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# Estimating the Electoral Consequences of Legislative Redistricting in France

Evripidis Bampis*

LIP6

Sorbonne Université

Paris, France

evripidis.bampis@lip6.fr

Thomas Ehrhard

Université Paris-Panthéon-Assas

Paris, France

thomas.ehrhard@gmail.com

Bruno Escoffier

LIP6

Sorbonne Université

Paris, France

Bruno.Escoffier@lip6.fr

Claire Mathieu

Institut de Recherche en Informatique

Fondamentale

CNRS and Université Paris Cité

Paris, France

clairemmathieu@gmail.com

Fanny Pascual

LIP6

Sorbonne Université

Paris, France

fanny.pascual@lip6.fr

David Saulpic

Institut de Recherche en Informatique

Fondamentale

CNRS and Université Paris Cité

Paris, France

david.saulpic@irif.fr

## Abstract

We study in this paper the influence that redistricting can have on the result of elections. First, we propose a definition of diversity of the distribution of legal electoral maps of a state, to capture the gerrymandering potential; then, for a specific electoral map of a state, we propose a definition of outliers (*positive outlier* or *negative outlier* with respect to a given political party), to identify the states where the electoral map ought to be audited.

We apply this approach to the French system, that has legal constraints very different from the U.S., much more granular and much less balanced. We first show that despite the legal constraints, there is some large diversity of maps, and as a consequence it is possible to draw the maps in order to favor one specific party in the aggregate (over all states). Then, we show that many states are positive or negative outliers, with a large imbalance between different parties. This calls for a specific audit of redistricting in those states, to examine whether the current map is indeed drawn in a way to advantage or disadvantage some party.

This is the first computer study of gerrymandering potential in France.

## CCS Concepts

* Applied computing → Law.

## Keywords

Redistricting, Gerrymandering, Audit electoral maps

## ACM Reference Format:

Evripidis Bampis, Thomas Ehrhard, Bruno Escoffier, Claire Mathieu, Fanny Pascual, and David Saulpic. 2025. Estimating the Electoral Consequences of Legislative Redistricting in France. In *The 2025 ACM Conference on Fairness, Accountability, and Transparency (FAccT '25), June 23–26, 2025, Athens,*

<figure><img src="image_2.png" /></figure>

This work is licensed under a Creative Commons Attribution 4.0 International License. FAccT '25, Athens, Greece

© 2025 Copyright held by the owner/author(s).

ACM ISBN 979-8-4007-1482-5/25/06

https://doi.org/10.1145/3715275.3732111

Greece. ACM, New York, NY, USA, 13 pages. https://doi.org/10.1145/3715275.3732111

## 1 Introduction

Many countries around the globe elect their members of parliament (MPs) based on some partition of the country into *districts*: there is a local election in each district, to choose its local MP. The choice of the districts boundaries may naturally impact the outcome of the election, as one could draw the boundaries based on the residential patterns of voters of districts in order to favor one result. The legal constraints on how these boundaries can be drawn are often the following: each district must (1) be connected, (2) be included into some *state* and (3) contain roughly the same number of people as the other districts in the state. The first two constraints ensure that the election is a local election, while the third represents the principle “one person, one vote”, to ensure that any vote across the country contributes the same fraction of MP.

Assume that the voting patterns of the smallest indivisible units (the *atomic units*) are known. Gerrymandering is the act of choosing a map, given the legal constraints, that favors one's own party. In the U.S., a large body of research has shown that gerrymandering is a common sport [18].

However, the U.S. case is particular in three aspects: first, the political landscape is strongly polarized around 2 parties, and therefore simulating elections is relatively easy; second, the map maker<fnref n="1" /> has a huge freedom in designing boundaries of the district, as the states are particularly big and the smallest atomic units are particularly small – a few hundred people, and, in average, 160 000 atomic units in each state. Last, the Supreme Court tends to set constraints based on individuals (see e.g. Gill v. Whitford, as explained in [20]) rather than global indicators.

Those strong features are not universally shared: for instance, in France, the smallest atomic unit is called a *canton*, and contains on average 30 000 inhabitants; and the state is a *département*, composed of 20 cantons on average. In such settings, it is not obvious that the map maker has enough room to gerrymander.

<footnote>*Authors are listed in alphabetic order.</footnote>

<footnote n="1">¹We call map maker any person or organization that has control over the redistricting process.</footnote>
Therefore, we first question the obvious, in non-U.S. cases: *Is gerrymandering at all possible?* And if so, to what extent?

An ideal gerrymandering algorithm would proceed as follows: for each legal map, compute the result of the election; then choose, among all legal maps, the one that most favors the algorithm's target party.<fnref n="2" /> Such an algorithm is not feasible, for three reasons: first, it might not be possible to go over the set of every legal maps; second, the votes of the smallest units might not be known exactly, so the result of the election cannot be computed; third, such an extreme algorithm would result in a public outcry.

Let us now discuss the question of estimating the votes of the smallest units. Such an estimate is country-dependent. In the U.S., this means having an estimate of the proportion of Democrats vs. Republicans in each of the smallest units, and that is typically estimated from the demographics and the past votes in the local voting station. In France, there are more than 2 main parties, so the election proceeds in two rounds. The local polling stations are subdivisions of the atomic units, so the past votes of those units are known exactly, regarding the first round. However, knowing those votes is not enough to fully simulate an election, as simulating the second round requires knowledge of the preferred candidate of each voter from among the subset of candidates that remain in the running for the second round of the election, but that subset is map-dependent. Estimating such a preferred candidate is a challenging question: to the best of our knowledge, it is currently not addressed in political science.

To bypass the difficulty of simulating the second round of elections, we propose to measure the *diversity* of the set of legal maps. We measure diversity in terms of the simpler quantity to estimate, namely the preferred party of voters – that they express in the first round of the election.<fnref n="3" /> A “diversity zero” should mean that, in any legal map, the result is the same. In that case, there is nothing to gerrymander, as the result is not influenced by the map – and, thus, the answer to our question is easy. On the other hand, “high diversity” should mean that there are many ways of drawing the map resulting in different results. Then, the choice of the map is more likely to influence the electoral result, and hence gerrymandering is possible.

Thus, we transform our initial question, and ask:

(Q1) How diverse is the set of electoral maps?

In case where the diversity of maps is high –and, therefore, we suspect that gerrymandering is possible – the natural next step is to examine whether a given map is the result of a gerrymandering process. Many metrics exist for this, as for instance partisan bias [19] or the efficiency gap [26]. However, they are mostly suited to bi-polarized scenario and first-past-the-post electoral system. In elections with more than 2 leading parties, with a two-round majority system, it is not clear how to use meaningfully those measures of gerrymandering – as we discuss further in Section 1.5.

In addition, most of the metrics suggested to evaluate gerrymandering have a common identified pitfall: “Although favoritism in a

legislative map might result from intentional partisan discrimination, it is widely understood that bias might also occur unintentionally because of the underlying residential patterns of Democratic and Republican voters" [5] (see also [1]).

Thus, our second question considers evaluating maps in a non bi-polarized scenario, with an emphasis on the residential patterns of voters: in this setting,

(Q2) How to evaluate whether a given electoral map is an outlier?

## 1.1 Our contributions

Our main goal is to evaluate the *diversity* of the set of legal maps, and the potentiality of gerrymandering. Given a notion of diversity, we also propose to evaluate whether a particular map is, informally, "in the average" or an "outlier". We propose ensemble methods: they are based on a large and representative set of legal maps, and study their diversity.

