# A Multi-Criteria Pen for Drawing Fair Districts: When Democratic and Demographic Fairness Matter

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## Abstract

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## 1. Introduction and Motivation

**Our contribution and paper outline**: Our framework extends the previous work on districting from the political science, applied mathematics, and operational research communities (see [25,26,27,28]), in several directions. First, and to the best of our knowledge, our approach is the first one to include demographic fairness criteria in the systematic design of districts. Second, it optimizes the five criteria simultaneously by constructing the so-called Pareto frontier; therefore, it provides a pool of efficient solutions to the decision-makers. This feature is crucial, since it helps decision-makers to make more informed decisions. Third, we implemented the proposed framework and tested the performance of a recent electoral law reform in Chile and the recent suggestion of the president of Chile to further change the electoral system. In both cases, the comparison between the results of the framework and the implemented and suggested changes shed some light on the interests and gerrymandering issues from each proposal, enabled us to audit the performance and criteria behind the decisions-makers that propel such changes. The results also show that our tool can design electoral systems that considerably improve the current system, both in classical measures of fairness as well in the newly introduced ones. The results that were associated with the suggestion of the Chilean president were published on the main online news portal in Chile on 16th June 2019 [29]. Furthermore, following this publication, a more complete report was delivered to governmental authorities as input for redrawing the current electoral system as part of the constitutional reform project.

## 2. Literature Review

#### 2.1. Malapportionment

#### 2.2. Political Districting: An Optimization Perspective

## 3. The Multi-Criteria Pen for FAIR Districting

**Phase I**: formulate and solve a MILP model that minimizes the total malapportionment of the system by optimally grouping territorial units (i.e., municipalities) and assigning the number of representatives to these groups. Following Chilean regulations, we ensure that electoral districts comply with the regional boundaries (i.e., a district only contains municipalities of a given territory). In this phase, we do not enforce contiguity or compactness and aim to find a fair distribution of districts to regions, and representatives to districts.**Phase II**: the second phase is only required when a region contains two or more districts.- Step 1: the solution from Phase I is repaired to obtain a solution that satisfies connectivity constraints. The resulting design corresponds to a feasible design for the electoral system.
- Step 2: a pool of efficient solutions, i.e., alternative districting plans, is sought by exploring the set of feasible solutions using a multi-objective optimization approach.

**Notation and preliminaries**: Let $V=\{1,\dots ,i,\dots ,n\}$ be the set of elementary territorial units (the municipalities in our case study), so that ${p}_{i}\in \mathbb{N}$ corresponds to the population or a measure of the population of the territorial unit $i\in V$, e.g., ${p}_{i}$ could correspond to the electoral roll of that unit (we will use this definition), and let $P={\sum}_{i\in V}{p}_{i}$. Likewise, let $({\alpha}_{i},{\beta}_{i})$ be the Cartesian coordinates of the geographical centroid of territorial unit $i\in V$; and let $\mathbf{d}$: $V\times V\to {\mathbb{R}}_{>0}$ be the distance function, so that ${d}_{ij}$, $i,j\in V{\mid}_{i\ne j}$, correspond to the distance between unit i and j. The proposed method considers the number of districts and the number of seats as input for the PDP, as these values are usually obtained through political discussions. Let $\lambda $ be the desired number of districts, and let $\sigma $ be the total number of seats. Complementary, let ${\sigma}^{-}$ and ${\sigma}^{+}$ be the minimum number and maximum number of seats to be assigned to any district, respectively.

#### 3.1. The Criteria of Democratic and Demographic Fairness

**Democratic fairness criteria**: For any given solution, the malapportionment of each district, say the district rooted at unit j, is given by

**Demographic fairness criteria**: One assumes that districts are territorially compact, i.e., to be geographically smooth. There are several alternatives to enforce such a behavior within districting plans. One approach is to seek territorial aggregations that are centered with respect to their geographical centroid (see [26,53,78,79,80] and the references therein).

#### 3.2. First Phase: An MILP Model for Minimum Malapportionment

#### 3.3. Second Phase: An Algorithm to Explore Pareto-Efficient Districting Plans

**Definition**

**1.**

## 4. Results and Discussion: The Chilean Case

#### 4.1. Brief Description of the Recent Electoral Reform in the Chilean Parliament

#### 4.2. Computing ${\mathit{MAL}}^{*}({\mathbf{y}}^{*},{\mathbf{x}}^{*},{\mathbf{e}}^{*})$ (First-Phase): Results and Discussion

#### 4.3. Exploring the Efficient Solutions (Second-Phase): Results and Discussion

#### 4.4. A New Redistricting Plan

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Summary of the malapportionment levels obtained for the districting design of a Chamber of Deputies considering 120 seats.

