[EM] Summary (and tweaking) of recent PR ideas inspired by Andy Jennings the two winner approval posting

[Forest Simmons]

Kristofer,

I'm glad you asked this question.  It shows keen insight and a desire to
explore the possibilities of this idea.

Let's call the space in question, "issue space." Then suppose that the
following condition is satisfied:  If a voter is closer to candidate X than
to candidate Y in issue space, then the voter will rank candidate X above
candidate Y (or else truncate both of them).

Under this condition the ballots that express a preference for X over any
other candidate are the ballots of the voters that are closer to X than to
any other candidate.

So if this condition holds, we don't need to know the precise distances
themselves to order to calculate the number of voters (or ballots) in the
respective Voronoi cells.

However, this condition involves several assumptions, including sincere
rankings.  It could be to our advantage to use a more subtle way of
estimating the number of voters/ballots in the respective Voronoi cells,
based on various possible metrics of distances between ballots.

In the case of range ballots, the L1 distance, or the L2 distance, both of
which reduce to the Hamming distance in the case of Approval ballots,
naturally suggest themselves.  However, neither of these is very
satisfactory, because a large clone set will exaggerate the influence of
that set on the metric.

There are ways to eliminate this clone dependence defect.  One way is to do
a "Singular Value Decomposition" of the matrix whose rows are the score
vectors from the ballots, and then use the eigen-vectors corresponding to
the significant singular values as a basis for the issue space, etc.

If you are interested, I could tell you about my idea of a "Poor Man's SVD"
that is computationally easier than the standard SVD, and even better
adapted to the task of elimination of clone dependence in this context.

It turns out that with either of these methods it is relatively easy to
pinpoint the positions of the respective candidates in issue space
independent of their publically announced personal score ballots.  This is
important for keeping insincere candidate posturing from manipulating the
results.

Furthermore, if this approach is taken, you can supplement (or even almost
entirely replace) the ballots with questionnaires on all of the issues, in
order to find the distances needed for defining the Voronoi cells.  Again
the distances in question need to be decloned.  But that is precisely the
problem solved by SVD analysis in the context of taxonomy, face
recognition, etc: in calculating the "distance" between faces, for example,
you take as many measurements of distances between different features of
the face as you desire.  Some of the these measurements will be highly
correlated.  We can think of these highly correlated measurements as
"clone" measurements.  The gold standard for "de-cloning" these sets of
measurements in this context is the SVD.

If the Voronoi cell members are calculated on the basis of both the
expressed preferences, and some subtle metric, then the degree of
concordance between the two results will be a measure of the sincerity of
the preference ballots.

I'm sure I have generated more questions than I have answered, but I think
they are mostly questions worth looking into.

My Best,

Forest


On Sun, Dec 6, 2015 at 1:22 AM, Kristofer Munsterhjelm <km_elmet at t-online.de
> wrote:

> On 12/06/2015 02:48 AM, Forest Simmons wrote:
> > The idea of convertin lottery methods into PR methods has been around
> > for a long time.  The obvious idea that has been tried over and over in
> > some form or another is to run the lottery on the entire set of ballots
> > with the entire set of candidates, and then elect the set of w
> > candidates with the greatest winning probabilities.  Since that doesn't
> > work very well, either we have resorted to allowing the winners to carry
> > weights with  them into the assembly, or we have gone back to sequential
> > methods with droop quotas, etc.
> >
> > I think a much better idea (that seems to have been entirely
> > over-looked) is to not run the lottery on the entire set of candidates,
> > but to run it on all of the subsets of size w,, and choose the most
> > satisfactory of these subsets.
> >
> > There are three things that wouold make a subset satisfactory:
> >
> > (1)  The  candidates in the set should not have too much difference in
> > the probabilities assigned by the lottery to their subset.  In other
> > words the least probability should be as large as possible.  In other
> > words, the entropy of the probability distribution should be as large as
> > possible.
> >
> > (2)  The candidates should have as high average ratings among their
> > supporters as possible.
> >
> > (3)   There should be few if any ballots in the set that truncate the
> > entire subset.
> >
> > My suggestions are an attempt to incorporate all of these ideals into a
> > way of comparing subsets of the appropriate size.
> >
> > Here's a simple example based only on ranked preference ballots that
> > illustrates the main idea:
> >
> > Suppose that we want to compare two subsets W and W' of the requisite
> > size w.
> >
> > Let b_i be the number of ballots that rank candidate c_i  of the set W
> > above every other candidate of W.  These numbers are the sizes of the
> > Voronoi (or Dirichlet) regions for the respective candidates of W
>
> In what space are these Voronoi regions embedded? (I assume you mean the
> Voronoi region for X is the region of points that are closer to X than
> they are to any other candidate point.)
>
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