A Solution to the Seat-Product Problem: The Square Root Rule of Elections

Seat-product models of the effective number of parties offer our best explanation for why party-systems fragment. Yet they rely on a simplification. Rather than model the effective number of parties writ large, they models this index using the largest party alone. This induces a positive bias, since small parties have a suppressive effect. I show that we can derive new seat-product models without need for this simplification. The key is to recognise that the effective number of parties is information theoretic in nature. This lets us use well-known information theoretic identities to avoid party shares altogether. Further, this process produces an interesting by-product: a new “root rule” of elections. Just like the cube root of a nation's population approximates the number of seats in its assembly, I show that the square root of the actual number of parties approximates the effective number of parties as well.