Region District (j) | MILP | # Seats | % Electoral Roll | % District Seats | MAL${}_{\mathit{j}}$ | (1) | (2) |
---|---|---|---|---|---|---|---|

Arica y Parinacota | 1 | 2 | 1.29 | 1.67 | 0.37 | 0.19 | |

Tarapacá | 2 | 2 | 1.70 | 1.67 | −0.03 | 0.02 | |

Antofagasta | 3 | 3 | 3.14 | 2.50 | −0.64 | 0.32 | |

Atacama | 4 | 2 | 1.63 | 1.67 | 0.04 | 0.02 | |

Coquimbo | 5 | 5 | 4.03 | 4.17 | 0.14 | 0.07 | |

Valparaíso | 6 | 2 | 1.63 | 1.67 | 0.04 | 0.05 | 0.04 |

7 | 8 | 6.63 | 6.67 | 0.04 | |||

8 | 3 | 2.49 | 2.50 | 0.01 | |||

Metropolitana | 9 | 8 | 6.68 | 6.67 | −0.01 | 0.10 | 0.07 |

10 | 8 | 6.74 | 6.67 | −0.07 | |||

11 | 8 | 6.69 | 6.67 | −0.02 | |||

12 | 7 | 5.88 | 5.83 | −0.05 | |||

13 | 8 | 6.70 | 6.67 | −0.03 | |||

14 | 8 | 6.66 | 6.67 | 0.01 | |||

O’Higgins | 15 | 2 | 1.77 | 1.67 | −0.11 | 0.09 | 0.11 |

16 | 4 | 3.39 | 3.33 | −0.06 | |||

Maule | 17 | 3 | 2.50 | 2.50 | 0.00 | 0.05 | 0.10 |

18 | 4 | 3.43 | 3.33 | −0.10 | |||

Bío-Bío | 19 | 3 | 2.88 | 2.50 | −0.38 | 0.25 | 0.38 |

20 | 2 | 1.61 | 1.67 | 0.06 | |||

21 | 7 | 5.80 | 5.83 | 0.03 | |||

22 | 2 | 1.65 | 1.67 | 0.02 | |||

Araucanía | 23 | 3 | 2.52 | 2.50 | −0.02 | 0.05 | 0.08 |

24 | 4 | 3.41 | 3.33 | −0.08 | |||

Los Ríos | 25 | 3 | 2.36 | 1.67 | −0.69 | 0.35 | |

Los Lagos | 26 | 6 | 4.93 | 5.00 | 0.07 | 0.04 | |

Aysén | 27 | 2 | 0.66 | 1.67 | 1.01 | 0.51 | |

Magallanes | 28 | 2 | 1.11 | 1.67 | 0.56 | 0.28 | |

120 | 100 | 100 | 2.39 |

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**Figure 2.**Districting designs for the Metropolitana region: Pursuing the best performance for each criterion.

**Figure 3.**Districting designs for the Metropolitana region: Opposite performances of compactness for ${\mathrm{MAL}}^{*}=0.36$.

**Table 1.**Comparison of the main characteristics of the electoral system valid from 1990 until 2015 and the reformed electoral system established in 2015 (see [39]). M stands for the number of representatives under election for the district.

1989–2015 System | Current System | ||
---|---|---|---|

Senate | # of districts | 19 | 15 |

# of seats | 38 | 50 | |

# of seats/district | 2 | from 2 to 5 | |

Allocation method | D’Hondt | D’Hondt | |

Chamber of Deputies | # of districts | 60 | 28 |

# of seats | 120 | 155 | |

# of seats/district | 2 | from 3 to 8 | |

Allocation method | D’Hondt | D’Hondt | |

Both cameras | Electoral Alliances | Allowed | Allowed |

Gender quotas | No | · Temporary until 2029 | |

· At least 40% of candidates must be of the each sex | |||

· Additional expenses reimbursement for elected women | |||

Candidates by list | M | M+1 |

Region | New District (j) | Previous Districts | # Seats | % Electoral Roll | % District Seats | MAL${}_{\mathit{j}}$ | (1) | (2) | (3) | (4) | (5) |
---|---|---|---|---|---|---|---|---|---|---|---|

Arica | 1 | 1 | 3 | 1.29 | 1.94 | 0.64 | 0.32 | ||||

Tarapacá | 2 | 2 | 3 | 1.70 | 1.94 | 0.24 | 0.12 | ||||

Antofagasta | 3 | 3, 4 | 5 | 3.14 | 3.23 | 0.09 | 0.05 | ||||

Atacama | 4 | 5, 6 | 5 | 1.63 | 3.23 | 1.60 | 0.80 | ||||

Coquimbo | 5 | 7, 8, 9 | 7 | 4.03 | 4.52 | 0.49 | 0.25 | ||||

Valparaíso | 6 | 10, 11, 12 | 8 | 5.29 | 5.16 | −0.13 | 0.22 | 0.31 | 1.36 × 10${}^{12}$ | 1.66 × 10${}^{5}$ | 1.57 × 10${}^{5}$ |

