[EM] The Sainte-Lague index and proportionality

[Kristofer Munsterhjelm]

On 07/08/2012 12:32 PM, Michael Ossipoff wrote:
>
> 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.

I imagine that the SLI would penalize bias more heavily than random 
inaccuracy. My intuition goes like this: there are only a few ways a 
method can be consistently biased (small-party bias or large-party 
bias), but there are many ways one might have random noise. Therefore, 
if you see bias of any given type in a seat distribution, then that 
would make you much less likely to think the distribution is a good fit 
to the voting distribution than if you saw just random noise of the same 
magnitude. In the same way, if the SLI measures goodness-of-fit between 
the distribution given by the votes and the seat distribution, adding 
consistent bias would produce a worse fit than would random noise of the 
same magnitude.

Chi-squared tests are also pretty good at distinguishing low quality 
pseudorandom number generators from better ones. When PRNGs fail, they 
usually fail by exhibiting bias. For instance, linear congruential 
generators exhibit bias where n sequential numbers fall on either of a 
small number of planes in n-dimensional space, where n depends on the 
generator.

So I think SLI would penalize bias pretty effectively. In any event, 
it's easy to check. Take the voting distribution, then add either 
consistent bias (correlation between s/q and q) to fix the RMSE of the 
result to a predetermined level. Then compare the SLI of the biased 
distribution from the SLI of the randomized one, and do this enough 
times in a Monte-Carlo fashion. If I'm right, the mean SLI should be 
worse for the distributions with bias than the ones with random noise.

> 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.

Now we just have to find a Webster analog of STV :-)

I have attempted to do so, but the methods are pretty complex. STV - or 
rather, the Droop proportionality criterion it meets - is more like LR.