Domination and Balance of Degressively Proportional Allocations

Abstract

According to the principle of degressively proportional allocation, if agents are ordered so that the sequence of their entitlements is increasing, then the quotients of goods apportioned to agents and entitlements of agents generate a nonincreasing sequence. In addition, an agent with a smaller entitlement cannot receive more goods than an agent with a greater entitlement and therefore degressive proportionality can be considered as a compromise between equality and proportionality. This compromise should guarantee that the interests of both low-entitlements and high-entitlements agents are protected. This paper considers the problem of domination of one group of agents over another in a degressively proportional allocation of goods. As a result, we conclude that there is no degressively proportional allocation rule such that its domination is finite with respect to both equal and proportional sequences of weight. However, it is possible to control both types of domination described.

Introduction

One of the most important elements in economics is the division of common resources (natural or produced) or distribution of imposed burdens. A condition of the well-balanced development of societies, which is also stable in the longer perspective, is control of domination by any division’s agents over other participants. The opportunity to gain an unbounded dominance may lead to marginalization of some groups and hence to a significant stratification of society. A precondition to sustainability is therefore the specification of transparent and acceptable principles of sustainable distribution which eliminate the chances of gaining too much domination. Thus a key element of a fair division problem is the determination of the amount of domination that is possible to be gained by some groups over others.

Based on egalitarian concepts, two fundamental rules of allocation have been developed. The first one is an absolute equality, when each agent, irrespective of its entitlements, is allocated the same amount of goods. The other one is the equality of representation (proportionality), where the share of a distributed good for each agent equals the share of its entitlements in the total entitlements of all agents. These principles are extreme cases of the concept of degressive proportionality. A well-known example of applying this concept is enacted in the Treaty of Lisbon as a legal basis of how seats in the European Parliament are distributed (Lamassoure & Severin, 2007). A degressively proportional allocation means that deputies from more populated countries represent more citizens than those from less populated ones. The intention of the authors of this concept was to provide a solution which could prevent the supremacy of the most populated or a group of the most populated countries in the European Parliament. Even if disproportions in population numbers would have increased, the rule was supposed to guarantee a proper, minimal representation to all members of the community, therefore preventing the most populated countries from an outright advantage in voting.

On the other hand, too much distance between proportionality and equality, favoring the latter, results in a significantly smaller number of citizens represented by a Member of European Parliament from a less populated country than by an MEP from a more populated country. As a result (see Theil, 1969; Theil & Schrage, 1977), the chance of hearing a case from the citizen of a smaller country is much greater than in the case of a citizen from a larger country. The limitation of greater distance from proportionality is justified by less privilege for the citizens of smaller states in regard to citizens of larger states. In this way, degressive proportionality as a principle of allocation should be seen as a means to not only prevent the domination of agents or groups of agents with significantly greater entitlements over the remaining participants of the division, but also to avoid excessive equalization of goods allocated to agents apart from the actual diversification of their entitlements.

Without additional assumptions, the principle of degressive proportionality also authorizes allocations that do not preserve monotonicity, that is, more populated countries can receive fewer mandates than less populated countries. This property is not desired and is rationally unacceptable. Therefore, in 2007, the European Parliament resolution on the composition of the European Parliament imposed an additional condition to disallow such situations. Consequently, the advantages granted to less populated countries cannot deprive any agent of the amount guaranteed by equal division when all agents are allocated the same amount of a good.

A principle formulated in such a general manner leads to many solutions, thus the main stream of research concerning degressive proportionality focused on the task of finding a rule allowing the indication of a unique solution for a given problem of allocation. The problem was originally formulated as a classical apportionment problem (an exhaustive description of the apportionment problem may be found, inter alia, in (Balinski & Young, 2001; Pukelsheim, 2017; Young, 1995), therefore, it is no wonder that most of the propositions which can be found in the literature basically modify the methods known in this area (see Cegiełka & Łyko, 2014; Charvát, 2019; Dniestrzański, 2014; Haman, 2017; Martínez-Aroza & Ramírez-González, 2008; Pukelsheim, 2010; Ramírez-González, 2012; Słomczyński & Życzkowski, 2012). These solutions generally indicate the degressively proportional sequences of real values and the methods of finding their complete representation. The main idea is how to select the real degressively proportional sequence that is unique and well defined in the case of applying the proportional rule (as in the so-called quotas). The selection rules for integer representation are typically obtained from a wide offer of propositions considered in a classical apportionment problem, such as the well-known Cambridge Compromise (Grimmett et al., 2011).

As is easily seen, the rounding itself, while determining integer representation, can result in a violation of the condition of degressive proportionality. This means that the principle is observed on the level of the sequence in real terms, but the allocation obtained from it does not necessarily satisfy the principle. This stream of research is referred to as unrounded degressive proportionality (see e.g., Cegiełka et al., 2018). Another group of research deals with finding the degressively proportional integer sequence directly. This stream of research is called rounded degressive proportionality (RDP). Findings from the classical apportionment problem are used here to a small degree, and other methods are also applied. The authors make use of the topological (Cegiełka et al., 2019; Łyko & Rudek, 2013, 2017) and ordering (Cegiełka et al., 2017, 2019; Dniestrzański & Łyko, 2014) properties of the set of all solutions. Their definite strength is that the obtained solutions satisfy the principle of regressive proportionality. Their weakness however, is in the insufficient transparency of algorithms in generating a final solution that is key with a view toward universal applications. Similarly important is a long-lasting approach based on well-known algorithms such as the methods of D’Hondt, Saint-Laguë, or Hare–Niemeyer.

