# Testing Penrose Limit Theorem: A case study of French local data

Zineb Abidi Perier, Vincent Merlin

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Zineb Abidi Perier, Vincent Merlin. Testing Penrose Limit Theorem: A case study of French local data. Mathematical Social Sciences, 2025, 137, pp.102418. 10.1016/j.mathsocsci.2025.102418. halshs-05090071

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Testing Penrose Limit Theorem:
A Case Study of French Local Data*†

Zineb Abidi Perier‡§

Vincent Merlin¶

Abstract.

In 1946, Penrose argued that in a weighted quota game, if the number of players is sufficiently large and the weight associated with the largest player is bounded, the Banzhaf/Penrose power of a player is approximately proportional to its weight. This conjecture is now known as the Penrose Limit Theorem (PLT). However, as the weight of the largest player increases and/or when it is surrounded by an ocean of small players, its Banzhaf/Penrose power approaches one, even if its own weight is far less than 50% of the total weight. This paper aims to empirically determine the conditions under which this assertion holds. Can we identify the threshold weight (as a percentage of the total sum of weights) below which the Penrose Limit Theorem applies? To address this question, we analyze a panel of 1,251 French intercommunal councils, where each town is represented by a given number of delegates. In particular, we compare the normalized Banzhaf index of the largest city to its weight in the council. As a consequence, we propose an alternative allocation rule that French law might adopt, considering factors such as the weight of the largest city in the council and the number of its members.

<footnote>*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>

<footnote>†We are grateful to the participants of the SING19 conference for their remarks. We also thank Stefan Napel and Frank Steffen for their comments.</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 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>
**Keywords:** Power Indices, Penrose Law, Banzhaf Index, Voting, Local Governance.

**JEL Numbers:** C71, D72, H77

**Highlights:**

* Empirical validation of Penrose's Limit Theorem in French inter-communal councils reveals its dependence on municipality size and council composition.

* The theorem holds as long as the size of the major player is bounded, and the number of players plays a role, too.

* Using a toy theoretical model, an original apportionment method is proposed to reduce the power of the largest municipality and promote fairer representation.

# 1 Introduction

It has been well known since the pioneering works of Penrose (1946, 1952) and Shapley and Shubik (1948), that a player's influence in a council or an assembly can differ significantly from the number of seats or delegates he controls. The theory of power indices is based upon the following ideas: the power of a player in an assembly should be measured by the number of coalitions in which he is pivotal (i.e., the number of coalitions where his support is necessary for a decision to pass). For example, in a parliament where Party A controls 49 seats, Party B controls 48 seats, and Party C controls 3 seats, any alliance of two parties can form a majority (51 seats). Although the three parties control each a different number of seats, they have the same a priori influence on decision-making, as any of them has the same probability of being pivotal. Hence, the Penrose measure of the power of a player is the number of times he is pivotal in a coalition, divided by the number of possible coalitions to which he may belong. Along the same lines, Shapley and Shubik (1948), using the same principles but attaching different weights to each possible coalition, developed their own power index.

In the sixties, Banzhaf (1965) rediscovered independently the notions developed by Penrose (1946, 1952) while analyzing the possible coalitions in the Nassau County Board. The representatives of Hempstead 1, Hempstead 2, North Hempstead, Oyster Bay, Glen Cove, and Long Beach controlled 31, 31, 21, 28, 2, and 2 votes, respectively. A simple calculation shows that
any coalition of two players from {Hempstead 1, Hempstead 2, Oyster Bay} could control a ma-
jority of votes, regardless of the opinions of North Hempstead, Glen Cove, and Long Beach.
These three latter cities are referred to as dummy players, as they have no influence on the final
decision. The non-normalized Banzhaf index is the same measure of power as the one proposed
by Penrose; the normalized Banzhaf index is the number of times a player is pivotal divided by
the total number of pivotal cases.

Although the notion of a pivotal voter (also called a swing voter, a critical voter, or a decisive
voter) is easy to understand, there is no alternative to enumerating all possible coalitions to
compute the power indices of a player in a game, which could quickly become difficult as the
number of players increases. However, in his booklet Penrose (1952) left us with some hope:
if the number of players is large enough, and if the number of delegates controlled by a given
player is not too high, the ratio between the voting power of any two players converges to the
ratio of their weights. This result is known as the Penrose Limit Theorem (PLT). In contrast,
in a game where a major player holds one-third of the delegates and is surrounded by an ocean
of minor players each represented by one delegate, the power of the major player tends toward
1 according to the Banzhaf/Penrose index, and toward $\frac{1}{2}$ for the Shapley-Shubik index.<fnref n="1" />

Though empirical studies of the US Electoral College (Owen, 1975) and the European Union
(Felsenthal and Machover, 2001, 2004) have already documented the over-representation of some
states in a federal union, it took some time for game theorists to extensively test the reach of the
Penrose Limit Theorem. Lindner and Machover (2004) consider an infinite chain of weighted
majority games; at each stage, a new player enters the game, with a predetermined weight. They
propose conditions on the distributions of weights that guarantee that the ratio between the
Banzhaf power between two players converges to their ratio of weight, as the number of players
increases. A particular case assumes only finitely many values for the weights, all of them being
positive integers, and such that the greatest common divisor of those values that occur infinitely
often is one. They also propose conditions under which the PLT holds for the Shapley-Shubik
index. Chang et al. (2006) complement this analysis through simulation tests, drawing weight
in the simplex through a uniform distribution and a poison distribution, from $n = 16$ to $n = 56$
players, for different values of the quota and for both the Banzhaf and Shapley-Shubik indices.
In contrast, Lindner and Owen (2007) displays some very specific cases where the PLT does not
hold. Boratyn et al. (2020) study the power of the largest player when the weights are drawn

<footnote n="1">¹This example is borrowed from Straffin’s chapter in the Handbook of Game Theory (Straffin, 1977b).</footnote>
from a uniform distribution in the simplex. Życzkowski and Słomczyński (2007, 2013) focus on
the role that the quota plays in ensuring PLT. Kurz (2018) explores conditions that limit the
distance between the vectors or weights and the vectors of power for the Banzhaf index, the
Shapley-Shubik index, and the nucleolus.

