Securing System Consistency: Coherence and Paradoxes

Abstract

This chapter assesses apportionment methods from an overall viewpoint as to whether vote weights and seat numbers always correspond in a fair manner. A new organizing principle turns out to be decisive: coherence. It demands that every solution for a general apportionment problem agrees with the solutions for all embedded subproblems. The Coherence Theorem states that an apportionment method is coherent if and only if it is compatible with a divisor method. The ground for the proof is prepared by showing that coherent methods are house size monotone and vote ratio monotone. In contrast, quota methods may produce non-monotonic results of a seemingly paradoxical nature.