# [EM] Simmons' multiwinner schemes based on ssf(x) ("Simmons staircase function")

Forest Simmons fsimmons at pcc.edu
Sat Dec 10 11:24:14 PST 2016

```Warren makes a good point here about the difficulty if not impossibility of
achieving "strong PR" with a distribution based method along the lines of
what I proposed.

After thinking about that I decided to try a linear decreasing density
function that does re-scale correctly, giving a Quality gauge

Q = integral over 0<x<1 of (1-x)G(x)

where G(x) is the percentage of ballot expectations greater than x.

It turns out that (as Warren sugests) this is precisely equivalent to a
method that is just a sum over sum function of the individual ballot

Q = 1 - Sum over all ballots B in beta of  E(B)-1)^2/#beta

If we restrict our search to methods of the type

Q = Integral over x from zero to one of f(x)*G(x)

then for f(x) to yield proportionality, it is necessary that when p+q=1,
then p*f(p)=q*f(q).  One way to achieve this is for f(p) = 1/p  .
Unfortunately, this is undefined at p = 0, and tends to give a divergent
integral.

Another way to achieve this is to use the constraint p + q = 1 to re-write
p*f(p)=q*f(q) as  (1-q)*f(p)=(1-p)*f(q), which will be satisfied as long as
f(x) = 2 - 2x, for example.

The condition for strong PR (given PR) is a rescaling condition:  for each
positive c less than one, there must exist c2 and c3 such that

f(x/c2+c)/c3 = f(x)

This works for f(x) = 2 - 2x, but unfortunately the PR property appears to
hold only for two factions: If we have three, and  if p + q + r = 1, then
we need pf(p)=qf(q)=r(f(r).  In this case the substitution trick no longer
works.  Is there another way to bypass this obstacle?

On Thu, Dec 8, 2016 at 8:55 AM, Warren D Smith <warren.wds at gmail.com> wrote:

> I should have been clearer about the complaint I had in mind.
> I'll now try.
>
> It seems to me that ANY quality-function Q of the form
>
> Q = integral(x=0 to 1)of   G( ssf(x), x )  dx
>
> where G(a,b) is any formula whatsoever,
> must fail to provide "strong PR."
>
> Why?  Well, if we add "commonly rated candidates" to the picture,
> that cuts the ssf(x) staircase into two pieces (cut at the common rating x)
> and rescales and repositions those two, with a new stairstep in between.
>
> To get strong PR, you need maximizing Q based on the  G(ssf(x), x)
> function
> to act the same on those two cut staircase pieces, as it does on the
> original whole
> staircase, so that we still get PR on the
> NON-commonly-rated-subset of the candidates
> just like we did before.  Since that is what "strong PR" means.
>
> Well, seems to me that simply does not happen... except for some
> trivial e.g. linear
> G formulae, which however failed to provide PR in the first place.
>
> You can also try using the inverse of the ssf(x) function in your
> definition of
> Q, and I tried a few games of that ilk, but it looked to me
> like you always lose.  I.e. always necessarily fail to achieve "strong PR."
>
> Am I confused?  I might be:  this argument I just made
> trying to refute you, is kind of a self-similarity argument.
> I.e. it argues that after the "cut the staircase into two then
> rescale & reposition pieces" operation, that maximizing Q
> has to behave the same, i.e. these operations need to
> be symmetries, sort of.  Then it says: they ain't.
>
> However, you could fight back by arguing that you
> are only going to care (for PR purposes) about approval-style
> range voting only.
> In which case, the whole self-similarity argument I was trying
> to make, is about intermediate scores (between 0 and 1,
> non-approval-style) whose existence you just deny.
>
> In that case, you might be ok, with the right G(a,b)  functions.
>
> --------
>
> Also, I'm not sure why writing Q's in this kind of way using integrals,
> is necessarily more desirable than the old ways we had of trying
> to write Q's using sums over all ballots and/or candidates,
> of functions of scores.
>
>
> --
> Warren D. Smith