# [EM] The Sainte-Lague index and proportionality

Michael Ossipoff email9648742 at gmail.com
Sun Jul 8 03:32:43 PDT 2012

```Kristofer:

When I read about it in the '80s, I, too, noticed the similiarity to
chi-squared.

Since we don't disagree about Sainte-Lague being the best, then the best I
get to do is quibble about _why_ it's the best.  :-)

The Sainte-Lague index doesn't seem to me to be the important thing. It's a
global measure, and that doesn't have the direct importance to individual
states, districts or parties, as compared to an individual measure.
...when we're talking about a measure of deviation of s/q values from what
they should ideally be.

For that, it seems to me that the important measure is also the simplest
and most obvious one, the one that results directly from SL's rounding-off:

Each party, state or district is as close as possible to its correct
proportional share.

But there is one global measure that I consider to be the most important.
The measure of correlation between q and s/q.

You know, as desirable as it is for everyone to be as close as possible to
their correct proportional share, _bias_ seems to me to be the really
important consideration in apportionment and PR. Size-bias.

Sainte-Lague Webster is the unbiased divisor method, and SL/Webster and
Largest Remainder are the unbiased methods, if, for any interval between
integers N and N+1,
every point in that interval is equally likely to be where we'd find a
party, district or state.

But that condition doesn't really obtain. Weighted Webster is more unbiased
than Webster.

I'm sure that SL/Webster is quite unbiased enough. And it's simple, and
it's well-precedented.  I don't propose any other method for apportionment
or PR.

Well, if splitting strategy were ever a problem, and if changing the 1st SL
denominator from 1 to 1.4 or 2 didn't satisfactorily get rid of the
problem, then I'd suggest Largest-Remainder as the solution. Raph suggested
2. I've read that Scandinavian countries that use Sainte-Lague use 1.4.  I
haven't heard of any actual problem with splitting strategy. People
sometimes write bad-examples of such strategy, but actually doing the
strategy would be difficult and dangerous. And even could be feasible and
problematic in ordinary SL, it would be much less so with the Modified SL
that uses 2 or 1.4 as the first SL denominator.

Ideally, in principle, in ordinary SL, splitting could give some people
double their rightful s/q. When the 1st SL denominator is changed from1 to
2, they can only get 4/3 of their rightful s/q by that strategy. ...for the
great difficulty and risk involved with that strategy.

Evidently, splitting strategy is not a problem for modified SL. I merely
mentioned Largest-Remainder as a solution in the event that splitting were
a problem in Sainte-Lague, and remained so in Modified Sainte-Lague.

Though I prefer SL/Webster, and consider it the best, Largest-Remainder
deserves credit for its advantages: Like SL, it's unbiased (under the
conditions that I specified), and it "satisfies quota"--if a party's number
of Hare quotas is between N and N+1, that party will get either N seats or
N+1 seats.

In Raph's bad-example, LR was unaffected by the splitting strategy, and
gave the same allocation as did SL under sincere voting.

Though I respect and praise LR's advantages, it seems that Sainte-Lague, or
at least Modified Sainte-Lague, doesn't really have a splitting problem,
and that SL is the best. Sainte-Lague should be the recommended allocation
method, for PR as well as for apportionment.

Mike Ossipoff

On Sat, Jul 7, 2012 at 11:56 AM, Kristofer Munsterhjelm <
km_elmet at lavabit.com> wrote:

> The Sainte-Laguë index is a measure of disproportionality that is
> minimized by Sainte-Laguë / Webster. (Michael Gallagher also recommended it
> as "the standard measure of disproportionality".)
>
> The Sainte-Laguë index is smiply the sum of, over all parties (or other
> distinct groups), (V_p - S_p)^2 / V_p, where V_p is the share of votes for
> party (or group) p, and S_p is its share of seats. If there were as many
> seats as voters, then V_p - S_p would be 0 and 0/x is 0 for any x != 0, so
> in the case of perfect proportionality, this index is 0.
>
> However, the case of perfect disproportionality shows a problem with this
> index. If there's a party who gets no votes whatsoever, then V_p is 0 and
> you get a division by zero. It's easy to, for this case, either say 0/0 = 0
> or just exclude zero-vote parties (as adding a party with no seats and no
> votes shouldn't have an effect), but if that party gets a seat, then the
> index resolves to infinity. It's pretty unlikely that a party with no votes
> would get a seat, but if a party with a low vote share would happen to get
> a seat, that could unbalance the index, so it'd be useful to find something
> that acts like the Sainte-Laguë index but handles those situations better.
>
> The expression of SUM over p, (V_p - S_p)^2 / V_p looks a lot like the x^2
> of the chi-squared test. If we multiply both V_p and S_p by the number of
> seats, we get a chi-squared test where the expected value is the number of
> seats the given party "ought" to have (in the ideal case), and the observed
> value is the number of seats it actually got -- although then the x^2 value
> is used directly instead of transformed into a p-value.
>
> And to my knowledge, the same problem exists in the context of chi-squared
> tests. There, they use rules of thumbs like "where there is only one degree
> of freedom, the approximation is not reliable if expected frequencies are
> below 10".
>
> One could go in two directions, then. First, that the Sainte-Laguë index
> is related to a chi-squared test of the probability that the seats were
> sampled from the distribution of ideal number of seats as given by the
> voters. Then, other ways of measuring goodness-of-fit might work where the
> Sainte-Laguë index itself fails. Perhaps an exact multinomial test would
> work for small assemblies. If one needs to have numbers similar to the
> Sainte-Laguë index, one could just reverse the final step of the
> chi-squared test (and go from p-value to x^2 rather than vice versa).
> Second, improvements to the chi-squared test could be used to improve the
> Sainte-Laguë index. Again, as an example, one could construct a
> "Sainte-Laguë G-index": 2 * SUM (over p) of ln(S_p/V_p), which is to the
> Sainte-Laguë index what the G-test is to the chi-squared test. Note,
> though, that this still has the original SLI's division-by-zero problem,
> and to get the same independence of no-vote parties, one'd have to set
> ln(0/0) = 0.
>
> (Usual disclaimer: I Am Not A Statistician.)
>
> ----
> Election-Methods mailing list - see http://electorama.com/em for list info
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20120708/5915f737/attachment-0004.htm>
```