Axioms and Divisor Methods for a Generalized Apportionment Problem with Relative Equality

Abstract

The allocation of seats in a legislative body to groups based on their size is a crucial issue in legal and political studies. However, recent findings suggest that an optimal allocation of seats may not be proportional to the size of the groups. For instance, the European Parliament (EP) utilizes a subproportional system known as degressive proportionality. Unfortunately, current apportionment methods for the EP lack a rigorous axiomatic analysis and fail to adequately address equality. Building upon recent research on equality in subproportional settings, this paper proposed a novel generalization of existing axioms and divisor methods for proportionality to encompass subproportionality with relative equality. Specifically, we consider a function $f\left(p\right)=a+b{p}^{\gamma }$ on the standard number of seats for a group of size p, where a, b and $\gamma$ are given non-negative constants, and a is an integer. This theory is exemplified through an empirical study focused on the EP.

1. Introduction

The optimal allocation of legislative seats to groups according to their population size or vote count is a persistently critical and often considered the most "susceptible" issue in legal and political studies [1]. Despite over two centuries of comprehensive exploration through debates (e.g., [1,2,3], reports (e.g., [4,5,6,7,8,9]), algorithmic solutions (e.g., [10,11,12,13,14,15]), mathematical theories (e.g., [16,17,18,19,20,21,22]), empirical studies (e.g., [23,24]), among other studies, a significant disparity still exists between theoretical constructs and real-world scenarios [24,25].

This paper focuses on mathematical theory and algorithms for the apportionment problem. Let $\mathbb{N}$, ${\mathbb{Z}}_{\ge 0}$, $\mathbb{R}$, and ${\mathbb{R}}_{\ge 0}$ denote the sets of natural numbers, non-negative integers, real numbers and non-negative real numbers, respectively. Given a number $S\in \mathbb{N}$ of seats and n groups $G_1,\dots ,G_n$ with size (population or vote count) $p_1,\dots ,p_n>0$, an apportionment is a vector $\mathit{x}=(x_1,\dots ,x_n)\in {\mathbb{Z}}_{\ge 0}^n$ such that ${\sum }_{i=1}^n{x}_i=S$, where $x_i\in {\mathbb{Z}}_{\ge 0}$ denotes the number of seats apportioned to group $G_i$.

Usually, we may want an apportionment satisfying certain requirements. The proportionality requirement, deemed essential for ensuring individual equality, has been a standard for over two centuries [26]. Individual equality means that the "weight of an individual", quantified by the proportion of seats to population (PSP), should be the same among all groups. This implies an “ideal” quota $x_i^{*}=\frac{p_{i}S}{{\sum }_{j=1}^n{p}_j}$ for all groups $G_i$. Since $x_i^{*}$ may not be a whole number, the (traditional) apportionment problem involves determining an integer apportionment that most closely reflects the proportionality (see, e.g., [6]).

While this may initially appear simple through an optimization approach, it involves several considerations. Firstly, there is no universally recognized inequality index to minimize—over 19 indices can be considered [27]. Different indices may yield distinct, typically non-unique, optimal apportionments, and it is beyond the capacity of the optimization theory to declare which index is the fairest in a societal context. In fact, the apportionment problem is often deemed political due to the lack of consensus on an inequality index [6].

Moreover, a black-box approach is typically not favored in practice, as non-experts tend to prefer understandable, simple mathematics. Consequently, instead of an apportionment derived through a complex, black-box optimization process, practical solutions necessitate a straightforward method that can be comprehended by non-experts. Therefore, various easy-to-understand methods have been proposed, such as Hamilton’s, Adam’s, Jefferson’s, Webster’s and Hill’s methods (see, e.g., [18,26]). All these methods aim to round fractional seats to integer seats, adhering to proportionality as closely as possible.

Nevertheless, none of these methods are without flaws. Hamilton’s method can lead to unexpected apportionments in certain circumstances, known as a paradox that arises from overlooking some basic principles of equality, as detailed in Chapters 7 and 8 of [16]. While the other methods mentioned above can prevent these known paradoxes ([16], p. 70), they offer no assurance that the assigned number of seats to a group $G_i$ remains closely (i.e., no more than one seat’s difference) aligned with $G_i$’s quota [16].

Given these challenges, it is natural to initially establish common sense principles or desirable properties as axioms to define “good” apportionment methods. This axiomatic approach is crucial for rigorous research. Balinski and Young have made significant contributions to this approach since 1982 [16], and it continues to be a dynamic area of study. More recently, Balinski and Ramirez argued that an apportionment method should meet four essential properties [20]. They further contended that Webster’s method outperforms other known methods, introducing additional properties to support their claim. While other proposed axiomatic systems [18,21] differ slightly, it is important to note that these axiomatic proposals also center around proportionality (see [18], p. 71).

This raises an intriguing question: why the emphasis on proportionality? For simplicity, let us consider apportionment based on population, not vote count, in the following. About 235 years ago, James Madison, often recognized as the “Father of the Constitution”, asserted that an appropriate number of seats should be subproportional to the population [1]. Madison even proposed an amendment to the US Constitution to enact subproportionality in the House’s size [25,28]. Subsequent researchers from diverse fields such as economics, social sciences, political sciences, mathematics and network sciences have found a large amount of theoretical and empirical evidence endorsing subproportionality [10,11,19,22,25,29,30,31,32,33,34,35]. All these findings suggest, explicitly or implicitly, a subproportional relationship for the apportionment problem (see [25] for a survey) that

$$x_i\propto p_i^{\gamma },i=1,2,\dots ,n,$$

(1)

for some constant $\gamma$ with $0<\gamma <1$. This clearly departs from the conventional assumption of proportionality, representing a fundamental shift in understanding.

