# The Power Index Curse Strikes Again:
## Over-representation of Major Cities in French Inter-communal Structures

Zineb Abidi Perier, Vincent Merlin

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*The power index curse strikes again:
Over-representation of major cities in french
inter-communal structures*

Zineb ABIDI PERIER

ERUDITE, Université Paris Est-Créteil, IST-PE and TEPP (FR CNRS 3435)

Vincent MERLIN

Univ Caen Normandie, CNRS, CREM - UMR6211, F-14000 Caen France

February 2025
# The Power Index Curse Strikes Again: Over-representation of Major Cities in French Inter-municipal Structures

Zineb Abidi Perier*†

Vincent Merlin‡§

## Abstract.

In 2020, the issue of improving local collaboration between French municipalities reached a milestone. From January of this year, every municipality would, for the first time, belong to one of 1,254 French Inter-municipality structures. This achievement was accompanied by precise rules describing the distribution of seats among the different cities of the inter-municipal councils that govern each structure. However, the fairness of seat allocations proposed by French law, based on principles of proportionality can be questioned. Indeed, since the pioneering contributions of Penrose (1946) and Banzhaf (1965), the theory of power indices has warned us that representing cities in proportion to their population would give too much power to major cities. We assess whether this phenomenon can also be observed in France since 2020, using a new database that accurately reports the number of seats per city. Though we encounter a large variety of situations across the country, our results show that the provisions in the seat allocation rules often lead to the over-representation of major cities. Alternative allocation rules are presented to reduce these inequalities in representation.

Keywords: Power Indices, Banzhaf Index, Voting, Local Governance, Apportionment

This research was funded in whole or in part by the French National Research Agency (ANR) as part of the Citizens project ANR-22-CE26-0019. For the purpose of its open access publication, the author who owns the rights applies a CC-BY Open Access License to any article accepted for publication resulting from this submission

<footnote>*ERUDITE, Université Paris Est-Créteil, IST-PE and TEPP (FR CNRS 3435)</footnote>

<footnote>†Corresponding author: Zineb Abidi Perier, 61 avenue du Général de Gaulle, Route de Choisy, Mail des mèches 94010 Créteil, France. E-mail: zineb.abidi@u-pec.fr.</footnote>

<footnote>‡Normandie Univ, UNICAEN, CNRS, CREM, 14000 Caen, France. E-mail: vincent.merlin@unicaen.fr.</footnote>

<footnote>§We are grateful to our research assistant Thibaud Rivet for helping us to manage the data.</footnote>
# 1 Introduction

Cooperation between French municipalities has a long history, with the first law encouraging such cooperation having been enacted in 1890. This law entitled municipalities to create inter-municipal groups of two forms, the Syndicat Inter-municipal à Vocation Unique (SIVU) and the Syndicat Inter-municipal à Vocation Multiple (SIVOM), which were the first Établissements Publics de Coopération Inter-municipal (EPCI). Since the 1990s, the French government has been actively encouraging cooperation among municipalities for the joint management of public goods and services. The development of cooperation between municipalities in France has been influenced by several laws over time, but one of the most significant laws for promoting inter-municipal cooperation is the Loi n° 2010-1563 de réforme des collectivités territoriales (hereafter the 2010 Law). This law made it compulsory for all municipalities to join an inter-communal structure by January 1st, 2014.<fnref n="1" /> This law also reduced flexibility in allocating the number of delegates and proposed an explicit criterion for delegate allocation. The 2010 law on territorial reform was supplemented by two additional laws in 2015: the Loi NOTRe (Nouvelle Organisation Territoriale de la République - New Territorial Organization of the Republic) and the Loi MAPTAM (Modernisation de l'action publique territoriale et d'affirmation des métropoles- Modernization of Public Territorial Action and Affirmation of Metropolitan Areas). Furthermore, these laws establish a council that governs the EPCIs, where each municipality is represented by a specific number of delegates directly elected by its voters. Therefore, by 2020, all municipalities in France were required to belong to one of the 1,254 EPCI. More importantly, the 2020 elections were also the first ones where the voters could directly elect their delegates in the EPCI.

The aim of this paper is to analyze the fairness of the allocation of seats among municipalities in the inter-municipal structure (EPCI) after the 2020 elections. For the first time, we are able to exploit a comprehensive dataset gathered by the French Interior government based on the results of these elections to get a precise picture of the delegates' allocations. In particular, we question to which extent the rules used to allocate the delegates among the municipalities, described by the 2010 Law and based on Jefferson's proportional principles, would favor larger cities.<fnref n="2" /> To achieve this, we analyze the fairness of the current apportionment of delegates among

<footnote n="1">¹With the exception of certain municipalities in the Île-de-France region, which were granted a derogation until January 1, 2016, to evaluate the formation of the Grand Paris Métropole, the rest of the communes were required to join an inter-communal structure by January 1, 2014.</footnote>

<footnote n="2">²Jefferson's method, also known as the D'Hondt method, is based upon the greatest divisors method. For</footnote>
French municipalities, drawing on the power index literature.

The social choice and game theory literature proposes several criteria for evaluating the fairness of seat allocations in a two-tier voting system, such as maximizing the total utility of the voters (Barberà and Jackson, 2006), equalizing the probability of success (Rae, 1969), and equalizing the probability of being pivotal (Banzhaf, 1965; Penrose, 1946). Following this pioneering work, this idea of equalizing the probability of being pivotal has been the most successful in the literature, leading to the development of the theory of power indices. As for empirical contributions, most of them have focused on legislative assemblies, such as recent analyses of the Greek Parliament by (Koki and Leonardos, 2019) or the governance structure of the Lebanese Republic by (?). The use of power index tools to analyze power relations in federal unions has also been widespread. One of the earliest contributions was the analysis of the US Electoral College (Owen, 1975).

Felsenthal and Machover (2001), Felsenthal and Machover (2004) and Laruelle and Widgrén (1998) have used power indices in the context of the European Union, while Leech (2002) has analyzed the governance of the International Monetary Fund, among others. Regarding the French case, Bisson et al. (2004) investigated the representation of municipalities in structures in the Basse-Normandie region, revealing that the distribution of voting power among municipalities often differs from their population weight. The authors proposed an algorithm to calculate a more equitable distribution of delegates that takes power indices into account.

In the same vein, Abidi et al. (2020) showed that the distribution of power was unexpectedly fair in many inter-municipal structures in Bretagne, Normandie, and Pays de Loire prior to the 2010 reform. Thanks to the 2020 election database, we are now able to examine the fairness of the distribution of delegates within EPCIs for all French municipalities and compare it to other scenarios on a larger scale.

The structure of this paper is as follows. Section 2 provides an overview of the French institutional context, while Section 3 explains the method used to measure voting power in inter-communal structures. Section 4 covers the data used and the empirical design adopted. Section 5 sets out the results of the study. Finally, Section 6 provides concluding remarks and highlights avenues for further research.

<footnote>more details about the apportionment methods, see Balinski and Young (2001).</footnote>
## 2 French institutional framework

Local government fragmentation is a significant challenge that has a profound impact on the efficient provision of local services. To address this issue, several countries, particularly in Europe, have implemented municipal consolidation programs aimed at providing joint local public goods. For instance, Denmark reduced the number of municipalities from 271 to 98 in 2007 through a process of mergers and consolidations (Blom-Hansen, 2010). Similarly, Sweden has reduced the number of municipalities from over 2,000 in the 1950s to 290, mainly through voluntary mergers (Stein, 2019). Meanwhile, other countries, such as France, Spain, and Italy, have implemented functional cooperation programs. This type of collaboration leads to the creation of a new tier of local government, known as the group level, where each municipality within a group or community is represented by a specified number of delegates in the governance council.