To generate the set of maps, we rely on the algorithm of [14] that allows to generate *all* valid legislative districts maps of France (that we simply call “legal map” in the following), in almost all states. For the sake of completeness, we explain in Appendix B its features, and provide some brief justification for the efficiency of the flood fill algorithm they used. Given a map, we simulate elections without complete information on the voters’ preferences: in two rounds systems, this entails some choices that we detail in Section 1.2 and Section 1.4. In particular, for two rounds systems, we only consider results of the first round and consider a winner any candidate reaching the second round.

*Answer to Question (Q1).* We evaluate, given a set of legal maps and voter’s preferences, the *diversity* of possible maps in terms of electoral results. We propose in Section 2 several related definitions of diversity, with the following objective in mind: a high diversity should mean that there are many maps with different outcomes.

This is related to gerrymandering for the following reason: if there are many maps with sensibly different outcomes, then gerrymandering is possible. On the other hand, when diversity is low, there is a common result to most the maps and gerrymandering is *less easy*. Indeed, in this case, we would find suspicious that the map maker chooses a map that does not yield the most common result. In some sense, gerrymandering is possible, but also highly visible.

We apply this methodology to France and its legislative elections of 2022 in Section 2. Our definitions of diversity are relative to a given party, and we evaluate the diversity of possible results for this party. We show that, in average over parties, 50% of the states have low diversity, while 33% have large diversity – and refer to Section 2 for precise definitions and comments on diversity. In other words, despite the legal constraints on redistricting in France, it seems in one third of the states possible to gerrymander the map – but, in half of the cases, the map has low influence on the result of elections.

*Answer to Question (Q2).* Last, we study the suspiciousness of a given map: our goal here is to leverage our diversity measures in order to indicate whether the map is an outlier. To do so, we rely on some form of statistical tests: we simulate elections on all the

<footnote n="2">²This ideal algorithm assumes that it is possible to predict exactly the votes. As this is not realistic, many strategies exist, such as *packing* or *cracking*, to ensure victory with some margin.</footnote>

<footnote n="3">³Voters actually vote for a candidate, which is assigned to a party. As we discuss in Section 1.2, we assume that voters vote for a party regardless of the candidate.</footnote>
maps, and check whether the map under scrutiny is an outlier for this test. Note that, for two rounds elections, this means simulating only the first round, as we discussed above. More specifically, we have a test for each party: if the map gives a better result for this party than most maps, then we call the map *positive outlier*; if it gives a worst result than most maps, we call it *negative outlier*. We propose to define “most” as 90% of the maps.

We note that outliers may have very good reason to exist, for instance based on the geographical properties of the map.<fnref n="4" /> However, we claim that all outliers maps *must be justified*. This is a new method for detecting possible gerrymandering, complementary to existing ones, that is well suited to multi-party elections.

In the case of France, we analyze, party-by-party and state-by-state, whether the current map is an outlier, based on the electoral results of the first round of 2022 legislative elections. Here, we find surprising results: the map is an outlier for at least one party in 44% of the states. Moreover, for each party we consider the difference between the number of states where the map is positive and negative outlier. This difference is -11% of the states for the left wing, -4% for the center, +5% for the right wing and +11% for the far right. Hence, for the 2022 elections, the current map seems to have noticeably favored the presence of far-right in second round over the left-wing – under assumptions we discuss in Section 1.2.

This global difference between parties stands in contrast with the U.S. case: [18] showed that, although the result in most U.S. states are outliers, the effect is mostly canceled at the national scale and there is no significant advantage for one party. Here, we find a significant difference, that calls for explanation.

Our methods – as most Ensemble methods – take the underlying residential patterns of voters into account, which addresses the issue raised by [1, 5]. Indeed, they are based on the set of all legal maps: instead of using the average vote per district in the full state (i.e., the ratio between the total number of votes for one party and the number of districts) we consider instead indicators based on the distribution of votes in legal maps. Introducing and leveraging this distribution is one of our key contributions.

## 1.2 Our method

To evaluate both diversity and the suspiciousness of a map, we rely on ensemble methods on which we attempt at simulating elections. We start from the knowledge of:

* a large set of legal maps: in the case of France, [14] generated all the maps (in most states). In other cases, we can rely on random walks such as ReCom [10] to generate a large set of balanced and compact maps.

* the voting preferences in each atomic unit.

Given the voting preferences in atomic units, we can easily simulate the outcome of an election in any virtual map, in first-past-the-post elections. For two rounds elections, it is unclear how to simulate the second round, in particular in an unstable electoral context such as France. Indeed, this would require the knowledge of the complete voting preferences of voters, to decide between the remaining candidates. While we can easily get access to the preferred party of any voter (this is public data, aggregated at the

level of atomic unit), it is impossible to guess the total order. As we do not wish to base our analysis on hypothetical data, we do not attempt to simulate the second round, and consider any party accessing the second round as a *winner*.

Our methods, both for diversity and for the identification of outliers, fix a party and rely on the histogram of number of wins – which counts, for each integer *k*, in how many maps the party wins *k* districts. There is therefore one histogram for each state and each party. Those can be either computed exactly (if we have access to all legal maps, as in our case study), or estimated from random samples.

As we formalize in Section 2, we base our diversity measures on the shape of the histogram: a spiked histogram means that there is a low diversity in the state, while a flat one indicate high diversity. Similarly, in Section 3 we base our definitions of outliers with respect to the histograms. More concretely, we build the histograms as follows. Our two assumptions (that we discuss further in Section 4) are the following:

(1) people vote for a party, not for a particular candidate, and

(2) the map does not influence the votes.

This allows to project what would be the result of the elections in a virtual map. Given a state, and the votes in each atomic unit of this state (we explain in Appendix C how we collected those for France), we can then simulate elections in each legal map merely by aggregating the votes of all atomic units in each district, see Section 1.4.

## 1.3 Our case study: France

We consider the 84 states (out of 96 in mainland France) for which [14] computed all legal maps. There are at most 12 districts for each of these states, with an average of 4.8 districts per state. This gives us a total list of 400 districts (out of 577 in total). Each state is made of atomic units called *cantons*. There are in average 20.9 cantons per state. The cantons of a state *S* and its number of districts *n*<sub>*D*</sub> are fixed. An *electoral map* of state *S* is a partition of the cantons into *n*<sub>*D*</sub> districts. A map is *legal* if each district is made up of a whole number of cantons in the state, if these cantons are connected, and if the difference between the population of a district and the average population of a district of the same state is no more than 20%.

For each state, [14] computed all the electoral maps that comply with the legal constraints: there is in average 113 540 maps per state, and the median is 2 303.

In France, legislative elections are held in two rounds. In a district, if a candidate obtained at the first round at least half of the votes and a fourth of registered voters, they are elected. Otherwise, the candidates who access to the second round are either:

* the two candidates with the most votes.

* all candidates who have obtained a number of votes larger than or equal to 12.5% of the number of registered voters, if there are more than two such candidates.

The second round is then a first-past-the-post election.

During the legislative elections of 2022, the candidates were divided into 4 main alliances: a left block, whose acronym is NUPES, a center-right block, ENS, the traditional right LR and the far-right RN. We call those alliances "parties" in the following. The outcome of those elections was the following. In the states we study, NUPES

<footnote n="4">⁴There can be a natural division in the state following a river or a mountain range, which is not taken into account in the legal requirement for validity of a district.</footnote>
was qualified to the second round in 258 districts, ENS in 299, LR in 63 and RN 178.<fnref n="5" />

To get a sense of the fragile equilibrium of the assembly, in 2023, a law to revoke the government was rejected with 278 in favor and 295 against: for this law to be accepted, it would have been enough to flip the opinion of only 9 MPs.