Subproportionality is notable not only because it aligns with findings from both empirical and theoretical studies, but also due to a recent shift from proportionality toward subproportionality in several institutions. For instance, the International Federation of Operational Research Societies [36], the European Parliament (EP) [37] and the United Nations Parliament Campaign [38], have all embraced subproportionality. Specifically, the EP’s adoption of a form of subproportionality, known as degressive proportionality, marks a significant departure from traditional practice.

Transitioning from proportionality to subproportionality presents a significant challenge. Degressive proportionality often faces criticism due to perceived “inequality”, given the widespread belief that, to ensure equality, PSP should be the same among all groups—even for a legislature that adopts a subproportionality (see [14]). Due to this, existing apportionment methods for the EP lack a discussion on the individual equality.

However, the belief that the PSP can be used to gauge individual equality is flawed. This is because that the PSP inherently assumes proportionality; therefore, it cannot be used to examine individual equality in a legislature that employs a subproportional approach. In response to this problem, we introduced a novel formulation of the apportionment problem that posits an explicit function $f=f\left(p\right)$ on a standard number of seats for a group of size p [25]. The traditional formulation is a special case with a function $f\left(p\right)\propto p$. We call this generalization the Generalized Apportionment Problem (GAP) and proposed a population seat index (PSI)

$$w(x,p)=\frac{f^{-1}\left(x\right)}{p}$$

(2)

to evaluate the weight (i.e., contribution) of an individual in assigning x seats to a group of size p, where $f^{-1}$ denotes the inverse function of f [25]. It is important to note that, without an explicit specification of f, it is impossible to evaluate the weight of an individual, since we do not know how people affect the number of seats (recall that the conventional definition of PSP implicitly assumes $f\left(p\right)\propto p$). In the proposed theory, $f^{-1}\left(x\right)$ signifies the equivalent group size that merits x seats. Therefore, the dimensionless quantity PSI (2) indicates the effective weight of an individual within the group. Consequently, we may consider that a relative individual equality is achieved when $w(x_i,p_i)=w(x_j,p_j)$, i.e.,

$$\frac{f^{-1}\left(x_i\right)}{p_i}=\frac{f^{-1}\left(x_j\right)}{p_j}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} i,j.$$

(3)

Notably, ref. [25] explored a standard function $f\left(p\right)=a+b{p}^{\gamma }$, where $a,b,\gamma \ge 0$ are given constants. The traditional (proportionality-oriented) apportionment problem is a special case of the GAP with $a=0$, $b>0$ and $\gamma =1$. Similarly, subproportionality (1) is also a special case with $a=0$, $b>0$ and $0<\gamma <1$. For this function $f\left(p\right)=a+b{p}^{\gamma }$, since $f^{-1}\left(x\right)={\left(\frac{x-a}{b}\right)}^{\frac{1}{\gamma }}$ (assuming $b,\gamma >0$), (3) implies that an apportionment $\mathit{x}=(x_1,x_2,\dots ,x_n)$ achieves relative individual equality if and only if

$$\frac{x_i-a}{p_i^{\gamma }}=\frac{x_j-a}{p_j^{\gamma }}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} .$$

(4)

Even though (4) is derived with the assumption that $b,\gamma >0$, it also applies to the case of the US Senate by letting $a=2$, $b=0$ and arbitrary $\gamma$. An interesting fact about (4) is that the value of b does not matter. Nevertheless, b is useful in other situations [25].

However, ref. [25] solely focused on a theoretical, fractional definition. It did not discuss how to find an (integer) apportionment aligning the equality formula (4). In this paper, we initially review existing axioms and methods for proportionality in Section 2. We then broaden these axioms to accommodate subproportionality aligning (4) in Section 3. Based on the developed theory, we extend divisor methods to handle the GAP with a standard function $f\left(p\right)=a+b{p}^{\gamma }$, where $a,b,\gamma \ge 0$ are constants, and a is an integer. Section 4 discusses the neglect of individual equality in the existing apportionment methods for the EP and presents empirical studies pertinent to the EP, proposing a function $f\left(p\right)=4+0.000042p^{0.8}$ with the proposed divisor methods. Finally, we conclude in Section 5.

2. Literature Review

Typically, an existing apportionment method consists of two stages: scaling and rounding. The scaling stage involves calculating a group’s share by scaling the group’s size by a common factor, like $\frac{S}{\sum p_j}$ (hence, its value may vary depending on the instance). Since this calculated share is usually not a whole number, the rounding stage then derives an integer share from the fractional share. Some methods may alternate between these two stages multiple times until a feasible apportionment is achieved.

Apportionment methods fall into two categories. Divisor methods employ fixed rounding rules and dynamic scaling rules, while quota methods feature fixed scaling rules and dynamic rounding rules [18]. Well-known divisor algorithms include Adam’s, Jefferson’s, Webster’s and Hill’s methods, each differing by its rounding rule (we will review them in Section 3). Conversely, Hamilton’s method is a notable quota algorithm (for more, see [39]). Further discussions can be found in [15,18].

