ABSTRACT
Previous chapters studied several systems for counting votes in which individual voters are entitled to the same number of votes. We also introduced weighted voting systems and a power index that measures the influence of an individual voter when participants have an unequal number of votes. This chapter examines procedures for assigning, hopefully in a fair way, the number of votes each participant in a weighted voting system should receive. In particular, this problem arises in the House of Representatives in assigning seats (and thus votes) to each state, based on each state’s population. The U.S. Constitution does not specify exactly how to do this. If we could allow fractional numbers of seats, then it would be a straightforward task to divide up the 435 House seats in a way proportional to population. The difficulty arises when we require, as is natural, that the number of seats allocated to each state be a whole number. As a result, different methods have competed over the past two centuries to be chosen as the apportionment method for the House of Representatives. These apportionment methods are applicable as well to any situation requiring a fair allocation of limited quantities of items (not necessarily votes) to different-sized groups, for example, distributing a fixed number of computers to the schools within a school district, based on each school’s student population. We also introduce certain paradoxes that arise in analyzing apportionment methods.
