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

[Kristofer Munsterhjelm]

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.