For studying the attributes of such methods, Balinski and Ramirez asserted that a substantial apportionment method should satisfy four essential properties [20]. Let $\mathit{p}=(p_1,p_2,\dots ,p_n)$ be the size vector of groups. For an apportionment method $\mathcal{A}$, let $\mathcal{A}(\mathit{p},S)$ denote the output of $\mathcal{A}$ for the size vector $\mathit{p}$ and S number of seats. Generally, $\mathcal{A}(\mathit{p},S)$ is a set of one or multiple apportionments. This is necessary as, for instance, when two groups of equal size need to share three seats, we want the method to offer two optimal apportionments ${\mathit{x}}_1^{*}=(1,2)$ and ${\mathit{x}}_2^{*}=(2,1)$ to facilitate a political decision. With the above notations in place, the four properties introduced by [20] can be expressed as follows.

Axioms by Balinski and Ramirez [20] for the proportionality:

• Anonymity: For any permutation $\sigma$ on $\mathit{p}$, $\mathcal{A}\left(\sigma \right(\mathit{p}),S)=\left\{\sigma \right(\mathit{x}\left)\right|\mathit{x}\in \mathcal{A}(\mathit{p},S)\}$, i.e., the output of $\mathcal{A}$ should not depend on the order of groups.
• Responsiveness: $\forall i,j$, $p_i>p_j\Rightarrow x_i\ge x_j$ for any $\mathit{x}=(x_1,x_2,\dots ,x_n)\in \mathcal{A}(\mathit{p},S)$.
• Scale-invariance: $\mathcal{A}(c\mathit{p},S)=\mathcal{A}(\mathit{p},S)$ for any constant $c>0$, where $c\mathit{p}=(c{p}_1,\dots ,c{p}_n)$.
• Exactness: $\mathcal{A}(\mathit{p},S)=\left\{\left({\overline{x}}_i\right)\right\}$ if ${\overline{x}}_i=\frac{p_{i}S}{{\sum }_{j=1}^n{p}_j}\in {\mathbb{Z}}_{\ge 0}$, $\forall i$.

Notice that $x_i=x_j$ is not required for $p_i=p_j$ as it may not be possible (e.g., when two groups of equal size share an odd number of seats). Nevertheless, we usually want $x_i$ to be close to $x_j$. For that purpose, the next axiom is introduced.

Balancedness axiom for the proportionality [18]:

• Balancedness: $\forall i,j$, $p_i=p_j\Rightarrow \left|x_i-x_j\right|\le 1$ for any $\mathit{x}=(x_1,x_2,\dots ,x_n)\in \mathcal{A}(\mathit{p},S)$.

Other axioms can be introduced as well [18,20,21]. Even though different terminologies are used, existing axiomatic systems all incorporate the four axioms put forth by Balinski and Ramirez. As such, this study expands on the four axioms proposed by [20] to also address subproportionality. We also take into account the Balancedness axiom, but the Fairness axiom from [21] is omitted due to a difficulty in proving it in our context.

In terms of subproportionality, as noted in the previous section, substantial evidence suggests that an optimal number of seats (representatives) for a group is often subproportional to the size of the group (for a comprehensive review, see [25]). Many methods have been suggested to implement degressive proportionality for the EP, including four methods that are based on simple mathematical principles: the Cambridge Compromise [40], Power Compromise [41], the parabolic method [42] and the 0.5-DPL Method [14], pp. 23–36. More intricate algorithms have also been proposed (see [43,44,45,46,47]).

However, none of the existing methods have been adopted by the EP. We believe that the reason for this is the lack of axiomatic analysis and guarantee on equality in the context of subproportionality. Particularly for the latter, degressive proportionality has often faced criticism for its perceived “inequality” since it requires the PSP of a more populous state to be smaller than that of a less populous state [37], which is seen as “unequal” under the traditional proportionality principle. However, individual equality in the context of subproportionality should not be assessed based on the PSP, as the PSP implicitly assumes proportionality. To address this issue, we proposed the index PSI (2) that can evaluate individual weight in subproportional settings [25]. The PSI is compatible with the traditional PSP with an $f\left(p\right)\propto p$ (as the case of (4) with $a=0$, $b>0$ and $\gamma =1$). To the best of our knowledge, our theory [25] is the first to provide a justification for subproportionality with respect to individual equality. This paper utilizes that theory to develop novel axioms and divisor methods for subproportionality with a focus on individual equality.

3. A Generalized Axiomatic System and a Framework of Generalized Divisor Methods

In this section, we generalize the aforementioned axioms to the GAP and propose a framework of generalized divisor methods.

Generalized Apportionment Problem (GAP):Given a number $S\in \mathbb{N}$ of seats, a vector $\mathit{p}=\left(p_i\right)\in {\mathbb{R}}_{\ge 0}^n$ of sizes of n groups and a bijective function $f=f\left(p\right):{\mathbb{R}}_{>0}\to {\mathbb{R}}_{\ge 0}$ on the standard number of seats for a group of size p, find an apportionment $\mathit{x}=\left(x_i\right)\in {\mathbb{Z}}_{\ge 0}^n$ such that it minimizes the difference between $\frac{f^{-1}\left(x_i\right)}{p_i}$ and $\frac{f^{-1}\left(x_j\right)}{p_j}$ for all $i,j$.

Analogous to the traditional apportionment problem (a special case of $f\left(p\right)\propto p$), this description of the GAP lacks a clear definition on the difference between $\frac{f^{-1}\left(x_i\right)}{p_i}$ and $\frac{f^{-1}\left(x_j\right)}{p_j}$. This flexibility is required by politics (before an inequality index can be officially decided). Therefore, it is useful to extend the aforementioned axioms and divisor methods.