## 1.4 Simulation of the election on a given (legal) map

In order to simulate the elections in France, we used the votes from 2022 legislative elections. We considered the 4 main parties: NUPES, ENS, LR and RN. These accounted for 83.25% of the votes and 98.88% of the candidates who went through to the second round. The details of these parties, and how we associated candidate to parties, are explained in Appendix C.

Since we assume people vote for a party and not a candidate, we were able to collect the votes per party at each voting station. We associated each voting station to the canton where it belongs: this gave us for each canton the number of votes for each party in the canton.

Therefore, for each district *d* of a map *M* generated by the algorithm of [14], and for each party *p*, we can compute the number of votes for *p* in the first round of district *d* by summing up the votes for party *p* in the cantons of *d*. Given the number of votes for each party in the districts of map *M*, we define the winners of the first round in map *M* using the second-round access rules explained in Section 1.3.

## 1.5 Related work

Our main research question relates to the possibility of gerrymandering, which, to the best of our knowledge, has not been much investigated. We are only aware of a study specific to a U.S. State, Massachussets, that shows that there is no way to draw a map with one district favoring Republicans – although they represent one third of the votes, and there are 9 districts – because of the way their vote is spread in the State [11]. Our work can be seen as an attempt to systematize and generalize this specific case study.

**Ensemble methods.** The starting point of our measures is the access to a large set of representative maps – in our case study of France, all the legal maps. In contexts other than France, there are efficient ways of generating such a representative set: [10] propose an efficient Markov chain to generate random maps, dubbed ReCom. It has been shown that, in some specific cases, this Markov chain samples proportionate to the spanning tree distribution [6], which is convenient as it enforces some notion of compactness. However, this implies the distribution is highly biased towards partitions with few atomic units on the periphery of each district [24]. This random walk is being used in some U.S. states to guide the future redistricting plan [7], and [23] provide a complete dataset of thousands of random maps in each of the U.S. states. Their generation is based on ReCom, in which they incorporate legal constraints others than balanceness and compactness, depending on specific jurisdiction (such as the necessity to ensure enough representation

for minorities). However, we argue that, in case where the number of atomic units in each state is not tremendously large (as opposed to the U.S. case), it may be better to sample uniformly at random among legal maps. Indeed, having large atomic units (and hence few blocs per district) naturally enforces some compactness; and, to evaluate the diversity of the legal maps, we find it better to consider all of them instead of only the ones with small periphery.

*Detecting Gerrymandering.* For instance, the Efficiency Gap, which measures the "wasted votes" between two parties, becomes unsuitable in a multiparty context where votes are distributed among several political formations, rendering the notion of "wasted votes" ambiguous. One of the most prominent measure is the partisan symmetry [17], which is specifically defined for 2-parties elections – and whose relevance and meaningfulness of partisan symmetry measures has been questioned in [9]. We refer to Appendix A for a more comprehensive discussion of those classical methods.

Recent studies have therefore highlighted the need for more sophisticated and adaptable methodologies to detect gerrymandering in multiparty and non-majoritarian systems. For example, Slomczyński et al. [25] introduced probabilistic models to address the challenges of detecting bias in multiparty elections, reflecting the need to move beyond binary-focused analyses. The method we propose to detect outliers follows the same goal, although with different models and techniques.

Related to our techniques, some methods to detect gerrymandering are based on ensembles of legal maps. As an example, [20] use them to give a gerrymandering score for each individual *v*: assuming *v* is in a district where its favorite candidate loses, the score is basically the fraction of maps from the sample where *v* is in a winning district. The authors comment on the advantages of this score: most notably, it is based on individuals only, and therefore may be better suited to the U.S. case because of *Gill v. Whitford* which said that: efficiency gap was merely a “single statewide measure of partisan advantage,” and thus the plaintiffs’ case was about “group political interests, not individual legal rights,” which the Court is not responsible for “vindicating.” Rather, “the Court’s constitutionally prescribed role is to vindicate the individual rights of the people appearing before it”. An additional contribution of this paper is to provide a “witness”, a map that has low score. This addressed the potential argument that high scores may necessary because of geography. This witness is built using a simulated annealing algorithm.

The index of [20] is quite similar to ours: however, we do not aim at showing that some individuals are advantaged or disadvantaged by the map, only to measure how much a *party* could coordinate nation-wise to bias the map. Our method also can present a witness, a map whose results are not *outliers* (as we define in Section 3).

This is not the only use of ReCom, which is now frequently used to assess gerrymandering in the U.S. states [15, 16, 22, 28].

**Other related works.** [8] studied the redistricting as an optimization problem (i.e., find the map with smallest population gap) using a graph theoretic approach. They obtained complexity results, as well as algorithms that compute the “best” map (in terms of balanceness) that are efficient when some graph parameters are small. They demonstrated the usefulness of their algorithm by computing the best map in most of states of France.

<footnote n="5">⁵After the second round, less relevant to our methodology, there were in total (out of the 577 districts of France), 151 MP from NUPES, 250 from ENS, 62 from LR and 89 from RN.</footnote>
## 2 Evaluation of the diversity of maps

Given the set of all legal maps computed using the algorithm of [14], or any sample of legal maps (generated by ReCom [10] or some other mean), our goal in this section is to evaluate the diversity of those maps. As explained in Section 1.2, we do so by examining the histograms of winners, for each state and party.

We believe that the diversity of those histograms answers Question (Q1), about the possibility of gerrymandering. In the extreme case where there is absolutely no diversity in the possible results – i.e., all maps yield the same result – then there is no possibility of gerrymandering. This happens, for example, if there is only one legal map, or if the winning party is the same in every atomic unit. In the less extreme case where the variety of result is limited, when at least 90% of the maps yield the same result, we believe gerrymandering is hard, in the sense that a map that deviates from the most frequent result is noticeable, and therefore should be clearly justified. At the other extreme, when the diversity is high and there are many maps yielding different results, then the map maker has some freedom of choice and has the possibility to select its best map.

We first derive histograms at the national level, i.e., when aggregating the results from all states. While this gives an idea of the total gerrymandering potential, this is (in our opinion) not informative enough, as it does not pin-point the precise locations where it may happen, and does not allow for a more fine-grained and specific analysis, that could confirm our infirm the gerrymandering potential. Hence, in a second part, we examine the results at a state level and define our main indicator for this granularity.

### 2.1 Diversity at the national level

A national map is obtained by choosing a legal map for each state. Summing the number of winners for each party over all states gives the nation-wide number of winners for each party. For each party $p$, we can therefore deduce various statistics:

* What is the the highest number of winners a party $p$ may obtain, nation-wide? That is computed by choosing in each state the legal map that gives the most winners for $p$.

* What is the lowest number of winners a party $p$ is guaranteed to have nation-wide? That is computed by choosing in each state the legal map that gives the fewest winners for $p$.

* How many winners might a party $p$ expect to have, nation-wide? Choosing in each state a random uniform map gives a distribution of the number of winners for $p$ nation-wide, for which we can calculate mean (summing the state-wide mean over all states) and variance (by independence, it suffices to sum the statewide variance over all states). Moreover, by taking an independent sample in each state, we can sample the nation-wide distribution; repeating the process yields an estimate of the nationwide distribution of the number of winners for $p$, plotted in Figure 1.

