On Ordering a Set of Degressively Proportional Apportionments

The most important problem in a practical implementation of degressive proportionality is its ambiguity. Therefore, we introduce an order relation on a set of degressively proportional allocations. Its main idea is to define greater allocations such that emerge from other after transferring a certain quantity of goods from smaller to greater entities contending in distribution. Thus, maximal elements in this ordering are indicated as the sought-after solution sanctioning boundary conditions as the only reason of moving away from the fundamental principle of proportionality. In case of several maximal elements the choice of one allocation remains an open issue, but the cardinality of the set from which we make a choice can be reduced significantly. In the best-known example of application of degressive proportionality, which is the apportionment of seats in the European Parliament, the considered set contains a maximal element. Thereby, there exits an allocation that is nearest to the proportional distribution with respect to transfer relation.