An Impossibility Theorem for Fair Bidimensional Representation: Towards a Biproportional Solution

Abstract

This paper concerns fair (proportional) representation of a population which is divided into categories according to two criteria as is the case, for example, when an electoral system takes into account the geographical as well as the political group of the elector. Our first model leads us to the conclusion that, on the one hand, there can be no fair representation — under our very weak conditions — as soon there are more than two constituencies or more than two parties, but on the other hand, a two by two situation can always be fairly represented. As these results are mainly negative, we go on to a second model and we give a necessary and sufficient condition for the existence of a biproportional delegation.

Several conversations with G. De Meur, X. Hubaut and Ph. Vincke stimulated a revision of a first draft of this paper, including shortcuts in the proof of our possibility theorem.