### 2.1 Different types of French EPCIs

With respect to the French case, France has the highest number of municipalities in Europe, with over 34,000 as of 2022.<fnref n="3" />

These municipalities encounter challenges in producing and offering certain public goods and services independently.<fnref n="4" /> Consequently, since 1999 the French government has been actively promoting the establishment of inter-municipal jurisdictions, in which each municipality should have at least one delegate and no single municipality can hold a majority of delegates. In 2020, the issue of improving local collaboration between French municipalities reached a milestone. From January of this year, every municipality would, for the first time, belong to one of 1,254 French Inter-municipality structures. It is important to note that in France there are four types of EPCI (Etablissement Public de Coopération Intercommunale): Communauté de Communes (CC), Communauté d'Agglomeration (CA), Communauté Urbaines (CU) and Métropoles (M). The main factor that differentiates these types of EPCI is their geographical scope, which is determined by the number of inhabitants that the EPCI members serve, and their competencies. As a general rule, the level of inter-communal cooperation and the geographical scope of the EPCI increase as one moves from the Community of Communes to the Metropolis (For more

<footnote n="3">³There are 17,000 municipalities in France with a population of fewer than 500 inhabitants.</footnote>

<footnote n="4">⁴For more details, the reader can refer to the local government website https://www.collectivites-locales.gouv.fr/.</footnote>
<table><caption>Table 1: The number of delegates assigned to the inter-communal structures by the 2010 law</caption><thead><tr><td>Population</td><td>Number of Delegates</td></tr></thead><tbody><tr><td>Less than 3,500 inhabitants</td><td>16</td></tr><tr><td>Between 3,500 and 4,999</td><td>18</td></tr><tr><td>Between 5,000 and 9,999</td><td>22</td></tr><tr><td>Between 10,000 and 19,999</td><td>26</td></tr><tr><td>Between 20,000 and 29,999</td><td>30</td></tr><tr><td>Between 30,000 and 39,999</td><td>34</td></tr><tr><td>Between 40,000 and 49,999</td><td>38</td></tr><tr><td>Between 50,000 and 74,999</td><td>40</td></tr><tr><td>Between 75,000 and 99,999</td><td>42</td></tr><tr><td>Between 100,000 and 149,999</td><td>48</td></tr><tr><td>Between 150,000 and 199,999</td><td>56</td></tr><tr><td>Between 200,000 and 249,999</td><td>64</td></tr><tr><td>Between 250,000 and 349,999</td><td>72</td></tr><tr><td>Between 350,000 and 499,999</td><td>80</td></tr><tr><td>Between 500,000 and 699,999</td><td>90</td></tr><tr><td>Between 700,000 and 1,000,000</td><td>100</td></tr><tr><td>More than 1,000,000</td><td>130</td></tr></tbody></table>

Source: www.collectivites-locales.gouv.fr

details, see Jaaidane and Larribeau (2023)).

## 2.2 The seat allocation rules

This change in the administrative structures governing municipalities was accompanied by precise rules describing the distribution of seats between the different cities of the inter-municipal councils that governed each structure. Table 1 displays the allocation of delegates to each EPCI based on its respective population size according to the 2010 Law. These rules can be sketched as follows:

1. The seat must be allocated according to Jefferson's apportionment method, that is, attributed according to the greatest divisor principle. However, no municipality should get
more than half of the seats.

2. If a municipality has received no seats, it receives an extra one, making the total number of delegates exceed the recommendations of Table 1.

3. Hence, if the total number of seats allocated to the inter-municipality council exceeds 30% of the number allocated by law (as indicated in Table 1), then it may allocate an additional 10% of seats among its members. The allocation of these extra seats must follow again Jefferson's principles.

4. Finally, the law enacted on March 9, 2015, authorized the implementation of a local agreement among the EPCI's members. They could increase the number of delegates by up to 25%.<fnref n="5" /> However, this agreement must meet three conditions: i) Each municipality must be granted at least one seat; ii) No single municipality may hold more than half of the total allocated seats; iii) The resulting representation resulting from the local agreement must not deviate by more than ±20% from the demographic weight of each municipality.

## 2.3 The French municipal and inter-municipal voting rules

The electoral processes for the municipal councils and the inter-municipal councils are linked. In fact, it is the same ballot that is used for the designation of the delegates in the municipal and in the inter-municipal council. Another key point is that electoral system is different, depending on the population size of the municipality.

### Municipalities with fewer than 1,000 inhabitants

Here, the electoral system is a majoritarian pluri-nominal voting one. The candidates may belong to a list, but do not have to. If *k* councilors are to be elected, a voter may vote for up to *k* candidates, as she wishes, following the recommendation of a list, or not.

To be elected in the first round, a candidate must obtain an absolute majority of the expressed votes as well as a number of votes equal to at least one-fourth of the total number of registered voters. If $0 < k' \le k$ seats need to be filled, a second round of voting occurs a week after.

<footnote n="5">⁵This local agreement is reserved for communities of communes and communities of agglomerations and must be the subject of concordant deliberations of at least two-thirds of the municipal councils representing more than half of the population, or half of the municipal councils representing 2/3 of the population. This majority must include the municipal council of the most populous municipality when its population represents more than one-quarter of the population of the EPCI.</footnote>
Then, a relative majority is sufficient, regardless of the number of voters, to get elected: the $k'$ candidates with the highest number of votes are elected.<fnref n="6" /> Next, the inter-municipal delegates are automatically designated in the order of the list established during the first municipal council meeting, after the election of the mayor and deputy mayors.

## Municipalities with more than 1,000 inhabitants

In this case, municipal councilors are elected through a two-round list system, without any addition, removal, or modification of the order of presentation.

In the first round of voting, if a list obtains an absolute majority of the valid ballots, regardless of the voter turnout, it is allocated a number of seats equal to half of the total number of seats, rounded up to the nearest integer. After this allocation, the remaining seats are distributed among all lists that have obtained at least 5% of the votes cast, according to proportional representation based on the highest averages. In the case that no list has obtained an absolute majority of the votes cast in the first round, a second round of voting takes place a week later. A list is qualified for the second round if it has received 10% of the votes; it can merge with another one if it has received 5% of the votes. The seat distribution in the second round is determined by a relative majority, following the same rules as the first round: 50% of the seats (rounded up) go the winner, and 50% are distributed among all the lists using Jefferson's method.

It is mandatory for community councilors to be drawn from the list of municipal councilors. They are elected through a similar voting system (50% of the seats for the winning list and 50% allocated through proportional representation, using Jefferson's method) and by using the ballot as for the municipal councilors.<fnref n="7" />

To give an example, Table 2 displays the results for the 2020 elections in Lille, which was one of the most closely contested elections this year. Ten lists were present at the first round, but only 3 of them passed the 10% threshold. None of them made an agreement with one of the three lists that obtained between 5% and 10% of the votes. In the second round, the Socialist

<footnote n="6">⁶In case multiple candidates receive an equal number of votes, the oldest candidate is elected.</footnote>

<footnote n="7">⁷There are two exceptions to these general rules. In case a municipality with more than 1,000 inhabitants is represented by three delegates in the inter-municipal council, one seat is directly attributed to the winning list, and two seats are apportioned according to the Jefferson method. In case of four delegates, two seats are attributed to the winning list, and two are apportioned according to the Jefferson method. These dispositions guarantee that a losing list that has obtained at list one third of the votes will be represented by one delegate in the inter-communal council.</footnote>
<table><caption>Table 2: 2020 Municipal Election in Lille</caption><thead><tr><th>Party<br>Leader</th><th>Socialists<br>Aubry</th><th>Greens<br>Baly</th><th>Center<br>Spielbout</th><th>Radical Left<br>Poix</th><th>Right<br>Daubresse</th><th>Ext Right<br>Cattelin</th><th>Others<br>4 lists</th></tr></thead><tbody><tr><th>First Round votes<br>(in %)</th><td>11,832<br>29.80</td><td>9,740<br>24.53</td><td>6,960<br>17.53</td><td>3,512<br>8.84</td><td>3,274<br>8.24</td><td>2,719<br>6.84</td><td>1,666<br>4.18</td></tr><tr><th>Second Round votes<br>(in %)</th><td>15,389<br>40.00</td><td>15,162<br>39.41</td><td>7,919<br>20.58</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><th>Municipality delegates (61)</th><td>43</td><td>12</td><td>6</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><th>Inter-municipal delegates (33)</th><td>24</td><td>6</td><td>3</td><td>—</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

list defeated the Greens by only 227 votes and 40% of the votes. Despite this razor thin margin, it obtained 31 seats out of 61 directly, plus 12 seats via the proportional part, for a total of 43. The same logic applies for the inter-communal council.<fnref n="8" />

This example illustrates that losing parties are clearly disadvantaged both in the municipal and inter-communal councils.

## 3 The measurement of voting power in inter-communal structures

Since the pioneering works of Penrose (1946, 1952), Shapley (1948) and Banzhaf (1965), the theory of power indices highlights that the number of seats a player controls in a parliament or council is not necessarily proportional to their actual power to influence the outcome of a vote or decision. Indeed, it offers precise mathematical methods for evaluating the relative power and influence of distinct players within a specific decision-making scenario. For instance, one criterion that can be utilized to allocate delegates in an inter-communal structure is to ensure that each citizen has an equal probability of being a swing voter.<fnref n="9" /> This means that the

<footnote n="8">⁸The *Métropole Européenne de Lille* groups together 95 municipalities, and 1,181,692 inhabitants. The smallest municipality is Warneton, with 241 inhabitants, while 234,482 inhabitants live in Lille, the largest city. The inter-communal council has 188 delegates in total. With 33 delegates, Lille controls approximately one-sixth of the delegates.</footnote>

<footnote n="9">⁹In the context of decision-making, a swing voter can be defined as a voter or player whose vote or action has the potential to change the outcome of a vote or decision. In other words, they are the ones who have the ability to tip the balance one way or the other. In the literature, swing voters are sometimes referred to as “decisive players” or “pivotal players”. These terms all refer to the same idea of an individual or a group whose vote or action is critical to the success or failure of a decision.</footnote>
probability of influencing the outcome of a vote should be the same for each citizen, regardless of which municipality the voter resides in.

