[EM] EM] Satisfaction Approval Voting - A Better Proportional Representation Electoral Method

[fsimmons at pcc.edu]

Note that the sum

1+1/3+…+1/(2n+1)

is the integral (with respect to t) from zero to one of the sum

1+t^2+…+t^(2n),

and that this integrand is a finite geometric sum with closed form

(1-t^(2n+1))/(1-t^2) .

So this is the appropriate integrand for a Sainte-Lague version that allows fractional values of n, i.e. that 
works with any kind of range ballot, not just approval.

[EM] Satisfaction Approval Voting - A Better Proportional Representation Electoral Method
Kristofer Munsterhjelm km-elmet at broadpark.no 
Sun May 23 02:03:51 PDT 2010 
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fsimmons at pcc.edu wrote:
> Satisfaction Approval Voting - A Better Proportional Representation Electoral Method
> 
> One way to generalize Proportional Approval voting to range ballots is by
> finding the most natural smooth extension of the function f that takes each
> natural number n to the sum
> 
> f(n) = 1 + 1/2 + ... + 1/n.
> 
> It turns out that we can extend f(n) to all positive real values of n via the
> integral
> 
> Integral from zero to one of (1-t^n)/(1-t) with respect to t
> 
> For PAV generalized to range ballots, first normalize the ratings to be between
> zero and one.

As might be obvious by my messages, I find Sainte-Lague of interest. 
What would the integral be for the corresponding "generalized divisor"
C/(n+C)?

If C is 1, we have D'Hondt. If C is 0.5, we have Sainte-Lague.

> Then for each proposed coalition C of k candidates (assuming there are to be k
> winners) and each range ballot r, let g(r,C) be f(S) where S is the sum of the
> ratings (according to r) of the alternatives in the coalition C.  Elect the
> coalition C with the greatest sum of g(r,C) over the range ballots r.

It would also be possible to do a sequential version, as with PAV.

> One should take as many nominations of winning coalitions as anybody wants to
> submit along with the results of SAV, sequential PAV, STV, and any other
> multiwinner method that can be computed from range style (including approval)
> ballots, and see which one of them has the highest generalized PAV score.

Or for that matter, determine its proportionality and BR as by my 
program. I haven't implemented Approval strategy into it yet, but the 
generator does rate each candidate (not just rank them), so the Range 
version could work.

How would you calculate the harmonic number for fractional values in a 
program? Perhaps the expansion:

ln(n) + gamma + 1/2n^-1 - 1/12n^-2 + 1/120n^-4

would be good enough, at least for test purposes.