[EM] A VSE-like proportionality measure, or the start of one

[Kristofer Munsterhjelm]

Someone on reddit said this list is too focused on single-winner 
methods, and there's been some talk about PR methods here, so I thought 
I'd add something... that ends up being related to my first post on the 
list.

The Sainte-Laguë index, which is optimized by Sainte-Laguë/Webster, is:

f_SLI = sum over parties i:
	(s_i - v_i)^2 / v_i

where s_i is the fraction of seats that party got, and v_i is the 
fraction of votes. (Or analogously: the fraction of seats is what we see 
- "observed" - and the fraction of the vote is what we would expect with 
perfect proportionality - "expected".) Fair enough, but it doesn't seem 
to be useful for non-party list methods.

But then I had a thought: we should be able to just do this in opinion 
space.

Suppose we have a spatial model where candidates and voters' positions 
in opinion space are drawn from the same distribution. Then for each 
opinion space point represented in the winner set, let
	s_i = the fraction of winners whose opinions have this coordinate in 
opinion space, and
	v_i = the probability density at this point.

In the traditional party list case, each candidate and voter occupies a 
position in opinion space corresponding to one of the parties, and each 
voter votes for the party whose opinion he holds. Thus
	v_i = share of voter sharing party i's position
	s_i = fraction of the winner set (seats) given to party i,

which reduces to the traditional Sainte-Laguë index.

Fans of Mike Ossipoff's divisor method might want to use the G-test instead:
	f_GT = 2 * sum over positions i:
		s_i * ln (s_i/v_i)

but it is not optimized by Sainte-Laguë in the party list case.

It's important that the v_i figures use the probability of the 
underlying distribution, not the share of voters who hold the opinion in 
question, because otherwise you could have a candidate who has an 
opinion none of the voters happen to have. Then the Sainte-Laguë index 
would be infinite. Call this the "zero opinion" problem.

===

Now, with a finite number of seats, it's impossible to obtain perfect 
proportionality. So why not use a VSE-like step to normalize the scores?

Let prop_i = (E[f_SLI(optimum)] - E[f_SLI(actual)]) / (E[f_SLI(optimum)] 
- E[f_SLI(random)])

Now prop_i = 0 corresponds to random winner (random winning set), and = 
1 corresponds to optimal.

A nice additional property is that in the single-winner case, with a 
normal distribution, the CW should optimize the measure, as a corollary 
of Black's singlepeakedness theorem.

===

This *almost* works for solving the problem of quantifying 
proportionality for methods like STV. One could imagine something like:

	- Ask a representative sample of the voters a bunch of political questions
	- Ask every candidate the same questions
	- calculate prop_i between the parliament and the inferred voter 
opinion space.

That would give a proportionality measure that works for any PR method 
at all, although it would jumble together voters not ranking/rating 
candidates in order, and disproportionality of the method itself.

But, alas, the "zero opinion" problem is still real, and the more 
questions your survey has, the worse it gets. I think the problem 
ultimately is that we're trying to make a one-sample test (do these 
chosen points represent the distribution) do the job of a two-sample 
test (are these and those points from the same distribution).

Empirically speaking, maybe one could get around this by smoothing the 
voter opinion distribution. But perhaps there's a better theoretically 
founded way to do it. There do, after all, exist two-sample chi-squared 
tests - maybe I should look more into them.

(It would also be pretty easy to augment this measure to include 
descriptive representation.)

===

In any case, this could be useful to evaluate PR methods' 
proportionality, and to check how much proportionality we lose by having 
small districts instead of one large district with something like STV. 
Ranking a thousand candidates may be impractical for real voters, but 
not for computer simulated ones :-)

It's not perfect. Suppose every voter but two has opinion x=0 on a 
single axis. Of the remaining two, one is at x=0.1 and one is at x=0.9 
Due to some kind of bias (ballot restrictions, funding problems, etc.), 
no candidate has this opinion. There are 10 candidates at x=0.1 and 10 
more at x=0.9. By the measure above, electing a council full of 
candidates at x=0.1 is just as proportional (not at all) as electing one 
full of candidates at x=0.9. This seems wrong. It can be dealt with by 
dividing voter space into a number of regions and then counting over 
each "bin", like a histogram... but then just where you put the bins 
will affect the outcome. Maybe there is a kernel density-like solution? 
I don't know it if there is.

Using a continuous distribution mitigates the problem because there's 
always some support and it's always decreasing away from where the voter 
opinions are concentrated[1]. But fully generally, this and the zero 
opinion problems are quite real.

The measure, like any straightforward proportionality measure, can also 
be criticized as quantifying the wrong thing: as Steve Eppley said in 
2008, (paraphrased) "why should I be interested if they have the same 
opinions as me, if they don't legislate the way I would?". In a similar 
way, it doesn't take into account coalition effects or kingmaker 
problems with low threshold PR methods, either.

-km

[1] In the limit - in the spatial model, a particular draw of voters 
might just happen to be placed elsewhere.