# The hidden curse of French inter-communal power

Zineb Abidi Perier, Vincent Merlin

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Zineb Abidi Perier, Vincent Merlin. The hidden curse of French inter-communal power. Social Choice and Welfare, 2025, 10.1007/s00355-025-01639-1. halshs-05374788

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# The Hidden Curse of French Inter-communal Power*

Zineb Abidi Perier†

Vincent Merlin‡

**Conflict of interest:** The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

## Abstract

In 2020, the issue of strengthening collaboration among French municipalities reached a turning point. As of January 2020, every municipality had to join one of the 1,254 French inter-municipal structures. This milestone was accompanied by precise rules governing the distribution of seats among municipalities within the inter-municipal councils that govern each structure. However, the fairness of seat allocation among municipalities, according to the rules proposed by French law and based on the principles of proportionality, can be questioned. Since the pioneering works of (Penrose, 1946) and (Banzhaf, 1965), power index theory has shown that allocating representation to municipalities in proportion to their population tends to grant excessive power to the larger members. In this paper, we examine whether this phenomenon of imbalance can

<footnote>*We are grateful to our research assistant, Thibaud Rivet, for his assistance in managing the data. This research was funded in whole or in part by the French National Research Agency (ANR) under the Citizens Project ANR-22-CE26-0019.</footnote>

<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 Mêches 94010 Créteil, France. E-mail: zineb.abidi@u-pec.fr.</footnote>

<footnote>‡Université de Caen Normandie, CNRS, CREM UMR 6211, F-14000, Caen, France. E-mail: vincent.merlin@unicaen.fr.</footnote>
also be observed in France since 2020, using a new database that provides precise information
on the number of seats allocated to each city. Even though we encounter significant variation
across the country, our findings show that the current rules often result in over-representation
of major cities. We also propose alternative allocation methods designed to mitigate these in-
equalities in representation.

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

JEL: D72, H77, C71.

# 1 Introduction

Cooperation among French municipalities has a long history. The first law encouraging such co-
operation was enacted in 1890, introducing two forms: the Syndicat Intercommunal à Vocation
Unique (SIVU, Single-Purpose Inter-municipal Association) and the Syndicat Intercommunal
à Vocation Multiple (SIVOM, Multi-Purpose Inter-municipal Association). These were the
first Établissements Publics de Coopération Intercommunale (EPCI, Public Establishments for
Inter-municipal Cooperation).<fnref n="1" /> Since the 1990s, the French government has actively encouraged
inter-municipal cooperation for the joint management of public goods and services. One of the
most significant laws in this process is the Loi n° 2010-1563 de réforme des collectivités territo-
riales (Act No. 2010-1563 on the Reform of Local Authorities, hereafter “the 2010 law”). This
law made it mandatory for all municipalities to join an inter-communal structure by January
1st, 2014.<fnref n="2" /> It also reduced flexibility in the allocation of council seats and introduced explicit

<footnote n="1">¹In the remainder of the paper, we use the acronym ‘EPCI’ to denote a French inter-municipal structure.</footnote>

<footnote n="2">²With the exception of certain municipalities in the Île-de-France region, which were granted a derogation until January 1, 2016, to allow time for the establishment of the Grand Paris Métropole, the rest of the communes</footnote>
criteria for determining the number of delegates per municipality. The 2010 law was supple-
mented 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 (Mod-
ernisation de l'Action Publique Territoriale et d'Affirmation des Métropoles – Modernization of
Public Territorial Action and Affirmation of Metropolitan Areas). These laws established coun-
cils governing the EPCIs, with each municipality represented by a specific number of delegates
directly elected by its voters. As a result, all municipalities in France were required to join one
of the 1,254 EPCIs by 2020. More importantly, the 2020 municipal elections marked the first
time that voters directly elected their delegates in the EPCIs.

This paper examines the fairness of seat allocation among municipalities within EPCIs following
the 2020 elections. For the first time, a comprehensive dataset enables a detailed analysis of
delegate allocation by merging the 2020 electoral datasets provided by the French Ministry of
the Interior and the Ministry of Local Government. In particular, we investigate the extent to
which the rules used to allocate the delegates among the municipalities, described by the 2010
law and based on Jefferson's principles of proportional apportionment, favor larger cities.<fnref n="3" /> To
do so, we assess the fairness of the current attribution of delegates among French municipalities,
drawing on the literature on power indices.

The literature on social choice and game theory offers several criteria for evaluating the fairness
of seat allocation in two-tier voting systems, including maximizing the total utility of voters
(Barberà and Jackson, 2006), and equalizing the probability of success (Rae, 1969), or equalizing
the probability of being pivotal (Banzhaf, 1965; Penrose, 1946). Among these, the probabil-
ity of being pivotal has been the most influential, leading to the development of power index
theory. Empirical applications of these indices have focused on legislative assemblies, such as

<footnote>were required to join an inter-municipal structure by January 1, 2014.</footnote>

<footnote n="3">³Jefferson's method, also known as the D'Hondt method, is based upon the principle of greatest divisors. For further details on seat allocation methods, see (Balinski and Young, 2001).</footnote>
recent analyses of the Greek Parliament (Koki and Leonardos, 2019) and the governance of the
Lebanese Republic (Diss and Zouache, 2015). Power index tools have also been used to analyze
power relations in federal systems, such as analyses of the US Electoral College (Owen, 1975),
the European Union (Felsenthal and Machover, 2001, 2004; Laruelle and Widgrén, 1998) and
the International Monetary Fund (Leech, 2002).

Research applying power indices to French EPCIs is relatively recent and often focuses on specific regions. These studies typically use the Shapley-Shubik and Banzhaf indices to reveal discrepancies between municipalities' population sizes and their actual voting power, highlighting persistent representational imbalances. For example, Bisson et al. (2004) investigated municipal representation in the EPCIs in the Basse-Normandie region and showed that the distribution of voting power among municipalities often diverges from their population weight. They proposed an algorithm to compute a more equitable distribution of delegates based on power indices.

Similarly, Abidi et al. (2020) found that before the 2010 reform, power distribution was relatively fair in many EPCIs in Brittany, Normandy, and Pays de la Loire but simulations of the reform indicated a general increase in power inequality. Blancard et al. (2020) examined how institutional changes affected voting power in municipalities in La Réunion. Dia and Kamwa (2019) conducted a similar analysis in Martinique and Guadeloupe. Barthélémy and Mathieu (2011) introduced a spatial dimension using the Shapley-Owen index, showing that geographic location can significantly influence power assessments. Although these studies provide valuable insights into the complexities of EPCI governance in France, most focus on the period before 2020 and on specific regions. Thanks to the 2020 election database, this paper extends the analysis of the fairness of the delegate distribution within EPCIs to all French municipalities and compares the observed distributions to alternative scenarios.

This paper is structured as follows. Section 2 provides an overview of the French institutional
context. Section 3 presents the methodology used to measure voting power in the EPCIS. Sec-
tion 4 describes the data used and the empirical design adopted. The results are detailed in
Section 5 and Section 6 concludes with a discussion and suggested avenues for further research.