3.1. A Generalized Axiomatic System for the GAP

Let $\mathcal{A}(\mathit{p},S,f)$ denote the output of an algorithm $\mathcal{A}$ for the GAP. The descriptions of the Anonymity, Responsiveness, Scale-invariance and Balancedness axioms can be generalized in a straightforward way. The Exactness axiom, however, needs a major revision. We observe that this axiom for the proportionality in fact requires an (integer) apportionment to meet the relative, individual equality, which depends on the standard function f (Section 1). Therefore, we extend and rename the Exactness axiom into a Relative-equality axiom.

Proposed generalized axioms for the GAP:

• Anonymity: For any permutation $\sigma$ on $\mathit{p}$, $\mathcal{A}\left(\sigma \right(\mathit{p}),S,f)=\left\{\sigma \right(\mathit{x}\left)\right|\mathit{x}\in \mathcal{A}(\mathit{p},S,f)\}$, i.e., the output of $\mathcal{A}$ should not depend on the order of groups.
• Responsiveness: $\forall i,j$, $p_i>p_j\Rightarrow x_i\ge x_j$ for any $\mathit{x}=(x_1,x_2,\dots ,x_n)\in \mathcal{A}(\mathit{p},S,f)$.
• Scale-invariance: $\mathcal{A}(c\mathit{p},S,f)=\mathcal{A}(\mathit{p},S,f)$, $\forall c>0$, where $c\mathit{p}=(c{p}_1,\dots ,c{p}_n)$.
• Balancedness: $\forall i,j$, $p_i=p_j\Rightarrow \left|x_i-x_j\right|\le 1$ for any $\mathit{x}=(x_1,x_2,\dots ,x_n)\in \mathcal{A}(\mathit{p},S,f)$.
• Relative-equality: $\mathcal{A}(\mathit{p},S,f)=\{\left(x_i^{*}\right)\in {\mathbb{Z}}_{\ge 0}^n\}$ if $\exists x_1^{*},x_2^{*},\cdots ,x_n^{*}\in {\mathbb{Z}}_{\ge 0}$ such that ${\sum }_{i=1}^n{x}_i^{*}=S$ and $\frac{f^{-1}\left(x_i^{*}\right)}{p_i}=\frac{f^{-1}\left(x_j^{*}\right)}{p_j}$, $\forall 1\le i,j\le n$. Or equivalently, if there exists a common factor w such that $f\left(w{p}_i\right)$ are whole numbers for all i and summing up to S, then those whole numbers are the apportionment. This w is the same $\frac{f^{-1}\left(x_i^{*}\right)}{p_i}$ for all i.

We note that $w(x_i^{*},p_i)=\frac{f^{-1}\left(x_i^{*}\right)}{p_i}$ in the Relative-equality axiom denotes the effective weight of an individual in allocating $x_i^{*}$ seats to size $p_i$. Our proposal is compatible with the traditional axioms with a function $f\left(p\right)\propto p$. This study considers a more general function $f\left(p\right)=a+b{p}^{\gamma }$ with some constants $a,b,\gamma$, where $a\ge 0$ denotes a pre-assigned, p-independent number of seats; $\gamma \ge 0$ denotes the scale factor of a standard number of representatives (seats) regarding to the group size p and $b\ge 0$ is the normalization factor. In practice, we may consider $p^{\gamma }$ as the total workload for p people and $\frac{1}{b}$ as the capacity of one representative. For this function, we can replace $\frac{f^{-1}\left(x_i^{*}\right)}{p_i}=\frac{f^{-1}\left(x_j^{*}\right)}{p_j}$ with Equation (4) in the Relative-equality axiom, with an advantage that it also covers the US Senate. We will discuss the case of the EP in the next section.

3.2. A Framework for Generalized Divisor Methods

For simplicity, we only consider the GAP with a function $f\left(p\right)=a+b{p}^{\gamma }$ for some constants $a\in {\mathbb{Z}}_{\ge 0}$ and $b,\gamma >0$. It is worth noting that a “recipe” to apply the existing theory with function $f\left(p\right)$ as the populations does not work in general, since existing divisor algorithms would divide $f\left(p\right)=a+b{p}^{\gamma }$ by a divisor $\lambda$, whereas the Relative-equality axiom asks to divide $b{p}^{\gamma }$ (without the constant term a). In fact, Equation (4) suggests a two-step divisor method to address this problem. In the first step, we allocate a seats to each group. And in the second step, we run a traditional divisor method with $S-na$ seats in total and entitlements $b{p}_i^{\gamma }$ for groups i. For completeness, we describe the detail in the following.

Following the notations in [20], let us call a function $d:{\mathbb{Z}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ a divisor function if $k\le d\left(k\right)\le k+1$ and $d\left(k\right)<d(k+1)$, $\forall k\in {\mathbb{Z}}_{\ge 0}$ (note that $d\left(k\right)$ can be a real number). For a divisor function d, a d-rounding rule is such a function ${[·]}_d:{\mathbb{R}}_{\ge 0}\to 2^{{\mathbb{Z}}_{\ge 0}}$ that, for any value $z\in {\mathbb{R}}_{\ge 0}$, calculates a set of one or two non-negative integers in the following way.

$${\left[z\right]}_d=\left\{\begin{array}{cc}\left\{0\right\},\hfill & 0\le z<d\left(0\right),\hfill \\ \{k+1\},\hfill & d\left(k\right)<z<d(k+1)forsomek,\hfill \\ \{k,k+1\},\hfill & otherwise(i.e.,z=d\left(k\right)forsomek\in {\mathbb{Z}}_{\ge 0}).\hfill \end{array}\right.$$

(5)

A framework for the two-step divisor methods is given in Algorithm 1. We remark that the value of b does not really matter, since $\lambda$ can be adjusted. Notice that existing theory on divisor methods (for proportionality) shows that a feasible $\lambda$ can be found inductively ([18], Section 4.5) or recursively ([16], Proposition 3.3). Moreover, any traditional divisor method for the $a=0$, $b>0$, $\gamma =1$ case can be used.