However, a common limitation of these analyses is that the weights are typically derived from
artificial statistical distributions. In contrast, our paper takes a novel approach by using a
large set of real data. Since 2014, by law, all French municipalities must belong to a so-called
*Etablissement Public de Cooperation Intercommunal* (hereafter EPCI), which can levy taxes
and have large prerogatives for the provision of local public goods. These EPCI are governed
by a council, where each municipality is represented by a number of delegates proportional to
its population. Hence, we consider the distribution of seats in the 1,251 French inter-communal
councils to test the PLT. This analysis also opens the door to designing a specific degressive
apportionment rule that could be used in the French institutional context and would limit the
weight of the major player.

Section 2 provides a formal presentation of the Banzhaf Power Index. Section 3 is dedicated to our dataset that comprises 1,251 French intermunicipal councils. In this section, we compute the Banzhaf index for all member cities and analyze the conditions under which the major cities are overrepresented. Section 4 introduces a toy model to facilitate the interpretation of the data. We subsequently return to the French institutional framework, proposing a practical apportionment rule to limit major cities' influence within inter-municipal councils. Finally, Section 5 concludes the paper.

## 2 Definition

Let $N$ be the set of players of cardinality $n$. A coalition $T$ is a non-empty subset of $N$. A vector $w = (w_1, \dots, w_i, \dots, w_n)$ assigns to each player $i$ a non negative real weight $w_i$. In most real-life applications, the weights are strictly positive integer numbers: they represent the number of delegates, mandates, or votes a player controls in a council. Let $q \in ]0, \sum_{i=1}^n w_i]$ be the quota, that is, the number of votes a coalition of players needs to pass a resolution. Typically, $q > \frac{1}{2} \sum_{i=1}^n w_i$. A decision process in a council or an assembly is then described by a weighted
quota game $G(N, w, q)$. It is represented by a characteristic function $v(T)$ defined by:

$$v(T) = \begin{cases} 1 & \text{if } \sum_{i \in T} w_i \ge q \\ 0 & \text{if } \sum_{i \in T} w_i < q \end{cases} \qquad (1)$$

A bill is passed if a coalition $T$ controls enough votes. Such coalitions are called winning coalitions. Notice that we can also define a weighted quota game by normalizing the sum of the weights to one. In this case, the weight represents the fraction of mandates that a player controls in an assembly, and the quota is the percentage of votes needed to pass a resolution. In this paper, we uniquely consider the weighted majority game, that the case where $q = \lceil \frac{1}{2} \sum_{i \in N} w_i \rceil$.

The non-normalized Banzhaf/Penrose power of voter $i$, $\beta_i$, is defined as the probability of influencing a decision either way; it is simply the probability of being decisive. It assumes that each player is equally likely to join or not a coalition to pass a bill, independently from other players' decisions.

$$\beta_i(v) = \sum_{T \subseteq N, i \in T} \frac{v(T) - v(T \setminus \{i\})}{2^{n-1}} \qquad (2)$$

where $2^{n-1}$ is the number of coalitions to which player $i$ may belong. The normalized Banzhaf index of player $i$ is the number of times player $i$ is swing, divided by the total number of swings, or simply:

$$\hat{\beta}_i(v) = \frac{\beta_i(v)}{\sum_{j \in N} \beta_j(v)} \qquad (3)$$

It derives the share of swing situations controlled by player $i$.

# 3 When does a player have too much power?

## 3.1 Data and empirical design

As mentioned in the introduction, the literature on Penrose's Limit Theorem in game theory has primarily focused on proposing scenarios in which it holds (Lindner and Machover, 2004; Chang et al., 2006) or on reverse design, which involves searching for the optimal game that could achieve a fair distribution of power according to a predetermined target (Boratyn et al., 2020; Kurz and Napel, 2014). However, to date, no study has utilized a large set of empirical data to test the validity range of Penrose's Limit Theorem.

To test the conditions under which Penrose's Limit Theorem holds, the authors gathered a
dataset of 1,254 French inter-communal structures encompassing 36,669 municipalities.<fnref n="2" /> By law, since January 1, 2014, each French municipality is required to belong to an inter-communal structure. More precisely, each inter-communal structure is governed by a council, wherein each municipality is represented by one or more delegates. Consequently, the voting system in this context resembles a two-tier system.

The rules governing the apportionment of delegates among the members of a French inter-municipal council and their designation are rather complex. These rules are presented in detail in Abidi Perier and Merlin (2025), where alternative methods for designing a fairer allocation of delegates are discussed. To summarize, the allocation rule combines Jefferson's apportion-ment mechanism with a series of provisions guaranteeing at least one delegate per municipality. Additional provisions allow limited flexibility for the members of the inter-communal structures to negotiate the final allocation of delegates among themselves. Ultimately, each municipality must have at least one delegate, no municipality may have fewer delegates than a less populated one, and no member should control more than half of the delegates. The composition of the inter-communal council used in this paper was set on August 31st, 2019, six months before the local elections of March 13th, 2020.