Finally, we can approximate the distribution by using the law of large numbers. Consider the case of France: there are 84 departements (states) in which [14] computed all legal maps. We use the Berry-Essen Theorem:

<figure><img src="image_3.png" /><figcaption>Figure 1: Histograms (filled bars) of the results out of 10 000 random samples, for each party, with the corresponding Gaussian distribution (thick lines).</figcaption></figure>

**THEOREM 2.1.** Let $X_1, X_2, \dots, X_n$ be $n$ independent random variables with $E(X_i) = 0$, $E(X_i^2) = \sigma_2 > 0$, and $E(|X_i|^3) = \rho_i < \infty$. Let

$$S_n = \frac{X_1 + X_2 + \dots + X_n}{\sqrt{\sigma_1^2 + \sigma_2^2 + \dots + \sigma_n^2}}$$

Then there exists an absolute constant $C_0 < 0.56$ such that for every $x$ we have the following upper bound:

$$|F_n(x) - \Phi(x)| < C_0 \frac{\sum_1^n \rho_i}{(\sum_1^n \sigma_i^2)^{3/2}},$$

where $F_n$ denotes the cumulative density function of $S_n$ and $\Phi(x)$ is the cumulative density function of the standard normal distribution.

To apply this Theorem to the case of France, we let $n = 84$ denote the number of states and $X_i$ be equal to $Y_i - E(Y_i)$, where $Y_i$ is the number of winners of party $p$ in state $i$ when the legal map for that state is random uniform.

The results are presented in Table 1. The observed distance from the normal distribution is $max_x |G_n(x) - \Phi(x)|$ where $G_n$ is the cumulative density function of the distribution obtained by taking 10000 samples of a national map. We also plot in Figure 1 the actual histograms, with the corresponding Gaussian density. We observe that the histograms almost match the Gaussian density, much better than what is predicted from Theorem 2.1.

#### 2.1.1 Comments.
We briefly comment the results of our case study. As a comparison point, we recall the specificity of France: our study concerns 400 districts, hence each party can win at most 400 times; and flipping the votes of 9 MP can change crucial votes (see section 1.3).

We point out that the difference between the minimum and maximum number of winners is extremely large, for all the parties. For LR for instance, its best (national) map triples its number of winners with respect to its worst map. Such a big difference was not necessarily expected, considering the large size (hence small number) of atomic units (cantons) in a district.

Compared to the max-min values, standard deviations are much smaller, not on the same order of magnitude. This can be also seen in the shape of histograms in Figure 1, where the extremal values do not even appear.

We can also compare those curves with the actual number of winners in the map actually used in 2022 are the following: NUPES won 258 times, ENS 299, LR 63 and RN 178. The results for NUPES, ENS and RN are all outside of the interval from the $3\sigma$ rule – NUPES
<table><caption>Table 1: For each 4 parties, the minimum and maximum number of wins it can obtain, nation-wide, with the expected value and standard deviation of the distribution of result. The distance from normal distribution is the maximum gap between the two cumulative density functions, and the upper-bound is computed using Theorem 2.1.</caption><thead><tr><th>party</th><th>min</th><th>max</th><th>E(S<sub>n</sub>)</th><th>standard deviation</th><th>observed distance from normal distribution</th><th>upper bound</th></tr></thead><tbody><tr><td>NUPES</td><td>207</td><td>333</td><td>281</td><td>4</td><td>0.052</td><td>0.14</td></tr><tr><td>ENS</td><td>243</td><td>360</td><td>312</td><td>3.9</td><td>0.048</td><td>0.14</td></tr><tr><td>LR</td><td>31</td><td>96</td><td>57</td><td>2.7</td><td>0.067</td><td>0.17</td></tr><tr><td>RN</td><td>101</td><td>229</td><td>160</td><td>4.1</td><td>0.044</td><td>0.13</td></tr></tbody></table>

much smaller, ENS smaller and RN much larger. LR, for its part, is outside of the $2\sigma$ interval, larger.

## 2.2 Diversity at a state level

The global analysis of the previous paragraph is rather crude, whereas we have enough information to perform a much finer analysis at the state level. In particular, it does not allow to inspect the reason for such a big variances – is it specific to some states, or uniformly spread? And, for the states with large variance, is there a specific reason for it?

To have a more precise analysis, we propose to measure diversity at a state level. In this case, we observe that the distribution in a single state has no reason to be Gaussian. And, indeed, we observe many states where distribution are either very flat or spiked (see e.g. Figure 7).

For convenience, we start by a few formal definitions. We fix one state with set of maps $M$ and focus on one party $p$. Given a district map $M \in \mathcal{M}$ for that state, we compute the number $w(p, M)$ of districts where party $p$ wins.<fnref n="6" /> We consider the vector of $w(M) = (w(p, M))_p$, that aggregates the number of wins for each party in map $M$. We consider the distribution of those winning vectors across all maps $M \in \mathcal{M}$.

The distribution is over a discrete set of values – the number of districts in the state (in France, in states we analyzed, this is at most 12). Our goal is to use this distribution to characterize the diversity of the set of legal maps. We propose two indicators based on this distribution. First, as we did for the nation-wide level, we consider the best and worst possible results for a party.

### 2.2.1 Extremal indicator.
For each party $p$, we consider the minimum and the maximum number of wins it can achieve in the state, i.e., $\min_M w(p, M)$ and $\max_M w(p, M)$. We define the extremal diversity as $\max_M w(p, M) - \min_M w(p, M)$.

This quantity tells us how much a completely biased map maker could influence the result. Indeed, having an extremal diversity equal to 0 for a party $p$ means that the number of second-round candidates from party $p$ is the same across all legal electoral maps $M$.<fnref n="7" />

We present the distribution of extremal diversities for each of the 4 main parties in France in Figure 2. We notice that the extremal diversity is often low – especially for party LR, where the extremal diversity is equal to 0 in almost half of the states. However, the extremal diversity can be as high as 6: this is the case for NUPES.

in State 62, where the number of NUPES candidates in a second round in a legal map of this state varies from 1 to 7.

*Pros and cons of the extremal diversity.* This indicator has the benefits of being extremely simple, and can be easily illustrated by providing the two maps yielding best and worst results. It also captures the magnitude of the potentiality of gerrymandering, which can be of interest.

However, it also suffers from several drawbacks. First, it does not capture the likelihood of the results: there are cases where extremal diversity is 2, but most maps attain the middle result – see as an example Figure 3.

In addition, we argue that the magnitude of potentiality of gerrymandering, at a state level, does not fully capture the potentiality of gerrymandering. Indeed, gerrymandering can be subtle, and even one more win in a state can have a dramatic impact overall (or, in practice, 9, as we discuss in Section 1.3).

For those reasons, we propose a different indicator, based on the flatness of the distribution instead of the magnitude of the results.

### 2.2.2 Mode indicator.
We observed empirically that for a given party, there is often a value that largely dominates the distribution: we therefore consider the mode of the distribution (which is the most frequent outcome), and its value (i.e. the proportion of maps achieving the mode). The histograms of values of mode is given in Appendix, in Figure 6. We propose the following definition:

*Definition 2.2 (Gerrymanderability).* Let $v$ be the value of the mode for a party $p$. We say that the state is:

* hardly gerrymanderable for $p$ when $v \ge 0.9$,

* possibly gerrymanderable for $p$ when $0.7 \le v < 0.9$,

* gerrymanderable for $p$ when $v < 0.7$.

This definition is meant in the following sense: if, in a state we class as hardly gerrymanderable, the map maker chooses a map with result different than the mode, this choice would be highly and easily questionable – as 90% of the maps give the same outcome, which is different from the one chosen by the map maker. On the other hand, in states that are gerrymanderable, there is not such a clear and expected result. The histogram of hardly gerrymanderable state is given in Figure 3. Figure 7 in appendix shows the other two cases.

