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

Ross Hyman rahyman at sbcglobal.net
Fri Dec 14 06:30:15 PST 2012

```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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