A Theory of Proportional Representation

Abstract

Integer proportional sharing problems mainly arise in electoral studies: apportionment of seats among states, or repartition of seats among the candidate lists in a ballot. This problem of fair sharing has given rise to various mathematical studies. These studies, culminating in Balinski and Young’s Fair Representation, Meeting the Ideal of One Man, One Vote [Yale University Press, New Haven, CT, 1982], fail to yield a global and satisfactory theory of proportional representation because they almost exclusively focus on a particular family of “multiplicative processes.” In order to build such a theory, several new concepts are introduced and two different approaches are considered to the problem, an algorithmic one and a geometric one. Both approaches prove the existence of a family of additive sharing processes in parallel to multiplicative ones, and the geometric approach allows both families to merge into a new wide family of processes containing all classical methods as particular cases. A series of properties is also studied that look desirable for such sharing processes. Introduction of an order structure over measure spaces illuminates classical results about some of these properties and allows the definition and study of a powerful tool for the comparison of sharing processes.