2 French institutional framework

The fragmentation of local government poses a major challenge, significantly affecting the effi-
cient provision of local services. It often results in inefficiencies, partly due to tax competition
among jurisdictions, which can trigger a “race to the bottom” (Tiebout, 1956; Oates, 1972;
Brennan and Buchanan, 1980). Recent research suggests that inter-municipal cooperation can
mitigate these effects (Di-Porto et al., 2017). Similarly, Abidi and Taugourdeau (2024) argue
that tax harmonization, when accounting for citizens’ preferences, can improve both equity
and efficiency. That is, several countries, particularly in Europe, have implemented municipal
consolidation programs to facilitate the joint provision of local public goods. For example, Den-
mark reduced its number of municipalities from 271 to 98 in 2007 through a process of merger
and consolidation (Blom-Hansen, 2010). In the same way, Sweden has reduced the number
of municipalities from more than 2000 in the 1950s to 290, mainly through voluntary mergers
(Stein, 2019). Meanwhile, other countries, such as France, Spain, and Italy, have implemented
a functional cooperation program (Hulst and van Montfort, 2007). 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

France has the highest number of municipalities in Europe, with more than 34,000 as of 2022. Among them, approximately 17,000 have fewer than 500 inhabitants.<fnref n="4" /> These small municipalities face difficulties in independently producing and providing certain public goods and services. Consequently, the French government has, since 1999, actively promoted the creation of inter-municipal jurisdictions governed by inter-municipal councils. Under this framework, each municipality must have at least one delegate, and no single municipality may hold the majority of the seats.

An important milestone in improving local collaboration among French municipalities was reached in 2020. As of January 2020, every municipality, for the first time, had to join one of 1,254 French EPCI structures. It is important to note that in France there are four types of EPCIs: Communauté de Communes (CC), Communauté d'Agglomération (CA), Communauté Urbaines (CU) and Métropoles (M) (See Table A1 in the Appendix). The main differences between these types of EPCIs are population size and the scope of their competencies. As a general rule, the degree of inter-municipal cooperation increases as one moves from Communauté de Communes to Métropoles. For further details, see (Jaaidane and Larribeau, 2023).

<footnote n="4">⁴For more details, the reader is can referred to the local government website https://www.collectivites-locales.gouv.fr/.</footnote>
## 2.2 Delegate Allocation Rules

<table><caption>Table 1: Number of Delegates Assigned to Inter-municipal Structures Under the 2010 Law</caption><thead><tr><td>Population</td><td>Number of Delegates</td></tr></thead><tbody><tr><td>Fewer than 3,500 inhabitants</td><td>16</td></tr><tr><td>3,500-4,999</td><td>18</td></tr><tr><td>5,000-9,999</td><td>22</td></tr><tr><td>10,000-19,999</td><td>26</td></tr><tr><td>20,000-29,999</td><td>30</td></tr><tr><td>30,000-39,999</td><td>34</td></tr><tr><td>40,000-49,999</td><td>38</td></tr><tr><td>50,000-74,999</td><td>40</td></tr><tr><td>75,000-99,999</td><td>42</td></tr><tr><td>100,000-149,999</td><td>48</td></tr><tr><td>150,000-199,999</td><td>56</td></tr><tr><td>200,000-249,999</td><td>64</td></tr><tr><td>250,000-349,999</td><td>72</td></tr><tr><td>350,000-499,999</td><td>80</td></tr><tr><td>500,000-699,999</td><td>90</td></tr><tr><td>700,000-1,000,000</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
The reform of municipal administrative structures was accompanied by specific rules governing
how delegates are distributed among the municipalities represented in each inter-municipal
council. Table 1 displays the number of delegates assigned to each EPCI according to population
size, as stipulated by the 2010 law. The allocation rules are as follows:<fnref n="5" />

1. The delegates must be allocated according to Jefferson's apportionment method (also known as the greatest divisor method). However, no one municipality may have more than half of the total seats.

2. If a municipality would otherwise have no delegate, it is granted an additional seat, so that the total number of delegates may exceed the baseline recommendations of Table 1.

3. If the total number of delegates allocated to an EPCI council exceeds the statutory number (as shown in Table 1) by more than 30%, the law allows for an allocation of an additional 10% of delegates. The allocation of these extra delegates must again follow Jefferson's principles. Although the rationale for this step is not explicitly stated in the legal text, we believe that the intention is to restore proportionality in favor of larger municipalities within each EPCI.

4. Finally, the law of 9 March 2015 authorizes the implementation of local agreements for Communautés de Communes and Communauté d'Agglomération (i.e. 1,219 out of 1,254 EPCIs).<fnref n="6" /> These municipalities may deviate from the total number of delegates obtained after the application of rules 1, 2, and 3 (within the limit of a 25% increase) and may

<footnote n="5">⁵For further details, see https://www.legifrance.gouv.fr/codes/article_lc/LEGIARTI000028534179/
2014-03-05.</footnote>

<footnote n="6">⁶This local agreement must be approved through concordant deliberations by at least two-thirds of the municipal councils representing more than half of the total population, or by half of the municipal councils representing 2/3 of the total population. The majority must include the municipal council of the most populous municipality if its population accounts for more than one-fourth of the total population of the EPCI.</footnote>
modify delegate assignment among cities, provided that: i) Each municipality receives at least one seat; ii) No single municipality holds more than half of the total allocated seats; and iii) The representation resulting from the local agreement does not deviate by more than ±20% from each municipality's demographic weight.

## 2.3 French Municipal and Inter-municipal Voting Rules

The electoral processes for municipal councils and EPCI councils are linked, as the same ballots are used to designate delegates for both councils. The electoral system depends on the size of the municipality's population.

### Municipalities with Fewer than 1,000 Inhabitants

In municipalities of fewer than 1,000 people, a majority system is used. Candidates may belong to a list, but this is not mandatory. If *k* councilors are to be elected, a voter can vote for up to *k* candidates, as he wishes, either following a list or choosing candidates independently. To be elected in the first round, a candidate must obtain an absolute majority of the votes cast, as well as a number of votes equal to at least one-fourth of the total number of registered voters. If some ($0 < k' \le k$) seats remain unfilled, a second round of voting occurs a week later. In this round, a relative majority is sufficient to be elected, regardless of the number of voters. The *k'* candidates with the highest number of votes are elected. In the event of a tie, the oldest candidate is elected. The first municipal council meeting elects the mayor and deputy mayors. EPCI delegates are selected from this list, following the order of election (mayor, first deputy mayor, second deputy mayor, etc.).
## Municipalities with More than 1,000 Inhabitants

In municipalities with more than 1,000 inhabitants, councilors are elected through a two-round list system, with no addition, removal, or modification of the order of presentation in the lists being allowed.

In the first round of voting, if a list obtains an absolute majority of valid ballots, regardless of voter turnout, it is allocated half of the total seats, rounded up. The remaining seats are distributed among all lists that have obtained at least 5% of votes, according to proportional representation based on the highest averages. If no list obtains an absolute majority in the first round, a second round of voting takes place a week later. A list qualifies for the second round if it has received at least 10% of the votes; a list that has received more than 5% of the votes can merge with another list. Seat distribution in the second round is determined by a relative majority and follows the same rules as the first round: 50% of the seats (rounded up) go to the winning list and 50% are distributed among all other lists using Jefferson's method.

