Disproportionality at Fair Elections Cannot Exceed 1/sqrt(2): A Note on the Gallagher Index and a New Normalised Measure

The Gallagher index of disproportionality is perhaps the most popular measure of electoral distortion in all of political science. While many assume that the index ranges from 0 (perfect proportionality) to 1 (perfect disproportionality), I prove that in any free and fair election its maximum possible value is bounded above by sqrt((N-1)/(2N)), reaching a maximum of 1/sqrt(2) where the number of parties, N, tends to infinity. This bound arises in the limiting case where all parties receive equal vote shares but only one secures representation. Building on this result, I then propose a new normalised version of the index that rescales disproportionality to lie between 0 and 1 under democratic conditions.