Table 1. Some well-known divisors methods with their divisor functions $d\left(k\right)$ (see, e.g., [16], p. 99).

	Adam’s	Jefferson’s	Webster’s	Hill’s
$d\left(k\right)$	k	$k+1$	$k+1/2$	$\sqrt{k(k+1)}$
i.e., rounding	up	down	off	at the geometric mean

lists some well-known ones.

Algorithm 1 The proposed framework for generalized divisor methods

Input: A number $S\in \mathbb{N}$ of seats, sizes $p_i$ for group i, $i=1,2,\dots ,n$, a standard function $f\left(p\right)=a+b{p}^{\gamma }$, where $a\in {\mathbb{Z}}_{\ge 0}$, $b>0$, and $\gamma >0$ are constants, $S>na$. Output: A set $\{\mathit{x}=(x_1,x_2,\dots ,x_n)\}\subseteq {\mathbb{Z}}_{\ge 0}^n$ of apportionment(s). 1: $\lambda =\frac{{\sum }_{i=1}^{n}b{p}_i^{\gamma }}{S-na}$ 2: $\mathcal{O}=\left\{(x_1,x_2,\dots ,x_n)|x_i\in {\left[\frac{b{p}_i^{\gamma }}{\lambda }\right]}_d,i=1,2,\dots ,n\right\}$ 3: while there is no $\mathit{x}=(x_1,x_2,\dots ,x_n)\in \mathcal{O}$ such that ${\sum }_{i=1}^n{x}_i=S-na$ do 4:    find a better $\lambda$ and let $\mathcal{O}=\left\{(x_1,x_2,\dots ,x_n)|x_i\in {\left[\frac{b{p}_i^{\gamma }}{\lambda }\right]}_d,i=1,2,\dots ,n\right\}$ 5: end while 6: output $\{(a+x_1,a+x_2,\dots ,a+x_n)\mid \mathit{x}=(x_1,x_2,\dots ,x_n)\in \mathcal{O}$, ${\sum }_{i=1}^n{x}_i=S-na$}.

It is worth mentioning that the House adopts a two-step method with $a=1$ and $\gamma =1$ [48]. As a result, Algorithm 1 serves as a direct generalization of this approach. Our primary contribution lies in justifying this procedure with respect to the individual equality, since the first step, which involves allocating an equal number of seats to all groups, is inconsistent with proportionality. With the equality theory presented in [25] (specifically, Equation (4)), we are able to address this discrepancy and provide a justification for such two-step methods.

Theorem 1.

Any generalized divisor method following Algorithm 1 satisfies the generalized Anonymity, Responsiveness, Scale-invariance, Balancedness and Relative-equality axioms.

Proof. 

The Anonymity axiom is evident from the property of the d-rounding rule. The Responsiveness and the Balancedness axioms hold since $p_i>p_j\iff p_i^{\gamma }>p_j^{\gamma }$ and $p_i=p_j\iff p_i^{\gamma }=p_j^{\gamma }$, respectively (note that $\gamma >0$ by assumption). The Scale-invariance axiom holds because $\frac{{\left(c{p}_i\right)}^{\gamma }}{\lambda }=\frac{p_i^{\gamma }}{{\lambda }^{\prime }}$ for a ${\lambda }^{\prime }=\lambda /c^{\gamma }$. For proving the Relative-equality axiom, we can use Equation (4) (notice that $b>0$). Therefore, we have proved the theorem. □

In the next section, we study existing methods for the EP with the Relative-equality axiom and use the proposed theory to study the apportionment issue for the EP.

4. An Empirical Study with the European Parliament (EP)

In this section, we delve into the apportionment issue specifically concerning the EP. The EU has established regulations specifying the minimum and maximum number of seats each Member State is entitled to in the EP, and it employs a subproportional approach known as the degressive proportionality [37]. According to Articles 1 and 2 of [37], the allocation of parliament seats is determined based on the populations of the Member States, as calculated by Eurostat. For this study, we utilize the population data from the 2011 census [49] to study the elections in 2020, and the population data from the 2021 census [50] to investigate the apportionment for the upcoming election in 2024.

The apportionment of seats in the EP is subject to a minimum number of 6 seats and a maximum number of 96 seats. Currently, there is no officially established standard function f or methodology for apportioning seats among the Member States [46]. Consequently, various methods have been proposed [40,41,42,51,52,53,54,55]. In Section 4.1, we provide an overview of some of these methods, particularly focusing on their lack of consideration for the Relative-equality axiom. In Section 4.2, we examine the apportionments for the EP in the 2020 election and analyze the upcoming 2024 election using the proposed generalized two-step methods, incorporating different rounding rules.