## 3.1 Definition of the Banzhaf index

Banzhaf's approach to measuring the power of a voter in an assembly is similar to that of Penrose (1946), which assesses a voter's ability to change the assembly decision by reversing their vote. A voting structure or cooperative game $G = (N, v)$ consists of a set of players $N = \{1, ..., n\}$ and a real-valued function $v$ defined on the subsets of $N$ to $\mathbb{R}_+$. A coalition $S$ is a non-empty subset of $N$. A simple game is a game where $v(S) \in \{0, 1\}$, where $v(S) = 0$ denotes a losing coalition and $v(S) = 1$ denotes a winning coalition. Let $\mathcal{W}$ denote the set of all winning coalitions, and let the sets not in $\mathcal{W}$ be referred to as losing coalitions. In other words, if all members of $S$ vote on an amendment, the amendment passes if $S$ is in $\mathcal{W}$ and is rejected otherwise. A player $i \in S$ is deemed decisive if her vote can change the outcome, meaning if $i$ votes "Yes," the decision is adopted, while if $i$ votes "No," the decision is rejected: $v(S) = 1$ and $v(S \setminus \{i\}) = 0$.

The non-normalized Banzhaf index of player $i$ is equivalent to the original Penrose index:

$$B_i(v) = \frac{\sum_{S \subseteq N, i \in S} [v(S) - v(S \setminus \{i\})]}{2^{n-1}} \quad (1)$$

where $2^{n-1}$ is the number of coalitions to which player $i$ may belong. This index gives the probability that a player $i$ is decisive. If the numerator of Eq. (1) is divided by the total numbers of swing players, we obtain the normalized Banzhaf power index, or simply:

$$NB_i(v) = \frac{B_i(v)}{\sum_{i \in N} B_i(v)} \quad (2)$$

A particular class of simple games is the weighted quota games $G[q, \tilde{a}]$. They are defined by a vector of positive weights, $\tilde{a} = (a_1, a_2, ..., a_n) \in \mathbb{R}_+^n$ and a quota $q \in [0, \sum_{i \in n} a_i]$. $a_i$ is the weight attached to player $i$, and a coalition $S$ is winning if and only if $\sum_{i \in S} a_i \ge q$.

## 3.2 Power in two-tier voting systems

The French inter-communal structure council provides an example of a two-tier voting system. At the first tier, citizens of each municipality elect delegates to represent them in the municipal council. At the second tier, the inter-municipal council is a college of elected members of the municipal council. Hence, any inter-communal structure council can be modeled as a weighted quota game, where $a_i$ is the number of delegates controlled by a municipality $i$, and $q$ is equal
to half of the weights (rounded up).

In an ideal two-tier system, when a proposition is discussed, each citizen would vote "Yes" or "No" and the delegates are expected to report the majority decision of their respective municipalities in bloc. This mechanism is the one used in the Electoral College for the election of the US President, and it serves as a basic model for the analysis of two-tier voting systems. Following the analogy with the US Electoral College, the probability that a voter is decisive in the union is equal to the probability that the state/municipality *j* that voter *i* belongs to is decisive in voting for a bill multiplied by the probability that the voter *i* is decisive in the election of the representatives inside his/her own state/municipality *j*.

Therefore, fairness can be achieved by applying probability theory. Penrose argues that in a model where voters must choose between two mutually exclusive proposals X and Y, and each vote is determined independently by randomly flipping a fair coin, and each municipality is represented by a weight $a_j$ at the federal level, the independence across voters entails the following assertion:

$$ 
\begin{array}{l}
\text{Probability that citizen } i \text{ in municipality } j \text{ is decisive in the assembly} = \\
\text{Probability that citizen } i \text{ is decisive in her municipality} \times \\
\text{Probability that municipality } j \text{ is decisive in the group}
\end{array}
\qquad (3)
$$

When dealing with large populations, the probability of a citizen being decisive in municipality *j* with $n_j$ inhabitants can be approximated by $\frac{\sqrt{2}}{\sqrt{\pi n_j}}$. Therefore, equity in terms of power between citizen *i* in municipality *j* and citizen *i'* in municipality *j'* can be achieved if:

$$ B_j \frac{\sqrt{2}}{\sqrt{\pi n_j}} = B_{j'} \frac{\sqrt{2}}{\sqrt{\pi n_{j'}}} \qquad (4) $$

$$ \Leftrightarrow \frac{B_j}{B_{j'}} = \frac{\sqrt{n_j}}{\sqrt{n_{j'}}} \qquad (5) $$

In other words, equity in the weighted games $G = [q, \tilde{a}]$ can be achieved if the ratio between the Banzhaf indices of two municipalities is proportional to the ratio of the square roots of their populations. Therefore, to achieve equity in terms of power, the normalized Banzhaf power of a municipality *j* should be as close as possible to its share of the population, measured in terms of square roots ($\frac{\sqrt{n_j}}{\sum \sqrt{n_{j'}}}$). The Quotient of a municipality *j* can be defined as:

$$ Q_j = N B_j \times \frac{\sum \sqrt{n_j}}{\sqrt{n_j}} \qquad (6) $$
This variable should ideally be close to 1 for each municipality. If it is greater than 1 (respectively less than 1), this indicates that the municipality is overweighted (respectively underweighted). To measure inequality in power distribution as determined by the Banzhaf index, following (Felsenthal and Machover, 2001; Abidi et al., 2020), we use the max-min index. This index represents the number of times the most powerful citizen is more powerful than the least powerful citizen.

$$max - min = \frac{Q_j^{max}}{Q_j^{min}} \qquad (7)$$

To give an example, $Q_j^{max} = 2$ means that the probability that the citizens in municipality j are pivotal is twice as high as it should be. Similarly, $Q_j^{min} = 0.5$ means that citizens of municipality j do not have as much of an opportunity to be decisive as they should. And finally, the value $max - min = 4$ indicates that a citizen in one city will encounter voting situations where he is pivotal far more frequently (4 times) than a citizen living another city.

Alternative measures were used to measure the inequality in terms of power. For example, Felsenthal and Machover (2001) propose different formulas to analyze the power inequality in the European Union, namely $Max(d) = Q_j^{max} - 1$ and $ran(d) = Q_j^{max} - Q_j^{min}$. They also suggest using conventional statistical indices, such as $\rho$ and $\chi^2$, to assess the degree of conformity between the power distribution and the vector of square roots of population. Though the authors do not offer any theoretical justification for using these different measures, the same conclusions were supported by all of statistical measures. We believe that $max - min$ index is the simplest to interpret.

## 3.3 A discussion of the Penrose scenario

Penrose (1946, 1952) was the first author to argue that states, cities, etc. should be represented by a number of delegates proportional to the square root of their population. The literature often names this recommendation the Penrose square root law. However, a crucial point in the demonstration comes from the fact that all the delegates vote in bloc. If we come back to the Métropole Européenne de Lille case presented in Table 2, this would mean that all 33 delegates from Lille would report the wishes of their citizens unanimously, whatever their political affiliation is. The player Lille would then have a weight 33 in the cooperative game that depicts the inter-communal council. There could be decisions where, indeed, all the delegates from a city defend the same interest. However, this is just one possible scenario. Let us mention two other possibilities.
We could consider that the three parties that obtained seats in the inter-communal council are three different players, *Lille*₁, *Lille*₂, and *Lille*₃ in the cooperative game that represents the inter-communal council. Hence, the number of players will increase, as each city that elects more than two delegates may be represented by two or more party lists. We may wish to estimate the Banzhaf power of each of these players in the inter-communal council, but this is a totally different game in terms of the number of players.

Another scenario is a variation of Penrose's one that takes into account the specific nature of French electoral law. Consider a case in which we have only two parties, *A* and *B*. We will show that if a voter is decisive in Lille, he will not decide the fate of all the 33 seats. Remember that 17 seats are given to the winner. If voter *i* is decisive, both parties have the same number of votes before he casts his vote, and the proportional part will give 8 seats for *A* and 8 seats for *B*. So, he will only decide whether the final distribution of seats will be 25 for *A* versus 8 for *B* or 8 for *A* versus 25 for *B*. He is only pivotal for half of the seats.<fnref n="10" /> That is, when a citizen is decisive, the weight attached to the player *Lille* should not be 33, but 17. This defines a new game with the same number of players but with weights divided by two, except for small municipalities (see the remark of footnote 7).

In this paper, we will only study the classical Penrose scenario. The description of the Lille Métropole case which takes into account all the details of French electoral law convinced us that it deserves a specific analysis.

# 4 Data and empirical design

## 4.1 Data description

To analyze power inequality in French inter-communal structures, we employ an extensive set of data from 1,254 EPCIs gathering 34,964 municipalities in France. Most of the data were collected via the Interior and Local Government Ministry, which manages a public database of the 2020 municipal election results for cities with more than 1,000 inhabitants. We combined this database with the one giving the list of all the inter-communal structures. When needed, the missing data for cities of fewer than 1,000 inhabitants were obtained by directly contacting the inter-communal structures or checking their website. For more detailed information about the database, the reader can refer to the companion data paper by Abidi Perier et al. (2025).