Notice that, in this paper, we are not interested in finding a fair representation for the citizens. In a two-tier voting system, when the conditions of the Penrose Limit Theorem are met, Penrose (1952) also proves that two citizens living in different jurisdictions will have a similar Banzhaf power if the number of delegates per jurisdiction is proportional to the square root of their population. We here consider the allocation of delegates as given, and the sole issue is to understand under which conditions, if there are any, the normalized Banzhaf power of a player that controls X% of the delegate is close to X%. The advantage of the 1,254 <fnref n="3" /> distributions of the delegates that we use here are that 1) they come from real cases, not artificial distributions <fnref n="4" /> 2) we can then design a new allocation rule where we would only give $\alpha X$ delegates, $\alpha \in [0; 1]$, to the main city, to effectively limit its share of power to X%.

<footnote n="2">²For more detailed information about the database, the reader can refer to the companion data paper by Abidi Perier et al. (2025).</footnote>

<footnote n="3">³Indeed, we only consider 1,251 cases, as three EPCIs, *La Métropole du Grand Paris*, *La Métropole de Lyon* and *La Métropole Aix Marseille Provence* are governed by specific rules.</footnote>

<footnote n="4">⁴Similarly, in social choice literature, libraries of real preferences profiles, such as Preflib https://preflib.simonrey.fr/ or Pabulib https://pabulib.org/ are used to estimate the 'real' probabilities of voting paradoxes.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Seat share versus share of power</figcaption></figure>

## 3.2 Assessing the power of the major city

We employ a simple measure to assess the alignment between the distribution of delegates and the distribution of power. The most straightforward metric is the ratio between the largest player's normalized Banzhaf power and its relative weight in the council, which should remain as close to one as possible. Although one could consider using an index comparing the distribution of delegates with the Banzhaf power distribution, it is evident that the Banzhaf power of the largest city is the primary source of inequality.

Figure 1 illustrates the evolution of power shares, as measured by the Banzhaf index, alongside the seat shares for the largest municipalities within each inter-communal structure. Unsurprisingly, the Banzhaf index for the largest player increases as its relative weight grows. Up to a delegate share slightly below 20%, the Banzhaf index appears to evolve in perfect proportion to the player's weight. However, between 20% and 50%, the normalized Banzhaf power frequently exceeds the seat percentage, occasionally reaching a value of 1. Furthermore, we observe increased variability beyond the 20% threshold. To provide greater precision, we computed the mean Banzhaf index at 1% intervals. Figure 2 reveals a discontinuity at a threshold of 17%. For EPCIs where the seat share lies within the interval [0.16, 0.17], the mean Banzhaf index for the largest player is 0.1472, whereas for the interval [0.17, 0.18], this value rises sharply to 0.1921. Notably, in the interval [0.2, 0.21], the mean Banzhaf index further increases to 0.2671. In the case of French inter-communal structures, this value can be interpreted as the threshold for significant shifts in power distribution.
<figure><img src="image_3.png" /><figcaption>Table 1: The distribution of EPCIs by quartile, based on the number of members.</figcaption></figure>

<table><thead><tr><th>Quartile</th><th>Nb EPCIs</th><th>Mean Nb Memb</th><th>Mean IB LC</th><th>Mean Wgt Seats LC</th><th>Mean Wgt Pop LC</th></tr></thead><tbody><tr><td>Q1:[2;14]</td><td>341</td><td>9.513</td><td>0.492</td><td>0.343</td><td>0.370</td></tr><tr><td>Q2:[15;23]</td><td>313</td><td>18.652</td><td>0.405</td><td>0.260</td><td>0.284</td></tr><tr><td>Q3:[24;37]</td><td>288</td><td>29.646</td><td>0.414</td><td>0.233</td><td>0.257</td></tr><tr><td>Q4:[38;158]</td><td>309</td><td>55.217</td><td>0.376</td><td>0.191</td><td>0.236</td></tr></tbody></table>

## 3.3 Segmenting the data by quartile

To delve further into our analysis, and more precisely to see if this is related to the number of municipalities forming an EPCI, we have segmented our EPCIs according to the number of municipalities they group using the quartiles. Table 1 presents the number of EPCIs (Nb EPCIs) in each quartile, along with the mean number of members (Mean Nb Memb), the mean normalized Banzhaf Index (Mean IB LC), and the weight in terms of seats (Mean Wgt Seats LC) and population (Mean Wgt Pop LC) for the largest city in each EPCI. We can notice that across all quartiles, the power measured by the Banzhaf Index for the largest cities significantly exceeds their weight in terms of both seats and population.

To further our analysis, we plot, in Figure 3, the share of power and the share of seats for the largest municipality in each EPCI, for each quartile. As we have fewer points, we chose a 2% interval to compute the mean values of the Banzhaf index In Figure 4, we display the mean values for the normalized Banzhaf index for each intervals, for each quartile.
<figure><img src="image_4.png" /><figcaption>Figure 4: Seat share versus Share of power (IB) of big cities-Quartile</figcaption></figure>

<figure><img src="image_5.png" /></figure>
We can first notice that the situation is quite different in the first quartile compared to the other ones. We observe that EPCIs with fewer than 14 municipalities exhibit a highly dispersed evolution of power compared to the share of seats. The other three quartiles show an S-shaped distribution for the Banzhaf index, which seems to shift to the left as *n* grows slightly.

In all cases, the PLT appears to hold when the share of delegates from the major municipality is sufficiently small. However, beyond a threshold close to 0.2, the power of the largest municipalities, as measured by the index, begins to increase more rapidly than their share of seats. To be more precise:

* In Q1, we first observe a mean value (0.1806) outside its interval for [0.16, 0.18].

