# [EM] Improving the Sainte-Laguë index

Kristofer Munsterhjelm km_elmet at t-online.de
Wed Sep 11 00:55:25 PDT 2013

```The Sainte-Laguë index is optimized by the Sainte-Laguë method. It is:

SUM over all parties p: (V_p - S_p)^2 / V_p

where V_p is the fraction of votes for a party, and S_p is the fraction
of seats. However, the score can range to infinity, so it's not clear
what it measures. Other indices measure disproportionality in percent
and so can't go beyond 100%.

But the Sainte-Laguë index looks very similar to the chi-square value
for goodness of fit:

x^2 = SUM over all entries x: (O_x - E_x)^2 / E_x

where O_x is observed and E_x is expected. Note that since (x-y)^2 =
(y-x)^2 this is equivalent to considering fraction of seats as O_x
(observed) and fraction of votes as expected (E_x). In other words, a
perfectly PR assembly would give exactly the same fraction of seats to
party P as the voters gave party P votes.

What does the Sainte-Laguë index measure? It gives a value on a
chi-square distribution according to how likely the assembly is to have
been drawn in an unbiased manner with respect to the vote fractions,
were the drawing random.

But the statistic itself usually isn't of interest. So that suggests
that one reverses the x^2, i.e. the Sainte-Laguë index, to get a
p-value. And that p-value *can* be interpreted, and does measure
something useful.

At least for large assemblies, drawing an assembly at random would often
give representative results, and some times unrepresentative ones. When
the assembly is unrepresentative, it is unlike what you would expect to
see when the assembly is drawn at random. Thus, if the assembly is
typical of something you would see at random, it is representative. The
value of a PR method, according to that interpretation, would lie in
always getting a representative assembly instead of getting one that
usually, but not always, is representative.

So in order to understand what the Sainte-Laguë index says, it appears
we should consider it as the result of a chi-square test and infer a
p-value from it.

The x^2 has limitations. It may err when there are few seats, or when
there are very many parties with little support each. But since we know
what we're looking for (a p-value of goodness of fit), we can instead
use something that provides it even in those cases: the G-test when the
expected (fraction of votes) numbers are low, and an exact multinomial
(binomial in a two-party case) test when there are few seats.

We can in any case use the G-test instead of the chi-square since (to my
knowledge) the former is strictly closer to the multinomial test than is
the latter. So an improved Sainte-Laguë index looks like

ISLI = 2 * SUM over all parties p: S_p * ln(S_p / V_p),

and will return the same thing the original Sainte-Laguë index does: a
value along the chi-square distribution. These values can be turned into
p-values by means of a chi-square distribution function with n-1 degrees
of freedom, where n is the number of parties.

Finally: the index (and the improved index) measures accuracy or
goodness of fit with respect to support by the voters. Since we use the
same fraction for voter support no matter the number of seats, the most
accurate method would be house monotone. I've already shown instances
where house monotonicity is not desirable, so in some sense, one could
say the index measures accuracy of the wrong thing, at least when there
are few seats.

A way of getting around this is to ask the method to optimize accuracy
of something that is closer to what we want. But "what we want" may not
be directly accessible. It's a relative quantity: "voter X is better
represented by Y than by Z". And so, finding out how to do that in as
good as possible a way is still open to research. The Sainte-Laguë (and
modified SLI) might give a good asymptotic result though (as number of
seats approach number of voters).

(In my disproportionality measurement program for individual candidate
multiwinner elections, I sidestepped this problem by giving each voter,
and candidate, hidden yes/no opinions. The voters would rank the
candidates so those closer in opinion to themselves came first, and then
the disproportionality was determined based on the distribution of
opinions, not candidates, in the assembly and among the people.)

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