Federica Ricca

Evaluation and optimization of electoral systems

Preface Part I. Classification and Evaluation of Electoral Systems. 1. The four phases of an electoral process 2. A unified description of electoral systems 3. Performance of an electoral system Part II. Designing Electoral Systems. 4. No electoral system is perfect 5. Basic properties for electoral formulas 6. Integer optimization approach 7. Rewarded and punished parties 8. Mixed electoral systems Part III. Designing Electoral Districts. 9. Traps in electoral district plans 10. Criteria for political districting 11. Indicators for political districting 12. Optimization Models Part IV. The Process of Electoral Reform: A Retrospective Critical View of a Political Scientist. 13. A difficult crossroad 14. The planning and politics of political reform Part V. A Short Guide to the Literature.

Local search algorithms for political districting

Electoral district planning plays an important role in a political election, especially when a majority voting rule is adopted, because it interferes in the translation of votes into seats. The practice of gerrymandering can easily take place if the shape of electoral districts is not controlled. In this paper we consider the following formulation of the political districting problem: given a connected graph (territory) with n nodes (territorial units), partition its set of nodes into k classes such that the subgraph induced by each class (district) is connected and a given vector of functions of the partition is minimized. The nonlinearity of such functions and the connectivity constraints make this network optimization problem a very hard one. Thus, the use of local search heuristics is justified. Experimentation on a sample of medium-large real-life instances has been carried out in order to compare the performance of four local search metaheuristics, i.e., Descent, Tabu Search, Simulated Annealing, and Old Bachelor Acceptance. Our experiments with Italian political districting provided strong evidence in favor of the use of automatic procedures. Actually, except for Descent, all local search algorithms showed a very good performance for this problem. In particular, in our sample of regions, Old Bachelor Acceptance produced the best results in the majority of the cases, especially when the objective function was compactness. Moreover, the district maps generated by this heuristic dominate the institutional district plan with respect to all the districting criteria under consideration. When properly designed, automatic procedures tend to be impartial and yield good districting alternatives. Moreover, they are remarkably fast, and thus they allow for the exploration of a large number of scenarios.

Weighted Voronoi region algorithms for political districting

Political districting on a given territory can be modelled as bi-objective partitioning of a graph into connected components. The nodes of the graph represent territorial units and are weighted by populations; edges represent pairs of geographically contiguous units and are weighted by road distances between the two units. When a majority voting rule is adopted, two reasonable objectives are population equality and compactness. The ensuing combinatorial optimization problem is extremely hard to solve exactly, even when only the single objective of population equality is considered. Therefore, it makes sense to use heuristics. We propose a new class of them, based on discrete weighted Voronoi regions, for obtaining compact and balanced districts, and discuss some formal properties of these algorithms. These algorithms feature an iterative updating of the distances in order to balance district populations as much as possible. Their performance has been tested on randomly generated rectangular grids, as well as on real-life benchmarks; for the latter instances the resulting district maps are compared with the institutional ones adopted in the Italian political elections from 1994 to 2001.

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 this paper is to 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. Besides 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 article, 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.

Political districting: from classical models to recent approaches

The political districting problem has been studied since the 60’s and many different models and techniques have been proposed with the aim of preventing districts’ manipulation which may favor some specific political party (gerrymandering). A variety of political districting models and procedures was provided in the Operations Research literature, based on single- or multiple-objective optimization. Starting from the forerunning papers published in the 60’s, this article reviews some selected optimization models and algorithms for political districting which gave rise to the main lines of research on this topic in the Operartions Research literature of the last five decades.

Extensive facility location problems on networks with equity measures

This paper deals with the problem of locating path-shaped facilities of unrestricted length on networks. We consider as objective functions measures conceptually related to the variability of the distribution of the distances from the demand points to a facility. We study the following problems: locating a path which minimizes the range, that is, the difference between the maximum and the minimum distance from the vertices of the network to a facility, and locating a path which minimizes a convex combination of the maximum and the minimum distance from the vertices of the network to a facility, also known in decision theory as the Hurwicz criterion. We show that these problems are NP-hard on general networks. For the discrete versions of these problems on trees, we provide a linear time algorithm for each objective function, and we show how our analysis can be extended also to the continuous case.

Comparing different metaheuristic approaches for the median path problem with bounded length

In this paper we consider the Bounded Length Median Path Problem which can be defined as the problem of locating a path-shaped facility that departures from a given origin and arrives at a given destination in a network. The length of the path is assumed to be bounded by a given maximum length. At each vertex of the network (customer-point) the demand for the service is given and the cost to reach the closest service-point is computed. The objective is to minimize the sum of these costs over all the customer-points in the network. We consider two local search metaheuristics, the well known Tabu search and the Old bachelor acceptance and we propose a comparative analysis of the performances of the two procedures. We implement streamlined versions of Tabu search and Old bachelor acceptance. Our main concern is to provide knowledge about the intrinsic strategy adopted by each metaheuristic and to know whether one is more appropriate than the other for our problem.

Minimax regret path location on trees

This work studies the problem of finding optimal paths with respect to the center, median and centdian objective functions, on networks with uncertain vertex weights that are given as intervals. Our approach looks for minimax regret paths which minimize the worst-case opportunity loss in the corresponding objective function. These problems are NP-hard on general graphs, therefore we study them on trees. We show that a discrete optimal path always exists for each of them, and provide polynomial time solution algorithms.

Bidimensional allocation of seats via zero-one matrices with given line sums

In some proportional electoral systems with more than one constituency the number of seats allotted to each constituency is pre-specified, as well as, the number of seats that each party has to receive at a national level. “Bidimensional allocation” of seats to parties within constituencies consists of converting the vote matrix V into an integer matrix of seats “as proportional as possible” to V, satisfying constituency and party totals and an additional “zero-vote zero-seat” condition. In the current Italian electoral law this Bidimensional Allocation Problem (or Biproportional Apportionment Problem—BAP) is ruled by an erroneous procedure that may produce an infeasible allocation, actually one that is not able to satisfy all the above conditions simultaneously.In this paper we focus on the feasibility aspect of BAP and, basing on the theory of (0,1)-matrices with given line sums, we formulate it for the first time as a “Matrix Feasibility Problem”. Starting from some previous results provided by Gale and Ryser in the 60’s, we consider the additional constraint that some cells of the output matrix must be equal to zero and extend the results by Gale and Ryser to this case. For specific configurations of zeros in the vote matrix we show that a modified version of the Ryser procedure works well, and we also state necessary and sufficient conditions for the existence of a feasible solution. Since our analysis concerns only special cases, its application to the electoral problem is still limited. In spite of this, in the paper we provide new results in the area of combinatorial matrix theory for (0,1)-matrices with fixed zeros which have also a practical application in some problems related to graphs.

The Forest Wrapping Problem on Outerplanar Graphs

In this paper we study the Forest Wrapping Problem (FWP) which can be stated as follows: given a connected graph G = (V, E), with |V| = n, let ?0 be a partition of G into K (not necessarily connected) components, find a connected partition ?* of G that wraps ?0 and has maximum number of components.The Forest Wrapping problem is NP-complete on grid graphs while is solvable in O(n log n) time on ladder graphs. We provide a two-phase O(n2) time algorithm for solving FWP on outerplanar graphs.