7 | 13, 14, 15 | 8 | 5.47 | 5.16 | −0.31 | ||||||

Metropolitana | 8 | 16, 20 | 8 | 7.02 | 5.16 | −1.86 | 4.55 | 1.86 | 1.54 × 10${}^{10}$ | 2.58 × 10${}^{6}$ | 9.69 × 10${}^{4}$ |

9 | 17, 18, 19 | 7 | 5.82 | 4.52 | −1.30 | ||||||

10 | 21, 22, 25 | 8 | 6.64 | 5.16 | −1.48 | ||||||

11 | 23, 24 | 6 | 4.79 | 3.87 | −0.92 | ||||||

12 | 26, 29 | 7 | 6.0 | 4.52 | −1.48 | ||||||

13 | 27, 28 | 5 | 4.16 | 3.23 | −0.93 | ||||||

14 | 30, 31 | 6 | 4.99 | 3.87 | −1.12 | ||||||

O’Higgins | 15 | 32, 33 | 5 | 2.96 | 3.23 | 0.27 | 0.33 | 0.38 | 3.21 × 10${}^{10}$ | 4.63 × 10${}^{4}$ | 4.82 × 10${}^{4}$ |

16 | 34, 35 | 4 | 2.20 | 2.58 | 0.38 | ||||||

Maule | 17 | 36, 37, 38 | 7 | 3.88 | 4.52 | 0.64 | 0.58 | 0.64 | 5.78 × 10${}^{10}$ | 9.47 × 10${}^{4}$ | 9.47 × 10${}^{4}$ |

18 | 39, 40 | 4 | 2.06 | 2.58 | 0.52 | ||||||

Bío-Bío | 19 | 41, 42 | 5 | 3.19 | 3.23 | 0.04 | 0.21 | 0.24 | 1.48 × 10${}^{11}$ | 1.00 × 10${}^{5}$ | 1.23 × 10${}^{5}$ |

20 | 43, 44, 45 | 8 | 5.40 | 5.16 | −0.24 | ||||||

21 | 46, 47 | 5 | 3.36 | 3.23 | −0.13 | ||||||

Araucanía | 22 | 48, 49 | 4 | 1.95 | 2.58 | 0.63 | 0.59 | 0.63 | 1.06 × 10${}^{11}$ | 1.82 × 10${}^{5}$ | 9.10 × 10${}^{4}$ |

23 | 50, 51, 52 | 7 | 3.98 | 4.52 | 0.54 | ||||||

Los Ríos | 24 | 53, 54 | 5 | 2.36 | 3.23 | 0.87 | 0.44 | ||||

Los Lagos | 25 | 55, 56 | 4 | 2.15 | 2.58 | 0.44 | 0.44 | 0.44 | 1.26 × 10${}^{11}$ | 1.63 × 10${}^{5}$ | 2.17 × 10${}^{5}$ |

26 | 57, 58 | 5 | 2.79 | 3.23 | 0.44 | ||||||

Aysén | 27 | 59 | 3 | 0.66 | 1.94 | 1.27 | 0.64 | ||||

Magallanes | 28 | 60 | 3 | 1.11 | 1.94 | 0.83 | 0.42 | ||||

155 | 100 | 100 | 9.96 |

Region | MILP District (j) | # Seats | % Electoral Roll | % District Seats | MAL${}_{\mathit{j}}$ | (1) | (2) |
---|---|---|---|---|---|---|---|

