Network flow methods for electoral systems

Researchers in the area of electoral systems have recently turned their attention to network flow techniques with the aim to resolve certain practically relevant problems arising in this area. The aim of the present paper is review some of this work, showing the applicability of these techniques even to problems of a very different nature. Major emphasis will be placed on “biproportional apportionment”, a problem that frequently arises in proportional electoral systems, but which in some countries is still ill-solved, or not dealt with rigorously, notwithstanding the availability of several sound solution procedures and their concrete application in some real-life elections. Beside biproportional apportionment, we shall discuss applications of network flows to problems such as vote transitions and political districting. Finally, we address the so-called “Give-up Problem”, which arises in the current elections for the Italian Parliament. It is related to the possible assignment of seats to multiple winners of a given party. Based on the results and techniques presented in this paper, it is fair to state that network flow models and algorithms are indeed very flexible and effective tools for the analysis and the design of contemporary electoral systems.