Apportionment methods for the House of Representatives and the court challenges

Four different methods have been used to apportion the seats in the United States House of Representatives among the states following the decennial census. The current method, the method of equal proportions, has been used for each census since 1940. In 1991, for the first time in U.S. history, the constitutionality of an apportionment method was challenged in court, by Montana and Massachusetts in separate cases. Montana proposed two methods as alternatives to equal proportions, the methods of harmonic means and smallest divisors, while Massachusetts proposed the method of major fractions. On March 31, 1992, in a unanimous decision, the U.S. Supreme Court upheld the constitutionality of equal proportions. This author wrote the declarations on the mathematical and statistical issues used by the defense in these cases. The declarations in the Massachusetts case contain several new theoretical and empirical results. This paper discusses the technical issues in these cases together with a brief history of the apportionment problem.