Minimax Multi-district Apportionments

The problem of seat apportionment in electoral systems turns out to be quite complex, since no apportionmentmethod exists which succeeds in verifying all the principal fairness criteria. Gambarelli in 1999 introduced an apportionment technique which is custom made for each case, respects Hare minimum, Hare maximum and Monotonicity and satisfies other criteria in order of preference. In this paper a generalization of that method is proposed, in order to extend it to the multi-district election case, where criteria should be respected at a global as well as at a local level. An existence theorem and a generating algorithm are supplied.