* In Q2, we first observe a mean value (0.1602) outside its interval for [0.14, 0.16].

* In Q3, we first observe a mean value (0.1422) outside its interval for [0.12, 0.14].

* In Q4, we first observe a mean value (0.1219) outside its interval for [0.10, 0.12].

These results indicate that the influence of the largest municipalities becomes more pronounced at lower values as *n* increases. At higher levels of seat share, the power measured by the indices approaches 1. These findings offer valuable insights into the dynamics of power distribution and decision-making processes within EPCI structures of varying sizes. They clearly suggest that smaller EPCIs may exhibit more proportional power-sharing, although the distribution becomes more dispersed beyond a certain threshold and depends heavily on the specific seat distribution of each EPCI. In contrast, larger EPCIs tend to have a more concentrated power structure, particularly once the major municipality surpasses a certain size threshold.

## 3.4 Segmenting the data by deciles

We can take our analysis a step further by segmenting the 1,251 EPCIs into 10 deciles. We can make an analysis similar to the one we made for quartiles, by computing the average value of Banzhaf index for each interval; Table 2 presents the same information as Table 1 but at the decile level. Once again, the power of large cities surpasses their weight across all deciles. Figure 5 illustrates the evolution of Banzhaf power relative to seat share as the size of the intercommunal council increases.

The main difference compared to the analysis by quartile is that the main of variability is displayed in the first decile. The mean Banzhaf measure in this decile is already superior to
the weight for the first interval we encounter in this decile, as 0.167857 does not lie within the [0.14; 0.16[ interval. This extreme variability in the first decile means that we can hardly consider that PLT holds for less than 9 players, but mostly because we have very few councils where the size of the largest player is inferior to 20%.

The S-shaped pattern starts to emerge as early as the second decile and becomes even more apparent in the following ones. In the case of the second decile, the first instance where the average Banzhaf index falls outside its expected interval is [0.18 : 0.2]. Once again, we can see that the value at which the Penrose Limit Theorem (PLT) holds tends to decrease as we move from D2 to D10. In D10, the mean value of the Banzhaf power is superior to the weight as soon as the weight is superior to 10%.

<table><caption>Table 2: The distribution of EPCIs by decile, based on the number of members.</caption><thead><tr><th>Decile</th><th>Nb EPCIs</th><th>Mean Nb Memb</th><th>Mean IB LC</th><th>Mean Wgt Seats LC</th><th>Mean Wgt Pop LC</th></tr></thead><tbody><tr><td>D1:[2;9]</td><td>153</td><td>6.582</td><td>0.518</td><td>0.383</td><td>0.414</td></tr><tr><td>D2:[9;12]</td><td>120</td><td>10.975</td><td>0.443</td><td>0.305</td><td>0.330</td></tr><tr><td>D3:[12;15]</td><td>108</td><td>14.074</td><td>0.459</td><td>0.298</td><td>0.316</td></tr><tr><td>D4:[15;19]</td><td>153</td><td>17.477</td><td>0.412</td><td>0.264</td><td>0.291</td></tr><tr><td>D5:[19;23]</td><td>120</td><td>21.367</td><td>0.414</td><td>0.257</td><td>0.280</td></tr><tr><td>D6:[23;27]</td><td>109</td><td>25.532</td><td>0.415</td><td>0.237</td><td>0.258</td></tr><tr><td>D7:[27;33]</td><td>119</td><td>30.462</td><td>0.426</td><td>0.238</td><td>0.261</td></tr><tr><td>D8:[33;42]</td><td>135</td><td>38.022</td><td>0.393</td><td>0.214</td><td>0.247</td></tr><tr><td>D9:[42;53]</td><td>115</td><td>47.417</td><td>0.377</td><td>0.199</td><td>0.239</td></tr><tr><td>D10:[53;158]</td><td>119</td><td>72.319</td><td>0.362</td><td>0.169</td><td>0.227</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Figure 5: Seat share versus share of power by deciles</figcaption></figure>
<table><caption>Table 3: The first intervals where the value of Banzhaf is greater than the share</caption><thead><tr><td>Decile</td><td>Interval</td><td>Mean IB</td></tr></thead><tbody><tr><td>D1:[2;9]</td><td>[0.14, 0.16[</td><td>0.167857</td></tr><tr><td>D2:[9;12]</td><td>[0.18, 0.2[</td><td>0.202197</td></tr><tr><td>D3:[12;15]</td><td>[0.16, 0.18[</td><td>0.181792</td></tr><tr><td>D4:[15;19]</td><td>[0.14, 0.16[</td><td>0.161871</td></tr><tr><td>D5:[19;23]</td><td>[0.12, 0.14[</td><td>0.141327</td></tr><tr><td>D6:[23;27]</td><td>[0.12, 0.14[</td><td>0.140312</td></tr><tr><td>D7:[27;33]</td><td>[0.12, 0.14[</td><td>0.149239</td></tr><tr><td>D8:[33;42]</td><td>[0.12, 0.14[</td><td>0.144125</td></tr><tr><td>D9:[42;53]</td><td>[0.1, 0.12[</td><td>0.120337</td></tr><tr><td>D10:[53;158]</td><td>[0.1, 0.12[</td><td>0.126909</td></tr></tbody></table>