We present in Table 2 the number of states falling into each category, for each political party. We note that a large number of states are hardly gerrymanderable, according to our definition. This indicate that, in those states, the geography and repartition of voters set strong constraints, and that the diversity of possible maps is limited. On the other hand, there is also a large number of states that are gerrymanderable, for which the conclusion is opposite.

<footnote n="6">⁶We recall briefly what we mean by “wins”. In a first-past-the-post election, it is easy to compute $w(p, M)$: it is simply the number of district where $p$ gets more votes than any other party. In two-round systems, we do not attempt at simulating a second round: instead, we say that $p$ wins if $p$ makes it to the second round.</footnote>

<footnote n="7">⁷This happens for instance for NUPES in State 5.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 2: Histograms of the extremal diversities for each of the 4 parties. The x-axis corresponds to values for extremal diversity; the y axis gives, for such a value x and for a given party, the number of cases (among the 84 states) where this party has extremal diversity x</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3: Histograms of the results for Ensemble (ENS) in State 39. For example, in 200 (out of the 222) legal maps of State 39, ENS obtains 2 candidates at the second round of the election.</figcaption></figure>

<table><caption>Table 2: For each of the 4 main parties in France, and for each category of Definition 2.2, the number and proportion of states in this category. This is out of 84 states.</caption><thead><tr><th></th><th>NUPES</th><th>ENS</th><th>LR</th><th>RN</th></tr></thead><tbody><tr><td>hardly gerrymanderable</td><td>38 (45%)</td><td>42 (50%)</td><td>57 (68%)</td><td>36 (43%)</td></tr><tr><td>possibly gerrymanderable</td><td>14 (17%)</td><td>10 (12%)</td><td>10 (12%)</td><td>13 (15%)</td></tr><tr><td>gerrymanderable</td><td>32 (38%)</td><td>32 (38%)</td><td>17 (20%)</td><td>35 (42%)</td></tr></tbody></table>

Our definition of gerrymanderability focuses on the value of the mode. This value is intuitively related to the number of districts in the state: the higher this number is, the higher the possible number of winners is, so the smaller the value of mode is expected to be, hence the more gerrymanderable the state should intuitively be. This is only partially true. On the one hand, as we can see in Figure 4 (right part), the average value of mode is essentially decreasing with the number of districts (especially considering that more than 2/3 of the states have at most 6 districts). On the other hand, as we can see in Figure 4 (left part), even for a high number of districts there are cases (sometimes many cases) with mode 1, so where the number of winners for a party does not depend on the map of the state.

### 2.2.3 Comments.
Our answer to the research Question (Q1), in the case of France, is nuanced: first, in a third of the states, there are many maps with different outcomes – hence, the map maker as some freedom to gerrymander. This number of state is large enough to strongly influence the assembly, as we discussed in Section 1.3. On the other hand, in half of the states, the map barely influences the results of the first round of 2022 legislative elections – which we translate as: in those states, gerrymandering is possible but also visible. This number is, in our opinion, surprisingly large.

In addition, we computed the number of hardly gerrymanderable states where the current map is distinct from the mode. We found

7 of them for NUPES, 15 for ENS, 11 for LR and 3 for RN. These case are suspicious: the actual result should be equal to the mode, or should have strong justification!

## 3 Measure of deviation

Once provided with a measure of diversity, as in Section 2, it is natural to evaluate the status of the maps currently in effect. The evaluation solely based on the notion of gerrymanderability has its limits: although it has a clear explanation in hardly gerrymanderable states, it is not clear what to infer for those that are gerrymanderable. Therefore, we present a notion based on an outlier analysis, that generalizes this previous idea. We sketched this, at a national level, in Section 2.1.1: we showed, using the $3\sigma$ rule, that the current results are clear outlier compared with the Gaussian distribution that we observe. At the level of a state, there is no Gaussian distribution anymore and the standard deviation loses parts of its meaning: instead, we propose to use quantiles. We start with the definition and application to France, and we discuss briefly the results in Section 3.2.

### 3.1 Outlier-based deviation

We propose an indicator that does not depend on the magnitude of a potential gerrymandering – as they might be small – but only on the cumulative distribution function of the legal maps. Formally:
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4: On the left: Each point corresponds to a pair (state,party), x-axis is the number of districts in the state, y-axis is the value of the mode of the pair. The size of the circle is proportional to the number of points with the same coordinates. On the right: x-axis is the number of circonscriptions, y-axis gives for a given x the average value of the mode over all pairs (state,party) where the state has x districts.</figcaption></figure>

Definition 3.1 (Outliers). Fix a party *p* and a map *M*. We say the map is:

* a positive outlier for *p* if the result on *M* is strictly better than on 90% of the legal maps.

* a negative outlier for *p* if the result on *M* is strictly worse than on 90% of the legal maps.

If a state is a positive outlier for a party, the result in that state is better than most of the maps and the party is advantaged by the map. On the other hand, a party that is a negative outlier in a state is disadvantaged by the map. Note that for a hardly gerrymanderable state, a map is an outlier if and only if it gives a result different from the mode: therefore, this notion of outliers generalizes the analysis we hinted in Section 2.2.3.

We computed, for France, the number of outliers for each party, based on the voting data from the first round of 2022 legislative elections. Results are given in Table 3.

<table><caption>Table 3: Number of outliers for each party, out of 84 states. The gap is the proportion of negative outliers minus proportion of positive outliers, and represents how often the party is advantaged by the map.</caption><thead><tr><th></th><th>NUPES</th><th>ENS</th><th>LR</th><th>RN</th></tr></thead><tbody><tr><td>positive</td><td>4</td><td>10</td><td>9</td><td>10</td></tr><tr><td>negative</td><td>13</td><td>13</td><td>5</td><td>1</td></tr><tr><td>Gap</td><td>-11%</td><td>-4%</td><td>5%</td><td>11%</td></tr></tbody></table>

## 3.2 Comments

Results of table 3 mean that, in the first round of the 2022 legislative elections, there are 4 states where the actual result for NUP is better than it would have been on 90% of the legal maps, and 13 states where the result is worse than on 90% of those maps.

We observe that the number of outliers differ greatly between different parties, and find a number of outliers states particularly

high. Our conclusion is that those states should be carefully examined, to understand whether the map was indeed gerrymandered or whether there are some other good reasons for it.

However, we emphasize that the results of the outlier-based analysis indicates that, under the assumption we made in Section 1.2, the current maps gave a strong advantage, in the first round of the elections 2022, to RN over NUPES. In our opinion, this is a strong motivation to further micro-analysis, to confirm or infirm those results and explain the large difference.

## 4 Discussion

We start by recalling our main results. We present a methodology that allows to quantify the diversity of legal redistricting maps. We do so under two assumptions: first, people vote for a party, not for a candidate; second, the map does not influence the result. Those assumptions are necessary to simulate elections in virtual maps, which is at the heart of our methods.

We analyze first the national level, for which the distribution of results follows closely a Gaussian distribution: computing its empirical average and standard deviation are enough to identify the distribution. Complementing this with the extremal values that can be attained allows to get a precise idea of the diversity of possible results, at a global scale. In addition, we propose to examine the state scale, for which the distribution does not follow a Gaussian distribution. Our notion of gerrymanderability allows to classify the states according to the shape of the distribution – with the idea that, if most maps give the same result, it is harder for a map maker to gerrymander without being easily spotted.