Making use of the principle of degressive proportionality is also justified by its positive impact on utilitarian welfare maximization, (e.g., Koriyama et al., 2013). On the other hand, there are no papers dealing with degressive proportionality from the viewpoint of the degree of domination of one agent or a group of agents. Considering the motivation of having legal power granted to the principle presented above, one has to assert that such a viewpoint is well-founded. Degressive proportionality is an intermediate solution between equality and proportionality. Equality obviously prevents domination of an agent or a coalition of agents whereas proportionality may lead, depending on entitlements, to an arbitrarily high dominance of one agent or a group of agents. Proportionality in contrast prevents an excessive equalization of the amounts of goods allocated to agents without taking account differences between the entitlements of agents.

The problem discussed in the article consists of indicating the amount of such domination in selected rules of degressively proportional allocation. The domination is identified with the maximal weighted ratio of the amount of goods allocated to any two subsets of agents. Weights in this case represent the shares of agents which can be considered (arbitrarily or by a group of agents) the fairest ones. A sequence of these weights is called a pattern. Due to the fact that degressive proportionality is a compromise between equality and proportionality, it is reasonable to examine domination in regarding an equal pattern (the weights of individual agents are equal) and a proportional pattern (the weights are proportional to entitlements).

The problem of balance and domination in relation to degressive proportional allocations has not been analyzed in the literature so far. Although the main idea of using such allocations is to protect agents with smaller entitlements from domination of agents with greater ones and vice versa, the potential scale of such domination has not been explored so far. This research gap is filled in the presented work. The main novelty of this work is the quantitative analysis of domination. Thanks to the formal, mathematical definition of this concept, it has been possible to study degressively proportional allocation rules from the perspective of maintaining a balance of benefits and losses resulting from the allocation, both for groups of agents with small and large entitlements. Thanks to this new approach, conclusions have been obtained that show to what extent a given rule can reconcile the conflicting interests of both groups and to what extent it is possible to control domination of one of them over the other.

The main results prove that domination of weak degressive proportional rule depends only on amounts of goods given to smallest and largest agent. Hence, protection against domination regarding an equal pattern as well as a proportional one is not possible. Nevertheless, some control of the domination effect is possible in regard to both patterns. In addition, the methods of such control are discussed in this article, both in a general case as well as for some actual allocation examples.

Degressive proportionality has been the subject of scientific research for only a dozen years, and many questions remain open. To the best of the authors’ knowledge, no studies analyzing the domination of degressively proportional allocations have been published so far. Therefore, the approach proposed by the authors is original and has a chance to initiate a research stream that can contribute to a more complete understanding of concept of domination in allocation problems.

Methods

Basic Definitions and Notation

In the article, we use the usual notions (compare e.g., Young, 1995). An allocation problem is a pair $(\mathit{p},h)$ , where $\mathit{p}=(p_1,p_2,\dots ,p_n)$ is a positive sequence of the entitlements of agents and $h>0$ is the amount of goods to be divided. An allocation is a nonnegative sequence $\mathit{s}=(s_1,s_2,\dots ,s_n)$ where $s_1+s_2+\dots +s_n=h$ . In the remainder we will assume that both sequence $\mathit{p}$ as well as its corresponding sequence $\mathit{s}$ are nondecreasing sequences. This conforms to a natural assumption that an agent with smaller entitlements may not be allocated more than an agent with greater entitlements. A set of indices of agents will be denoted by $N=\{1,2,\dots ,n\}$ . In order to precisely distinguish a sequence from a set and point on a plane, we denote a sequence by a letter in bold and denote a set and a point by a capital letter. Additionally, for the sake of simplicity, instead of “agent with greater entitlements” we shall write “greater agent,” and instead of “agent with smaller entitlements” we shall write “smaller agent.”

An allocation rule is a function $\mathrm{\phi }:(\bigcup _{n\in \mathbb{N}}{\mathbb{R}}_{+}^n\times {\mathbb{R}}_{+})\to \bigcup _{n\in \mathbb{N}}{\mathbb{R}}_{+}^n$ , that is, a function that assigns a unique solution $\mathit{s}=\mathrm{\phi }(\mathit{p},h)\in {\mathbb{R}}_{+}^n$ to every allocation problem $(\mathit{p},h)\in {\mathbb{R}}_{+}^n\times {\mathbb{R}}_{+}$ .

A rule of equal allocation and a proportional rule are the classical examples of allocation rules. In an equal allocation, which is a solution of a rule of equal allocation, every agent receives the same amount of goods. In a proportional allocation, which is a solution of a proportional rule, every agent receives the amount that is proportional to its entitlements. There is exactly one equal allocation and one proportional allocation for every allocation problem, that is, if $(\mathit{p},h)$ is an allocation problem, then a proportional allocation is allocation ${\mathit{s}}_{\mathrm{prop}}(\mathit{p},h)={\mathit{s}}_{\mathrm{prop}}=\frac{h}{p}\mathit{p}$ , where $p=p_1+p_2+p_n$ , and an equal allocation is allocation ${\mathit{s}}_{\mathrm{equal}}(\mathit{p},h)={\mathit{s}}_{\mathrm{equal}}=\frac{h}{n}(1,1,..,1)$ .