Arica y Parinacota | 1 | 3 | 1.29 | 1.94 | 0.64 | 0.32 | |

Tarapacá | 2 | 3 | 1.70 | 1.94 | 0.23 | 0.12 | |

Antofagasta | 3 | 4 | 3.14 | 2.58 | −0.56 | 0.28 | |

Atacama | 4 | 3 | 1.63 | 1.94 | 0.30 | 0.15 | |

Coquimbo | 5 | 6 | 4.03 | 3.87 | −0.16 | 0.08 | |

Valparaíso | 6 | 8 | 5.42 | 5.16 | −0.26 | 0.22 | 0.26 |

7 | 8 | 5.33 | 5.16 | −0.17 | |||

Metropolitana | 8 | 3 | 2.01 | 1.94 | −0.07 | 0.36 | 0.17 |

9 | 8 | 5.17 | 5.16 | −0.01 | |||

10 | 7 | 4.68 | 4.52 | −0.17 | |||

11 | 7 | 4.68 | 4.52 | −0.17 | |||

12 | 7 | 4.64 | 4.52 | −0.12 | |||

13 | 8 | 5.17 | 5.16 | −0.01 | |||

14 | 6 | 3.92 | 3.87 | −0.05 | |||

15 | 6 | 3.93 | 3.87 | −0.06 | |||

16 | 8 | 5.23 | 5.16 | −0.07 | |||

O’Higgins | 17 | 8 | 5.16 | 5.16 | 0.00 | 0.00 | |

Maule | 18 | 3 | 1.97 | 1.94 | −0.03 | 0.06 | 0.10 |

19 | 6 | 3.97 | 3.87 | −0.10 | |||

Bío-Bío | 20 | 7 | 4.52 | 4.52 | −0.01 | 0.16 | 0.24 |

21 | 7 | 4.60 | 4.52 | −0.08 | |||

22 | 4 | 2.82 | 2.58 | −0.24 | |||

Araucanía | 23 | 6 | 3.91 | 3.87 | −0.04 | 0.06 | 0.08 |

24 | 3 | 2.01 | 1.94 | −0.08 | |||

Los Ríos | 25 | 3 | 2.36 | 1.94 | −0.42 | 0.21 | |

Los Lagos | 26 | 7 | 4.93 | 4.52 | −0.42 | 0.21 | |

Aysén | 27 | 3 | 0.66 | 1.94 | 1.27 | 0.64 | |

Magallanes | 28 | 3 | 1.11 | 1.94 | 0.83 | 0.42 | |

155 | 100 | 100 | 3.29 |

**Table 4.**Current system v/s Our method: Relative differences (in %) in the performance of the five criteria.

Region | #Dist | #Seats | $\Delta \mathbf{MAL}$ | $\Delta {\mathbf{MAL}}_{{\mathit{j}}^{*}}$ | $\Delta \mathrm{D}$ | $\Delta \mathrm{G}$ | $\Delta T$ |
---|---|---|---|---|---|---|---|

Valparaíso | 2 | 16 | 0% | 29% | 68% | 94% | 354% |

Metropolitana | 9 | 60 | 92% | 94% | 75% | −54% | 80% |

Maule | 2 | 9 | 89% | 90% | 17% | 34% | 70% |

Bío-Bío | 3 | 18 | 19% | 53% | 17% | 0% | 76% |

Araucanía | 2 | 9 | 90% | 90% | 19% | 5% | 95% |

**Table 5.**Trade-offs of global democratic fairness (in %): Differences when pursuing compactness, global demographic fairness, and local demographic fairness.

Region | $\Delta min\mathbf{MAL}{\mid}_{{\mathrm{D}}^{*}}$ | $\Delta min\mathbf{MAL}{\mid}_{{\mathrm{G}}^{*}}$ | $\Delta min\mathbf{MAL}{\mid}_{{T}^{*}}$ | $\Delta min\mathbf{D}{\mid}_{{\mathbf{MAL}}^{*}}$ | $\Delta min\mathbf{G}{\mid}_{{\mathbf{MAL}}^{*}}$ | $\Delta max\mathbf{T}{\mid}_{{\mathbf{MAL}}^{*}}$ |
---|---|---|---|---|---|---|

Valparaíso | −1452% | −935% | 0% | −105% | −1481% | 0% |

Metropolitana | −4008% | −16% | −1085% | −160% | 0% | −17% |

Maule | −61% | 0% | 0% | −2% | 0% | 0% |

Bío-Bío | −1058% | −208% | 0% | −4% | 0% | 0% |

Araucanía | −699% | 0% | 0% | 0% | 0% | 0% |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Álvarez-Miranda, E.; Campos-Valdés, C.; Quiroga, M.M.; Moreno-Faguett, M.; Pereira, J.
A Multi-Criteria Pen for Drawing Fair Districts: When Democratic and Demographic Fairness Matter. *Mathematics* **2020**, *8*, 1404.
https://doi.org/10.3390/math8091404

**AMA Style**

Álvarez-Miranda E, Campos-Valdés C, Quiroga MM, Moreno-Faguett M, Pereira J.
A Multi-Criteria Pen for Drawing Fair Districts: When Democratic and Demographic Fairness Matter. *Mathematics*. 2020; 8(9):1404.
https://doi.org/10.3390/math8091404

**Chicago/Turabian Style**

Álvarez-Miranda, Eduardo, Camilo Campos-Valdés, Maurcio Morales Quiroga, Matías Moreno-Faguett, and Jordi Pereira.
2020. "A Multi-Criteria Pen for Drawing Fair Districts: When Democratic and Demographic Fairness Matter" *Mathematics* 8, no. 9: 1404.
https://doi.org/10.3390/math8091404