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). In some cases, this is a problem, since it makes it difficult to compare disproportionality across time and space, especially given rising fragmentation. As such, I propose a new normalised index that runs from 0 to 1, no matter the size of the party system.