Network flow methods for electoral systems

Abstract

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.

Introduction to Computational Social Choice

“…This area of research, which is sometimes seen as being located at the borderline between social choice theory and political science, gives rise to a variety of computational problems. Algorithms for districting are reviewed by Ricca et al (2013) (see also the works of Pukelsheim et al (2012), Ricca et al (2007), and Hojati (1996) for technical contributions to this field). Algorithms for apportionment are discussed by Balinski and Demange (1989), , and Lari et al (2014).…”

Multi-attribute proportional representation

“…Ding and Lin [8] studied a game-theoretic model for a party-list proportional representation system under specific assumptions, and show that computing the Nash equilibria of the game is hard. Also related is the computation of bi-apportionment (assignment of seats to parties within regions), investigated in a few recent papers [22,23,14]. Constrained approval voting (CAP) [4,20] is probably the closest work to our setting (MAPR).…”

Network models and biproportional rounding for fair seat allocations in the UK elections

“…An electorate might want to consider a number of these criteria simultaneously in which case a multi-objective approach would be better. Finally, we note the paper of (Pukelsheim et al 2012) as an excellent review of network modelling approaches for various electoral problems including seat allocation and political districting; the work of (Gaffke and Pukelsheim 2008a, b) on treating the fairness problem by convex integer optimization and duality to structure algorithms; and Pretolani's quadratic knapsack approach to apportionment (Pretolani 2014).…”

“…In particular, we choose to interpret electoral systems as instances of network flow, taking care when translating continuous models to the integer problem underlying the allocation of MPs. Pukelsheim et al (2012) provides an excellent review of network models in this area for the interested reader. In our work, we look at a wide range of objective functions that we can attempt to optimize, chosen to promote criteria that seem reasonable for PR to achieve.…”