Problem Statement

Consider two disjoint non-empty subsets of $N$ , that is, $A,B\subset N,A\cap B=\varnothing$ . Notice, that we do not assume that sets $A$ and $B$ are a partition of $N$ . Let $\mathit{s}=(s_1,s_2,\dots ,s_n)$ be an allocation for $(\mathit{p},h)$ . Let $\mathit{w}=(w_1,w_2,\dots ,w_n)$ be a positive nondecreasing sequence, such that $w_1+w_2+\dots +w_n=1$ , that will be called a pattern or a normed pattern allocation. A pattern reflects a sense of fairness, that is, in this case allocation $h\mathit{w}$ will be defined as the fairest allocation with respect to $\mathit{w}$ in the set of all allocations. A domination of A over B with respect to sequence $\mathit{w}$ is

$$\mathrm{domi}{\mathrm{n}}_{\mathit{w}}(A,B)=\frac{\sum _{i\in A}{s}_i}{\sum _{i\in A}{w}_i}/\frac{\sum _{j\in B}{s}_j}{\sum _{j\in B}{w}_j}.$$

A domination of allocation $\mathit{s}$ with respect to pattern $\mathit{w}$ is a maximal domination for every pair of disjoint subsets $A$ , $B$ of $N$ , that is

$$\mathrm{domi}{\mathrm{n}}_{\mathit{w}}\left(\mathit{s}\right)=\underset{A,B}{max}\mathrm{domi}{\mathrm{n}}_{\mathit{w}}(A,B).$$

A domination of an allocation can be interpreted as a discrepancy of allocation $\mathit{s}$ with an allocation considered as the fairest with respect to pattern $\mathit{w}$ , that can be helpful for example to reduce the set of feasible allocations. If $\mathrm{domi}{n}_{\mathit{w}}(A,B)=\frac{\sum _{i\in A}{s}_i}{\sum _{i\in A}{w}_i}/\frac{\sum _{j\in B}{s}_j}{\sum _{j\in B}{w}_j}=k$ , to $\frac{\sum _{i\in A}{s}_i}{\sum _{i\in A}{w}_i}=k·\frac{\sum _{j\in B}{s}_j}{\sum _{j\in B}{w}_j}$ , a więc $\frac{\sum _{i\in A}{s}_i}{\sum _{j\in B}{s}_j}=k·\frac{\sum _{i\in A}{w}_i}{\sum _{j\in B}{w}_j}$ . So, domination of $A$ over $B$ determines how much ratio of the sum of goods allocated to agents from set $A$ to sum of goods allocated to agents from set $B$ differs from analogous ratio of the sum of goods allocated to both sets in the allocation considered to be the fairest, represented by $\mathit{w}$ . Value of $\mathrm{domi}{n}_{\mathit{w}}(A,B)>1$ can be interpreted in such a way that the agents from set $A$ , compared to the agents from set $B$ , received more than they should, and therefore the allocation is negatively unfair from the point of view of the agents from set B. $\mathrm{domi}{n}_{\mathit{w}}(A,B)<1$ can be interpreted similarly. The division will be fair with respect to sets $A$ and $B$ if $\mathrm{domi}{n}_{\mathit{w}}(A,B)=1$ . Of course, $\mathrm{domi}{n}_{\mathit{w}}(A,B)=\frac{1}{\mathrm{domi}{n}_{\mathit{w}}(B,A)}$ .

• Example 1. (Determining $\mathrm{domi}{n}_{\mathit{w}}(A,B)$ i $\mathrm{domi}{n}_{\mathit{w}}\left(\mathit{s}\right)$ ) Table 1 shows an example of determining dominance for the allocation of 60 goods between three agents and a equal pattern, taking $N=\{1,2,3\}$ , $w=(\frac{1}{3},\frac{1}{3},\frac{1}{3}),h=60$ and $\mathit{s}=(10,20,30)$ .

Table 1.

Determining Domination for Allocation $\mathit{s}=(10,20,30)$ and Equal Pattern.

$A$	$B$	$\sum _{i\in A}{s}_i$	$\sum _{i\in A}{w}_i$	$\frac{\sum _{i\in A}{s}_i}{\sum _{i\in A}{w}_i}$	$\sum _{j\in B}{s}_j$	$\sum _{j\in B}{w}_j$	$\frac{\sum _{j\in B}{s}_j}{\sum _{j\in B}{w}_j}$	$\mathrm{domi}{\mathrm{n}}_{\mathit{w}}(A,B)$
{1}	{2}	10	1/3	30	20	1/3	60	1/2
{1}	{3}	10	1/3	30	30	1/3	90	1/3
{1}	{2,3}	10	1/3	30	50	2/3	75	2/5
{2}	{1}	20	1/3	60	10	1/3	30	2
{2}	{3}	20	1/3	60	30	1/3	90	2/3
{2}	{1,3}	20	1/3	60	40	2/3	60	1
{3}	{1}	30	1/3	90	10	1/3	30	3
{3}	{2}	30	1/3	90	20	1/3	60	3/2
{3}	{1,2}	30	1/3	90	30	2/3	45	2
{1,2}	{3}	30	2/3	45	30	1/3	90	1/2
{1,3}	{2}	40	2/3	60	20	1/3	60	1
{2,3}	{1}	50	2/3	75	10	1/3	30	5/2

The calculations in Table 1 show that ${\mathrm{domin}}_{\mathit{w}}\left(\mathit{s}\right)=3$ (bolded row).

