# [EM] The Sainte-Lague index and proportionality

Kristofer Munsterhjelm km_elmet at lavabit.com
Tue Jul 10 03:17:46 PDT 2012

```On 07/09/2012 06:33 AM, Michael Ossipoff wrote:

> SL/Webster minimizes the SL index, right? It's known that Webster has
> _no_  bias if the distribution-condition that I described obtains--the
> uniform distribution condition.

> I'm not a statistician either, and so this is just a tentative
> possibility suggestion: What about finding, by trial and error, the
> allocation that minimizes the calculated correlation measure. Say, the
> Pearson correlation, for example. Find by trial and error the allocation
> with the lowest Pearson correlation between q and s/q.

> For the goal of getting the best allocation each time (as opposed to
> overall time-averaged equality of s/q), might that correlation
> optimization be best?

Sure, you could empirically optimize the method. If you want
divisor methods can have it so you just have to find the right parameter
for the generalized divisor method:

f(x,g) = floor(x + g(x))

where g(x) is within [0...1] for all x, and one then finds a divisor so
that x_1 = voter share for state 1 / divisor, so that sum over all
states is equal to the number of seats.

We may further restrict ourselves to a "somewhat" generalized divisor
method:

f(x, p) = floor(x + p).

For Webster, p = 0.5. Warren said p = 0.495 or so would optimize in the
US (and it might, I haven't read his reasoning in detail). Also, I think
that the bias is monotone with respect to p. At one end you have

f(x) = floor(x + 0) = floor(x)

which is Jefferson's method (D'Hondt) and greatly favors large states.
At the other, you have

f(x) = floor(x + 1) = ceil(x)

which is Adams's method and greatly favors small states.

If f(x, p) is monotone with respect to bias as p is varied, then you
could use any number of root-finding algorithms to find the p that sets
bias to zero, assuming your bias measure is continuous. Even if it's not
continuous, you could find p so that decreasing p just a little leads
your bias measure to report large-state favoritism and increasing p just