Voting power and proportional representation of voters

We prove that for the proportional representative election system if parties’ sizes are uniformly distributed on the simplex, the expected ratio of a party size to its political power, measured by the Shapley–Shubik index, converges to 1, as the number n of parties increases indefinitely. The rate of convergence is high and it is of the magnitude of 1/n. Empirical evidence from the Netherlands elections supports our result. A comparison with the Banzhaf index is provided.