EPCI delegates are selected from the list of municipal councilors, following the same voting system: 50% of the seats go to the winning list and 50% are allocated through proportional representation, using Jefferson's method, with the same ballot used for municipal councilors.<fnref n="7" />

Table 2 shows an example based on the results of the 2020 municipal elections in Lille, one of the most contested elections of the year. Ten lists competed in the first round, but only three passed the 10% threshold. None merged with one of the three lists that obtained between 5%

<footnote n="7">⁷There are two exceptions to these general rules. If a municipality with more than 1,000 inhabitants is represented by three delegates in the EPCI council, one seat is directly attributed to the winning list, and the other 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 Jefferson's method. These dispositions ensure that any losing list obtaining at least one third of the votes will be represented by one delegate in the inter-municipal council.</footnote>
and 10% of the votes. In the second round, the Socialist list defeated the Greens by only 227 votes, receiving 40% of the total votes.

Despite this razor-thin margin, the Socialist list was allocated 31 of the 61 seats directly, plus 12 seats via proportional representation, for a total of 43 seats. The same logic applies to the inter-municipal council, demonstrating that losing parties are clearly disadvantaged in both municipal and EPCI councils.<fnref n="8" />

<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><td>First Round votes<br>(in %)</td><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><td>Second Round votes<br>(in %)</td><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><td>Municipality delegates (61)</td><td>43</td><td>12</td><td>6</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Inter-municipal delegates (33)</td><td>24</td><td>6</td><td>3</td><td>—</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

## 3 The Measurement of Voting Power in Inter-municipal Structures

From its inception, the theory of power indices has highlighted that the number of seats a player holds in an assembly is not necessarily proportional to their actual influence on the outcome of a vote (Penrose, 1946, 1952; Shapley and Shubik, 1954; Banzhaf, 1965). This theory employs

<footnote n="8">⁸The Métropole Européenne de Lille comprises 95 municipalities with a total population of 1,181,692 inhabitants. The smallest municipality is Warneton, with 241 inhabitants, while 234,482 inhabitants live in Lille, the largest municipality. The EPCI council has 188 delegates in total. With 33 delegates, Lille controls approximately one-sixth of the delegates.</footnote>
rigorous mathematical methods to evaluate the relative power and influence of players within a specific decision-making scenario. For example, in an inter-municipal context, one possible criterion for allocating delegates is to ensure that each citizen has an equal probability of being a swing voter, i.e., a voter or player whose vote or decision has the potential to change the outcome of a vote or decision. In other words, swing voters 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'. All these terms refer to the same idea of an individual or a group whose vote or action is critical to the success or failure of a proposal. This means that the probability of influencing the outcome of a vote should be the same for every citizen, regardless of the municipality in which the voter resides.

## 3.1 Definition of the Banzhaf Index

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 one in which $v(S) \in \{0, 1\}$, where $v(S) = 0$ denotes a losing coalition, and $v(S) = 1$ denotes a winning coalition. Furthermore, in a simple voting game, we assume that $v(\emptyset) = 0$, $v(N) = 1$, and that the monotonicity condition holds:

$$ \forall S \subseteq T \subseteq N, v(S) \le v(T). $$

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 considered decisive if their vote can change the outcome, that is, if $i$ votes 'Yes', the decision is adopted and 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 the player *i* may belong.

This index gives the probability that the player *i* is decisive. If the numerator of Eq. 1 is divided by the total number of swing players, we obtain the normalized Banzhaf power index, defined as:

$$NB_i(v) = \frac{B_i(v)}{\sum_{t=1}^{n} B_t(v)} \quad (2)$$

A particular class of simple games is the weighted quota games, denoted $G[q, \tilde{a}]$ class (also known in the literature as “weighted games”, “weighted majority games” or “weighted voting games”). They are defined by a vector of positive weights, $\tilde{a} = (a_1, a_2, \dots, a_n) \in \mathbb{R}_+^n$ and a quota $q \in ]0, \sum_{i \in N} a_i]$. Each player *i* is assigned a weight $a_i$, and a coalition *S* wins if and only if $\sum_{i \in S} a_i \ge q$.

The concept of weighted quota games can be used to model the decision process for the French EPCI. It operates as a two-tier voting system. In the first tier, the citizens of each municipality elect representatives to their respective municipal councils. In the second tier, the EPCI council functions as a college of delegates representing these municipal councils. Hence, each EPCI council can be modeled as a weighted quota game, where $a_i$ is the number of delegates controlled by municipality *i* and the quota *q* is equal to half the weights (rounded up).

## 3.2 Measuring Power in Two-Tier Voting Systems

In an ideal two-tier system, when a proposition is discussed, each citizen would vote “Yes” or “No” and each delegate would then report the majority decision of their municipality as a single block vote. A classic example of such a structure is the U.S, Electoral College used in
presidential elections, which often serves as a model for the analysis of two-tier voting systems.

Following this analogy, the probability that a voter is decisive in the union is equal to the probability that the state/municipality *j* to which the voter *i* belongs is decisive in voting for a bill multiplied by the probability that the voter *i* is decisive in the election of representatives inside their own state/municipality *j*.

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

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

When dealing with large populations, the probability that a citizen is 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:

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

Equation 4 should remain true for all pairs $(j, t) \in N$ to achieve equity among all citizens in all municipalities. Hence:

$$ \frac{\sum_{t=1}^{n} B_t}{B_j} = \frac{\sum_{t=1}^{n} \sqrt{n_t}}{\sqrt{n_j}} \qquad (5) $$

$$ \Leftrightarrow \frac{1}{NB_j} = \frac{\sum_{t=1}^{n} \sqrt{n_t}}{\sqrt{n_j}} \qquad (6) $$

$$ \Leftrightarrow NB_j = \frac{\sqrt{n_j}}{\sum_{t=1}^{n} \sqrt{n_t}} \qquad (7) $$
In other words, to achieve equity in voting power across all citizens, the normalized Banzhaf power of a municipality j ($NB_j = \frac{B_j}{\sum_{t=1}^n B_t}$) should be as close as possible to its share of the population, measured in terms of square roots ($\frac{\sqrt{n_j}}{\sum_{t=1}^n \sqrt{n_t}}$). The Quotient for municipality j can be defined as:

$$Q_j = NB_j \times \frac{\sum_{t=1}^{n} \sqrt{n_t}}{\sqrt{n_j}} \qquad (8)$$

Ideally, this variable should be close to 1 for each municipality. A value greater than 1 indicates that the municipality is overweighted, while a value below 1 implies underrepresentation. To measure inequality in the distribution of voting power as determined by the Banzhaf index, following Felsenthal and Machover (2001) and Abidi et al. (2020), we use the max-min index. This index expresses the ratio of the power of the most powerful citizen to that of the least powerful citizen.

$$max - min = \frac{Q_j^{max}}{Q_j^{min}} \qquad (9)$$

For example, $Q_j^{max} = 2$ implies that citizens of municipality j are twice as likely to be pivotal as they should be. Similarly, $Q_j^{min} = 0.5$ indicates that the likelihood that citizens of municipality j will be pivotal is half of the fair level. And finally, the value $max - min = 4$ indicates that the difference between the most favored and least favored municipalities, in terms of deviation from the square-root benchmark, is fourfold.