<footnote n="10">¹⁰If the number of seats for the proportional part is odd, he would also be decisive for an extra seat.</footnote>
These data give the exact number of delegates that represent each city in its inter-municipal Council. With these data, we derive the Banzahf index of each municipality within its council and assess whether the distribution of power is fair, by computing the *max – min* index for each inter-municipality. We propose the results not only for the observed data but also for three other possible allocations of the seats. Note that we will not analyze three specific EPCIs, where different rules apply: these are Métropole du Grand Paris, which consists of 131 municipalities, Métropole de Lyon, which consists of 59 municipalities, and Métropole d'Aix-Marseille-Provence, which consists of 92 municipalities.[11]

## 4.2 Four scenarios for the allocation of delegates

### Scenario O: the Observed distribution of delegates

This benchmark scenario represents the current distribution of seats assigned to each municipality of the EPCI to which they belong, observed in 2020. Note that the law of March 9, 2015, authorized the establishment of a local agreement, allowing the number of seats to be increased (or decreased) by a maximum of 25%. The agreement must still meet three conditions: i) Each municipality have at least one seat, ii) No municipality may have more than half of the seats, iii) the representation resulting from the local agreement cannot deviate by more than ± 20% from the demographic weight of the municipality.[12]

Hence, in our database, we will encounter some EPCIs that stuck exactly to the initial rules, while some others altered the final allocation of delegates via local agreements. This provision leads us to propose an alternative to the observed scenario, to check whether a local agreement has been implemented or not.

### Scenario PLJ: Proportional distribution per capita using the total number of EPCI delegates assigned by the 2010 law using the Jefferson rule

This scenario uses the distribution of seats according to Table 1 and assumes that no agreement had been made between the EPCIs' members. To simulate the seat allocations, we rely on the Jefferson apportionment rule at each step. Delegates are allocated one by one, the next seat being given to the municipality with the greatest ratio ($\frac{n_i}{a_i+1}$). The process stops when all the

<footnote>[11]For further information regarding seat allocation, please refer to Appendix III.</footnote>

<footnote>[12]Unless the distribution of common law already deviates by ± 20% and the new distribution makes it possible to come closer to it, and except in cases where 2 seats are attributed to a commune when common law would lead to it being attributed only one seat.</footnote>
seats have been allocated. Notice that a city cannot have more than half of the seats. If this is the case, the extra seats go to the city with the second largest ratio ($\frac{n_i}{a_i+1}$). At the end of the process, if a city has received no seats, it gets one extra seat automatically. The 2020 Law has an extra provision: if the number of extra seats has increased the total number of seats by 30%, 10% extra seats are automatically allocated to the council. The allocation of these extra seats is governed by the same Jefferson rule, with the limit that no city controls more than half of the seats.

Hence, this scenario simulates the strict allocation of the Law without any agreement. Here the comparison of this scenario and scenario O will allow us to assess the impact of local agreements on both on the total number and the allocation of delegates. We will also check whether or not giving more autonomy to the EPCIs via a local agreement reduces the inequalities in terms of power.

## Scenario SRLJ: Population square root population criterion using the total number of EPCI delegates assigned by the 2010 law using the Jefferson rule

The objective of this scenario is to compare the reality with the theoretical recommendation of the Penrose Square Root Rule. It assumes that apportionment of delegates is made according to the population square root of each municipality. Hence, for each municipality $i$ belonging to $EPCI_j$, the number of delegates is roughly proportional to:

$$ \frac{\sqrt{n_i}}{\sum_{t \in EPCI_j} \sqrt{n_t}} $$

To be more precise, the same rules as in Scenario PLJ are met: the initial number of seats to allocate is drawn from Table 1, we use a first Jefferson apportionment, add extra seats to city with no delegates, and at the end could add 10% more delegates if needed.

## Scenario PLH: Proportional distribution per capita using the total number of EPCI delegates assigned by the 2010 law using the Hamilton rule

One particularity of French law is that it is based on the Jefferson's greatest divisor principle, which by definition favors the biggest city. The legislature then needed to add many different provisions to ensure that each city had at least a delegate and that the largest cities have the 'right' number of delegates. However, one may wonder whether the Hamilton greatest remainder principle would perform better in terms of municipalities' representations: this is why we introduced this scenario.
If $\bar{a}$ delegates are to be apportioned in inter-municipality $j$, we first compute the divisor

$$d = \frac{\sum_{i=1}^{n} n_i}{\bar{a}}$$

that is, the number of inhabitants that guarantees one delegate. Each municipality should then get a number of delegates equal to:

$$a_i = \lfloor \frac{n_i}{d} \rfloor$$

and the remaining seats are allocated to the municipalities with the highest remainders.

Hence, we adapt the French dispositions by using Hamilton instead of Jefferson throughout the process:

* No city should control more than 50% of the seats. If it is the case, we limit its number of seats to 50%, and allocate the remaining seats to all the other cities.

* If a city has no seat at the end of the process, it gets automatically an extra seat.

* If the number of seats is increased by more than 30%, we add 10% extra seats that are allocated again according to Hamilton's principle, and barring any city from controlling a majority of seats.

Note that this process may not be the unique way to integrate the Hamilton principle while respecting the French electoral laws but we believe that it is faithful way to do so.

### 4.2.1 Scenario SRLH: Population square root population criterion using the total number of EPCI delegates assigned by the 2010 law using the Hamilton rule

This scenario assumes that each municipality has a number of delegates proportional to its population square root, and uses the same rules as in Scenario PLH.

The objective of this scenario is to test whether a change in the apportionment rule would limit the creation of extra seats, and lead to a fairer distribution of power.

## 4.3 Comparing the allocation of seats across scenarios

In order to compare the number of seats assigned by each scenario to EPCIs, we compute the differences in seats between the three scenarios and the observed one (scenario O). Table 3 provides the difference in the number of delegates between scenario O (the observed number of delegates) and scenario PLJ (simulated with the Jefferson method), scenario SRLJ (simulated with the square root population and Jefferson), scenario PLH (simulated with the Hamilton
<table><caption>Table 3: Difference in the number of delegates between scenario O (the observed number of delegates) and our four scenarios</caption><thead><tr><th>Interval</th><th>O - PLJ</th><th>O - SRLJ</th><th>O - PLH</th><th>O - SRLH</th></tr></thead><tbody><tr><td>[-100%, -50%[</td><td>0</td><td>0</td><td>3</td><td>0</td></tr><tr><td>[-50%, -25%[</td><td>5</td><td>4</td><td>9</td><td>0</td></tr><tr><td>[-25%, -20%[</td><td>1</td><td>0</td><td>1</td><td>0</td></tr><tr><td>[-20%, -15%[</td><td>6</td><td>2</td><td>4</td><td>0</td></tr><tr><td>[-15%, -10%[</td><td>11</td><td>3</td><td>5</td><td>1</td></tr><tr><td>[-10%, -5%[</td><td>51</td><td>8</td><td>8</td><td>0</td></tr><tr><td>[-5%, -0%[</td><td>104</td><td>3</td><td>3</td><td>2</td></tr><tr><td><strong>Total Negative</strong></td><td><strong>178</strong></td><td><strong>20</strong></td><td><strong>33</strong></td><td><strong>3</strong></td></tr><tr><td><strong>Equal</strong></td><td><strong>655</strong></td><td><strong>43</strong></td><td><strong>48</strong></td><td><strong>1</strong></td></tr><tr><td>]0%, +5%[</td><td>104</td><td>52</td><td>90</td><td>0</td></tr><tr><td>] + 5%, +10%[</td><td>80</td><td>79</td><td>159</td><td>0</td></tr><tr><td>] + 10%, +15%[</td><td>79</td><td>145</td><td>218</td><td>1</td></tr><tr><td>]15%, 20%[</td><td>61</td><td>230</td><td>346</td><td>3</td></tr><tr><td>]20%, 25%[</td><td>75</td><td>180</td><td>158</td><td>15</td></tr><tr><td>]25%, 50%[</td><td>19</td><td>473</td><td>199</td><td>259</td></tr><tr><td>]50%, 100%[</td><td>0</td><td>29</td><td>0</td><td>969</td></tr><tr><td><strong>Total Positive</strong></td><td><strong>418</strong></td><td><strong>1,188</strong></td><td><strong>1,170</strong></td><td><strong>1,247</strong></td></tr><tr><td>Total</td><td>1,251</td><td>1,251</td><td>1,251</td><td>1,251</td></tr><tr><td>Mean Diff</td><td>53,75 -52,66</td><td>53,75-44,01</td><td>53,75-46,76</td><td>53,75-31,17</td></tr></tbody></table>

method)), scenario SRLH (simulation with the square root population and the Hamilton method) respectively.

