Apportionment with parity constraints

In the classic apportionment problem, goal is to decide how many seats of a parliament should be allocated each party as result an election. The divisor methods solve this problem by defining notion proportionality guided some rounding rule. Motivated recent challenges in context electoral apportionment, we consider question allocate under parity constraints between candidate types (e.g., equal number men and women elected) while at same time satisfying proportionality. We study two different approaches question. first provide theoretical analysis recently devised mechanism based on greedy approach. then propose analyze that follows idea biproportionality introduced Balinski Demange. contrast with biproportional method Demange, ruled levels proportionality: Proportionality satisfied level parties means method, used candidates type party. A typical benchmark two-dimensional fair share (a.k.a matrix scaling), which corresponds ideal fractional solution. lower bounds distance these solutions, explore their consequences apportionment.