• Example 2. Another interesting example of the usage of domination is comparison of the dominance of each individual agent relative to the set of all other agents. Such domination may reflect the strength of a single agent compared to the others. An example of an allocation problem where such an approach may be applicable is the problem of allocation of slots in the FIFA World Cup tournament between the individual member confederations of FIFA resulting from the team classification method proposed by the Bureau of the FIFA Council (FIFA, 2017), which is to be in force from 2026. We have: FIFA={OFC, CONMEBOL, CONCACAF, AFC, CAF, UEFA} (where acronyms stand for confederations: OFC—Oceanian, CONMEBOL—South American, CONCACAF—North, Central American and Caribbean, AFC—Asian, CAF—African and UEFA—European), $h=48$ , $\mathit{s}=(1\frac{1}{3},6\frac{1}{3},6\frac{2}{3},8\frac{1}{3},9\frac{1}{3},16)$ . Can it be said that any single confederation is privileged over the rest? Of course, the answer to this question requires prior determination of the criterion of “fairness” expressed by a specific pattern. Let’s assume three different variants of criteria. The first is that the criterion of equality was followed, that is, that each federation should receive the same number of slots. In the second one, only the number of national teams associated with individual confederations eligible to participate in the World Cup was taken into account. The third variant is that the number of individual federations and the sporting level of the federation measured by the number of teams in the top 50 of the FIFA/Coca-Cola ranking were taken into account to the same extent (FIFA, 2023). This corresponds to the following patterns: (a) ${\mathit{w}}_{\mathrm{equal}}=\frac{1}{6}(1,1,1,1,1)$ , b) ${\mathit{w}}_{\mathrm{num}}=\frac{1}{207}(8,10,35,46,54,54)$ , c) ${\mathit{w}}_{2\mathrm{factors}}=\frac{1}{2}{\mathit{w}}_{\mathrm{numb}}+\frac{1}{2}(0,0.16,0.1,0.08,0.14,0.52)$ . The dominations for individual federations in relation to all others are presented in Table 2.

• Fact 1. For all positive real numbers $a_1,a_2,\dots ,a_n,b_1,b_2,\dots ,b_n$ we have $\frac{a_1}{b_1}\ge \frac{a_2}{b_2}\ge \dots \ge \frac{a_n}{b_n}\Rightarrow \frac{a_1}{b_1}\ge \frac{\sum _{i=1}^n{a}_i}{\sum _{i=1}^n{b}_i}\ge \frac{a_n}{b_n}.$

Proof. Condition $\frac{a_1}{b_1}\ge \frac{a_2}{b_2}\ge \dots \ge \frac{a_n}{b_n}$ implies $\frac{a_1}{b_1}\ge \frac{a_n}{b_n}$ , $\frac{a_2}{b_2}\ge \frac{a_n}{b_n}$ , …, $\frac{a_{n-1}}{b_{n-1}}\ge \frac{a_n}{b_n}$ and $\frac{a_n}{b_n}\ge \frac{a_n}{b_n}$ , equivalently $a_1{b}_n\ge a_n{b}_1$ , $a_2{b}_n\ge a_n{b}_2$ , …, $a_{n-1}{b}_n\ge a_n{b}_{n-1}$ and $a_n{b}_n\ge a_n{b}_n$ . Adding these inequalities we get $(a_1+a_2+\dots +a_n)b_n\ge a_n(b_1+b_2+\dots +b_n)$ , which is equivalent to $\frac{\sum _{i=1}^n{a}_i}{\sum _{i=1}^n{b}_i}\ge \frac{a_n}{b_n}$ . Appropriately rearranging other inequalities, resulting from this condition: $\frac{a_1}{b_1}\ge \frac{a_1}{b_1}$ , $\frac{a_1}{b_1}\ge \frac{a_2}{b_2}$ , …, $\frac{a_1}{b_1}\ge \frac{a_n}{b_n}$ and similarly rearranging as above, we get $\frac{a_1}{b_1}\ge \frac{\sum _{i=1}^n{a}_i}{\sum _{i=1}^n{b}_i}$ .

• Proposition 1. For allocation $\mathit{s}$ and pattern $\mathit{w}$ , $\mathrm{domi}{n}_{\mathit{w}}\left(\mathit{s}\right)=\underset{i\in N}{max}\frac{s_i}{w_i}/\underset{i\in N}{min}\frac{s_i}{w_i}\ge 1$ .

Proof. Every sequence of quotients $s_i/w_i$ can be arranged in nondecreasing order, hence by Fact 1 we have $\underset{k\in A}{min}\frac{s_k}{w_k}\le \frac{\sum _{i\in A}{s}_i}{\sum _{i\in A}{w}_i}\le \underset{j\in A}{max}\frac{s_j}{w_j}$ and $\underset{k\in B}{min}\frac{s_k}{w_k}\le \frac{\sum _{j\in B}{s}_i}{\sum _{j\in B}{w}_i}\le \underset{j\in B}{max}\frac{s_j}{w_j}$ for any $A,B\subset N$ .

• Proposition 2. For allocation $\mathit{s}$ and pattern $\mathit{w},\mathrm{domi}{n}_{\mathit{w}}\left(\mathit{s}\right)=1\iff \mathit{s}=h\mathit{w}$ .

