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2006-18

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

by Gaffke, N., Pukelsheim, F..


Series: 2006-18, Preprints

MSC:
90C10 Integer programming
90B80 Discrete location and assignment

Abstract:
When the seats in a parliamentary body are to be allocated proportionally to vote counts or population numbers, divisor methods form the primary class to carry out the apportionment. We present a unified characterization of divisor methods, based on primal and dual optimization problems. Our approach embraces pervious and impervious divisor methods, and vector and matrix problems. The essential quantities turn out to be the scores, that is, the quotients of the growth points of the underlying rounding function and the input weights. While scores afford a sequential assignment of seats for vector problems, this is not so for matrix problems. Yet scores continue to play a central role, in that the goal function of the primal optimization problem turns out to be a cumulative product of consecutive scores. The apportionment results are determined by multipliers, which in turn appear as the variables of the dual optimization problem.

Keywords:
Apportionment methods - BAZI - Elementary vectors - Hagenbach-Bischoff method - Iterative proportional fitting procedure - Jefferson method - New Zurich apportionment procedure - Probability distribution with given marginals - Rational probabilities - Sainte-Lague method - Separation theorems - Transportation problems - Webster method