US vs. European apportionment practices: The conflict between monotonicity and proportionality

To ensure equal representation, the voting districts of a country must be more or less of the same size. Designing such voting districts, however, is not an easy task due to the fact that voting districts are encompassed in administrative regions. Since the respective share of an administrative region, i.e.\ the number of seats its entitled to based on its population, is not necessarily an integer number, it is hard to distribute the seats in a fair way. The arising fair distribution problem is called the apportionment problem. Proportionality of the allocation is the most important, but not the only factor of a fair solution. Monotonicity related difficulties, administrative and demographic issues make the problem more complex. We provide an overview of the classical apportionment methods as well as the Leximin Method - a new apportionment technique designed to comply with the recommendation made by the Venice Commission. We discuss the properties of apportionments and test the most prominent methods on real data.