4.1. On Existing Apportionment Methods for the EP

The Cambridge Compromise is the most famous proposal for the apportionment issue of the EP. Let $x+Y=\{x+y\mid y\in Y\}$ and $min\{x,Y\}=\{min\{x,y\}\mid y\in Y\}$ for a real x and a set Y of reals. The Cambridge Compromise can be expressed as follows.

$${\mathcal{A}}_C(\mathit{p},S)=\left\{\left(x_i\right)\in {\mathbb{Z}}_{\ge 0}^n|\sum _{i=1}^n{x}_i=S,x_i\in min\left\{96,{\left[5+\frac{p_i}{\lambda }\right]}_d\right\},\forall i\right\},$$

where $\lambda >0$ is a parameter to be found and Adam’s is recommended for the d-rounding rule [40]. This method is equivalent to employing a standard function $f\left(p\right)=5+bp$ for some constant $b>0$ but with an upper bound 96. Due to this upper bounding, it does not provide a guarantee for the Relative-equality axiom in general.

The Power Compromise [41] is a variant of the Cambridge Compromise. It removes the explicit upper bounding by replacing $p_i$ with $p_i^E$ for some parameter E common to all groups such that $96\in 5+{\left[\frac{max{\left\{p_i\right\}}^E}{\lambda }\right]}_d$. If E is a constant, then this method belongs to our proposal. Unfortunately, the value of E depends on the instance. Thus, it is impossible to calculate the PSI, and there is no need to discuss the Relative-equality axiom.

The parabolic method [42] can be expressed as follows.

$${\mathcal{A}}_P(\mathit{p},S)=\left\{\left(x_i\right)\in {\mathbb{Z}}_{\ge 0}^n|\sum _{i=1}^n{x}_i=S,x_i\in {\left[6+\frac{90(x_i^{*}-m)}{M-m}+\lambda (x_i^{*}-m)(x_i^{*}-M)\right]}_d,\forall i\right\},$$

where $\lambda >0$ is a parameter to be found, $x_i^{*}=\frac{p_{i}S}{\sum p_j}$, $m=min\left\{x_i^{*}\right\}$, $M=max\left\{x_i^{*}\right\}$, and Webster’s and Jefferson’s are recommended for rounding. As $x_i^{*}$ depends on the population of other States, it is impossible to calculate the PSI for this method.

The 0.5-DPL Method ([14], pp. 23–36), introduces a quota $k\times \left(\frac{0.5p_i}{{\sum }_{j=1}^n{p}_j}+\frac{0.5p_i^{1/2}}{{\sum }_{j=1}^n{p}_j^{1/2}}\right)\times S$ for group i with Adam’s rounding rule, where k is a parameter to be found. It also introduces the following constraint ([14], p. 30) on the value of k in order to satisfy the minimum (six) and maximum (ninety-six) requirements of seats [37].

$$\sum _{i=1}^n\mathrm{mid}\left\{6,k\left(\frac{0.5p_i}{{\sum }_{j=1}^n{p}_j}+\frac{0.5p_i^{1/2}}{{\sum }_{j=1}^n{p}_j^{1/2}}\right),96\right\}=S.$$

Again, it is impossible to calculate the PSI for this method, and there is no need to discuss the Relative-equality axiom. There are many other methods for the apportionment of the EP [13,43,44,45,46,47,51,52,54,55]. None of these methods demonstrate a specific consideration for individual equality in their design (partly because the EP has not decided on a standard), except for proportionality, which cannot be applied to a subproportional context.

4.2. An Empirical Study Related to the EP

To study the apportionment for the EP in 2020 election and the upcoming 2024 election with a focus on individual equality, we employ Algorithm 1 with a function $f\left(p\right)=a+b{p}^{\gamma }$, where $a\in {\mathbb{Z}}_{\ge 0}$, $b,\gamma >0$ are some constants to be decided. Additionally, we utilize four d-rounding rules as shown in Table 1. These choices serve as the basis for our analysis.

To find constants a, b and $\gamma$, we considered the following settings. We try $a=0,1,2,3,4,5,6$ since the minimum number of seats is 6. To find appropriate values of $\gamma$ (which decides how subproportional the method is), for each a we applied a log-scale linear regression with the populations of the 2011 census [49] and the statutory number of seats in 2020 [56]. The results are shown in

Table 2. Log-scale linear regression results on $\gamma$ (population data: [49]; seats data: [56]). For the case of $a=6$, States with 6 seats were excluded.

a	0	1	2	3	4	5	6
$\gamma$	$0.56$	$0.60$	$0.64$	$0.70$	$0.78$	$0.91$	$1.00$
b	$2.7\times {10}^{-3}$	$1.4\times {10}^{-3}$	$6.2\times {10}^{-4}$	$2.3\times {10}^{-4}$	$5.7\times {10}^{-5}$	$6.2\times {10}^{-6}$	$1.2\times {10}^{-6}$
adj. R${}^2$	$0.96$	$0.96$	$0.97$	$0.98$	$0.99$	$0.99$	$0.97$

. It can be observed that, depending on the value of a, $0.56\le \gamma \le 1$. Since the results of regression are usually ad hoc (i.e., sensitive to the data), we decided to investigate only $\gamma =0.5,0.6,0.7,0.8,0.9,1.0$ to avoid over-fitting. Of course, further studies can explore a higher number of digits.