Alternative measures have also been used to assess inequality in voting power. For example, Felsenthal and Machover (2001) propose different formulas, $Max(d) = Q_j^{max} - 1$ and $ran(d) = Q_j^{max} - Q_j^{min}$ to analyze power inequality in the European Union. They also suggest using conventional statistical indices, such as $\rho$ and $\chi^2$, to assess the degree of conformity between the observed power distribution and the square root vector of the population. Although the authors
did not offer any theoretical justification for using these different measures, they ultimately
reached the same conclusions for all the measures they proposed. We consider the *max – min*
index to be most straightforward measure of inequality in voting power, as its interpretation is
simple.

# 4 Data and Empirical Design

## 4.1 Data Description

To analyze power inequality in French EPCIs, we employ an extensive dataset covering 1,254 EPCIs and encompassing 34,964 municipalities in France. Most of the data were collected from the Ministries of the Interior and Local Government. They provide a public database containing the results of the 2020 municipal elections for cities with more than 1,000 inhabitants. This database was combined with official lists of all EPCIs. When necessary, missing information for municipalities with fewer than 1,000 inhabitants was obtained by directly contacting the EPCI or checking their websites. For more detailed information on the database, refer to the companion data paper by Abidi Perier et al. (2025). The dataset includes the exact number of delegates representing each municipality in its respective EPCI council. Using these data, we derive the Banzhaf index for each municipality within its council and assess whether the distribution of voting power is fair by calculating the *max–min* index for each EPCI. The results are presented not only for the observed data but also for four alternative seat allocation scenarios. It should be noted that the Métropole du Grand Paris (131 municipalities), the Métropole de Lyon (59 municipalities), and the Métropole d'Aix-Marseille-Provence (92 municipalities) are excluded from our analysis, as they are governed by different rules.<fnref n="9" />

<footnote n="9">⁹As Paris, Lyon and Marseilles are the three largest cities in France in terms of population, they are subdivided into districts called "arrondissements". Paris was first divided into districts during the French Revolution. Its current division into 20 arrondissements was established in 1859. Lyon was initially divided into five arrondissements.</footnote>
## 4.2 Four Scenarios for the Allocation of Delegates

### Scenario O: Observed Distribution of Delegates

This scenario serves as the reference point, corresponding to the actual distribution of delegates observed in 2020. It reflects the current distribution rules for delegates, including the provisions introduced by the law of 9 March 2015, which authorized local agreements, allowing for a maximal variation of +25% in the total number of seats. Note that an agreement may also reduce the total number of seats. The agreement must still meet three conditions:

i) Each municipality must have at least one seat, ii) No municipality may hold more than half of the seats, iii) The representation resulting from the local agreement must not deviate by more than ± 20% from the demographic weight of the municipality.<fnref n="10" /> Hence, the database includes some EPCIs that adhered exactly to the initial rules, while others altered the final allocation of delegates through local agreements. To check whether local agreements were implemented or not, we propose several alternative allocation scenarios.

ments in 1852, with the current nine arrondissements introduced in 1964. Marseille's 16 arrondissements were created in 1946. This administrative level is the basis for the election of EPCI delegates. Electoral districts correspond to either a single arrondissement or to a group of arrondissements. As a result, the assumption that a city's delegates will vote as a unified block is not applicable in these cases. For example, Paris is divided into 20 arrondissements and 17 electoral districts. The first four arrondissements together form one electoral district, which appoints three delegates to the Métropole du Grand Paris council, while the most populous, the 15th arrondissement, appoints 7 delegates on its own.

<footnote n="10">¹⁰Unless the distribution under common law already deviates by ± 20% and the new distribution reduces this deviation, and except in cases where two seats are attributed to a municipality that would otherwise be entitled to only one under common law.</footnote>
Scenario PJ : Proportional Distribution Per Capita Using the Jefferson Rule

This scenario simulates a strict application of the legal rules, without any local agreement.

Comparing this scenario with Scenario O allows us to assess the impact of local agreements both on the total number of delegates and on their distribution across municipalities. Theoretically, Macé and Treibich (2021) suggest that when small entities join a federal union, it is rational for them to be overrepresented. Thus, comparing Scenarios O and PJ enables us to evaluate whether granting more autonomy to EPCIs through local agreements helps reduce inequalities in voting power. Recall that only Communautés de Communes and Communautés d'Agglomération can make an agreement. Therefore, the O and PJ scenarios are identical for the 21 Métropoles and the 14 Communautés Urbaines.

In this scenario, the delegates are distributed according to Table 1, assuming that no agreement
was made between the EPCI members. The Jefferson apportionment rule applies at each step.
The delegates are assigned one by one to the municipality with the highest ratio $\frac{n_i}{a_i+1}$, where
$n_i$ is the number of inhabitants of the municipality $i$ and $a_i$ is the weight of the municipality $i$.
The process continues until all delegates are allocated.

A municipality cannot hold more than half of the total seats. If this occurs, the extra seats
are assigned to the city with the next-highest ratio ($\frac{n_i}{a_i+1}$). At the end of the process, any
municipality without delegates automatically gets one additional seat. The 2020 law further
stipulates that if the total number of seats is increased by 30%, an additional 10% of seats
must be added and are allocated again according to the Jefferson rule, still ensuring that no
municipality controls more than half of the seats.
## Scenario SRJ: Square Root of Population Rule Using the Jefferson Method

This scenario compares the observed situation with the theoretical recommendation of the Penrose Square Root Rule (see Eq. 7), which assumes that delegates are allocated according to the population square root of each municipality. Thus, for each municipality $i$ belonging to $EPCI_j$, the number of delegates should be roughly proportional to:

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

The same procedural rules as in Scenario PJ are followed. The initial number of seats to be allocated is drawn from Table 1. The Jefferson apportionment rule is applied, and then additional seats are added for municipalities with no delegate. If the total number of seats increases by more than 30%, an additional 10% of delegates are added and distributed (again using the Jefferson method).

## Scenario PH: Proportional Distribution Per Capita Using the Hamilton Rule

We introduced this scenario to test whether Hamilton's greatest remainder principle could achieve a fairer distribution of delegates across municipalities. This alternative is included because French law relies on Jefferson's greatest divisor principle, which inherently favors the largest municipality. Legislators therefore have had to add several provisions to ensure that municipalities have at least one delegate and that larger cities have a 'fair' amount of representation. It is worth examining whether using the Hamilton principle, which favors small cities, might be a better way of achieving the goal of fairness in a more direct and straightforward manner. If $\bar{a}$ delegates are to be apportioned in $EPCI_j$, we first compute the divisor, i.e., the number of inhabitants necessary to be guaranteed one delegate.

$$ d = \frac{\sum_{t=1}^{n} n_t}{\bar{a}} $$
Each municipality should then receive a number of delegates equal to:

$$a_t = \lfloor \frac{n_t}{d} \rfloor$$

and the remaining seats are allocated to the municipalities with the largest remainders. In this way, we adapt the French dispositions by substituting Hamilton's principle for Jefferson's while maintaining the key constraints:

* No municipality may control more than 50% of the seats. If necessary, any seats above 50% will be allocated to other municipalities.