We apply those definitions to the case of France, where we observe a large extremal diversity at a national level – with results that can vary by a factor 3 – but a standard deviation at a scale comparable to that of the tightest votes. This comparison indicates that the variety (following the Gaussian distribution) is not so large. However, the actual results fall outside of the $3\sigma$ (for NUPES, ENS and RN) or $2\sigma$ (for LR) interval: the actual map is therefore not “typical” from the distribution we drawn.
We propose then to examine the results at a smaller scale, with the notion of gerrymanderability (see Definition 2.2). At the state level, we find that half of the states are hardly gerrymanderable, in average over all parties, and one third are gerrymanderable.

This gives a nuanced answer to our initial question (Q2): at the national level, there is a large extremal diversity of results, but any "typical" map falls into a small interval. Although we should note that this interval is of the same order as the gap in the tightest votes in the assembly. Second, at the state level, we show that there is a tiny diversity in half of the states, which is quite surprising to us, as we would have expected all states to be vulnerable to gerrymandering.

To answer Question (Q2), we present a definition of outliers based on the distribution of results, for each party and state. Although we believe that this definition is quite strict – an outlier is in the first or last decile of the distribution – we found a surprisingly large number of outliers, as we comment in Section 3.2. This provides a hint towards where the map has been potentially gerrymandered, and calls for specific explanations.

*Limits of our assumptions.* Our model assumes that people vote for parties, not for specific local candidates; and, in addition, that the votes do not depend on the map. This is not perfectly true in practice: there may exist a well-known local candidate with a strong local base, who is particularly attractive to voters from a particular district; or a candidate may have done a very successful campaign, increasing the vote for their party in their specific district only. However, the current tendency of France political system tends to rub out those local effect, as documented in [4]. We note moreover that those are common assumptions in the literature on gerrymandering, that requires an estimate on the voters' behavior.

*The second round.* Another limit is that we do not simulate second round – hence, we do not know who actually wins the election at the end, and we cannot formally conclude anything about whether gerrymandering happened or not. Simulating the second round in practice, in a country where the votes are quite fickle as in France, requires to make too many approximations. In addition, we believe that, to evaluate the diversity of the map and capture faithfully the diversity of the several parties, it is best to use result of first rounds only. Nonetheless, we point that our methodology would stay valid, if one is willing to make the required approximations.

## Conclusion

We answer our two initial questions by introducing some new notions, based on an ensemble of legal redistricting maps. We present a classification of states to answer whether gerrymandering is possible, in multi parties elections. We apply it to France, for the legislative elections of 2022, and show that half of the states are "hardly gerrymanderable" while a third are "gerrymanderable" – a magnitude that may have a strong impact on result of parliament votes. We also propose a definition of positive and negative outliers, with the intent of characterizing whether a given map advantages or disadvantages a party in a state. Here, our application to France hints that, for the 2022 legislative elections, the current map advantages the right and far-right, in the sense that it allows more

candidates from the far right to qualify to the second round than a typical map. On the other-hand, it disadvantages the left parties.

However, our conclusion from this case study is not that far-right is advantaged other the left, because of limits inherent to our data and methodology. Instead, we pinpoint precise states that are suspicious: we do not pretend that those states have indeed been gerrymandered, but we suggest to investigate them with more caution.

Besides the case study to France, we believe that our methodological contribution is an original contribution to the burgeoning field of analyses redistricting processes outside of the U.S. This is indeed the first time, to our knowledge, that the possibility of gerrymandering is questioned, and the first generic methodology to evaluate the diversity of legal redistrictings. This opens an interesting research directions: we generated the set of legal maps using the algorithm from [14], which allows to generate all legal maps in France, but not in all states. For the remaining ones – as well as other countries – is appears necessary to sample a random map, following some "nice" distribution. Given the legal constraints in different countries, it is not clear what this nice distribution should be, and this is worth investigating – together with finding an algorithm that actually samples from the chosen distribution.

## Acknowledgments

This work was [partially] funded by the grant ANR-19-CE48-0016 from the French National Research Agency (ANR).

## References

[1] Pankaj K. Agarwal, Shao-Heng Ko, Kamesh Munagala, and Erin Taylor. 2022. Locally Fair Partitioning. In *Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelfth Symposium on Educational Advances in Artificial Intelligence, EAAI 2022 Virtual Event, February 22 - March 1, 2022*. AAAI Press, 4752-4759. doi:10.1609/AAAI.V36I5.20401

[2] Sebastian Bervoets and Vincent Merlin. 2012. Gerrymander-proof representative democracies. *International Journal of Game Theory* 41 (2012), 473–488.

[3] André Blais. 2006. The causes and consequences of the cumul des mandats. *French Politics* 4 (2006), 266–268.

[4] Julien Boelaert, Sébastien Michon, and Étienne Ollion. 2017. *Métier: député. Enquête sur la professionnalisation de la politique en France*. Raisons d'agir.

[5] Barry Burden and Corwin Smidt. 2020. Evaluating Legislative Districts Using Measures of Partisan Bias and Simulations. *SAGE Open* 10, 4 (December 2020), 2158244020981054

[6] Moses Charikar, Paul Liu, Tianyu Liu, and Thuy-Duong Vuong. 2022. On the Complexity of Sampling Redistricting Plans. *CoRR* abs/2206.04883 (2022). doi:10.48550/ARXIV.2206.04883 arXiv:2206.04883

[7] Matt Chen. 2021. Tufts research lab aids states with redistricting process. *The Tuft Daily* (2021). https://www.tuftsdaily.com/article/2021/04/tufts-research-lab-aids-states-with-redistricting-process

[8] Vincent Cohen-Addad, Philip N. Klein, Dániel Marx, Archer Wheeler, and Christopher Wolfram. 2021. On the Computational Tractability of a Geographic Clustering Problem Arising in Redistricting. In *2nd Symposium on Foundations of Responsible Computing (FORC 2021) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 192)*, Katrina Ligett and Swati Gupta (Eds.). Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 3:1–3:18. doi:10.4230/LIPIcs.FORC.2021.3

[9] Daryl DeFord, Natasha Dhamankar, Moon Duchin, Varun Gupta, Mackenzie McPike, Gabe Schoenbach, and Ki Wan Sim. 2023. Implementing partisan symmetry: Problems and paradoxes. *Political Analysis* 31, 3 (2023), 305–324.

[10] Daryl DeFord, Moon Duchin, and Justin Solomon. 2021. Recombination: A Family of Markov Chains for Redistricting. *Harvard Data Science Review* 3, 1 (mar 31 2021). https://hdsr.mitpress.mit.edu/pub/1ds8ptxu.

[11] Moon Duchin, Taissa Gladkova, Eugene Henninger-Voss, Ben Klingensmith, Heather Newman, and Hannah Wheelen. 2019. Locating the representational baseline: Republicans in Massachusetts. *Election Law Journal: Rules, Politics, and Policy* 18, 4 (2019), 388–401.
[12] Thomas Ehrhard. 2015. Des élections disputées? Découpage électoral et compétition électorale dans les circonscriptions législatives, en France, entre 1997 et 2012. *Cahiers du CEVIPOL* 2 (2015).

[13] Thomas Ehrhard. 2017. *Le découpage électoral sous la Ve République. Intérêts parlementaires, logiques partisanes*. Classiques Garnier.

[14] Thomas Ehrhard, Solal Attias, Evripidis Bampis, Vincent Cohen-Addad, Bruno Escoffier, Claire Mathieu, Fanny Pascual, Adèle Pass-Lanneau, and David Saulpic. 2022. Découpage électoral des circonscriptions législatives en France. *Revue française de science politique* 72, 3 (2022), 333–364.

