Some Properties of Divisor Methods for Legislative Apportionment and Proportional Representation

Some rather elegant properties of linear divisor methods are derived and used to establish upper and lower bounds on the possible variation between apportionment and exact quota entitlement. A probability distribution is derived for this variation, and it is shown that the probability of the variation exceeding one seat is very small with the major fractions linear divisor method.A less rigorous analysis of the nonlinear equal proportions method shows that in practice it is very similar to the major fractions method, but with a very slight bias in favor of small parties (or states). It is concluded that there is no “best†apportionment method, but a knowledge of the properties of the various methods enables a political choice of the most appropriate method.