Generalized harmonic voting with better proportionality scores

Since Harmonic seems to be better than psi according to the 
proportionality measure I've been using, maybe it'd be useful to try to 
figure out what's going on.

What's the difference between the two? Say the winner set is [A, B, C]. 
Psi doesn't care about whether you really like A and loathe B and C, or 
if your opinions are meh about all three. That is,

	psi(1+0+0) = psi(1/3+1/3+1/3).

d'Hondt Harmonic gives the former set of ratings a score of 1, and the 
latter a score of 1/3 + 1/2 * 1/3 + 1/3 * 1/3 = 1/3 + 1/6 + 1/9 = 11/18, 
which is less. It values highly rated candidates more.

Maybe valuing such candidates more is part of what makes its PR better.

-

On his page about "linear PR methods", Warren Smith defined this class 
of methods:

Suppose a certain set of candidates are the winners and we want to know 
the quality (or score) of that set. One way to do that is to sum up 
scores per ballot, so that one ballot doesn't interfere with another. So 
let score(winners) = sum over voters v: score(winners, v).

For some voter v, let the jth highest rated candidate among the proposed 
winners be have rating R_v,j (my notation) scaled so that 0 is minimum 
and 1 is maximum. Then let the voter's contribution to the toal score be

score(winners, v) = F( sum j=1..|W|: R_v,j * A_j )

where F is some function, W is the set of winners (so |W| is the number 
of winners), and A_1,... is a (possibly infinite) sequence of constants.

The idea is to set the constants A_j so that if the voting is 
approval-style, then F is always some sum of divisors, e.g. for d'Hondt:
	1		if the voter approved of one of the winners
	1 + 1/2		if the voter approved of two
	1 + 1/2 + 1/3	if the voter approved of three, etc.

so F(A_1) = 1, F(A_1 + A_2) = 1 + 1/2, F(A_1 + A_2 + A_3) = 1 + 1/2 + 
1/3, etc;

or for Sainte-Laguë: 1 if the voter approved of one, 1 + 1/3 for two, 1 
+ 1/3 + 1/5 for three, etc.

The funny part is that for any monotone function that grows to +infinity 
(i.e. unbounded positive range), there exist appropriate A values to 
realize this kind of approval-style PR. A silly example with d'Hondt and 
the standard normal quantile (inverse pdf):
	We want A_1 to be set so that q(A_1) = 1
		So A_1 = p(1) ~= 0.841
	We want A_2 to be set so that q(A_1 + A_2) = 3/2
		So A_1 + A_2 = p(3/2) ~= 0.9332
		So A_2 = 0.9332 - 0.841 ~= 0.0918
	etc.

With harmonic voting, F(x) = x. A_j is just 1, 1/2, 1/3, etc. (for d'Hondt).

So let's consider F(x) = x^k for some exponent k. The lower k is, the 
less of a difference there is between a loved/hated arrangement and an 
everything-meh one. For something like F(x) = sqrt(x), a rating of 0.5 
becomes a Harmonic equivalent of sqrt(0.5) ~= 0.71; while with F(x) = 
x^2, the 0.5 becomes 0.25.
	This is not *entirely* true since later ratings are scaled by A_j. But 
for F(x) = x^k, A_1 = 1.

When I did that, it turned out k ~= 2 gave the best results for two 
seats and d'Hondt, with a somewhat smaller value for Sainte-Laguë. I 
haven't checked more than two, because I thought it would be interesting 
to be piecewise polynomial; it might be that the proper exponent for the 
third candidate may be different than the one for the second.

The VSE results I found were, for 2 seats, 10 candidates, 360 voters, 
1440 iterations:
         Name          Proportionality type   k     VSE
         Polynomial    d'Hondt (d=1)          0.01  0.322
         Polynomial    d'Hondt (d=1)          0.5   0.508
         Polynomial    d'Hondt (d=1)          1.0   0.862
         Polynomial    d'Hondt (d=1)          1.5   0.936
         Polynomial    d'Hondt (d=1)          2.0   0.956
         Polynomial    d'Hondt (d=1)          2.5   0.946
         Polynomial    d'Hondt (d=1)          3.0   0.926

         Polynomial    Sainte-Laguë (d=0.5)   0.01  0.322
         Polynomial    Sainte-Laguë (d=0.5)   0.5   0.671
         Polynomial    Sainte-Laguë (d=0.5)   1.0   0.922
         Polynomial    Sainte-Laguë (d=0.5)   1.5   0.952
         Polynomial    Sainte-Laguë (d=0.5)   1.66  0.953
         Polynomial    Sainte-Laguë (d=0.5)   2.0   0.946
         Polynomial    Sainte-Laguë (d=0.5)   2.5   0.924
         Polynomial    Sainte-Laguë (d=0.5)   3.0   0.903

         Sequential P. d'Hondt (d=1)          0.5   0.522
         Sequential P. d'Hondt (d=1)          1.0   0.779
         Sequential P. d'Hondt (d=1)          2.0   0.868
         Sequential P. d'Hondt (d=1)          3.0   0.851

         Sequential P. Sainte-Laguë (d=0.5)   0.5   0.622
         Sequential P. Sainte-Laguë (d=0.5)   1.0   0.796
         Sequential P. Sainte-Laguë (d=0.5)   2.0   0.839
         Sequential P. Sainte-Laguë (d=0.5)   3.0   0.816

(Egalitarian/maxmin was much worse, so I haven't included them here; 
they reached a maximum of around 0.35.)

I then plotted the VSE results for different values of delta (divisor 
method it reduces to) and exponent/polynomial degree. I've attached the 
plots to this post - the dotted lines indicate polynomial degree (k) of 
1 and delta of 0.5, i.e. Sainte-Laguë harmonic. There's an odd almost 
delta-irrelevant increase in VSE for sequential polynomial right above 
degree 1. Generally, the sequential methods do somewhat worse than the 
global ones, which fits with the pattern for Harmonic itself. Note also 
the change of scale for maxmin's gradient - I had to set the max to 0.5 
or it'd be near-impossible to see the pattern.

On the one hand, it's interesting that pushing the polynomial above 
linear can improve Harmonic. But on the other, I dont't see any obvious 
way to describe the maxima theoretically, or any theory that could 
explain why they're located there. It kinda feels like a curve fit: 
something closer to the best fit of a method to an approach that doesn't 
directly align with the method itself. The proportionality metric lends 
itself to clustering/assigning groups of voters "their" candidate, and 
Harmonic isn't a clustering-like method.

Perhaps there's an explanation, though, and I'm just missing it :-)

-km

----
Election-Methods mailing list -- [email protected] - see https://electorama.com/em for list info