[15] Gregory Herschlag, Han Sung Kang, Justin Luo, Christy Vaughn Graves, Sachet Bangia, Robert Ravier, and Jonathan C Mattingly. 2020. Quantifying gerrymandering in north carolina. *Statistics and Public Policy* 7, 1 (2020), 30–38.

[16] Gregory Herschlag and Jonathan C Mattingly. 2021. Methods used to Analyze 2020 North Carolina State Congressional Redistricting Landscape. (2021).

[17] Jonathan N Katz, Gary King, and Elizabeth Rosenblatt. 2020. Theoretical foundations and empirical evaluations of partisan fairness in district-based democracies. *American Political Science Review* 114, 1 (2020), 164–178.

[18] Christopher T Kenny, Cory McCartan, Tyler Simko, Shiro Kuriwaki, and Kosuke Imai. 2023. Widespread partisan gerrymandering mostly cancels nationally, but reduces electoral competition. *Proceedings of the National Academy of Sciences* 120, 25 (2023), e2217322120.

[19] Gary King and Robert X Browning. 1987. Democratic representation and partisan bias in congressional elections. *American Political Science Review* 81, 4 (1987), 1251–1273.

[20] Jerry Lin, Carolyn Chen, Marc Chmielewski, Samia Zaman, and Brandon Fain. 2022. Auditing for gerrymandering by identifying disenfranchised individuals. In *Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency*. 1125–1135.

[21] Igor Mandric, Igor Roșca, and Radu Buzatu. 2020. Gerrymandering and fair districting in parallel voting systems. *arXiv preprint arXiv:2002.06849* (2020).

[22] Cory McCartan and Kosuke Imai. 2023. Sequential Monte Carlo for sampling balanced and compact redistricting plans. *The Annals of Applied Statistics* 17, 4 (2023), 3300–3323.

[23] Cory McCartan, Christopher T. Kenny, Tyler Simko, George Garcia III, Kevin Wang, Melissa Wu, Shiro Kuriwaki, and Kosuke Imai. 2022. Simulated redistricting plans for the analysis and evaluation of redistricting in the United States. *Scientific Data* 9, 689 (2022).

[24] Ariel D. Procaccia and Jamie Tucker-Foltz. 2022. Compact Redistricting Plans Have Many Spanning Trees. In *Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022*, Joseph (Seffi) Naor and Niv Buchbinder (Eds.). SIAM, 3754–3771. doi:10.1137/1.9781611977073.148

[25] Wojciech Slomczyński, Dariusz Stolicki, and Stanisław Szufa. 2023. Nonparametric Detection of Gerrymandering in Multiparty Elections. *arXiv preprint arXiv:2306.03085* (2023).

[26] Nicholas O Stephanopoulos and Eric M McGhee. 2015. Partisan gerrymandering and the efficiency gap. *U. Chi. L. Rev.* 82 (2015), 831.

[27] Dariusz Stolicki, Wojciech Slomczyński, and Jarosław Flis. 2019. A Probabilistic Model for Detecting Gerrymandering in Partially-Contested Multiparty Elections. *Transactions on Computational Collective Intelligence XXXIV* (2019), 1–20.

[28] Samarth Wahal. [n. d.]. Quantifying Gerrymandering using Markov Chain Monte Carlo Algorithms. ([n. d.]).

# A More Related work

## A.1 Classical methods to detect gerrymandering.

Just as the Efficiency Gap, the Partisan Bias, which assesses the distortion between votes and seats won by two parties lose some of its relevance in systems where political alliances evolve between electoral rounds [27]. Similarly, the Mean-Median Difference, based on a binary distribution of votes, fails to capture the complex electoral dynamics of multiparty systems where alliances and coalitions can shift between rounds [25]. The Polsby–Popper Test, which analyzes district compactness, is less useful in contexts where electoral boundaries follow small and strict administrative rules, limiting geographic manipulation [2], like in the French case [13], or for the Moldovan parliamentary elections of 2019 [21]. The Declination and the Lopsided-Means Test, both based on asymmetric vote distributions, are similarly ill-suited for multiparty systems where voting behaviors are more fragmented. Collectively, these tools, designed

for single-round systems with two parties, fail to reliably detect potential electoral distortions in multiparty, two-round systems.

## A.2 On the hardness of studying gerrymandering in France

In France, the structural constraints inherent to its multipartite and two-round electoral system introduce additional complexity in the detection and analysis of gerrymandering. The two-round single-member majoritarian voting system fosters an electoral dynamic, characterized by the possibility of strategic withdrawals and alliances between rounds, which complicates the assessment of partisan distortions and the simulation of hypothetical second-round outcomes. This institutional design inherently encourages inter-party agreements aimed at maximizing electoral success, thereby obscuring potential manipulations in the translation of votes into seats and the identification of partisan electoral determinism. Furthermore, redistricting in France are rare and generally occurred in response to constitutional constraints, resulting in only minimal updates to the electoral map [13]. The observed lack of electoral competitiveness in certain districts is often less a consequence of opportunistic partisan gerrymandering than a product of historical political entrenchment and institutionalized local dynamics. These entrenched strongholds, commonly referred to as “electoral fiefs”, are further solidified by the practice of “cumul des mandats” [3]. While analyzing the competitiveness of constituencies before and after redistricting can provide insights into the short-terms effects of redistricting changes [12], such analyses remain largely empirical and their conclusions are subject to debate, particularly when considered the electoral outcomes over a long period. This limitation was notably evident following the 2010 redistricting, where the unchanged electoral map produced four different outcomes in the French National Assembly in 2012, 2017, 2022, and 2024. This variability highlights the inherent challenges of drawing strong conclusions about the effects of redistricting in systems characterized by multiparty competition and evolving electoral coalitions.

We note, however, that our study of diversity of the set of electoral maps stays valid: it gives a precise idea of the potential election results that could have happened in 2022.

# B Computing the set of legal maps

For the sake of completeness, we briefly describe the algorithm used in [14] to generate the legal maps of France. Using publicly available data,<fnref n="8" /> they built in each state (département in France) the adjacency map of each atomic unit (canton).

The algorithm to build the maps of a given state proceeds district by district. For a given district *i*, the algorithm takes as input all possible (partial) maps for the first *i* − 1 districts. For each such (partial) map *S*, it will compute all the possibilities for district *i*. This is done by a recursive depth-first-search algorithm, which basically works as follows:

(1) *S* being fixed, all the possibilities for district *i* starts with the same canton (to avoid having solutions that are the same wit respect to permutations of districts);

<footnote n="8">⁸From the open platform for French public data data.gouv.</footnote>
(2) given a canton *j* that is added to district *i*, the recursion considers the set $N(j) = \{k_1, k_2, \dots, k_p\}$ of yet unassigned cantons adjacent to *j*, and makes *p* recursive calls, where the *z*-st call is to put $k_z$ in district *i* and $k_1, \dots, k_{z-1}$ not in district *i* (to avoid duplicates in the set of solutions). Of course lower and upper bounds in population are considered to add legal maps and to stop the recursive calls.

Also, the algorithm does this for all the districts but the last one. As the last one contains all the remaining cantons, there is no branching there, and the algorithm simply tests connectivity and population for this last district.

The initial canton added to district *i* (point (1) above) affects the efficiency of the algorithm. The heuristic starts with the canton that has lowest degree, in the subgraph induced by the yet unassigned cantons.