## O versus PLJ

It is observed that slightly over half of the EPCIs (655/1251) have an exact match between the observed total number of delegates and the simulated count. More surprisingly, we observe that 178 EPCIs display an observed total number of delegates smaller than the number that is proposed by the law, while 418 EPCIs have a higher number. This suggests that approximately half of the inter-communal structures have utilized the opportunity to modify the number of
<table><caption>Table 4: CC de la Vallée du Gapeau (Departement 83)</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Observed</th><th>PLJ</th></tr></thead><tbody><tr><td>Belgentier</td><td>2,461</td><td>3</td><td>2</td></tr><tr><td>Solliès-Ville</td><td>2,494</td><td>3</td><td>2</td></tr><tr><td>Solliès-Toucas</td><td>5,827</td><td>6</td><td>6</td></tr><tr><td>La Farlède</td><td>9,195</td><td>8</td><td>11</td></tr><tr><td>Solliès-Pont</td><td>11,310</td><td>10</td><td>13</td></tr><tr><td><strong>Total</strong></td><td><strong>31,287</strong></td><td><strong>30</strong></td><td><strong>34</strong></td></tr><tr><td><strong>Difference in the number of seats</strong></td><td></td><td><strong>-11.76%</strong></td><td></td></tr></tbody></table>

delegates by approximately ± 25%. However, upon examining Table 3, it is apparent that only 23 EPCIs have decreased their number of departed delegates by more than 10%. Among them, only six EPCIs have reduced their number of delegates by more than 25%. These particular EPCIs are characterized by a very small membership size, ranging from 3 to 6 members. This unique circumstance justifies an exception being made in regard to their number of delegates.

Table 4 displays a case with a limited number of members, where the negotiations aimed at reducing the totam number of seats and diminishing the number of seats of the larger municipalities.

Meanwhile, some other EPCIs used the authorized margin of negotiation to augment their number of delegates. Namely, 418 of 1,251 EPCIs have a higher total number of seats. Different patterns can be observed. In the first pattern, EPCIs aim to distribute more influence (and, consequently, seats) to medium-sized municipalities. An illustration of this is evident in Table 5, where the number of seats for the largest municipality is reduced by 1, while the seats for medium-sized municipalities are increased. A second pattern emerges in EPCIs encompassing a large number of municipalities, such as the example of EPCI CA du Bassin d'Aurillac in Table 6, which includes 25 municipalities. Here, we observe an increase of 1 seat for the three largest municipalities.

As expected, the scenarios using the square root of population and those based on the Hamilton method generate fewer extra seats compared to the pure Jefferson method and the observed number of seats. While the average number of seats per council was almost the same in the Observed and PLJ scenarios, using the populations' square root with Jefferson reduces the average
<table><caption>Table 5: CC du Pays de Duras (Departement 47)</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Number of Seats</th><th>PLJ</th></tr></thead><tbody><tr><td>Saint-Géraud</td><td>97</td><td>1</td><td>1</td></tr><tr><td>Sainte-Colombe-de-Duras</td><td>106</td><td>1</td><td>1</td></tr><tr><td>Esclottes</td><td>153</td><td>1</td><td>1</td></tr><tr><td>Baleyssagues</td><td>181</td><td>1</td><td>1</td></tr><tr><td>Auriac-sur-Dropt</td><td>183</td><td>1</td><td>1</td></tr><tr><td>Savignac-de-Duras</td><td>221</td><td>1</td><td>1</td></tr><tr><td>Saint-Astier</td><td>227</td><td>1</td><td>1</td></tr><tr><td>Soumensac</td><td>240</td><td>2</td><td>1</td></tr><tr><td>Saint-Jean-de-Duras</td><td>254</td><td>2</td><td>1</td></tr><tr><td>Pardaillan</td><td>311</td><td>2</td><td>1</td></tr><tr><td>Villeneuve-de-Duras</td><td>324</td><td>2</td><td>1</td></tr><tr><td>Monteton</td><td>333</td><td>2</td><td>1</td></tr><tr><td>Saint-Pierre-sur-Dropt</td><td>341</td><td>2</td><td>1</td></tr><tr><td>Loubès-Bernac</td><td>418</td><td>2</td><td>2</td></tr><tr><td>Saint-Sernin</td><td>442</td><td>2</td><td>2</td></tr><tr><td>Lévignac-de-Guyenne</td><td>677</td><td>3</td><td>3</td></tr><tr><td>Duras</td><td>1,305</td><td>6</td><td>7</td></tr><tr><td><strong>Total</strong></td><td><strong>5,813</strong></td><td><strong>32</strong></td><td><strong>27</strong></td></tr><tr><td><strong>Difference in the number of seats</strong></td><td></td><td><strong>+18.52%</strong></td><td></td></tr></tbody></table>

size of the council by almost 8 seats. A similar result is observed when we use the Hamilton apportionment method, minus 7 seats on average. If the objective of the legislator were to reduce the size of councils, the combination of the population's square root and Hamilton would achieve this goal, with an average size of the council reduced by almost 23 delegates! Therefore, almost all EPCIs have an observed number of seats higher compared to these scenarios, with only a few rare exceptions.

It is important to note that these general comments apply all over France, regardless of the region or department to which they belong. As shown in Figure 1, several EPCIs have maintained the same number of seats allocated by law without utilizing the ± 25% range authorized by the
<table><caption>Table 6: CA du Bassin d'Aurillac (Departement 15) - 25 municipalities</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Number of Seats</th><th>PLJ</th></tr></thead><tbody><tr><td>Vezels-Roussy</td><td>133</td><td>1</td><td>1</td></tr><tr><td>Saint-Cirgues-de-Jordanne</td><td>144</td><td>1</td><td>1</td></tr><tr><td>Mandailles-Saint-Julien</td><td>190</td><td>1</td><td>1</td></tr><tr><td>Teissières-de-Cornet</td><td>298</td><td>1</td><td>1</td></tr><tr><td>Lascelle</td><td>305</td><td>1</td><td>1</td></tr><tr><td>Laroquevieille</td><td>357</td><td>1</td><td>1</td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr><tr><td>Jussac</td><td>2,091</td><td>2</td><td>2</td></tr><tr><td>Ytrac</td><td>4,396</td><td>5</td><td>4</td></tr><tr><td>Arpajon-sur-Cère</td><td>6,387</td><td>7</td><td>6</td></tr><tr><td>Aurillac</td><td>26,828</td><td>27</td><td>26</td></tr><tr><td><strong>Total</strong></td><td><strong>55,265</strong></td><td><strong>68</strong></td><td><strong>60</strong></td></tr><tr><td><strong>Difference in the number of seats</strong></td><td></td><td></td><td><strong>+13.33%</strong></td></tr></tbody></table>

law (Figure 1-left).<fnref n="13" /> It is noteworthy that EPCIs in western regions (Brittany and Pays de la Loire) seem to have utilized the plus 25% margin to the greatest extent. This is the only regional pattern we could detect. Figure (1-right) and Figure 2 clearly demonstrate that scenarios SRLJ, PHL and SRLH result in a lower number of seats.

# 5 Assesing Inequality in Terms of Power

## 5.1 The impact of local agreements: O versus PLJ

In section 3.2, we introduced the Quotient $Q_j = NB_j \times \frac{\sum \sqrt{n_j}}{\sqrt{n_j}}$. Ideally, it should be equal to one in any municipality: to achieve equity in terms of power, the normalized Banzhaf power of a municipality $j$ should be as close as possible to its share of the population, measured in terms of square roots. Hence, the ratio $max - min = \frac{Q_j^{max}}{Q_j^{min}}$ measures the inequality in terms of power between the least favored and the most favored citizen in a given inter-communal structure. Table 7 presents the results of the max-min ratios for the five scenarios. Following

<footnote n="13">¹³Remember that the cases of Paris, Lyon, and Marseille are not included in our analysis, creating blank regions in the maps.</footnote>
<table><caption>Table 7: The distribution of the max-min ratios according to scenario O and our 4 scenarios</caption><thead><tr><th>Max-Min ratio</th><th>O</th><th>PLJ</th><th>SRLJ</th><th>PLH</th><th>SRLH</th></tr></thead><tbody><tr><td>“1-2"</td><td>19</td><td>9</td><td>221</td><td>16</td><td>140</td></tr><tr><td>“2-5"</td><td>530</td><td>477</td><td>922</td><td>680</td><td>570</td></tr><tr><td>“5-10"</td><td>303</td><td>330</td><td>92</td><td>291</td><td>393</td></tr><tr><td>“10-100"</td><td>239</td><td>251</td><td>11</td><td>187</td><td>135</td></tr><tr><td>“100-inf"</td><td>160</td><td>184</td><td>5</td><td>77</td><td>13</td></tr><tr><td>incl. "infinity"</td><td>13</td><td>7</td><td>5</td><td>16</td><td>13</td></tr><tr><td>Total</td><td>1,251</td><td>1,251</td><td>1,251</td><td>1,251</td><td>1,251</td></tr></tbody></table>

<table><caption>Table 8: The evolution of the max-min ratios across scenarios</caption><thead><tr><th>Scenarios</th><th>O-PLJ</th><th>O-SRLJ</th><th>O-PLH</th><th>O-SRLH</th></tr></thead><tbody><tr><td>Negative</td><td>685</td><td>75</td><td>357</td><td>437</td></tr><tr><td>Equal</td><td>316</td><td>34</td><td>46</td><td>12</td></tr><tr><td>Positive</td><td>250</td><td>1,142</td><td>848</td><td>802</td></tr><tr><td>Total</td><td>1,251</td><td>1,251</td><td>1,251</td><td>1,251</td></tr></tbody></table>

the methodology outlined in Abidi et al. (2020), we have classified the ratios into five categories:

1: $1 \leq max - min \leq 2$. This corresponds to situations where the deviations of the power of a citizen from one are limited, in the $\pm 33\%$ range. For example, $Q_j^{min} > 0.66$ and $Q_j^{max} < 1.33$ falls in this category.

