Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems

When the seats in a parliamentary body are to be allocated proportionally to some given weights, such as vote counts or population data, divisor methods form a prime class to carry out the apportionment. We present a new characterization of divisor methods, via primal and dual optimization problems. The primal goal function is a cumulative product of the discontinuity points of the rounding rule. The variables of the dual problem are the multipliers used to scale the weights before they get rounded. Our approach embraces pervious and impervious divisor methods, and vector and matrix problems.