Seat biases of apportionment methods for proportional representation

In proportional representation systems, an important issue is whether a given apportionment method favors larger parties at the expense of smaller parties. For an arbitrary number of parties, ordered from largest to smallest by their vote counts, we calculate (apparently for the first time) the expected differences between the seat allocation and the ideal share of seats, separately for each party, as a function of district magnitude, with a particular emphasis on three traditional apportionment methods. These are (i) the quota method with residual fit by greatest remainders, associated with the names of Hamilton and Hare, (ii) the divisor method with standard rounding (Webster, Sainte-Laguë), and (iii) the divisor method with rounding down (Jefferson, Hondt). For the first two methods the seat bias of each party turns out to be practically zero, whence on average no party is advantaged or disadvantaged. On the contrary, the third method exhibits noticeable seat biases in favor of larger parties. The theoretical findings are confirmed via empirical data from the German State of Bavaria, the Swiss Canton Solothurn, and the US House of Representatives.