Proof. It follows from Fact 1 that condition $\mathrm{domi}{n}_{\mathit{w}}\left(\mathit{s}\right)=1$ is satisfied if and only if $\underset{i\in N}{max}\frac{s_i}{w_i}=\underset{j\in N}{min}\frac{s_j}{w_j}$ , which holds if and only if $\frac{s_1}{w_1}=\frac{s_2}{w_2}=\dots =\frac{s_n}{w_n}=c$ , that is, if $\mathit{s}=c\mathit{w}$ , where $c$ can be an arbitrary nonnegative constant. Evidently a sequence $c\mathit{w}$ is an allocation if and only if $c=h$ , that is, if $\mathit{s}=h\mathit{w}$ .

As regards any allocation, one can discuss domination and a sense of dual balance, with respect to normed pattern allocation. Balance can be considered as an opposite to domination. Therefore, one can define an allocation balance coefficient as $\mathrm{balanc}{\mathrm{e}}_{\mathit{w}}\left(\mathit{s}\right)=\frac{1}{\mathrm{domi}{\mathrm{n}}_{\mathit{w}}\left(\mathit{s}\right)}.$ The range of this coefficient varies from 0 to 1. Value 1 means an ideal balance with respect to sequence $\mathit{w}$ . Values close to 0 indicate a significant imbalance, a disparity to the pattern.

Table 2.

Determining the Domination of One Confederation Over All Other Confederations Associated With FIFA With Different Patterns of Distribution of Places for the FIFA World Cup From 2026.

Confederation	Domination with respect to
	Equal pattern	Pattern ${\mathit{w}}_{\mathrm{num}}$	Pattern ${\mathit{w}}_{2\mathrm{factors}}$
OFC	0.142857	0.710714	1.45
CONMEBOL	0.76	2.9944	1.307369
CONCACAF	0.806452	0.792627	1.037528
AFC	1.05042	0.735294	1.180178
CAF	1.206897	0.683908	0.962899
UEFA	2.5	1.416667	0.780624

It follows from Proposition 2 that only an allocation proportional to normed pattern allocation has domination equal to 1. This situation can be interpreted as complete conformity to a pattern and balance of allocation and thus has no privileges for any group of agents with respect to another one. In any other case one can always indicate two subsets of agents, with one dominating over the other one.

A significant property of domination and balance coefficient follows from Propositions 1 and 2. Given an allocation that is considered by all parties as the fairest one, we can reduce the set of feasible allocations to those for which value $\mathrm{domi}{\mathrm{n}}_{\mathit{w}}(\mathit{s})$ , and thereby, $\mathrm{balanc}{\mathrm{e}}_{\mathit{w}}(\mathit{s})$ , is closest to 1. This can be applied especially when the allocation mostly desired by a significant part of agents for some reason does not belong to the set of feasible allocations. In particular, a normed coefficient $\mathrm{balanc}{\mathrm{e}}_{\mathit{w}}(\mathit{s})$ can be useful in such comparisons. Both coefficients are easy to compute and have a simple interpretation, therefore they can be used by agents as tools of reducing the set of feasible allocations. It can be less complicated and thus easier to understand than in case of other more intricate indicators (see Charman, 2017; Cegiełka et al., 2021).

A normed pattern allocation can be defined for a given problem of allocation $(\mathit{p},h)$ or generally for all such problems, that is, for an allocation rule. The examples of such generally defined patterns are sequences with equal elements or proportional to respective elements of sequence $\mathit{p}$ . Hence, we consider patterns ${\mathit{w}}_{\mathrm{equal}}=\frac{1}{n}(1,1,..,1)$ (equal pattern) and ${\mathit{w}}_{\mathrm{prop}}=\mathit{p}/\sum _i{p}_i$ (proportional pattern). In such cases it is possible to define a domination of an allocation rule with respect to $\mathit{w}$ . Let $\mathrm{\phi }$ be an allocation rule such that $\mathrm{\phi }(\mathit{p},h)=\mathit{s}$ . Then a domination of a rule $\mathit{\phi }$ with respect to ${\mathit{w}}_{\mathrm{equal}}$ is called the value $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=\underset{(\mathit{p},h)}{sup}\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}(\mathit{s})$ . On the other hand, value $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\underset{(\mathit{p},h)}{sup}\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}(\mathit{s})$ is called a domination of rule $\mathrm{\phi }$ with respect to ${\mathit{w}}_{\mathrm{prop}}$ . One has to emphasize that supremum in the definition of domination of an allocation rule concerns the set of all possible sequences $\mathit{p}$ of dimension at least 2. Moreover, pattern ${\mathit{w}}_{\mathrm{prop}}$ depends on sequence $\mathit{p}$ , therefore taking the supremum with respect to $\mathit{p}$ one is taking the supremum with respect to ${\mathit{w}}_{\mathrm{prop}}$ .

