<div dir="ltr"><div><div><div><div><div><div>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.<br><br></div>After thinking about that I decided to try a linear decreasing density function that does re-scale correctly, giving a Quality gauge<br><br></div>Q = integral over 0<x<1 of (1-x)G(x)<br><br></div>where G(x) is the percentage of ballot expectations greater than x.<br><br></div>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 expectations, one that we had already tried, namely<br><br></div>Q = 1 - Sum over all ballots B in beta of E(B)-1)^2/#beta<br><br><br></div><div>If we restrict our search to methods of the type<br><br></div><div>Q = Integral over x from zero to one of f(x)*G(x)<br><br></div><div>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.<br><br></div><div>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<br></div><div>f(x) = 2 - 2x, for example.<br><br></div><div>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<br><br></div><div>f(x/c2+c)/c3 = f(x)<br></div><div><br></div><div>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?<br></div><div><br><br></div><div><div><div><div><div><div><div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Dec 8, 2016 at 8:55 AM, Warren D Smith <span dir="ltr"><<a href="mailto:warren.wds@gmail.com" target="_blank">warren.wds@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">I should have been clearer about the complaint I had in mind.<br>
I'll now try.<br>
<br>
It seems to me that ANY quality-function Q of the form<br>
<br>
Q = integral(x=0 to 1)of G( ssf(x), x ) dx<br>
<br>
where G(a,b) is any formula whatsoever,<br>
must fail to provide "strong PR."<br>
<br>
Why? Well, if we add "commonly rated candidates" to the picture,<br>
that cuts the ssf(x) staircase into two pieces (cut at the common rating x)<br>
and rescales and repositions those two, with a new stairstep in between.<br>
<br>
To get strong PR, you need maximizing Q based on the G(ssf(x), x) function<br>
to act the same on those two cut staircase pieces, as it does on the<br>
original whole<br>
staircase, so that we still get PR on the<br>
NON-commonly-rated-subset of the candidates<br>
just like we did before. Since that is what "strong PR" means.<br>
<br>
Well, seems to me that simply does not happen... except for some<br>
trivial e.g. linear<br>
G formulae, which however failed to provide PR in the first place.<br>
<br>
You can also try using the inverse of the ssf(x) function in your definition of<br>
Q, and I tried a few games of that ilk, but it looked to me<br>
like you always lose. I.e. always necessarily fail to achieve "strong PR."<br>
<br>
Am I confused? I might be: this argument I just made<br>
trying to refute you, is kind of a self-similarity argument.<br>
I.e. it argues that after the "cut the staircase into two then<br>
rescale & reposition pieces" operation, that maximizing Q<br>
has to behave the same, i.e. these operations need to<br>
be symmetries, sort of. Then it says: they ain't.<br>
<br>
However, you could fight back by arguing that you<br>
are only going to care (for PR purposes) about approval-style<br>
range voting only.<br>
In which case, the whole self-similarity argument I was trying<br>
to make, is about intermediate scores (between 0 and 1,<br>
non-approval-style) whose existence you just deny.<br>
<br>
In that case, you might be ok, with the right G(a,b) functions.<br>
<br>
--------<br>
<br>
Also, I'm not sure why writing Q's in this kind of way using integrals,<br>
is necessarily more desirable than the old ways we had of trying<br>
to write Q's using sums over all ballots and/or candidates,<br>
of functions of scores.<br>
<span class="gmail-m_-5728686397716368630HOEnZb"><font color="#888888"><br>
<br>
--<br>
Warren D. Smith<br>
<a href="http://RangeVoting.org" rel="noreferrer" target="_blank">http://RangeVoting.org</a> <-- add your endorsement (by clicking<br>
"endorse" as 1st step)<br>
</font></span></blockquote></div><br></div></div></div></div></div></div></div></div></div>