# [EM] an entropy formula for the effective number of parties

Kristofer Munsterhjelm km_elmet at lavabit.com
Sun Dec 16 03:38:30 PST 2012

```On 12/14/2012 05:15 AM, Ross Hyman wrote:
> Here is a physics alternative to the "effective number of parties"
> http://en.wikipedia.org/wiki/Effective_number_of_parties
>
> Based on the concept of entropy, a sensible formula for the
> effective  number of parties = exp(-sum_i P_i log(P_i))
>
> where P_i is the portion of the votes or portion of seats for party i.  sum_i P_i =1.
>
> It is sensible because for an election where n parties get 1/n of
> the  vote each and the rest of the parties get zero votes, the effective
> number of parties from the entropy formula is n.

In the latest (unpublished) version of my voting methods analysis
program, I used an exact multinomial goodness-of-fit test for
quantifying proportionality.

(In the program's model, each voter and candidate has a binary vector of
yes/no opinions and rank candidates with lesser Hamming distance above
those with greater. The proportionality is then the goodness of fit of
the distribution of yes-es of the candidates elected to the yes-es of
the voters participating.)

Perhaps something like it could be used for ENPP. I'm not entirely sure
how it'd work, though, and it might not be "number of political parties"
anymore.

I'm mentioning this because I think there's some kind of relation
between the Sainte-Lague index and the chi-squared test. They certainly
look alike. Thus, an "improved" Sainte-Lague index could use the G-test,
and the G-test looks somewhat like an entropy calculation.

Since I first used SLI, then "G-test SLI", then an exact multinomial
test for my program, I thought that something similar might be
applicable to measuring party-wise proportionality. But I can't quite
see what it would be. Perhaps my ideas would be useful to you though :-)

```