* Any municipality with no seat at the end of the process automatically receives an additional seat.

* If the number of seats increases by more than 30%, an additional 10% of seats are allocated using Hamilton's principle, while ensuring no municipality controls a majority.

Note that this approach is not the only way to incorporate the Hamilton principle while complying with French electoral law. However, it provides a consistent and faithful interpretation that adheres to the step-by-step logic of the existing legal framework.<fnref n="11" />

## Scenario SRH: Square Root of Population Criterion Using the Hamilton Rule

As with the previous case, the objective of this scenario is to test whether changing the apportionment rule would reduce the number of extra seats and lead to a fairer distribution of power. This scenario assumes that each municipality has a number of delegates proportional to the square root of its population and follows the same allocation rules as in Scenario PH.

<footnote n="11">¹¹Alternatively, the Hamilton method could be applied directly to the final number of seats, subject to the constraints that no single municipality may hold more than 50% of the seats and that each municipality must have at least one delegate.</footnote>
## 4.3 Comparing the Allocation of Seats across Scenarios

To compare the number of seats assigned to EPCIs in each scenario, Table 3 presents the difference in the number of delegates between the observed scenario O and each of the four simulated alternatives. Figure 1 also presents these differences for scenario PJ (Jefferson method), scenario SRJ (square root of population + Jefferson), scenario PH (Hamilton method) and scenario SRH (square root of population + Hamilton). Figure 1 highlights a clear and strong asymmetry in the impact of each scenario on delegate allocation.

<table><caption>Table 3: Differences in the Number of Delegates Between Scenario O (Observed Number of Delegates) and the Four Simulated Scenarios</caption><thead><tr><th>Interval</th><th>O - PJ</th><th>O - SRJ</th><th>O - PH</th><th>O - SRH</th></tr></thead><tbody><tr><th scope="row">Total Negative</th><td>47</td><td>3</td><td>17</td><td>1</td></tr><tr><th scope="row">Total Equal</th><td>249</td><td>1</td><td>55</td><td>3</td></tr><tr><th scope="row">Total Positive</th><td>955</td><td>1,247</td><td>1,179</td><td>1,247</td></tr><tr><th scope="row">Total</th><td>1,251</td><td>1,251</td><td>1,251</td><td>1,251</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>Figure 1: Distribution of Delegate Differences by Scenario</figcaption></figure>
## O versus PJ

Approximately 20% of the EPCIs (249 out of 1251) have an exact match between the observed number of delegates and the simulated count (Table 3). More surprisingly, we observe that 47 EPCIs have fewer delegates than prescribed by the law, while 955 EPCIs have more than the prescribed number. This suggests that around 80% of the inter-municipal structures implemented a local agreement that modified the total number of delegates and their allocation. These aggregate observations underscore the extent to which EPCIs have made use of the discretionary margin afforded by law. To illustrate this dynamic, Table 4 examines a specific case, the CC de la Vallée du Gapeau, where negotiations focused on reducing the total number of delegates and limiting the representation of the larger municipalities. Meanwhile, 955 out of 1,251 EPCIs used the negotiation margin to increase their number of delegates. Different patterns can be observed. In the first, EPCIs aim to give more influence by distributing more seats to medium-sized municipalities.

<table><caption>Table 4: CC de la Vallée du Gapeau (Département 83) - 5 Municipalities</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Observed</th><th>PJ</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>Total</td><td>31,287</td><td>30</td><td>34</td></tr><tr><td>Difference in the number of seats</td><td></td><td>-11.8%</td><td></td></tr></tbody></table>

In the CC du Pays de Duras (Table 5), the number of delegates assigned to the largest municipal
pality is reduced by one, while those of medium-sized municipalities is increased. However, in some cases, the effects on power distribution are less obvious. In the CA du Bassin d'Aurillac in Table 6, the ten largest municipalities gain between one and three additional delegates. Under the local agreement, the power of the smallest village declines, the influence of medium-sized municipalities increases, and it is unclear whether the power of the largest municipality rises or declines.

<table><caption>Table 5: CC du Pays de Duras (Département 47) - 17 Municipalities</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Number of Seats</th><th>PJ</th></tr></thead><tbody><tr><td>Saint-Géraud</td><td>97</td><td>1</td><td>1</td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</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></td><td><strong>+18.5%</strong></td></tr></tbody></table>
<table><caption>Table 6: CA du Bassin d'Aurillac (Département 15) - 25 Municipalities</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Number of Seats</th><th>PJ</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>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr><tr><td>Crandelles</td><td>867</td><td>1</td><td>1</td></tr><tr><td>Reilhac</td><td>1,136</td><td>2</td><td>1</td></tr><tr><td>Saint-Simon</td><td>1,178</td><td>2</td><td>1</td></tr><tr><td>Vesac</td><td>1,241</td><td>2</td><td>1</td></tr><tr><td>Sansac de Marmiesse</td><td>1,394</td><td>2</td><td>1</td></tr><tr><td>Saint-Paul-des-Landes</td><td>1,581</td><td>2</td><td>1</td></tr><tr><td>Naucelles</td><td>2,080</td><td>2</td><td>1</td></tr><tr><td>Jussac</td><td>2,091</td><td>2</td><td>1</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>5</td></tr><tr><td>Aurillac</td><td>26,828</td><td>27</td><td>24</td></tr></tbody><tfoot><tr><td><strong>Total</strong></td><td><strong>55,265</strong></td><td><strong>68</strong></td><td><strong>55</strong></td></tr><tr><td><strong>Difference in the number of seats</strong></td><td></td><td><strong>+23.6%</strong></td><td></td></tr></tfoot></table>

## O versus PH, SRJ and SRH

As expected, scenarios that use the square root rule and those based on the Hamilton method generate fewer additional seats compared to the pure Jefferson method and the observed number of seats. Compared to the observed situation, the total number of delegates decreases in 1,247 cases for the SRJ scenario, in 1,179 cases under PH and in 1,247 cases under SRH. If the
legislator's objective were to reduce the size of councils, the combination of the square root rule and the Hamilton method would be most effective in achieving this goal.

# 5 Assessing Inequality in Terms of Power

The objective of this section is to determine which scenario produces the least inequality in terms of the distribution of power among municipalities. Ideally, the Quotient $Q_j = NB_j \times \frac{\sum_{l=1}^{n} \sqrt{n_l}}{\sqrt{n_j}}$ should equal one for every municipality. To achieve equity in terms of power, the normalized Banzhaf power of municipality $j$ should be as close as possible to its share of the population, measured using the square root rule. Thus, the ratio $max-min = \frac{Q_j^{max}}{Q_j^{min}}$ measures the inequality in power between the least favored and most favored citizen in a given inter-communal structure.

Table 7 presents the results of the max-min ratios for the five scenarios. Following Abidi et al. (2020), these ratios are classified into five categories.

1. $1 \leq max - min \leq 2$. This corresponds to situations where the power deviations among citizens are limited to the ±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 range of variation increases to within ±66%. For example, an inter-communal structure with $Q_j^{min} > 0.33$ and $Q_j^{max} < 1.66$ falls into this category.