Likewise, we can talk about the rule balance coefficient. It suffices to define $\mathrm{balanc}{\mathrm{e}}_{{\mathit{w}}_{\mathrm{equal}}}(\mathrm{\phi })$ and $\mathrm{balanc}{\mathrm{e}}_{{\mathit{w}}_{\mathrm{prop}}}(\mathrm{\phi })$ as $\mathrm{balanc}{\mathrm{e}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=\underset{(\mathit{p},h)}{inf}\mathrm{balanc}{\mathrm{e}}_{{\mathit{w}}_{\mathrm{equal}}}(\mathit{s})$ and respectively, $\mathrm{balanc}{\mathrm{e}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\underset{(\mathit{p},h)}{inf}\mathrm{balanc}{\mathrm{e}}_{{\mathit{w}}_{\mathrm{prop}}}(\mathit{s})$ . The rule balance coefficient that equals zero indicates that following rule φ can lead to an extreme imbalance with respect to the pattern. In another, dual approach, we can say that this rule represents an unrestricted domination. Given equal allocation rule $\mathrm{\phi }$ we have $\mathrm{balanc}{\mathrm{e}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=0$ , whereas for proportional allocation rule $\mathrm{\phi }$ we have $\mathrm{balanc}{\mathrm{e}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=0$ . Therefore, this coefficient can serve to analyze the rules of degressively proportional allocation as regards their balance with respect to the two extreme patterns determined by equal and proportional normed pattern allocations.

Since we are interested in the domination of degressively proportional allocation rules, let us define such. We say that $\mathit{s}$ is a degressively proportional allocation with respect to sequence $\mathit{p}$ if sequence $\mathit{s}/\mathit{p}=(\frac{s_1}{p_1},\frac{s_2}{p_2},\dots ,\frac{s_n}{p_n})$ is nonincreasing and rule $\mathrm{\phi }$ satisfies the condition of degressive proportionality if $\mathrm{\phi }(\mathit{p},h)$ is a degressively proportional allocation for all $(\mathit{p},h)$ . By DP we shall denote, depending on the context, degressively proportional allocations and rules which satisfy the condition of degressive proportionality.

Notice that, in contrast to equal and proportional allocations, there may exist more than one degressively proportional allocation $\mathit{s}$ for a given allocation problem $(\mathit{p},h)$ . Hence, many various allocation rules may exist which lead to DP allocations.

It follows from Proposition 1 that if $\mathit{s}$ is a degressively proportional allocation with respect to sequence $\mathit{w}$ , then coefficient $\mathrm{domi}{n}_{\mathit{w}}\left(\mathit{s}\right)$ depends exclusively on the amount of goods and on the ratio of entitlements of the greatest agent to entitlements of the smallest agent. Corollaries 1 and 2 describe these relationships in detail.

• Corollary 1. If allocation $\mathit{s}$ is degressively proportional with respect to pattern $\mathit{w}$ , then $\mathrm{domi}{n}_{\mathit{w}}\left(\mathit{s}\right)=\frac{s_1}{w_1}/\frac{s_n}{w_n}$ .

Proof. The condition of degressive proportionality of $\mathit{s}$ with respect to $\mathit{w}$ indicates that $\frac{s_1}{w_1}\ge \frac{s_2}{w_2}\ge \dots \ge \frac{s_n}{w_n}$ , hence $\underset{i\in N}{max}\frac{s_i}{w_i}=\frac{s_1}{w_1}$ and $\underset{j\in N}{min}\frac{s_j}{w_j}=\frac{s_n}{w_n}$ . Applying Proposition 1 proves the thesis.

Let us note that merely assuming monotonicity of allocation $\mathit{s}$ with respect to sequence $\mathit{w}$ is not sufficient for Corollary 1 to hold. For example, if $\mathit{s}=(50,52,60)$ , and $\mathit{w}=(0.28,0.35,0.37)$ then $\mathit{s}/\mathit{w}\approx (179,149,162)$ , thus indicating that $\mathrm{domi}{n}_{\mathit{w}}\left(\mathit{s}\right)=\frac{s_1}{w_1}/\frac{s_2}{w_2}\ne \frac{s_1}{w_1}\frac{s_3}{w_3}$ .

• Corollary 2. If allocation $\mathit{s}$ is degressively proportional with respect to sequence $\mathit{p}$ , then

• (b) $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathit{s}\right)=\frac{s_1}{s_n}·\frac{p_n}{p_1}$ ;

• (c) $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathit{s}\right)·\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathit{s}\right)=\frac{p_n}{p_1}$ .

Proof. Since allocation $\mathit{s}$ is degressively proportional with respect to $\mathit{p}$ , there is (1) $s_1\le s_2\le \dots \le s_n$ and (2) $\frac{s_1}{p_1}\ge \frac{s_2}{p_2}\ge \dots \ge \frac{s_n}{p_n}$ . In view of ${\mathit{w}}_{\mathrm{equal}}=(\frac{1}{n},\frac{1}{n},..,\frac{1}{n})$ and by condition (1) as well as Proposition 1 we get $\underset{i\in N}{max}\frac{s_i}{1/n}/\underset{j\in N}{min}\frac{s_j}{1/n}=\frac{s_n}{s_1}$ , which proves (a). By condition (2), the form of proportional sequence and by Proposition 1 we get $\underset{i\in N}{max}\frac{s_i}{p_i/p}/\underset{j\in N}{min}\frac{s_j}{p_j/p}=\frac{s_1}{p_1/p}/\frac{s_n}{p_n/p}=\frac{s_1}{s_n}·\frac{p_n}{p_1}$ . Inequality $\frac{s_1}{s_n}·\frac{p_n}{p_1}\ge 1$ is equivalent to inequality $\frac{s_1}{p_1}\ge \frac{s_n}{p_n}$ , which is held by property (2). Therefore (b) is proved. Equality (c) simply results from (a) and (b).

