Note on axiomatic properties of apportionment methods for proportional representation systems

Apportionment methods are used in proportional representation systems for the apportionment of parliamentary seats among political parties proportionately to their vote counts, or for the allocation of parliamentary seats between geographical districts proportionately to their population figures. From an axiomatic viewpoint apportionment methods ought to satisfy six basic principles: anonymity, balancedness, concordance, decency, exactness, and fairness. It is well-known that the first two principles are implied by the last four. In this note it is shown that the last four principles are logically independent of each other.