3. $5 < max - min \leq 10$. Differences 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 is likely to be 10 times less (or more) pivotal than another.

5. $max - min > 100$. This situation is clearly inequitable. In this category, the $max - min$ ratio may even tend toward infinity, whenever a municipality is a so called dummy player.
in the inter-communal council, that is, when it is never pivotal.

<table><caption>Table 7: Distribution of the Max-Min Ratios Across Scenario O and the Four Alternative Scenarios</caption><thead><tr><th>Category</th><th>Max-Min ratio</th><th>O</th><th>PJ</th><th>SRJ</th><th>PH</th><th>SRH</th></tr></thead><tbody><tr><td>Category 1</td><td>[1-2[</td><td>20</td><td>7</td><td>140</td><td>16</td><td>140</td></tr><tr><td>Category 2</td><td>[2-5[</td><td>528</td><td>458</td><td>573</td><td>678</td><td>567</td></tr><tr><td>Category 3</td><td>[5-10[</td><td>303</td><td>326</td><td>418</td><td>291</td><td>389</td></tr><tr><td>Category 4</td><td>[10-100[</td><td>235</td><td>238</td><td>108</td><td>188</td><td>143</td></tr><tr><td>Category 5</td><td>[100-∞[</td><td>165</td><td>222</td><td>12</td><td>78</td><td>12</td></tr><tr><td></td><td>including “∞”</td><td>9</td><td>6</td><td>8</td><td>3</td><td>3</td></tr><tr><td>Total</td><td></td><td>1,251</td><td>1,251</td><td>1,251</td><td>1,251</td><td>1,251</td></tr></tbody></table>

## 5.1 The Impact of Local Agreements: O versus PJ

It is recalled in (Abidi et al., 2020) that before 2014, EPCIs had considerable freedom in how delegates were allocated among their members. The only constraints were that each municipality must have at least one delegate, no municipality can hold a majority of delegates, and larger municipalities could not have fewer delegates than smaller ones. Within these limits, any two-thirds majority of municipalities (including the largest) could approve any agreement. Abidi et al. (2020) showed that in the regions of Brittany, Normandy and Pays de Loire, many EPCIs collectively adopted an allocation of delegates that led to a fair distribution of power.

In 2020, however, the 2010 Law introduced more restrictions on possible agreements. Comparing the Observed and PJ scenarios allows us to assess whether local agreements reduced inequality in voting power. Table 7 indicates that equity among citizens is achieved in very few situations when the 2010 Law is strictly applied. Local agreements only marginally improve the situation.
Scenarios O and PJ have only 20 and 7 EPCIs, respectively, with a max-min ratio within the range [1-2].

Similarly, scenarios O and PJ have the largest number of EPCIs with a max-min ratio greater than 10, with some being even greater than 100. Specifically, the current distribution of seats includes 235 EPCIs in which the most powerful citizen has at least ten times more power than the least influential. In addition, there are 165 EPCIs with a max-min ratio greater than 100. In 9 EPCIs, one municipality and its citizens have no influence at all on the decision! The situation is slightly worse under the PJ scenario, with more EPCIs in categories 3, 4 and 5 and six EPCI councils with dummy players.

Can we therefore conclude that the Observed scenario is always preferable to the PJ scenario? The results differ across EPCIs, reflecting different goals when making local agreements. For example, in the Communauté de Communes Cœur de Perche (see Table 9) members chose to make a local agreement giving one additional seat to the four most populous municipalities, increasing the total number of delegates from 28 to 32. As a consequence, the max-mix ratio rose from 3.234 under the PJ scenario to 3.721 in the observed situation. In contrast, in the Communauté des Communes Pays de Dol et de la Baie du Mont Saint-Michel (see Table 10), the agreement reduced the number of delegates of the main municipality from 10 to 8, while eight medium-sized municipalities got an extra seat. Consequently, the max-min index dropped sharply from 10.14 to 3.25.
<table><caption>Table 8: The scenarios with the smallest max-min ratios across EPCI</caption><thead><tr><th>Result Observed vs :</th><th>PJ</th><th>SRJ</th><th>PH</th><th>SRH</th></tr></thead><tbody><tr><td>Negative</td><td>618</td><td>421</td><td>351</td><td>438</td></tr><tr><td>Equivalent</td><td>400</td><td>16</td><td>56</td><td>16</td></tr><tr><td>Positive</td><td>233</td><td>814</td><td>844</td><td>797</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>

<figure><img src="image_3.png" /><figcaption>Figure 2: Evolution of the max-min ratios across scenarios</figcaption></figure>
<table><caption>Table 9: CC Cœur du Perche (Département 61, 12 Municipalities)</caption><thead><tr><th>Municipalities</th><th>Population</th><th>Observed</th><th>PJ</th><th>SRJ</th><th>PH</th><th>SRH</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>3.72</td><td>3.23</td><td>2.02</td><td>3.20</td><td>2.20</td></tr></tbody></table>
<table><caption>Table 10: 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>PJ</th><th>SRJ</th><th>PH</th><th>SRH</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>Cherrueix</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-Broladre</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>
Can we assert that, in general, local agreements were mostly used to limit the number of seats, and consequently, the power of the largest municipality, as observed in the CC Pays de Dol et de la Baie du Mont Saint-Michel? To gain a broader perspective, Table 8 and Figure 2 compare each EPCI individually to determine whether the max-min ratio increases or decreases between the Observed scenario and the PJ, SRJ, PH, and SRH scenarios, respectively. The max-min index is lower in 618 cases in the observed situation compared to the PJ scenario, indicating that local agreements have reduced inequality in those cases. Conversely, a strict application of the law would have produced more equitable outcomes than letting municipalities reach an agreement in 233 cases. In line with Abidi et al. (2020), our findings suggest that, more frequently, local agreements reduce inequality in terms of power within the councils.

## 5.2 Hamilton as a Remedy?

The Hamilton apportionment rule inherently favors smaller municipalities in the allocation of delegates. It can potentially reduce the size of the councils, and could increase the value of the $Q_j$ ratio for small municipalities. In fact, the PH scenario outperforms both the O and PJ scenarios in that it has the highest number of EPCIs with a max-min ratio below 5, indicating that, when proportional distribution per capita of seats is applied, adopting the Hamilton method could mitigate voting power inequalities among municipalities within the same EPCI. However, as shown in Table 8, the picture remains nuanced. While the value of the max-min ratio improves for 2/3 of the EPCIs (844), the situation deteriorates for 351 of them.

## 5.3 Comparison with the Square Root Law

The SRJ and SRH scenarios align with the normative recommendations of Penrose's square root rule. They significantly reduce power inequalities. Nevertheless, there are still some EPCIs where the largest municipality is overrepresented, even when the square root of population is
used to allocate delegates. Specifically, 108 EPCIs in the SRJ scenario and 143 EPCIs in the SRH scenario still have a max-min ratio greater than 10 (see Table 7). Overall, it can be inferred that the combination of the number of delegates allocated according to the 2010 law (Table 1) along with an allocation of delegates proportional to the square root of the population using the Jefferson method yields the most equitable outcomes in terms of reducing power inequalities among French municipalities. This approach outperforms the SRH scenario in achieving a higher degree of power equity. However, a few instances of dummy players (3 under SRJ and 3 under SRH) are still observed.