Notice that the value of $b>0$ can be arbitrary for a divisor method since the divisor $\lambda$ will be adjusted accordingly. Therefore, we use $b=1$ in empirical studies and provide a value based on normalization later. Hence, we investigate all the $7\times 1\times 6\times 4=168$ combinations of $a\in \{0,1,2,3,4,5,6\}$, $b\in \left\{1\right\}$, $\gamma \in \{0.5,0.6,0.7,0.8,0.9,1.0\}$ and four d-rounding rules (Table 1) to find good apportionments for the upcoming 2024 election based on the 2021 census [50]. To compare the degree of inequality, we use the PSI (see Equation (2)) to calculate the (effective) weight of individuals and employ an inequality index

$$\frac{{\mathrm{PSI}}_{max}}{{\mathrm{PSI}}_{min}}=\frac{max\left\{w(x_i,p_i)\right\}}{min\left\{w(x_j,p_j)\right\}}=\frac{max\left\{\frac{{(x_i-a)}^{1/\gamma }}{p_i}\right\}}{min\left\{\frac{{(x_j-a)}^{1/\gamma }}{p_j}\right\}}.$$

(6)

We remark that other indices can also be developed. We employ this one for the compatibility with the most commonly used index ${\mathrm{PSP}}_{max}/{\mathrm{PSP}}_{min}=max\{x_i/p_j\}/min\{x_j/p_j\}$ which is a special case of (6) with $a=0$ and $\gamma =1$. For comparison, the latter is also provided.

Firstly, let us assume that the membership, total seat number, minimum and maximum seats remain unchanged. Let $x_{min}$ and $x_{max}$ denote the minimum and the maximum allocated seats, respectively. We are interested in those apportionments that maximize $x_{max}-x_{min}$, i.e., most closely to the minimum and maximum seat requirements.

Table 3. Results on the apportionments for election 2024 of the EP with all the same requirements.

a	$\mathit{\gamma }$	Rounding Rule	${\mathit{x}}_{min}$	${\mathit{x}}_{max}$	$\frac{{\mathbf{PSI}}_{max}}{{\mathbf{PSI}}_{min}}$	$\frac{{\mathbf{PSP}}_{max}}{{\mathbf{PSP}}_{min}}$
6	0.9	Adam’s	7	96	1.88	11.68
4	0.8	Webster’s	6	95	1.77	10.12

presents the results. Only two combinations $(a,\gamma )=(6,0.9),(4,0.8)$ meet our interest, where the latter shows a lower inequality degree. Notice that the proposed PSI-based inequality index is less than 2, whereas the conventional PSP-based inequality index is more than 10.

Next, we study the case that the membership remains unchanged, any Member State with increased population does not lose its current seats, any Member State with decreased population does not gain more seats and the total number of seats is between the current 705 and the upper limit 750. The results are shown in

Table 4. Results on the apportionment for election 2024 of the EP where we assume that the membership remains unchanged, any Member State with increased population does not lose a seat, any Member State with decreased population does not gain more seats and the total number of seats is between 705 and 750. These are the combinations that satisfy all the requirements.

a	$\mathit{\gamma }$	Rounding Rule	S	${\mathit{x}}_{min}$	${\mathit{x}}_{max}$	$\frac{{\mathbf{PSI}}_{max}}{{\mathbf{PSI}}_{min}}$	$\frac{{\mathbf{PSP}}_{max}}{{\mathbf{PSP}}_{min}}$
4	0.8	Webster’s	709	6	96	1.77271	10.0132
4	0.8	Webster’s	710	6	96	1.77271	10.0132
4	0.8	Webster’s	711	6	96	1.53206	10.0132
4	0.8	Webster’s	712	6	96	1.53206	10.0132
4	0.8	Hill’s	711	6	96	1.53206	10.0132
4	0.8	Hill’s	712	6	96	1.53206	10.0132
4	0.8	Hill’s	713	6	96	1.53206	10.0132

. It shows that only the setting $a=4$ and $\gamma =0.8$ can provide feasible solutions that meet all our interests. We remark that if one of the aforementioned constraints is relaxed, there exist many more feasible solutions. Therefore, we focus on $a=4$ and $\gamma =0.8$ because it leads to feasible apportionments that satisfy more strict requirements, including the minimum and maximum numbers of seats, and with a small inequality index.

From the results in Table 3 and Table 4, we find that $a=4$ and $\gamma =0.8$ are reasonable values. For the value of b, we can set it to $b=0.000042$ by a normalization $b=\frac{S-na}{\sum p_i^{\gamma }}$ for $a=4$, $\gamma =0.8$, $S=705$ and the 2011 population $p_i$. We also confirmed that this value of b is almost the same for the 2021 population data. Therefore, we propose a standard function $f\left(p\right)=4+0.000042p^{0.8}$ for the apportionment of the EP with the proposed two-step divisor methods. The rounding rule, the minimum and maximum number of seats and the total number of seats can be flexibly decided according to further requirements by the EP.

We also compare the results with the optimal apportionments that minimizes the inequality index (6), which were found by solving Mixed Integer Linear Programming (MILP) problems with Gurobi Optimizer 10.0.1. See

Table 5. Apportionment comparison for election 2024 of the EP. Columns $p_{2011}$ and $p_{2021}$ are the population in the 2011 and 2021 census, respectively. Column s${}_{2020}$ are the current seats. For $S=705$, 709, we use function $f\left(p\right)=4+0.000042p^{0.8}$ with Webster’s rounding as shown in Table 3 and Table 4, respectively. Columns opt${}_{705}$ and opt${}_{709}$ show the apportionments that minimize the inequality index (6).

