Seat biases in proportional representation systems with thresholds

In proportional representation systems, apportionment methods are used to convert the number of votes of a party into the number of seats allocated to this party. An interesting characteristic of any such method are the seat biases, that is, the expected differences between the actual seat allocation and the ideal share of seats, separately for each party, when parties are ordered from largest to smallest. For electoral 'systems with a threshold, that is, with a minimum percentage of votes that parties must reach in order to be eligible to participate in the apportionment process, we show that seat biases decrease from their maximum to zero, as the threshold increases from zero to its maximum, and that all seat biases decrease linearly.