2: $2 < max - min \leq 5$. The variation range has doubled, up to $\pm 66\%$. For example, an inter-communal structure with $Q_j^{min} > 0.33$ and $Q_j^{max} < 1.66$ falls in this category.

3: $5 < max - min \leq 10$. The difference between $Q_j^{min}$ and $Q_j^{max}$ becomes progressively significant. For example, a situation where $Q_j^{min} > 0.3$ and $Q_j^{max} < 3$ falls in this category.

4: $10 < max - min \leq 100$. In this case, a citizen may be pivotal 10 times less frequently than he should be, and another pivotal 10 times more frequently than he should be.

5: $max - min > 100$. We can hardly pretend that the situation is satisfactory in terms of equity among citizens. In this category, we also found a few cases where the ratio goes to infinity, if one city is a dummy player in the inter-communal structure.
<table><caption>Table 9: Difference in Max-Min index between scenarios O and PLJ</caption><thead><tr><th>Seats of the major city (O-PLJ)</th><th>Same</th><th>Decrease</th><th>Increase</th><th>Total</th></tr></thead><tbody><tr><td>Equal</td><td>7</td><td>0</td><td>2</td><td>9</td></tr><tr><td>Negative</td><td>7</td><td>326</td><td>22</td><td>355</td></tr><tr><td>Positive</td><td>3</td><td>13</td><td>38</td><td>54</td></tr><tr><td>Total</td><td>17</td><td>339</td><td>62</td><td>418</td></tr></tbody></table>

According to Table 7, Scenarios O and PLJ have only 19 and 9 EPCIs, respectively, with a max-min ratio in the range "1-2". This indicates that equity among citizens is achieved in very few situations. In contrast, Scenarios O and PLJ have the highest number of EPCIs with a max-min ratio higher than 10 and even higher than 100, compared to the other three scenarios. Specifically, the current distribution of seats includes 239 EPCIs where the most powerful citizen has at least ten times the power of the least powerful one. Additionally, there are 160 EPCIs with a max-min ratio higher than 100. We can even find 13 inter-communal structures where a city and its citizen do not have any influence at all on the decision<fnref n="14" />. The situation is even slightly worse for the PLJ scenario, with more EPCIs in categories 3, 4 and 5. We also observe 7 inter-communal councils with dummy players.<fnref n="15" /> In Figure 3 and 4 in Appendix IV, notice that we cannot detect any regional pattern regarding the inequalities.

Can we say that the Observed scenario is always better than the PLJ scenario? The situation indeed differs across the EPCIs, as they have pursued different goals when making a local agreement. Table 8 compares the EPCIs one by one, to check whether the max-min ratio increases or decreases when we move from the Observed scenario to PLJ, SRLJ, PLH and SRLH, respectively.

To give an example, Communauté de Communes Cœur de Perche (see Table 10), an EPCI with 12 members, displays a situation where its members chose to make a local agreement, and to give one extra seat to the four most populous cities, raising the total number of delegates from 28 to 32. As a consequence, the max-mix ratio rises from 3.234 to 3.721 when we move from

<footnote n="14">¹⁴We give the list here: CA Cannes Pays de Lérins, CA du Grand Verdun, CA La Riviéra du Levant, CC Aveyron Bas Ségala Viaur, CC de la Sologne des Rivières, CC de la Vallée de Chamonix-Mont-Blanc, CC de l'Est Guyanais, CC des Portes de Romilly sur Seine, CC des Savanes, CC des sources du lac d'Annecy, CC Gartempe - Saint Pardoux, CC Les Portes Briardes Entre Villes et Forêts, CC Rhône Lez Provence.</footnote>

<footnote n="15">¹⁵CC Lyon-Saint-Exupéry en Dauphiné CC du Val de l'Eyre, CA du Centre Littoral, CA Cannes Pays de Lérins, CA du Grand Verdun, CA La Riviéra du Levant, CC Les Portes Briardes Entre Villes et Forêts.</footnote>
the strict application of the law (PLJ) to the observed situation. In table 8 we observe that 250 EPCIs display a similar pattern. In the same table, we observe that using a local agreement has a positive impact, by reducing the max-min ratio, in 685 cases. To give an example, in the Communauté de Communes Pays de Dol et de la Baie du Mont Saint-Michel, an EPCI with 13 municipalities (see Table 11), the agreement was used to reduce the number of seats of the major city from 10 to 8, while 8 medium-sized cities got an extra seat. As a consequence, the max-min index decreased from 10.14 to 3.25 when the municipalities enforced a local agreement.

All in all, we have more situations (685) where the max-min increases when we move from the Observed scenario to the strict application of the law (PLJ). Can we assert that the local agreements were mostly used to limit the number of seats and power of the major city, as in CC Pays de Dol et de la Baie du Mont Saint-Michel? Table 9 focuses on the 418 EPCIs identified in Table 3 where the number of seats increased. We clearly observed that 355 of those EPCIs used the agreement to reduce the number of seats of the major city; in the meantime, for 326 of these EPCIs, the max-min index decreased when we moved from the PLJ scenario to the Observed scenario. Symmetrically, out of the 54 EPCIs where the number of seats for the major city increased due to a local agreement, we observed an increase in the max-min ratio for 38 of them.

All these elements confirm that more inter-communal structures used local agreements to limit the number of seats of the largest city, and its power as well.
<table><caption>Table 10: CC Coeur du Perche (Département 61, 12 municipalities)</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Observed</th><th>PLJ</th><th>SRLJ</th><th>PLH</th><th>SRLH</th></tr></thead><tbody><tr><td>Saint-Cyr-la-Rosiere</td><td>236</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Saint-Germain-des-Grois</td><td>245</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Moutiers-au-Perche</td><td>398</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>La Madeleine-Bouvet</td><td>418</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Verrieres</td><td>428</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Saint-Pierre-la-Bruyere</td><td>444</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Cour-Maugis sur Huisne</td><td>634</td><td>1</td><td>1</td><td>2</td><td>1</td><td>1</td></tr><tr><td>Berd'huis</td><td>1,142</td><td>3</td><td>3</td><td>3</td><td>3</td><td>1</td></tr><tr><td>Bretoncelles</td><td>1,517</td><td>4</td><td>3</td><td>3</td><td>3</td><td>2</td></tr><tr><td>Remalard en Perche</td><td>1,993</td><td>6</td><td>5</td><td>4</td><td>4</td><td>2</td></tr><tr><td>Perche en Noce</td><td>2,130</td><td>6</td><td>5</td><td>4</td><td>5</td><td>2</td></tr><tr><td>Sablons sur Huisne</td><td>2,171</td><td>6</td><td>5</td><td>4</td><td>5</td><td>2</td></tr><tr><td>Total</td><td></td><td>32</td><td>28</td><td>26</td><td>27</td><td>16</td></tr><tr><td>Max-min ratio</td><td></td><td>3.721</td><td>3.234</td><td>2.024</td><td>3.202</td><td>2.200</td></tr></tbody></table>