State	${\mathit{p}}_{2011}$	${\mathit{p}}_{2021}$	s${}_{2020}$	$\mathit{S}=705$	opt${}_{705}$	$\mathit{S}=709$	opt${}_{709}$
Malta	417,432	519,562	6	6	6	6	6
Luxembourg	512,353	643,941	6	6	6	6	6
Cyprus	840,407	921,033	6	6	7	6	7
Estonia	1,294,455	1,319,629	7	7	8	7	8
Latvia	2,070,371	1,893,223	8	8	9	8	9
Slovenia	2,050,189	2,108,912	8	9	9	9	9
Lithuania	3,043,429	2,810,761	11	10	10	10	10
Croatia	4,284,889	3,871,833	12	12	12	12	12
Ireland	4,574,888	5,105,761	13	14	14	14	14
Slovakia	5,397,036	5,449,270	14	14	14	14	15
Finland	5,375,276	5,533,179	14	14	15	14	15
Denmark	5,560,628	5,840,045	14	15	15	15	15
Bulgaria	7,364,570	6,519,789	17	16	16	16	16
Austria	8,401,940	8,964,889	19	19	19	19	19
Hungary	9,937,628	9,685,409	21	20	20	20	20
Portugal	10,562,178	10,343,066	21	21	21	21	21
Sweden	9,482,855	10,452,262	21	21	21	21	21
Greece	10,816,286	10,481,735	21	21	21	21	21
Czechia	10,436,560	10,524,167	21	21	21	21	22
Belgium	11,000,638	11,554,767	21	23	23	23	23
The Netherlands	16,655,799	17,475,443	29	30	30	30	30
Romania	20,121,641	19,053,815	33	32	32	32	32
Poland	38,044,565	37,019,327	52	51	51	52	51
Spain	46,815,910	47,400,798	59	62	61	62	61
Italy	59,433,744	59,030,133	76	73	72	74	72
France	64,933,400	65,471,806	79	79	77	80	78
Germany	80,219,695	83,239,650	96	95	95	96	96
Inequality index (6)	-	-	1.77	1.77	1.40	1.77	1.39

. It can be observed that the optimization approach, which minimizes the inequality, can be paradoxical. For both 705 and 709 seats, the seats assigned to Latvia increases while the population decreases, whereas the seats assigned to France decrease while the population increases. To avoid this paradox, it is possible to include more constraints in the MILP. However, this approach is limited, since there may exist paradoxes that are difficult to address. Nevertheless, the apportionments found by the proposed function $f\left(p\right)=4+0.000042p^{0.8}$ and the two-step divisor methods have the same inequality index compared to the current apportionment.

Table 6. Apportionment comparison based on the data of Table 3 of [14], p. 33. Column p is from Table 1 of [14], p. 29, and the total number of seats is 701. Our method uses the proposed function $f\left(p\right)=4+0.000042p^{0.8}$ with Adam’s rounding.

State	p	0.5-DPL	Power	Parabolic	Cambridge	Our Method
Malta	434,403	6	6	6	6	6
Luxembourg	576,249	6	7	7	6	6
Cyprus	848,319	6	7	7	7	7
Estonia	1,315,944	6	8	8	7	8
Latvia	1,968,957	7	9	9	8	9
Slovenia	2,064,188	7	9	9	8	9
Lithuania	2,888,558	9	10	10	9	10
Croatia	4,190,669	11	12	12	11	13
Ireland	4,664,156	12	13	12	11	13
Slovakia	5,407,910	13	14	13	12	14
Finland	5,465,408	14	14	13	12	14
Denmark	5,700,917	14	14	14	13	15
Bulgaria	7,153,784	16	16	15	15	17
Austria	8,711,500	18	18	17	17	19
Hungary	9,830,485	20	19	19	18	20
Sweden	9,998,000	20	20	19	18	21
Portugal	10,341,330	21	20	20	19	21
Czechia	10,445,783	21	20	20	19	21
Greece	10,793,526	21	21	20	19	22
Belgium	11,289,853	22	21	21	20	22
The Netherlands	17,235,349	29	28	28	27	29
Romania	19,759,968	32	31	31	31	32
Poland	37,967,209	53	51	53	54	51
Spain	46,438,422	62	60	62	65	60
Italy	61,302,519	77	76	77	83	73
France	66,661,621	82	81	83	90	78
Germany	82,064,489	96	96	96	96	91
Inequality index (6)	-	3.03	2.28	2.45	2.24	1.69

compares the results of the proposed method and the four methods introduced in Section 4.1. It only shows the apportionment obtained by our method with Adam’s rounding rule, since it achieves the minimum inequality index among the four rounding rules. It can be seen that our method performs the best among the five methods, though the maximum number of seats is not satisfied. We note that the result of the proposed method matches the optimal solution given by the MILP.

5. Conclusions

In this article, we proposed a novel generalization of existing axioms and divisor methods for proportionality to also encompass subproportionality. This is achieved by introducing a function $f=f\left(p\right)$ on the standard number of seats for a group of size p. Notably, the novel Relative-equality axiom reveals that the conventional Exactness axiom, which pertains to proportionality, is fundamentally about ensuring individual equality. To correctly assess the degree of inequality in subproportional settings, we have utilized a recent theory proposed by [25], which advocates for the use of the PSI (Population Seat Index) instead of the conventional PSP (proportion of seats to population), where the latter is valid only for proportionality. We have applied these principles to study the apportionment of the EP and have observed that existing methods for the EP lack both axiomatic analysis and focus on individual equality. Finally, through an extensive empirical study, we have found a reasonable recommendation, i.e., a function $f\left(p\right)=4+0.000042p^{0.8}$ and the two-step divisor methods, for the 2024 EP election that meets the requirements of the EP and aims to minimize the degree of individual inequality.

This study has some limitations. The inequality index (6) is compatible with the most commonly used index for proportionality but it includes only two States and assumes $x_j>a$ for all j. It is sensible to ${\mathrm{PSI}}_{min}$, and is not affected by the allotted size of many States. In the future, a comprehensive study on other indices may be interesting.