Intuitively, this reduces the number of times the algorithm generates partial maps that cannot lead to legal (global) maps. It builds a district 'from the boarder', leaving highly connected subgraphs for the other districts. If one would start with high degree/central cantons, the district under construction would break the graph into several small connected components, that may not fit the population bounds, see below for an illustration. More specifically, if the graph is a tree, one should start from a leaf in order to reduce the risk that, once the current district is removed from the graph, the connected component becomes too small.

Let us illustrate the interest of considering low degree vertices when branching in the algorithm that generates all (legal) maps. We consider the state called "Corse du Sud" in France, which contains 11 cantons. The adjacency graph of these cantons is given in Figure 5. Let us imagine, for the sake of the illustration, that we have to make 2 districts that contain, for population balance reasons, either 5 or 6 cantons. Considering then the constraint of connectivity, the only legal map is the one where one district is made of cantons $C_1$ to $C_5$, and the other district made of cantons $C_6$ to $C_{11}$.

In the algorithm, suppose that we start building the first district with canton $C_5$ (highest degree vertex). The algorithm builds all possibilities for the first district, i.e., all connected sets of 5 or 6 cantons including $C_5$. It can be checked that there are 64 possibilities - among which 63 cannot lead to a legal map (in our case, leading to a non connected second district), as there is a unique legal map.

Suppose instead that the algorithm starts with canton $C_{11}$ which has degree 1. Then there are only 9 possibilities to make the first district (connected, with 5 or 6 cantons, including $C_{11}$), so only 8 cases that cannot lead to a legal map. This is much smaller than starting with $C_5$, the algorithm generates much less partial maps that cannot be extended to (legal) global maps.

## C Simulation of the French legislative elections and data sources

In order to simulate the elections in France, we used the votes from 2022 legislative elections. During these elections, in each district, the "ministère de l'intérieur" assigns each candidate one of 16 political nuances.<fnref n="9" /> If a candidate is nominated or supported by a single

political alliance that has its own "nuance", the candidate is assigned the nuance specific to that alliance. A candidate who has not received nomination or support, or who has received nomination or support from a party that does not have its own nuance, may be assigned a "nuance of tendency" (e.g. EXG, for Extreme Left, or DVD, for small right-wing parties).

We focused on the 4 main alliances: NUPES (a union of left parties), Ensemble (central, the presidential coalition), LR (traditional right) and RN (far right). LR was the main party of an alliance called UDC: we assimilated all parties of UDC to LR<fnref n="10" />.

In addition to alliances, some candidates where supported by one of the main alliances, although their official party was different (for instance, a small local one). We assigned manually candidate to an alliance if the candidate was elected and seated with the alliance or if there is evidences in the media that the alliance supported the candidate. The candidates that we manually affiliated to a party are given in Table 4.

Because of the alliances, we could not directly use the result of elections for each canton as given in https://www.data.gouv.fr/fr/datasets/elections-legislatives-des-12-et-19-juin-2022-resultats-du-1er-tour/#/resources/2c98645b-c345-4250-bb90-2ba3ec20434e. Indeed, in this file, some candidates have been relabeled not always in accordance with the alliances. Therefore, we collected the votes per candidate at each voting station from https://www.data.gouv.fr/fr/datasets/election-presidentielle-des-10-et-24-avril-2022-resultats-definitifs-du-1er-tour/, and associated each voting station to a canton using the database of https://www.insee.fr/fr/information/3539086. Those contained typos and missing data that we corrected by hand.

Combining those databases gave us the following data: for each atomic unit (canton) and each of the 4 main alliance, as well as for every political nuance that is not part of an alliance, the number of votes for this alliance (or political nuance) in the first round of the 2022 legislative election.

For the vast majority of the districts (382 of the 400 studied districts, i.e., 95, 5% of the cases), there was one candidate of each of these 4 main alliances in the first round of the election. In rare cases, there is in a district no candidate for one of the four alliances. In this case, we either identified a candidate supported by the alliance, even if they did not have the alliance label at the time of the election (the list of these candidates is given in Table 4), or we counted 0 voter for the alliance in this canton. In the latter case, which concerns 4, 5% of the districts, the other parties running and politically close to the alliance not present usually obtained scores lower than 5%.

We used this data with the list of legal maps from [14] to simulate the result of the first round in each legal map: this is the basis of our analysis from Section 2 and Section 3.

<footnote n="9">⁹These political nuances are available here: https://www.archives-resultats-elections.interieur.gouv.fr/resultats/legislatives-2022/nuances.php.</footnote>

<footnote n="10">¹⁰This alliance contains the candidates of parties LR, UDI, and LC. There are also a few candidates officially indicated 'without label' who are in fact supported by the UDC alliance. All the candidates supported by the UDC alliance have been found on https://fr.wikipedia.org/wiki/%C3%89lections_1%C3%A9gislatives_fran%C3%A7aises_de_2022 (and the pages dedicated to each department).</footnote>
<figure><img src="image_8.png" /><figcaption>Figure 5: Adjacency graph of the 11 cantons of the state “Corse du Sud” of France.</figcaption></figure>

<table><caption>Table 4: List of candidates that we have manually assigned to an alliance that supports them.</caption><thead><tr><td>ID of the state (département)</td><td>ID of the district (circonscription)</td><td>name of the candidate</td><td>alliance assigned to the candidate</td></tr></thead><tbody><tr><td>1</td><td>5</td><td>Abad</td><td>ENS</td></tr><tr><td>7</td><td>1</td><td>Saulignac</td><td>NUPES</td></tr><tr><td>7</td><td>1</td><td>Valla</td><td>LR</td></tr><tr><td>22</td><td>2</td><td>Desbois</td><td>LR</td></tr><tr><td>24</td><td>4</td><td>Dubois</td><td>ENS</td></tr><tr><td>29</td><td>2</td><td>Larsonneur</td><td>ENS</td></tr><tr><td>31</td><td>8</td><td>Aviragnet</td><td>NUPES</td></tr><tr><td>34</td><td>6</td><td>Ménard</td><td>RN</td></tr><tr><td>54</td><td>5</td><td>Potier</td><td>NUPES</td></tr><tr><td>55</td><td>1</td><td>Pancher</td><td>LR</td></tr><tr><td>57</td><td>3</td><td>Colin-Oesterlé</td><td>LR</td></tr><tr><td>59</td><td>21</td><td>Descamps</td><td>LR</td></tr><tr><td>59</td><td>18</td><td>Bricout</td><td>LR</td></tr><tr><td>64</td><td>3</td><td>Habib</td><td>ENS</td></tr><tr><td>67</td><td>8</td><td>Sander</td><td>LR</td></tr><tr><td>77</td><td>3</td><td>Septiers</td><td>ENS</td></tr><tr><td>78</td><td>7</td><td>Brosse</td><td>LR</td></tr><tr><td>78</td><td>12</td><td>Moutot</td><td>LR</td></tr><tr><td>85</td><td>4</td><td>Besse</td><td>LR</td></tr><tr><td>88</td><td>3</td><td>Naegelen</td><td>LR</td></tr><tr><td>91</td><td>2</td><td>Dugoin-Clement</td><td>LR</td></tr><tr><td>91</td><td>8</td><td>Dupont-Aignan</td><td>RN</td></tr><tr><td>91</td><td>9</td><td>Hidri</td><td>LR</td></tr></tbody></table>

## D Additional figures

We finally provide two additional figures, illustrating further our definition of gerrymanderable states.
<figure><img src="image_9.png" /><figcaption>Figure 6: Histogram giving, depending on the value of mode (x-axis, width of bars = 0.02) the number of pairs (state, party) with this value of mode.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 7: On the left: A possibly gerrymanderable state, RN in state 51. On the right: a gerrymanderable state, LR in state 52.</figcaption></figure>