Axiomatic Districting

In a framework with two parties, deterministic voter preferences and a type of geographical constraints, we propose a set of simple axioms and show that they jointly characterize the districting rule that maximizes the number of districts one party can win, given the distribution of individual votes (the \optimal gerrymandering rule"). As a corollary, we obtain that no districting rule can satisfy our axioms and treat parties symmetrically.

A computational approach to unbiased districting

In the context of discrete districting problems with geographical constraints, we demonstrate that determining an (ex post) unbiased districting, which requires that the number of representatives of a party should be proportional to its share of votes, turns out to be a computationally intractable (NP-complete) problem. This raises doubts as to whether an independent jury will be able to come up with a “fair” redistricting plan in case of a large population, that is, there is no guarantee for finding an unbiased districting (even if such exists). We also show that, in the absence of geographical constraints, an unbiased districting can be implemented by a simple alternating-move game among the two parties.

The political districting problem: A survey

Computer scientists and social scientists consider the political districting problem from different viewpoints. This paper gives an overview of both strands of the literature on districting in which the connections and the differences between the two approaches are highlighted.

Optimal partisan districting on planar geographies

We show that optimal partisan districting in the plane with geographical constraints is an NP-complete problem.

Axiomatic districting

In a framework with two parties, deterministic voter preferences and a type of geographical constraints, we propose a set of simple axioms and show that they jointly characterize the districting rule that maximizes the number of districts one party can win, given the distribution of individual votes (the "optimal gerrymandering rule"). As a corollary, we obtain that no districting rule can satisfy our axioms and treat parties symmetrically.

The Multiple Hierarchical Legislatures in a Representative Democracy: Districting for Policy Implementation

We build a multiple hierarchical model of a representative democracy in which, for instance, voters elect county representatives, county representatives elect district representatives, district representatives elect state representatives, and state representatives elect a prime minister. We use our model to show that the policy determined by the final representative can become more extreme as the number of hierarchical levels increases because of increased opportunities for gerrymandering. Thus, a sufficiently large number of voters gives a district maker an advantage, enabling her to implement her favorite policy. We also show that the range of implementable policies increases with the depth of the hierarchical system. Consequently, districting by a candidate in a hierarchical legislative system can be viewed as a type of policy implementation device.

Clustering of Territorial Areas: A Multicriteria Districting problem

The success of regional development policies depends on the homogeneity of the territorial units. This paper aims to propose a framework for obtaining homogenous territorial clusters based on a Pareto frontier considering multiple criteria related to territories’ endogenous resources, economic profile and socio-cultural features. This framework is developed in two phases. First, the criteria correlated with development at the territorial unit level are determined through statistical and econometric methods. Then, a multi-criteria approach is developed to allocate each territorial unit (parishes) to a territorial agglomerate, according to the Pareto frontier established.

Measures in Sectorization Problems

Sectorization means dividing a whole into parts (sectors), a procedure that occurs in many contexts and applications, usually to achieve some goal or to facilitate an activity. The objective may be a better organization or simplification of a large problem into smaller sub-problems. Examples of applications are political districting and sales territory division. When designing/comparing sectors some characteristics such as contiguity, equilibrium and compactness are usually considered. This paper presents and describes new generic measures and proposes a new measure, desirability, connected with the idea of preference.

Sectorization: measures and an electromagnetism based approach

Sectorization means dividing a set of basic units into sectors or parts, a procedure that occurs in several contexts, such as political, health and school districting, social networks and sales territory or airspace assignment, to achieve some goal or to facilitate an activity. This presentation will focus on three main issues: Measures, a new approach to sectorization problems and an application in waste collection. When designing or comparing sectors different characteristics are usually taken into account. Some are commonly used, and they are related to the concepts of contiguity, equilibrium and compactness. These fundamental characteristics will be addressed, by defining new generic measures and by proposing a new measure, desirability, connected with the idea of preference. A new approach to sectorization inspired in Coulomb’s Law, which establishes a relation of force between electrically charged points, will be proposed. A charged point represents a small region with specific characteristics/values creating relations of attraction/repulsion with the others (two by two), proportional to the charges and inversely proportional to their distance. Finally, a real case about sectorization and vehicle routing in solid waste collection will be mentioned.

A redistricting problem applied to meter reading in power distribution networks

The capacitated redistricting problem (CRP) has the objective to redefine, under a given criterion, an initial set of districts of an urban area represented by a geographic network. Each node in the network has different types of demands and each district has a limited capacity. Real-world applications consider more than one criteria in the design of the districts, leading to a multicriteria CRP (MCRP). Examples are found in political districting, sales design, street sweeping, garbage collection and mail delivery. This work addresses the MCRP applied to power meter reading and two criteria are considered: compactness and homogeneity of districts. The proposed solution framework is based on a greedy randomized adaptive search procedure and multicriteria scalarization techniques to approximate the Pareto frontier. The computational experiments show the effectiveness of the method for a set of randomly generated networks and for a real-world network extracted from the city of São Paulo. © 2013 Elsevier Ltd.