## 5.2 Hamilton as a remedy?

Regarding the PLH scenario, it is noteworthy that it outperforms the O and PLJ scenarios in the sense that it has the highest number of EPCIs with a max-min ratio below 5. This suggests that seat allocation using the Hamilton method performs better than the current Jefferson method used by French municipalities. As depicted in Figure 4, it is notable that the PLH scenario exhibits a lower number of EPCIs with a max-min ratio exceeding 100. However, the number of EPCIs with a dummy player is quite high (16). This result indicates that if municipalities intend to persist in utilizing proportional distribution per capita of seats, adopting the Hamilton method could serve as a means (solution) to mitigate inequalities in terms of voting power among municipalities within the same EPCI. However, Table 8 indicates that while the value of the max-min ratio improves for 2/3 of the EPCIs (848), the situation deteriorates for 357 of them, making the whole picture quite contrasted.
<table><caption>Table 11: CC du Pays de Dol et de la Baie du Mont Saint-Michel (Département 35, 19 municipalities)</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Observed</th><th>PLJ</th><th>SRLJ</th><th>PLH</th><th>SRLH</th></tr></thead><tbody><tr><td>Vieux-Viel</td><td>317</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Saint-Georges-de-Gréhaigne</td><td>374</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Broualan</td><td>382</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Saint-Marcan</td><td>457</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Sains</td><td>490</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Sougeal</td><td>581</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Trans-la-Forêt</td><td>587</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Roz-sur-Couesnon</td><td>1,015</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Le Vivier-sur-Mer</td><td>1,049</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Cherrieux</td><td>1,129</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Mont-Dol</td><td>1,131</td><td>2</td><td>1</td><td>2</td><td>1</td><td>1</td></tr><tr><td>Saint-Brolade</td><td>1,154</td><td>2</td><td>1</td><td>2</td><td>1</td><td>1</td></tr><tr><td>Le Boussac</td><td>1,204</td><td>2</td><td>2</td><td>2</td><td>2</td><td>1</td></tr><tr><td>Roz-LAndrieux</td><td>1,367</td><td>2</td><td>2</td><td>2</td><td>2</td><td>1</td></tr><tr><td>Epiniac</td><td>1,450</td><td>2</td><td>2</td><td>2</td><td>2</td><td>1</td></tr><tr><td>Baguer-Pican</td><td>1,694</td><td>3</td><td>2</td><td>2</td><td>2</td><td>1</td></tr><tr><td>Baguer-Morvan</td><td>1,738</td><td>3</td><td>2</td><td>2</td><td>2</td><td>1</td></tr><tr><td>Pleine-Fougères</td><td>2,014</td><td>4</td><td>3</td><td>2</td><td>3</td><td>1</td></tr><tr><td>Dol-de-Bretagne</td><td>5,958</td><td>8</td><td>10</td><td>4</td><td>7</td><td>2</td></tr><tr><td>Total</td><td>24,091</td><td>41</td><td>35</td><td>30</td><td>32</td><td>20</td></tr><tr><td>Max-min ratio</td><td></td><td>3.25</td><td>10.14</td><td>2.03</td><td>4.47</td><td>2.52</td></tr></tbody></table>

## 5.3 Comparison with the square root law

Finally, the SRLJ and SRLH scenarios are in line with the normative recommendations of Penrose. They reduce inequality drastically in terms of power, but there are still a few EPCIs where the central city is *too big*, even if we use the population square root to allocate the number of delegates. Specifically, there are 16 EPCIs in the SRLJ scenario and 148 EPCIs in the SRLH scenario where the max-min ratio exceeds 10. Hence, it can be inferred that the combination of the number of delegates allocated by the 2010 law (in Table 1) along with an allocation of
delegates proportional to the square root of the population, following the Jefferson method, yields the most favorable outcomes in terms of reducing power inequalities among French municipalities. As depicted in Figure 7, it is evident that the SRLH scenario exhibits a larger number of instances where the voting power is relatively equal (less than 5). This observation indicates a higher degree of power equity in comparison to the SRLJ scenario. However, we still observe a few instances where there exist dummy players (5 for SRLJ, and 13 for SRLH).

# 6 Conclusion

This research aims to analyze the distribution of the voting power of citizens within the French inter-municipal structures. To achieve this, we have taken advantage of the availability of a comprehensive database that provides the allocation of seats to each municipality in the councils that govern the inter-communal structures. This has allowed us to investigate the representativeness of municipalities and to compare the existing distribution of seats with four simulated scenarios. The first two scenarios, namely PLJ and SRLJ, employ the Jefferson method and allocate seats based on the criteria of proportional representation to population and to the square root of population, respectively. In contrast, the other two scenarios, PLH and SRLH, utilize the same criteria but employ the Hamilton method.

Our primary finding indicates that the current allocation, known as “Scenario O,” which follows the principle of Jefferson proportional representation based on existing populations and where municipalities were able to negotiate seat distributions among themselves while adhering to specific rules, exhibits lower levels of inequality compared to the PLJ scenario in 685 cases out of 1,251. The PLJ scenario, which applies a similar seat allocation method but excludes any negotiation possibilities, results in higher levels of inequality. This observation suggests that negotiations have played a role in mitigating inequalities among municipalities belonging to the same inter-municipal structure in some instances. However, the allocation of more seats in the observed scenario than in the PLJ scenario also provides a greater opportunity for the largest municipality to acquire additional delegates in some cases and increases the inequality in 250 EPCIs.

We also find that by combining the criterion of proportionality to the population with the utilization of the Hamilton method, a significant reduction in inequalities in terms of voting power is observed. And, as expected, scenarios based on the criterion of the square root of
population, namely SRLJ and SRLH, yield better results in terms of voting power. Indeed, our
data confirm the so called Penrose square root law.

However, it may be difficult to impose representation based on the square root of populations
in modern democracies, where the “one man-one vote” principle is widely accepted. We should
also add that the justification of the square root principle is based on a specific probabilistic
assumption to model the voting behavior, the adequacy of which to real election data has been
contested by Gelman et al. (2004). Hence, if French municipalities opt to retain population
proportion as the basis for allocating seats among municipalities within the same inter-municipal
structure, the most effective approach for reducing power inequalities is then to combine this
criterion with the Hamilton method.

Alternatively, to escape the curse of over-representation of major cities in inter-municipal councils, a solution could be to divide them into several electoral jurisdictions, each one sending a limited number of delegates to the inter-municipal council, as is the case for Paris, Lyon and Marseilles. To give an example, Lille could be split into 3-4 jurisdictions, each one sending less than 10 delegates to the inter-municipal council. However, this solution would create other issues, such as the possibilities of gerrymandering and inversed majorities.

Our paper has limitations that should be addressed in future work. First, like many other
studies, we use the bloc voting hypothesis, assuming that delegates of a municipality vote as a
unified bloc, meaning they vote for the same decision within the inter-municipal council. This
assumption can be relaxed in future research by considering alternative voting behaviors, such
as delegates voting according to their political affiliations. Finally, another more ambitious and
challenging extension is to develop a seat allocation method that effectively minimizes inequality
among municipalities within the same inter-municipal structure, regardless of the size, i.e., the
number of municipalities constituting the structure.
References

Zineb Abidi, Matthieu Leprince, and Vincent Merlin. Power inequality in inter-municipal structures: The simulated impact of a reform in the case of the municipalities in western france.
*Économie & prévision*, 217(1):65–87, 2020.

Z. Abidi Perier, V. Merlin, and T. Rivet. The repartition of delegates in french inter-communal structures in march 2020, 2025. https://doi.org/10.34847/nkl.cdd239fj

Michel Balinski and H. Peyton Young. *Fair representation: Meeting the ideal of one man, one vote*. Brookings Institution Press, Washington, D.C., 2nd edition, 2001.

John F. Banzhaf. Weighted voting doesn't work: A mathematical analysis. *Rutgers Law Review*, 13(2):317-343, 1965.

Salvador Barberà and Matthew O. Jackson. On the weights of nations: Assigning voting weights in a heterogeneous union. *Journal of Political Economy*, 114(2):317–339, 2006.

Franck Bisson, Jean Bonnet, and Dominique Lepelley. The determination of the number of delegates inside the inter-communal structures: An application of the banzhaf power index.
*Revue d'Économie Régionale & Urbaine*, 2:259–282, 2004.

Jens Blom-Hansen. Municipal amalgamations and common pool problems: The danish local government reform in 2007. *Scandinavian Political Studies*, 33(1):51–73, 2010.
URL https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9477.2009.00239.x
https://doi.org/10.1111/j.1467-9477.2009.00239.x.

Dan S Felsenthal and Moshé Machover. The treaty of nice and qualified majority voting. *Social Choice and Welfare*, 18(3):431–464, 2001.

Dan S Felsenthal and Moshé Machover. Analysis of qm rules in the draft constitution for europe proposed by the european convention 2003. *Social Choice and Welfare*, 23(1):1–20, 2004.

Andrew Gelman, Jonathan N Katz, and Joseph Bafumi. Standard voting power indexes don't work: an empirical analysis. *British Journal of Political Science*, 34(4):657–674, 2004.

Touria Jaaidane and Sophie Larribeau. The effects of inter-municipal cooperation and central grant allocation on the size of the french local public sector. *European Journal of Political Economy*, 76:102271, 2023. ISSN 0176-2680. URL https://www.sciencedirect.com/science/article/pii/S0176268022000714. https://doi.org/10.1016/j.ejpoleco.2022.102271.
Christina Koki and Spyros Leonardos. Coalitions and voting power in the greek parliament of 2012: A case study. *Homo Oeconomicus: Journal of Behavioral and Institutional Economics*, 35(4):295-313, 2019.

Annick Laruelle and Mika Widgrén. Is the allocation of voting power among eu states fair? *Public Choice*, 94(3-4):317-339, 1998.

Dennis Leech. Voting power on the governance of the international monetary fund. *Annals of Operations Research*, 109(1-4):373-395, 2002.

Guillermo Owen. Evaluation of a presidential election game. *American Political Science Review*, 69:947-953, 1975.

Lionel S Penrose. The elementary statistics of majority voting. *Journal of the Royal Statistical Society*, 109(1):53-57, 1946.