# 4 Coping with the large player

As discussed in Section 3, there are instances where the weight of the largest municipality becomes excessively large, absorbing all opportunities to act as a swing player in a weighted majority game. Consequently, when a central planner, aims to allocate a share of power to the largest player proportional to its weight, X%, it should assign Y% of the delegates to him, with $Y \le X$. More generally, if the planner's objective is to distribute the power among the n players according to a target vector $t = (t_1, ..., t_i, ..., t_n)$, the objective of so-called inverse power index problem is to find the vector of weights $w = (w_1, ..., w_i, ..., w_n)$ and the quota q such that $\hat{\beta} = (\hat{\beta}_1, ..., \hat{\beta}_i, ..., \hat{\beta}_n)$ and $t = (t_1, ..., t_i, ..., t_n)$ are as close as possible according to a certain metric.<fnref n="5" /> There are two ways to solve this problem. The first one is to propose a general heuristic method. For example, Życzkowski and Słomczyński (2007, 2013) suggested keeping the weights unchanged but considering a quota $q^* = \frac{1}{2} (1 + \sqrt{\sum_{i=1}^{n} w_i^2})$. The other solution is to propose a search algorithm (see for example Laruelle and Widgrén (1998); Leech (2002a); Fatima et al. (2008)) that will search for the optimal weighted quota games. Kurz and Napel (2014) propose a comparison of heuristic methods and search algorithms for up to 20 players.

<footnote n="5">⁵Notice that the same issue can also be addressed for other power indices, such as the Shapley-Shubik index, the nucleolus, etc.</footnote>
We will focus on a simple heuristic rule, where $Y = \alpha X$ and $\alpha \in ]0, 1]$. The parameter $\alpha$ represents the discount factor that should be applied to limit the power of the largest player. Can a simple rule be used to derive the right value for $\alpha$? To address this issue, we will first introduce a toy model that may provide an initial insight.

## 4.1 A toy model for Banzhaf power: Theory

We consider a specific weighted quota game with a large number of players, $n$. The game consists of $n$ players, where the first $n-1$ players each have a weight of 1, while player $n$ has a weight of $K \in \mathbb{N}$, where $1 \le K \le n-1$. In this section, we analytically examine the evolution of the large player's power as their weight increases from 1 to $K$, for different values of $n$, specifically with respect to the Banzhaf index. This analysis will first enable us to identify a range in which the Penrose Limit Theorem can be considered valid, and second, to propose a discount factor for the large player's weight when it becomes excessively high.

To simplify the analysis, we will normalize the sum of the weights to 1. Let $S = \frac{K}{K+n-1}$ be the relative weight of player $n$. The relative weight of a small player $i = 1, \dots, n-1$ is now $\frac{1-S}{n-1} = \frac{1}{K+n-1}$. By definition, we have $\frac{1-S}{n-1} \le S \le 0.5$. Our focus is exclusively on the majority game.

To evaluate the Banzhaf power of the largest player, we will use the normal approximation developed by (Leech, 2002b), which has also been employed by (Feix et al., 2004) and (Lepelley et al., 2011) to analyze the referendum paradox. This method relies on the principle that a normal distribution can approximate the sum of independent random variables when the number of variables is sufficiently large.

More precisely, the Banzhaf index is based on a probabilistic model, which asserts that each player votes 'yes' (1) or 'no' (0) with equal probability, independently from the other players (see Straffin (1977a) and Andjiga et al. (2003) for more on the probabilistic interpretation of power indices). Let $x_i$ be the random variable that describes the behavior of a small player, with mean $m_i$ and standard deviation $\sigma_i$. The variable $x_i$ takes values 0 with probability $\frac{1}{2}$, and value $\frac{1-S}{n-1}$ with probability $\frac{1}{2}$. We can easily compute the mean and the variance for the variable $x_i$:

$$m_i = \left( \frac{1}{2} \frac{1-S}{n-1} \right) \quad \text{and} \quad \sigma_i^2 = \frac{1}{4} \left( \frac{1-S}{n-1} \right)^2$$
Let us define $x = \sum_{i=1}^{n-1} x_i$, the random variable that will give the sum of the votes in favor of a proposal by the $n-1$ small players and $h_{n-1}(x)$ be the distribution function of $x$. As all the variables $x_i$ are independent, using the Central Limit Theorem, the distribution function $h_{n-1}(x)$ can be approximated by a normal law with mean $\bar{m} = \sum_{i=1}^{n-1} m_i$ and variance $\sigma^2 = \sum_{i=1}^{n-1} \sigma_i^2$ if $n$ is large enough.

$$ \bar{m} = \sum_{i=1}^{n-1} m_i = \frac{1-S}{2} \quad \text{and} \quad \sigma^2 = \sum_{i=1}^{n-1} \sigma_i^2 = \frac{(1-S)^2}{4(n-1)} $$

Using the Central Limit Theorem, the distribution $h_{n-1}(x)$ can be approximated by a normal distribution with parameters $\bar{m}$ and variance $\sigma^2$.

$$ h_{n-1}(x) \approx \frac{1}{\sqrt{2\pi}} \frac{1}{\sigma} \exp \left( -\frac{(x - \bar{m})^2}{2\sigma^2} \right) \quad (4) $$

$$ \approx \frac{\sqrt{2}}{\sqrt{\frac{\pi (1-S)^2}{n-1}}} \exp \left( -2 \frac{\left(x + \frac{S-1}{2}\right)^2 (n-1)}{(1-S)^2} \right) \quad (5) $$

