[EM] Improving the Sainte-Laguë index

[Kristofer Munsterhjelm]

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.)