Lionel S. Penrose. *On the Objective Study of Crowd Behavior*. H.K. Lewis and Co, 1952.

Douglas Rae. Decision rules and individual values in constitutional choice. *American Political Science Review*, 63:40-56, 1969.

Shubik Martin Shapley, Lloyd .S. A method for evaluating the distribution of power in a committee system. *American Political Science Review*, 48:787-792, 1948.

Johan Stein. The striking similarities between northern norway and northern sweden. *Arctic Review on Law and Politics*, 10:79-102, 2019.
# Appendix I: Number of EPCIs according to their statute

<table><caption>Table 12: Number of EPCIs according to their statute</caption><thead><tr><th>On January 1st, 2020</th><th>2016</th><th>2017</th><th>2018</th><th>2019</th><th>2020</th></tr></thead><tbody><tr><td>Métropoles</td><td>13</td><td>14</td><td>21</td><td>21</td><td>21</td></tr><tr><td>Urban Communities (CU)</td><td>11</td><td>15</td><td>11</td><td>13</td><td>14</td></tr><tr><td>Community of Agglomeration (CA)</td><td>196</td><td>219</td><td>222</td><td>223</td><td>222</td></tr><tr><td>Community of communes (CC)</td><td>1,842</td><td>1,018</td><td>1,009</td><td>1,001</td><td>997</td></tr><tr><td>Lyon Métropole</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Number of isolated municipalities</td><td>27</td><td>5</td><td>4</td><td>4</td><td>4</td></tr><tr><td>Number of EPCI with fiscal autonomy</td><td>2,062</td><td>1,266</td><td>1,263</td><td>1,258</td><td>1,254</td></tr><tr><td>Number of municipalities</td><td>35,885</td><td>35,416</td><td>35,357</td><td>34,970</td><td>34,968</td></tr></tbody></table>

Source: DGCL, Banatic

# Appendix II: The Paris, Lyon, and Marseille Metropoles

The voting rules are different in three EPCIs: Métropole du Grand Paris (gathering 131 municipalities), Métropole de Lyon (gathering 59 municipalities), and Métropole d'Aix-Marseille-Provence (gathering 92 municipalities). These metropolises have a unique characteristic: the central city is subdivided into arrondissements (districts). This is the level that is considered in the election of inter-municipal delegates. Each arrondissement or group of arrondissements has a specific number of delegates. As a result, the assumption that the city's delegates will vote as a unified block is not applicable in these cases. For instance, Paris is divided into 20 arrondissements and 17 groups. The first four arrondissements form a group with 3 delegates, while the remaining arrondissements have a fixed number of seats allocated to them (Table 13). Marseille, on the other hand, is divided into 16 arrondissements, with every two arrondissements forming a group and having a specific number of delegates assigned to them, as presented in Table 14. The case of Lyon is slightly more intricate. The city is comprised of 9 arrondissements, divided into 5 groups, each with a specific number of delegates. Notably, the third arrondissement is divided into two parts. The eastern part has 11 delegates assigned to it, while the western part forms a group with the sixth arrondissement as displayed in Table 15.
<table><caption>Table 13: Paris (Métropole du Grand Paris)</caption><thead><tr><th>Groups</th><th>Arrondissement</th><th>Population</th><th>Number of seats</th></tr></thead><tbody><tr><td rowspan="4">Group 1</td><td>1st</td><td>16,395</td><td rowspan="4">3</td></tr><tr><td>2nd</td><td>21,042</td></tr><tr><td>3rd</td><td>34,389</td></tr><tr><td>4th</td><td>28,370</td></tr><tr><td>Group 2</td><td>5th</td><td>59,631</td><td>1</td></tr><tr><td>Group 3</td><td>6th</td><td>41,976</td><td>1</td></tr><tr><td>Group 4</td><td>7th</td><td>52,193</td><td>1</td></tr><tr><td>Group 5</td><td>8th</td><td>37,368</td><td>1</td></tr><tr><td>Group 6</td><td>9th</td><td>60,071</td><td>1</td></tr><tr><td>Group 7</td><td>10th</td><td>90,836</td><td>2</td></tr><tr><td>Group 8</td><td>11th</td><td>147,470</td><td>4</td></tr><tr><td>Group 9</td><td>12th</td><td>141,287</td><td>4</td></tr><tr><td>Group 10</td><td>13th</td><td>183,399</td><td>5</td></tr><tr><td>Group 11</td><td>14th</td><td>136,941</td><td>4</td></tr><tr><td>Group 12</td><td>15th</td><td>235,178</td><td>7</td></tr><tr><td>Group 13</td><td>16th</td><td>168,554</td><td>5</td></tr><tr><td>Group 14</td><td>17th</td><td>168,737</td><td>5</td></tr><tr><td>Group 15</td><td>18th</td><td>196,131</td><td>5</td></tr><tr><td>Group 16</td><td>19th</td><td>188,066</td><td>5</td></tr><tr><td>Group 17</td><td>20th</td><td>196,739</td><td>6</td></tr><tr><td>Total</td><td></td><td>2,204,773</td><td>60</td></tr></tbody></table>
<table><caption>Table 14: Marseille (Métropole d'Aix-Marseille-Provence)</caption><thead><tr><th>Groups</th><th>Arrondissement</th><th>Population</th><th>Number of seats</th></tr></thead><tbody><tr><td rowspan="2">Group 1</td><td>1st</td><td>39,265</td><td rowspan="2">5</td></tr><tr><td>7th</td><td>34,596</td></tr><tr><td rowspan="2">Group 2</td><td>2nd</td><td>23,898</td><td rowspan="2">5</td></tr><tr><td>3th</td><td>52,277</td></tr><tr><td rowspan="2">Group 3</td><td>4th</td><td>49,636</td><td rowspan="2">4</td></tr><tr><td>5th</td><td>45,098</td></tr><tr><td rowspan="2">Group 4</td><td>6th</td><td>40,734</td><td rowspan="2">6</td></tr><tr><td>8th</td><td>82,155</td></tr><tr><td rowspan="2">Group 5</td><td>9th</td><td>76,053</td><td rowspan="2">6</td></tr><tr><td>10th</td><td>59,131</td></tr><tr><td rowspan="2">Group 6</td><td>11th</td><td>57,519</td><td rowspan="2">6</td></tr><tr><td>12th</td><td>61,931</td></tr><tr><td rowspan="2">Group 7</td><td>13th</td><td>91,959</td><td rowspan="2">12</td></tr><tr><td>14th</td><td>60,968</td></tr><tr><td rowspan="2">Group 8</td><td>15th</td><td>79,092</td><td rowspan="2">5</td></tr><tr><td>16th</td><td>16,009</td></tr><tr><td>Total</td><td></td><td>870,321</td><td>49</td></tr></tbody></table>

<table><caption>Table 15: Lyon (Métropole de Lyon)</caption><thead><tr><th>Groups</th><th>Arrondissement</th><th>Population</th><th>Number of seats</th></tr></thead><tbody><tr><td rowspan="3">Group 1 : Lyon-Center</td><td>1st</td><td>29,403</td><td rowspan="3">11</td></tr><tr><td>2nd</td><td>30,318</td></tr><tr><td>4th</td><td>35,683</td></tr><tr><td>Group 2 : Lyon-East</td><td>3th</td><td>101,938</td><td>7</td></tr><tr><td rowspan="2">Group 3 : Lyon-West</td><td>5th</td><td>49,576</td><td rowspan="2">11</td></tr><tr><td>9th</td><td>52,849</td></tr><tr><td rowspan="2">Group 4 : Lyon-North</td><td>6th</td><td>52,622</td><td rowspan="2">9</td></tr><tr><td>3th</td><td>??,???</td></tr><tr><td rowspan="2">Group 5 : Lyon-South</td><td>7th</td><td>84,310</td><td>8</td></tr><tr><td>8th</td><td>85,980</td><td>9</td></tr><tr><td>Total</td><td></td><td>522,679</td><td>55</td></tr></tbody></table>
<figure><img src="image_2.png" /><figcaption>Figure 1: Number of delegates: Scenario O VS Scenario PLJ and Scenario PLH</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Number of delegates: Scenario O VS Scenario PLH and Scenario SRLH</figcaption></figure>

Appendix III: Comparison of the number of delegates
<figure><img src="image_4.png" /><figcaption>Figure 4: Geographic distribution of the max-min index in EPCIs according to Scenario PLJ and PLH</figcaption></figure>

<figure><img src="image_5.png" /></figure>

# Appendix IV: Geographic representations of max-min indices.
<figure><img src="image_6.png" /><figcaption>Figure 6: The disparity between the max-min ratio of scenarios O and, respectively, scenarios PLJ and SRLJ</figcaption></figure>

<figure><img src="image_7.png" /></figure>

Appendix V: Comparisons of the max-min ratios
<figure><img src="image_8.png" /><figcaption>Figure 7: The disparity between the max-min ratio of scenarios O and, respectively, scenarios PLH and SRLH</figcaption></figure>