With weight $S$, the major player is pivotal when the variable $x$ lies between $\underline{x} = 0.5 - S$ and $\bar{x} = 0.5$. If the variable $x$ is below $\underline{x}$, there are not enough votes to pass a resolution even if player $n$ joins the coalition of small voters who have already voted in favor of a resolution; if $x$ is above $\bar{x}$, the small players have already reached the majority threshold without the help of the player $n$. We obtain an estimation of the non-normalized Banzhaf power of the major player as a function of $S$ and $n$, $\beta_n(S, n)$:

$$ \beta_n(S, n) \approx \int_{\underline{x}}^{\bar{x}} \frac{\sqrt{2}}{\sqrt{\frac{\pi (1-S)^2}{n-1}}} \exp \left( -2 \frac{\left(x + \frac{S-1}{2}\right)^2 (n-1)}{(1-S)^2} \right) dx \quad (6) $$

Now, consider the evolution of the power of a small player. Without loss of generality, we will derive the Banzhaf power of player 1. Let $y = \sum_{i=2}^{n-1} x_i$ be the variable that describes the total number of votes cast by players 2 to $n-1$ in favor of a resolution, and let $g_{n-2}(y)$ be its distribution function. The mean and variance of variable $y$ are respectively:

$$ \bar{m}' = \sum_{i=1}^{n-2} m_i = \frac{(1-S)(n-2)}{2(n-1)} \quad \text{and} \quad \sigma'^2 = \sum_{i=1}^{n-2} \sigma_i^2 = \frac{(n-2)}{4} \left( \frac{1-S}{n-1} \right)^2 $$

Which leads to:

$$ g_{n-2}(y) \approx \frac{1}{\sqrt{2\pi}} \frac{1}{\sigma'} \exp \left( -\frac{(x - \bar{m}')^2}{2\sigma'^2} \right) \quad (7) $$
The power of player 1 will then depend upon whether the major player, $n$, has backed the proposition or not. If it has not backed the proposition, we need to integrate $g_{n-2}(y)$ between $y_1 = 0.5 - \frac{1-S}{n-1}$ and $\bar{y}_1 = 0.5$. If the major player has already backed the proposition, we need to integrate $g_{n-2}(y)$ between $y_2 = 0.5 - S - \frac{1-S}{n-1}$ and $\bar{y}_2 = 0.5 - S$. Hence, we get an estimation of the non normalized power of player 1 as a function of $S$ and $n$:

$$ \beta_1(S, n) \approx \frac{1}{2} \int_{0.5 - \frac{1-S}{n-1}}^{0.5} g_{n-2}(y) dy + \frac{1}{2} \int_{0.5 - S - \frac{1-S}{n-1}}^{0.5-S} g_{n-2}(y) dy \quad (8) $$

At last, we can estimate the over-evaluation of the major player $n$ by computing its share of normalized Banzhaf power as a function of $S$:

$$ \hat{\beta}_n(S, n) \approx \frac{\beta_n(S, n)}{\beta_n(S, n) + (n-1)\beta_1(S, n)} \quad (9) $$

Before going further, we can test the validity of our approximation. Table 4 computes the difference between our approximations and the real values$^6$, for Banzhaf's non-normalized and normalized versions, for $n = 20$ and $n = 100$. First, it is worth noticing that while it is recommended to have $n \ge 30$ to apply the Central Limit Theorem, the approximations are good even for a smaller number of players, $n = 20$. However, as we reach $K = 12$ (which corresponds to $S = \frac{12}{31} \approx 0.387$) our estimation overestimates the Banzhaf power of a small player by 17.8%. The quality of the approximation deteriorates as $K$ grows further. In the real data, there are very few opportunities for a small player to be swing compared to the large one, and our technique cannot cope with such low numbers. The same pattern can be observed with $n = 100$. The estimation remains very correct till $K = 40$ (that is, $S = \frac{40}{139} \approx 0.288$) but overestimates the probability of being swing by a factor 3 for $K = 60$. However, as those numbers are extremely low, they have only a small impact on the probability of the large player being decisive.

Going back to the estimation of the Banzhaf power of the largest player proposed by equation 9, we derive it for different coalition sizes $S$, with $n = 20, 50, 100, 200$. Figure 6 displays the four resulting curves and compares them to the diagonal. We observe that the power distribution follows an S-shaped curve, which shifts to the left as $n$ increases, with the rightmost curve corresponding to $n = 20$ and the leftmost to $n = 200$. These results are consistent with the observations in our data set, as shown in Figure 2.