It turns out that there is no degressively proportional allocation rule for which both domination with respect to equal allocation as well as to proportional allocation would be finite.

• Proposition 3. For all degressively proportional allocation rules $\mathrm{\phi }$ holds $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=\infty$ or $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\infty$ .

Proof. By Corollary (2c), the properties of supremum and since $\underset{\mathit{p}}{sup}\frac{p_n}{p_1}=\infty$ we obtain $\underset{\mathit{p}}{sup}\frac{p_n}{p_1}=\underset{(\mathit{p},h)}{sup}(\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)·\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right))=\infty \iff \underset{(\mathit{p},h)}{sup}\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=\infty$ or $\underset{(\mathit{p},h)}{sup}\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\infty$ .

It follows from Proposition 3 that for rules satisfying the condition of DP it is not possible to restrict domination at the same time with respect to equal allocation and to proportional allocation. For example, in the case of seat allocation rules in the European Parliament, if a given rule ensures that the quotient of the number of seats allocated to the greatest and the smallest state does not exceed a certain positive number, then an infinite increase in the quotient of populations in the greatest and the smallest state should result in an infinite increase in the quotient of the number of citizens represented by one MEP from the smallest country and the greatest country, and vice versa.

In order to avoid the privileged treatment of any agent during the phase of selecting an allocation rule, one should only consider rules with both dominations comparable, that is, infinite. This is for example the property of allocation rules which apply power functions $\mathrm{\phi }(\mathit{p},h)=\mathit{s}=\frac{h}{\sum _{i=1}^n{p}_i^{\beta }}(p_1^{\beta },p_2^{\beta },\dots ,p_n^{\beta })$ , where $\beta \in (0,1)$ (see e.g., Cegiełka et al., 2021; Słomczyński & Życzkowski, 2012). Indeed, by Corollary 2 we have $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=\underset{(\mathit{p},h)}{sup}\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\underset{\mathit{p}}{sup}\frac{p_n^{\beta }}{p_1^{\beta }}=\underset{\mathit{p}}{sup}{\left(\frac{p_n}{p_1}\right)}^{\beta }=\infty$ and $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\underset{(\mathit{p},h)}{sup}\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\underset{\mathit{p}}{sup}\frac{p_1^{\beta }}{p_n^{\beta }}·\frac{p_n}{p_1}=\underset{\mathit{p}}{sup}{\left(\frac{p_n}{p_1}\right)}^{1-\beta }=\infty$ .

Nevertheless, even with such restrictions, we still deal with some privilege. Taking the quotient of both dominations considered, we obtain

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\underset{(\mathit{p},h)}{sup}\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)}{\underset{(\mathit{p},h)}{sup}\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)}=\frac{\underset{\mathit{p}}{sup}{\left(\frac{p_n}{p_1}\right)}^{\beta }}{\underset{\mathit{p}}{sup}{\left(\frac{p_n}{p_1}\right)}^{1-\beta }}=\underset{\mathit{p}}{sup}{\left(\frac{p_n}{p_1}\right)}^{2\beta -1}}\\ \multicolumn{1}{c}{=\{\begin{array}{c}0,\beta \in (0,\frac{1}{2})\hfill \\ \multicolumn{1}{c}{1,\beta =\frac{1}{2}}\\ \multicolumn{1}{c}{\infty ,\beta \in (\frac{1}{2},1)}\end{array}}\end{array}$$

This means, however, that for a power rule allocation with an exponent unequal to 1/2, both dominations are infinite, but the greater the quotient of agents’ entitlements, the sooner some agents may gain over others following this solution. As a consequence, an unlimited and uneven increase of agents’ entitlements might influence all agents participating in the division to disapprove the rule with the properties described above. Free from such potential inconvenience is the rule which satisfies the following condition for any $(\mathit{p},h)$

$$\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\kappa \mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right),$$

where $\kappa >0$ .

Coefficient $\kappa >1$ , in the case of allocating seats in the European Parliament can be interpreted as a kind of compensation for the smallest states. If we denote quotient $\frac{\mathrm{\phi }(\mathit{p},h)\left(p_n\right)}{\mathrm{\phi }(\mathit{p},h)\left(p_1\right)}$ by $b$ , then even if the greatest state would be allocated $b$ times more seats, but a citizen of the smallest country will exert $b\kappa$ times stronger influence on the composition of the European Parliament than a citizen of the largest country. In turn, coefficient $\kappa <1$ can be interpreted as compensation for the greatest countries. If we denote quotient $\frac{\mathrm{\phi }(\mathit{p},h)\left(p_1\right)/p_1}{\mathrm{\phi }(\mathit{p},h)\left(p_n\right)/p_n}$ by $c$ , then a citizen of the smallest country will exert $c$ times stronger influence on the composition of the European Parliament than a citizen of the greatest state, but instead the greatest country will get $c/\kappa$ times more seats. Owing to that, the rules of this type can be more easily accepted by all agents participating in the division. Moreover, it turns out that one can correctly calculate domination with respect to both considered patterns.

• Proposition 4. If degressively proportional allocation rule $\mathrm{\phi }$ satisfies condition $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\kappa \mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)$ then $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\sqrt{\kappa \frac{p_n}{p_1}}$ and $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\sqrt{\frac{1}{\kappa }\frac{p_n}{p_1}}$ .

