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

Ross Hyman rahyman at sbcglobal.net
Sat Dec 15 12:00:03 PST 2012

```Two formulas, Wildgen's index of hyperfractionalization and Molinar's index are mentioned without providing the formulas.  Does anybody know what they are?  The entropy formula pops up in many places under different names so I wouldn't be surprised if one of these formulas was the entropy formula I presented.

For Laakso-Taagepera:
n_a = 2, n_b = 8/3 , n_c =4
n_b/n_a = 4/3 , n_c/n_b = 3/2
For Golosov:
n_a= 2,  n_b = 15/7  , n_c = 4
n_b/n_a = 15/14,  n_c/n_b = 28/15

--- On Fri, 12/14/12, Jameson Quinn  wrote:

From: Jameson Quinn
Subject: Re: [EM] an entropy formula for the effective number of parties
To: "Ross Hyman"
Cc: election-methods at electorama.com
Date: Friday, December 14, 2012, 9:59 AM

How would that work using the other formula?
(I know I'm kinda just being lazy here, but I think other people would be interested.)
Jameson

2012/12/14 Ross Hyman

example using entropy formula

two parties that split the vote equally:
1/2 , 1/2     effective number of parties n_a=2
One of these parties divides equally:
1/2, 1/4, 1/4  effective number of parties n_b= 2sqrt(2)
now the other party also divides equally:

1/4, 1/4, 1/4, 1/4 effective number of parties n_c= 4

n_b/n_a = n_c/n_b = sqrt(2)

Splitting one party has the same effect on the ratio, regardless if the other party has split or not.

--- On Fri, 12/14/12, Ross Hyman wrote:

Consider that there are a number of parties, with the ith party having
vote fraction P_i.  Now consider that you can divide the parties into
two, left parties and right parties. Call the vote fraction for the left
parties P_L and the vote fraction for the right parties P_R.  Use the
effective number of parties formula to determine the effective number of
left parties N_L and the effective number of right parties N_R.  Using
the entropy formula,
if only N_L changes, the ratio of the new number of total effective
parties over the old number of total effective parties depends only on
P_L, the new N_L and the old N_L.  It will not depend at all how the
right parties divide up their votes.  No other formula will do this.

Interesting. When is it different from the other formula?

Jameson

Here is a physics alternative to the "effective number of parties" formulas mentioned on the Wikipedia page:

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.

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