# [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
population-pair monotonicity, then your task becomes much easier: only
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
a little leads your bias measure to report small-state favoritism.

```

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