Speaker: Daniele Pretolani, University of Modena and Reggio Emilia

Title: Title: Finding and evaluating apportionments via the Gini index

Abstract:
A proportional apportionment is a fair distribution of parliamentary seats among parties – or equivalently among states in a federal system, as in the US House of Representatives and in the EU Parliament. Since distortion is unavoidable when mapping votes into seats, the goal of proportional apportionment methods is to minimize disproportionality, that is, unequal distribution of political representation among citizens. Several indexes have been proposed to measure disproportionality, and typically it can be shown that a certain index (or class of indexes) is minimized by a corresponding (traditional or ad-hoc) apportionment method. The Gini index is a classical measure of inequality, often used in economics and social sciences, and also adopted as a tool for comparing and evaluating apportionments. Some Authors also suggested the idea of a proportional method that aims at minimizing the resulting Gini index. Such method may be useful for evaluation purpouses, if not for actual apportionment. We pursue this idea in this work.
In particular, we consider quotient methods, where the (fractional) natural quota of each party must be rounded either up or down. We show that the rounding of quotas yielding minimum Gini index can be found solving a particular case of quadratic knapsack, a widely studied combinatorial optimization problem. Preliminary computational results, including real cases from the EU Parliament and the US House of Representatives, show that the method is effective, since the instances to solve are rather easy. Extending our approach beyond quotient methods turns out to be a nontrivial task, since the quadratic knapsack structure is lost, and a larger and more complicated model arises. The problem is greatly simplified, however, for the EU Parliament, where the “degressive proportionality” condition introduced by current agreements (Treaty of Lisbon) turns the model into a linear one. During the talk, we shall report on some ongoing research along these lines.

Organizer: CORAL