## 5.4 A Closer Look at Dummy Players

We conclude our analysis with a closer look at dummy players i.e., municipalities that have a delegate but no effective influence on council decisions. Table 11 presents the nine observed cases and the corresponding distribution of delegates. The CA du Grand Verdun is not included in Table 11 as this is an exception. Five of its members are symbolic municipalities with zero inhabitants, destroyed during World War I. Although these municipalities remain legally recognized and have designated mayors, their presence artificially inflates the *max-min* index to infinity despite the absence of true dummy players. The lists of councils with dummy players in the other scenarios are provided in Appendices A2 to A5. It should be noted that the French law did not foresee the possibility of councils with dummy players and proposes no mechanism to prevent them. Even more strikingly, some local agreements led to the creation of councils with dummy players, situations which could have been avoided by strictly applying the law (Scenario PJ). It is noticeable that most councils with a dummy player are small, comprising four to six members.

However, although the existence of dummy players is possible under French law, it appears to be less common than the theoretical models would predict. The works of (Barthélémy et al., 2013;
Barthélémy and Martin, 2021) studied the conditions around the existence of dummy players in the weighted majority game. First, they established exact conditions on player weights in the weighted majority games that ensure the existence of a dummy player in systems with $n = 4$, 5, and 6 players. Second, assuming that all possible distributions of delegates are equally likely, they derived the apriori probability of having at least one dummy player for different values of $\sum_{i=1}^{n} a_i$, the total number of delegates. Their analysis excludes the possibility of a player controlling more than half of the delegates. Table 12 displays some of the results (Barthélémy et al., 2013) for n players and total delegates $\bar{a} = \sum_{i=1}^{n} a_i$, and compares them with the observed data. The observed values are clearly lower than the theoretical ones.

<table><caption>Table 11: EPCIs with dummy players under scenario O</caption><thead><tr><td>Name (and Département Number)</td><td>Members</td><td>Delegate Vector</td><td>Majority</td></tr></thead><tbody><tr><td>CA Cannes Pays de Lérins (06)</td><td>5</td><td>(29,16,9,7,1)</td><td>32/62</td></tr><tr><td>CA La Riviéra du Levant (971)</td><td>4</td><td>(17,15,8,1)</td><td>21/41</td></tr><tr><td>CA du Centre Littoral (976)</td><td>6</td><td>(22,12,9,4,1,1)</td><td>25/48</td></tr><tr><td>CC de l'Est Guyanais (973)</td><td>4</td><td>(11,7,1,1,1)</td><td>11/20</td></tr><tr><td>CC du Pays d'Urfé (42)</td><td>11</td><td>(6,4,2,...,2,1)</td><td>14/27</td></tr><tr><td>CC Les Portes Briardes Entre Villes et Forêts (77)</td><td>5</td><td>(17,7,7,6,1)</td><td>20/38</td></tr><tr><td>CC du Sud Est du Pays Manceau (72)</td><td>5</td><td>(12,9,5,5,2)</td><td>14/26</td></tr><tr><td>CC Lyon Saint-Exupéry en Dauphiné (38)</td><td>6</td><td>(11,6,5,5,2,2)</td><td>16/31</td></tr><tr><td>Métropole de Toulouse (31)</td><td>37</td><td>(67,8,5, ..., 1)</td><td>67/133</td></tr></tbody></table>

The players underlined in the delegate number vector are dummy players.

The authors could not find information about CC de l'Est Guyanais to confirm that the main city has a majority of delegates. Toulouse is a specific case, being a Métropole.
<table><caption>Table 12: The probability of dummy players, theory and data</caption><thead><tr><th rowspan="2"></th><th colspan="4">Theoretical Values</th><th>Observed</th></tr><tr><th>a&#772; = 15</th><th>a&#772; = 30</th><th>a&#772; = 60</th><th>Limit</th><th>values</th></tr></thead><tbody><tr><th>4 cities</th><td>0.434</td><td>0.308</td><td>0.387</td><td>1/2 = 0.500</td><td>2/32 = 0.062</td></tr><tr><th>5 cities</th><td>0.260</td><td>0.282</td><td>0.385</td><td>35/60 = 0.530</td><td>3/23 = 0.130</td></tr><tr><th>6 cities</th><td>0.181</td><td>0.100</td><td>0.223</td><td>155/312 = 0.497</td><td>2/25 = 0.080</td></tr></tbody></table>

# 6 Conclusion

This study sets out to analyze the distribution of citizens' voting power within French EPCIs. The first objective was to build a comprehensive database detailing the allocation of seats to each municipality in the councils that govern the inter-communal structures. This dataset enabled an in-depth assessment of the representativeness of municipalities and allowed us to compare the current distribution of seats with four alternative simulated scenarios.

The first two scenarios, namely PJ and SRJ, employ the Jefferson apportionment method, with seat allocation based on the criteria of proportional representation to population and the square root of population, respectively. The other two scenarios, PH and SRH, use the Hamilton apportionment method, under the same respective criteria.

The observed situation, Scenario O, follows the principle of Jefferson proportional representation based on current population figures. It allows municipalities to negotiate seat distributions among themselves while adhering to specific rules. In contrast, the PJ scenario applies a similar seat allocation method but excludes any negotiation, enforcing a strict interpretation of the 2010 Law. The current allocation exhibits lower levels of inequality than the PJ scenario in 618 cases out of 1,251. This observation suggests that negotiations have played a role in
mitigating inequalities among municipalities belonging to the same EPCI. This finding aligns
with the theoretical predictions (Macé and Treibich, 2021), who argue that smaller players
tend to obtain more decision-making weight than their proportional share when they join a
union. However, the allocation of more seats in the observed scenario than in the PJ scenario
also allows larger municipalities to acquire additional delegates and has increased inequality in
233 EPCIs. In nine cases, local agreements did not eliminate the presence of dummy players.
Combining the population-proportional representation with the Hamilton method helps reduce
inequalities in voting power. As expected, scenarios based on the square root of population,
namely SRJ and SRH, yield great equity of voting power and are in line with the theoretical
model (penrose1946).

To go a step further, the works of (Życzkowski and Słomczyński, 2013) and (Feix et al., 2007) suggested that in addition to using weights proportional to the square root of the population, a superquota should also be used to increase inequality in terms of power. More precisely, the Banzhaf and Penrose indices assume that each player votes independently ‘yes’ or ‘no’ with equal probability $\frac{1}{2}$. When the number of voters becomes large, and no player overwhelmingly dominates in terms of weight, the total number of ‘yes’ votes can be approximated by a normally distributed random variable.

In (Feix et al., 2007) it is demonstrated that the optimal quota, one that brings the Banzhaf power of a player closer to its weight is a superquota equal to $(\mu + \sigma)$ where $\mu$ is the mean of the random variable and $\sigma$ its standard deviation. (Słomczyński and Życzkowski, 2010) referred to the method that uses weights proportional to the square root of the populations and the optimal superquota as the Jagiellonian Compromise. They advocated its adoption for the Council of Ministers of the European Union. In the EU example they present, the quota should be equal to 61.6% of the votes. This critical threshold balances equity and functionality, ensuring that
each citizen's vote carries approximately equal weight in collective decisions.

