Disproportionality is Sensitive to the Number of Parties: A Note on the Gallagher Index and a New Normalised Measure

The Gallagher index is the most popular measure of disproportionality in political science. While many assume that it can range from 0 (perfect proportionality) to 1 (perfect disproportionality), I prove that its maximum value is constrained by the size of the party system. For any election which features N_V effective vote-winning parties, the Gallagher index cannot exceed √(N_V + 1)/(2N_V). Given that party-system fragmentation has risen across developed democracies, researchers may sometimes wish to distinguish how disproportional an election was from how disproportional it could have been, given the size of the party system. As such, I propose a complementary index that runs from 0 to 1, no matter the degree of fragmentation.