<footnote>$^6$To obtain exact values, we used the online simulator proposed by Dennis Leech on the webpage https://homepages.warwick.ac.uk/ecaee/.</footnote>
<table><caption>Table 4: Validity of the approximation</caption><thead><tr><th rowspan="3">n</th><th rowspan="3">k</th><th colspan="4">Large Player</th><th colspan="4">Small Players</th></tr><tr><th colspan="2">Abs. Banzhaf</th><th colspan="2">Rel. Banzhaf</th><th colspan="2">Abs. Banzhaf</th><th colspan="2">Rel. Banzhaf</th></tr><tr><th>real</th><th>est.</th><th>real</th><th>est.</th><th>real</th><th>est.</th><th>real</th><th>est.</th></tr></thead><tbody><tr><td rowspan="5">20</td><td>2</td><td>.352394</td><td>.353644</td><td>.100000</td><td>.100233</td><td>.166924</td><td>.167081</td><td>.047368</td><td>.047356</td></tr><tr><td>4</td><td>.640717</td><td>.641204</td><td>.217391</td><td>.218854</td><td>.121399</td><td>.120453</td><td>.041190</td><td>.041112</td></tr><tr><td>8</td><td>.936432</td><td>.933542</td><td>.601266</td><td>.601640</td><td>.032684</td><td>.032532</td><td>.020986</td><td>.020966</td></tr><tr><td>12</td><td>.995575</td><td>.994094</td><td>.943925</td><td>.934463</td><td>.003113</td><td>0.003669</td><td>.002951</td><td>.003449</td></tr><tr><td>16</td><td>.999924</td><td>.999758</td><td>.998697</td><td>.996728</td><td>.000069</td><td>.000172</td><td>.000069</td><td>.000172</td></tr><tr><td rowspan="5">100</td><td>2</td><td>.159178</td><td>.160099</td><td>.020000</td><td>.020005</td><td>.078785</td><td>.078427</td><td>.009899</td><td>.009799</td></tr><tr><td>10</td><td>.685120</td><td>.687553</td><td>.124929</td><td>.125518</td><td>.048474</td><td>.047901</td><td>.008839</td><td>.008744</td></tr><tr><td>20</td><td>.956125</td><td>.956637</td><td>.478748</td><td>.482465</td><td>.010515</td><td>.010261</td><td>.005265</td><td>.005175</td></tr><tr><td>40</td><td>.999954</td><td>.999946</td><td>.998057</td><td>.997843</td><td>.000020</td><td>.000021</td><td>.000020</td><td>.000021</td></tr><tr><td>60</td><td>1.000000</td><td>.999999</td><td>1.000000</td><td>.999999</td><td>2.7 × 10<sup>-10</sup></td><td>7 × 10<sup>-10</sup></td><td>.000000</td><td>2.7 × 10<sup>-10</sup></td></tr><tr><td>98</td><td>1.000000</td><td></td><td>1.000000</td><td></td><td>3.1 × 10<sup>-30</sup></td><td>3.1 × 10<sup>-30</sup></td><td></td><td></td></tr></tbody></table>

<table><caption>Table 5: Thresholds for 20% excess of power</caption><tr><td>n</td><td>20</td><td>50</td><td>100</td><td>200</td></tr><tr><td>S*</td><td>.1575</td><td>.1004</td><td>.0714</td><td>.0508</td></tr></table>

We also observe clearly that beyond a certain point, $S^*$, the power of the largest player begins to diverge from its weight. Figure 6 shows the ratio between the power of the largest player and its weight, with the right-hand side offering a zoomed-in view of the interval [0; 0.1]. It is immediately apparent that for $n = 100$ and $n = 200$, the ratio diverges rapidly. Table 5 presents the values of $S^*$ at which the ratio reaches 1.2, indicating a 20% excess in power. This table highlights the significant role of the number of players, as the largest player can already attain a 20% power excess at $S^* = 5\%$ when $n = 200$.

## 4.2 Coping with the large player: Empirical test

The next question is to determine the factor $\alpha(n)$ we should use to discount the weight of the major in the majority game so that its share of power is equal to $S$. Said differently, we wish to find the factor $\alpha(n)$ such as:

$$ \hat{\beta}_n(\alpha S) = S $$
<figure><img src="image_7.png" /><figcaption>Figure 7: Excess of power of the largest player as a function of $S$ for $n = 20, 50, 100, 200$ (Left); Zoom for $S < 10\%$ (right).</figcaption></figure>

<figure><img src="image_8.png" /></figure>

We ran a Python program to solve the equation for some values of $S$ and $n$. We display our results in Table 7. Note that as long as $\alpha S$ remains lower than 0.5, we can use the formulas to compute $\alpha$, even if $S > 0.5$. Though the discount factor could be small for $S = 0.05$, we notice it decreases rapidly as both $n$ and $S$ increase. To give an example, at $S = 0.3$ and $n = 160$, the number of delegates attached to the major player should be reduced by almost 60%! Table 6 displays the case of Grand Besançon Métropole (68 members); all the municipalities have one delegate, except Saint-Vit, which has two delegates, and Besançon, which has 55 delegates, making a total of 123 delegates. If our objective is to give Besançon a share of Banzhaf power in line with its true weight (that is $\frac{55}{123} \approx 0.447$), we should use a discount factor close to 0.4. It would only be represented by 23 delegates, making a total of 91 delegates.
<table><caption>Table 6: CU Grand Besançon Métropole - 68 members</caption><thead><tr><th>Municipalities</th><th>Seats</th><th>% Seats</th><th>New seats</th><th>Banzhaf</th><th>New Banzhaf</th></tr></thead><tbody><tr><td>Champoux</td><td>1</td><td>0.0081</td><td>1</td><td>1.13 × 10<sup>-12</sup></td><td>0.0017</td></tr><tr><td>⋮</td><td>1</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr><tr><td>64 cities</td><td>1</td><td>⋮</td><td>1</td><td>1.13 × 10<sup>-12</sup></td><td>0.0017</td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr><tr><td>Saone</td><td>1</td><td>0.0081</td><td>1</td><td>1.13 × 10<sup>-12</sup></td><td>0.0017</td></tr><tr><td>Saint-Vit</td><td>2</td><td>0.0162</td><td>2</td><td>1.35 × 10<sup>-12</sup></td><td>0.0032</td></tr><tr><td>Besançon</td><td>55</td><td>0.447</td><td>23</td><td>1</td><td>0.8784</td></tr><tr><td>Total</td><td>123</td><td>1</td><td>91</td><td>1</td><td>1</td></tr></tbody></table>

Then, for all values of $n \ge 50$, we compute a discount factor and apply it to our dataset.<fnref n="7" /> By filtering our data, we identify 155 EPCIs with a membership of 50 or more. Figures 9 and 8 illustrate the relationship between the target ratio (the share of seats divided by the power measured by the Banzhaf index) using the observed and adjusted seat shares, respectively. It can be observed that the ratio hovers around 1 when applying the discount factor to the number of seats, whereas the ratio based on the observed number of seats revolves around 2.5. This finding suggests that the power of large cities in intercommunal structures (EPCIs) is significantly greater than their weight in terms of seat share (i.e., population share). Conversely, when applying the discount factor, this power-to-seat ratio approaches 1 for the majority of the studied EPCIs.