However, it may be difficult to implement representation based on the square root of populations in modern democracies, where proportionality to population is a deeply ingrained principle. Moreover, the justification for the square root principle is based on specific probabilistic assumptions. (Gelman et al., 2004) contested the validity of these hypotheses by comparing them with U.S. electoral data. Therefore, if French municipalities opt to retain population proportionality as the basis for seat allocation among municipalities within the inter-municipal structures, the most effective approach to reducing power inequalities is to combine this criterion with the Hamilton method. Empirically, using superquotas may also enhance fairness, though no theoretical model has endorsed this proposition.

Beyond empirical studies on power indices, a complementary area of research explores normative apportionment methods designed to achieve more equitable distributions of seats. For example, (Kurz and Napel, 2014) compared different methods (including the Jageillonian Compromise and search-based algorithms) to identify the fairest distribution of delegates in terms of power. To progressively narrow the possible allocations of delegates, (Gambarelli, 1999) proposed using different criteria sequentially. He narrows the realm of possible allocations by applying fundamental fairness criteria (such as Hare minimum and maximum, monotonicity, and symmetry). Then, using a lexicographic equity principle inspired by the Nucleolus, he compares the remaining distributions with regard to their capacity to reduce inequality among districts. These procedures can be directly implemented in inter-communal settings as well. (Gambarelli and Palestini, 2007) later extended this framework to multi-district systems, addressing complex fairness constraints at both local and aggregate levels of representation.

A remaining ambitious challenge is to develop a general seat allocation method that effectively minimizes inequality among municipalities within the same inter-municipal structure, regardless
of the number of constituent municipalities. Alternatively, to address the persistent problem
of overrepresentation of large cities in inter-municipal councils, one possible solution could be
to divide the largest municipality into several electoral jurisdictions, each one sending a limited
number of delegates to the inter-municipal council. For example, Lille (see Table 2) could be
split into four jurisdictions, each sending fewer than ten delegates to the inter-municipal council.

This paper also has certain limitations that should be addressed in future work. As in numerous
previous studies on this topic, we use the block voting assumption, whereby all delegates from
a municipality vote as a unified bloc in the same way within the inter-municipal council. This
assumption could be relaxed in future research by considering alternative voting behaviors, such
as delegates voting according to their political affiliations.

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## Appendix I: Number of EPCIs by Statute

<table><caption>Table A1: Types of EPCIs and Number of Municipalities, 2016-2020</caption><thead><tr><th>Type of EPCIs</th><th>2016</th><th>2017</th><th>2018</th><th>2019</th><th>2020</th></tr></thead><tbody><tr><td>Metropoles</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>Number of isolated municipalities<sup>a</sup></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>

<footnote><sup>a</sup> “Isolated municipalities” are communes that do not belong to any inter-municipal structure. Under French law, such non-membership is allowed only in narrowly defined situations (e.g., specific insular or mountainous areas, certain arrangements with overseas territories, or temporary administrative transitions such as ongoing mergers). These communes are included in the total number of municipalities but excluded from the EPCI counts.</footnote>
# Appendix II: EPCIs with Dummy Players

<table><caption>Table A2: EPCIs with Dummy Players under Scenario PH</caption><thead><tr><td>Name (and Département Number)</td><td>Members</td><td>Delegate Vector</td><td>Majority</td></tr></thead><tbody><tr><td>CA Cannes Pays de Lérins (06)</td><td>5</td><td>(26,15,8,7,1)</td><td>29/57</td></tr><tr><td>CA La Riviéra du Levant (971)</td><td>4</td><td>(16,15,8,1)</td><td>21/40</td></tr><tr><td>CA Var Estérel Méditérranée (83)</td><td>5</td><td>(22,15,6,4,1)</td><td>25/48</td></tr><tr><td>CA du Centre Littoral (976)</td><td>6</td><td>(21,11,9,5,1,1)</td><td>25/48</td></tr><tr><td>CC de l'Ouest Guyanais (973)</td><td>8</td><td>(19,6,5,4,4,3,1,1)</td><td>22/43</td></tr><tr><td>CC Les Portes Briardes Entre Villes et Forêts (77)</td><td>5</td><td>(17,7,7,6,1)</td><td>20/38</td></tr><tr><td>CC Lyon Saint-Exupéry en Dauphiné (38)</td><td>6</td><td>(11,6,5,5,2,1)</td><td>16/30</td></tr><tr><td>CC du Val de l'Eyre (33)</td><td>5</td><td>(11,8,8,2,1)</td><td>16/30</td></tr></tbody></table>

<table><caption>Table A3: EPCIs with Dummy Players under Scenario PJ</caption><thead><tr><td>Name (and Département Number)</td><td>Members</td><td>Delegate Vector</td><td>Majority</td></tr></thead><tbody><tr><td>CA Cannes Pays de Lérins (06)</td><td>5</td><td>(26,15,8,7,1)</td><td>29/57</td></tr><tr><td>CA La Riviéra du Levant (971)</td><td>4</td><td>(17,15,8,1)</td><td>21/41</td></tr><tr><td>CA du Centre Littoral (976)</td><td>6</td><td>(22,11,9,5,1,1)</td><td>25/49</td></tr><tr><td>CC Les Portes Briardes Entre Villes et Forêts (77)</td><td>5</td><td>(17,7,7,6,1)</td><td>20/38</td></tr><tr><td>CC Lyon Saint-Exupéry en Dauphiné (38)</td><td>6</td><td>(11,6,5,5,2,1)</td><td>16/30</td></tr><tr><td>CC du Val de l'Eyre (33)</td><td>5</td><td>(11,9,8,1,1)</td><td>16/30</td></tr></tbody></table>
<table><caption>Table A4: EPCIs with Dummy Players under Scenario SRJ</caption><thead><tr><td>Name (and Département Number)</td><td>Members</td><td>Delegate Vector</td><td>Majority</td></tr></thead><tbody><tr><td>CA de la Riviéra du Levant (971)</td><td>4</td><td>(6,5,4,1)</td><td>9/16</td></tr><tr><td>CC Méditéranée Portes des Maures (83)</td><td>6</td><td>(4,3,3,3,1,1)</td><td>9/16</td></tr><tr><td>CC Rhône Lez Provence (84)</td><td>5</td><td>(7,3,3,2,1)</td><td>9/16</td></tr></tbody></table>

<table><caption>Table A5: EPCIs with Dummy Players under Scenario SRH</caption><thead><tr><td>Name (and Département Number)</td><td>Members</td><td>Delegate Vector</td><td>Majority</td></tr></thead><tbody><tr><td>CA de la Riviéra du Levant (971)</td><td>4</td><td>(6,5,4,1)</td><td>9/16</td></tr><tr><td>CC Méditéranée Porte des Maures (83)</td><td>6</td><td>(3,3,3,3,3,1)</td><td>9/16</td></tr><tr><td>CC Rhône Lez Provence (84)</td><td>5</td><td>(6,3,3,3,1)</td><td>9/16</td></tr></tbody></table>