We analyze properties of apportionment functions in context of the problem of allocating seats in the European Parliament. Necessary and sufficient conditions for apportionment functions are investigated. Some exemplary families of apportionment func
tions are specified and the corresponding partitions of the seats in the European Parliament among the Member States of the European Union are presented. Although the choice of the allocation functions is theoretically unlimited, we show that the constraints are so strong that the acceptable functions lead to rather similar solutions.
The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favour of the square root voting system, where th
e voting weights of member states are proportional to the square root of their population are discussed and a link between this solution and the random walk in the one-dimensional lattice is established. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold q for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic union of M states and derive in this case an explicit approximate formula for the level of the optimal threshold: q simeq 1/2+1/sqrt{{pi} M}. The prefactor 1/sqrt{{pi}} appears here as a result of averaging over the ensemble of unions with random populations.
