...@pcc.edu wrote:
Trying to build a metric from a set of ranked ballots is fraught with
difficulties, and your outline of a procedure for doing it is
interesting to me.
The simplest, least sophisticated idea I have so far that seems to
have some use is to define the distance between two ca
2011/7/13 Kristofer Munsterhjelm
> fsimm...@pcc.edu wrote:
>
> Of course if we have a multiwinner method, we don't want all of the
>> winners concentrated in the center of the population. That's why we
>> have Proportional Repsentation.
>>
>> Also the purpose of stochastic single winner methods
Bob Richard wrote:
On 7/13/2011 11:14 AM, fsimm...@pcc.edu wrote:
Jameson, I'm surprised that you consider a Condorcet method to be too
extremist or apt to suffer center
squeeze.
Think Yee diagrams; all Condorcet methods yield identical diagrams,
while center squeeze shows up
clearly in met
fsimm...@pcc.edu wrote:
Of course if we have a multiwinner method, we don't want all of the
winners concentrated in the center of the population. That's why we
have Proportional Repsentation.
Also the purpose of stochastic single winner methods ("lotteries") is
to spread the probability around
2011/7/13
> Jameson, I'm surprised that you consider a Condorcet method to be too
> extremist or apt to suffer center
> squeeze.
>
Hmm... you're right, I hadn't recognized that your "remove one of closest
pair" method was Condorcet-compliant, as any pairwise method.
>
> Think Yee diagrams; all
On 7/13/2011 11:14 AM, fsimm...@pcc.edu wrote:
Jameson, I'm surprised that you consider a Condorcet method to be too extremist
or apt to suffer center
squeeze.
Think Yee diagrams; all Condorcet methods yield identical diagrams, while
center squeeze shows up
clearly in methods that allow it.
Jameson, I'm surprised that you consider a Condorcet method to be too extremist
or apt to suffer center
squeeze.
Think Yee diagrams; all Condorcet methods yield identical diagrams, while
center squeeze shows up
clearly in methods that allow it.
Of course if we have a multiwinner method, we do
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