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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation>, as the number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>n</mi> </math> </EquationSource> </InlineEquation> of parties increases indefinitely. The rate of convergence is high and it is of the magnitude of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\frac{1}{n}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </math> </EquationSource> </InlineEquation>. Empirical evidence from the Netherlands elections supports our result. A comparison with the Banzhaf index is provided. Copyright Springer-Verlag Berlin Heidelberg 2014