However, there are two important caveats. First, the reduction tends to cause the ratio to fall below one, particularly for values of $S \in [0.1, 0.2]$. We identified 18 EPCIs with a ratio of 0.8, none of which exhibited such low values prior to applying the discount factor. Secondly, while we observed 99 EPCIs in the real data with a ratio greater than 1.2, there remain specific cases where this ratio still exceeds 1.2 after reducing the influence of the major player; specifically, 13 EPCIs fall into this category. Indeed, our heuristic method is not sufficient to address these cases. For example, the case of Besançon is an outlier observable at the top-right end of Figure 9. Although we reduced its number of delegates by 60%, it still controls 87% of swing possibilities (see Table 6). This highlights that the complexities of computed power in weighted quota games

<footnote n="7">⁷Note that we choose $n \ge 50$ to ensure it is sufficiently large to allow the use of the normal distribution approximation.</footnote>
<table><caption>Table 7: The discount factor $\alpha$ as a function of $n$ and $S$</caption><thead><tr><th>S</th><th>n=20</th><th>n=50</th><th>n=100</th><th>n=160</th><th>n=200</th></tr></thead><tbody><tr><td>0.05</td><td>0.9967</td><td>0.9652</td><td>0.9292</td><td>0.8946</td><td>0.8747</td></tr><tr><td>0.1</td><td>0.9470</td><td>0.8722</td><td>0.7941</td><td>0.7312</td><td>0.6992</td></tr><tr><td>0.15</td><td>0.8807</td><td>0.7743</td><td>0.6774</td><td>0.6077</td><td>0.5744</td></tr><tr><td>0.2</td><td>0.8126</td><td>0.6897</td><td>0.5883</td><td>0.5197</td><td>0.4880</td></tr><tr><td>0.25</td><td>0.7497</td><td>0.6205</td><td>0.5205</td><td>0.4553</td><td>0.4257</td></tr><tr><td>0.3</td><td>0.6941</td><td>0.5642</td><td>0.4677</td><td>0.4064</td><td>0.3789</td></tr><tr><td>0.35</td><td>0.6459</td><td>0.5180</td><td>0.4259</td><td>0.3682</td><td>0.3425</td></tr><tr><td>0.4</td><td>0.6042</td><td>0.4798</td><td>0.3919</td><td>0.3376</td><td>0.3135</td></tr><tr><td>0.45</td><td>0.5683</td><td>0.4478</td><td>0.3640</td><td>0.3126</td><td>0.2899</td></tr><tr><td>0.5</td><td>0.5372</td><td>0.4207</td><td>0.3406</td><td>0.2918</td><td>0.2704</td></tr><tr><td>0.55</td><td>0.5102</td><td>0.3977</td><td>0.3210</td><td>0.2745</td><td>0.2541</td></tr><tr><td>0.6</td><td>0.4869</td><td>0.3781</td><td>0.3043</td><td>0.2598</td><td>0.2403</td></tr></tbody></table>

cannot always be resolved through normal approximations.

<figure><img src="image_9.png" /><figcaption>Figure 8: Seat share versus target ratio based on the population share of municipalities using the observed seats</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 9: Seat share versus target ratio based on the population share of municipalities using the adjusted seats</figcaption></figure>

# 5 Conclusion

This paper empirically investigates the conditions under which Penrose's Limit Theorem applies within the context of French inter-communal councils and proposes a theoretical model utilizing the Banzhaf index to reduce the power of the largest municipality. Our analysis, based on a dataset comprising 1,251 councils, reveals that the applicability of the theorem is contingent upon several factors, notably the size of the largest municipality relative to the council and the overall number of municipalities within the council.

First, the analysis by deciles reveals that we need at least 10 members for the PLT to be satisfied in general. Second, for the remaining deciles, the major player must typically be below 20% for the PLT to apply. Interestingly, this maximum threshold decreases as the number of players increases. In D10, where councils range from 53 to 158 members, the size of the largest player should not exceed 10% for the PLT to hold.

Our findings underscore the importance of considering these factors when designing allocation rules for inter-communal councils. To mitigate the excessive power of the largest municipality, particularly in larger councils, it is necessary to implement degressive apportionment rules that allocate delegates in a manner that limits the influence of the largest entity.

While this paper offers valuable insights into the dynamics of power distribution in French inter-communal councils, further research is needed to explore potential allocation rules aimed
at limiting the over-representation of larger cities in councils with different big cities. The
allocation rule we propose, based on the application of the law of large numbers to define a
table of discount factors, represents a first step. Though it improves the situation, it sometimes
misses its target.

A similar analysis can be leveraged in other countries with inter-communal governance structures where municipalities send delegates to councils. For instance, in Italy, frameworks like "Unioni di Comuni" and "Città Metropolitane" ensure fair municipal representation (Bianchi and Pompili, 2004; Lippi, 2011). Similarly, Belgium's "Inter-communales" assign council seats based on population or contributions (Deschamps and De Rynck, 2008), and in Japan, municipal mergers and cooperation agreements also follow structured representation criteria (Tanaka, 2007). These examples demonstrate how voting power indices can be applied across different institutional and geographic settings.

Future research should aim to understand whether the weight of the second-largest municipality also plays a role and whether this factor can be included in the determination of the optimal discount factor. It could also be extended to the analysis of the Shapley-Shubik index, which we did not consider in this analysis.
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