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

Forest Simmons fsimmons at pcc.edu
Sat Dec 5 17:48:41 PST 2015


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

Similarly, let b'_i be the number of ballots that rank candidate c'_i  of
the set W' above every other candidate of W'.

Let m = min(b_i) and m'= min (b'_i).

If m > m', then W is better than W' in the sense of (1) above.  Since top
ratings from the ballots are used preferentially, (2) is given some
support.  And the bigger the smallest Voronoi Region, the more likely the
Voronoi Regions will come close to covering the voters, fulfilling
condition (3).

If you can picture the Voronoi regions in your mind, then you can
understand everything I do about this and similar methods based on random
ballot lotteries.

Obviously the method is computationally hard (because of the number of
subsets of size w).  But given a few million such subsets found by other
means, it can quickly compute the m's and put them in order.



On Sat, Dec 5, 2015 at 2:31 AM, Kristofer Munsterhjelm <km_elmet at t-online.de
> wrote:

> On 12/05/2015 01:09 AM, Forest Simmons wrote:
> > I have tweaked things slightly in order to make things slightly simpler
> > and to make sure that when Range style ballots are used the method
> > reduces to standard Range in the one winner case.
> >
> > The general method takes any proportional lottery based on score/range
> > (including approval) style ballots and converts it into a PR election
> > method.
> >
> > Recall that a lottery method is a system of assigning probabilities to
> > candidates.  A lottery method is "proportional" if it assigns
> > probabilities in proportion to the respective faction sizes when all
> > faction members vote with single-minded and exclusive loyalty in favor
> > of their favorite.
>
> Proportionality is a good start, but I'd think one would also need
> proportionality among sets so that the voters aligned with multiple
> candidates don't have to coordinate which candidate to vote on.
>
> One possible DSV way of defining this would be: consider an approval
> election where each voter shows exclusive loyalty to some subset of the
> candidates. Then select the candidates so that the chance of the voter
> getting someone from his subset is maximized, where every voter is
> considered of equal importance.
>
> But I imagine this could be difficult for lottery methods since there
> are many different assignments that meet the property above. For instance:
>
> 10: A B (faction A)
> 10: C B (faction C)
>
> where both the lottery {0% A, 100% B, 0%C} and the lottery {50% A, 0% B,
> 50% C} gives each faction the same chance of getting one of the
> candidates of the favored subset.
>
> I guess that's part of why say, the Droop proportionality criterion,
> also involves the number of seats, not just the number of candidates.
>
> Does your method meet some form of set proportionality? If so, which?
>
> > Where else can we find suitable proportional lotteries?
> >
> > Andy Jennings found that he could take any sequential PR method and
> > convert it into a proportional lottery by cloning all of the candidates
> > as many times as needed, and allowing a huge number of winners.  Then
> > the candidate lottery probabilities are proportional to the number of
> > clones they have in the winning circle.
> >
> > In fact it turned out that Andy;s first application of that technique
> > generated the Nash Lottery by use of sequential PAV adapted to range
> > ballots.
> >
> > More recently, Andy and I have seen that there are many ways to convert
> > practlcally any single winner method into a sequential PR method.
> >
> > Putting all of this together we have the following diagram:
> >
> > single winner -> sequential PR -> Lottery -> multi-winner PR
>
> Andy's way of turning any sequential PR method into a proportional
> lottery sounds quite a bit like how you could turn multiwinner PR into a
> party list method.
>
> Say you have a multiwinner method X and want to turn it into a party
> list method. Let the voters rate or rank each party as if they were
> candidates; then, if the number of seats is s, clone each candidate
> (representing a party) s times. Finally, run an s-seat election using
> the multiwinner method in question.
>
> Andy's method seems to be what you'd get if you let s go to infinity and
> let the method be sequential (I'm guessing to ensure convergence). In
> turn, that suggests that the diagram above could be expanded into:
>
> single winner -> sequential PR -> lottery -> party list PR, multi-winner PR
>
> To go from a lottery to party list PR, let each party's support be its
> probability of being drawn. Then use Webster (or some other party list
> method of choice) to quantize the probabilities into seat numbers. This
> won't be a perfect party list PR method because some parties could end
> up having no seats yet, by having nonzero probability, exclude other
> parties (e.g. LCR in a one-seat situation).
>
> Another option is to just repeatedly draw from the lottery (with
> replacement) until all s seats are filled, but that has greater variance
> than using a party list method for the quantization.
>
> In a similar way, you could also go from lottery to a weighted
> multiwinner method. Select the s candidates with greatest probability,
> elect them, and let their weights in the assembly be proportional to
> their probabilities. But I suspect (as I've been trying to get to in my
> clone posts) that there's an inherent cloning problem to this kind of
> system, and the strategic equilibrium becomes Droop quota PR. That is,
> unless there's some way of preventing a large faction from masquerading
> as multiple smaller factions and filling up more of the s slots. It
> might be better to hold the sum of weights constant and let s vary than
> holding s constant and letting the sum of weights vary.
>
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