• Proof. By Corollary 2 (c) we have $\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\frac{p_n}{p_1}/\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)$ , which proves the thesis after substitution the condition of assumption and simple calculations.

From a practical viewpoint the most interesting rules are, of course, those which satisfy the condition of compensation with $\kappa =1$ . In this case the advantages gained by the greatest states are identical to the benefits of citizens from the smallest countries. Moreover, it can be easily proved that such rules, for each $(\mathit{p},h)$ , minimize the sum of dominations $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)+\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)$ . We call such rules balanced in the sense of domination. One has to note that the only consistent rule (on consistent rules see for example (Chun, 1999; Thomson, 2011, 2015) that is balanced in the sense of domination is the square root rule, that is, the above-described power rule with exponent $\beta =1/2$ . One can therefore consider the rule of this type as the most fair in the context of domination. It is noteworthy that many other authors in various ways have also been arguing for regarding the square root method as the most fair. For example, Penrose employs an assumption of voters’ independent decisions (Penrose, 1946). Theil and Schrage minimize the average number of citizens represented by the parliament members from each party (Theil & Schrage, 1977). Dniestrzański examines the conditions to be satisfied by the derivative of the allocation function in degressively proportional allocation (Dniestrzanski, 2016).

The following section presents the examples of DP allocation rules for which one of the dominations with respect to ${\mathit{w}}_{\mathrm{equal}}$ or with respect to ${\mathit{w}}_{\mathrm{prop}}$ is finite. It has to be emphasized that the domination of any degressively proportional allocation with respect to a sequence $\mathit{w}$ depends exclusively on the extreme values of allocation and on this sequence. Owing to this, in the following discussion any sequence $(p_1,p_2,\dots ,p_n)$ is represented by sequence $(p_1,p_n)$ . Certainly, in this approach one sequence $(p_1,p_n)$ represents more than just one sequence $(p_1,p_2,\dots ,p_n)$ . Thus allocation problems with equal ratios of entitlements of the greatest and smallest agents are equated. Such an approach in a sense introduces a restriction of the extreme values of the so-called weak proportionality (see Theil, 1969), which indicates that the ratio of the amount of goods allocated to any two agents depends on the ratio of their entitlements.

Empirical Verification

Let $\mathrm{\phi }$ be a degressively proportional allocation rule, let $n,n^{*}\in \mathbb{N}\backslash \{1\}$ , $\mathit{p}=(p_1,p_2,\dots ,p_n)$ , ${\mathit{p}}^{*}=(p_1^{*},p_2^{*},\dots ,p_{n^{*}}^{*})$ be the sequences of entitlements and let $(\mathit{p},h)$ , $({\mathit{p}}^{*},h^{*})$ be the allocation problems. We say that rule $\mathrm{\phi }$ satisfies the condition of generalized weak proportionality (GWP) if

$$\frac{p_n}{p_1}=\frac{p_{n^{*}}^{*}}{p_1^{*}}\Rightarrow \frac{\mathrm{\phi }(\mathit{p},h)\left(p_n\right)}{\mathrm{\phi }(\mathit{p},h)\left(p_1\right)}=\frac{\mathrm{\phi }({\mathit{p}}^{*},h^{*})\left(p_{n^{*}}^{*}\right)}{\mathrm{\phi }({\mathit{p}}^{*},h^{*})\left(p_1^{*}\right)}.$$

Dominations of GWP rules with respect to ${\mathit{w}}_{\mathrm{equal}}$ and with respect to ${\mathit{w}}_{\mathrm{prop}}$ depend exclusively on quotient $a=\frac{p_n}{p_1}$ . A function $f$ such that $f\left(a\right)=f\left(\frac{p_n}{p_1}\right)=\frac{\mathrm{\phi }(\mathit{p},h)\left(p_n\right)}{\mathrm{\phi }(\mathit{p},h)\left(p_1\right)}=b$ is connected with each rule. Let us note that function $f$ reduces the set of feasible allocation rules to rules which stipulate that each agent may be allocated at most $f(a)$ times more than any other agent. It is easily checked, that each DP rule satisfying the condition of weak proportionality, that is, $\frac{\mathrm{\phi }(\mathit{p},h)\left(p_i\right)}{\mathrm{\phi }(\mathit{p},h)\left(p_j\right)}=g\left(\frac{p_i}{p_j}\right)$ for a certain multiplicative function $g$ , also satisfies the GWP condition.

Considering only rules satisfying the GWP condition we can write the dominations of the rule with respect to equal allocation and proportional allocation as $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=\underset{a\to \infty }{lim}f\left(a\right),$ and $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\underset{a\to \infty }{lim}\frac{a}{f\left(a\right)}$ . Moreover, one can easily prove that rules of this type satisfy condition $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\kappa \mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)$ if and only if $f\left(a\right)=\sqrt{\frac{a}{\kappa }}$ .

Let us consider sequence $\mathit{p}=(p_1,p_2,\dots ,p_n)$ and its representation $(p_1,p_n)$ . Hereafter we shall also write $p_n=a{p}_1$ , where $a\ge 1$ . The consequence of $\mathit{s}$ being DP with respect to $\mathit{p}$ is that if we write $s_n=b{s}_1$ then $1\le b\le a$ . Projection of the set of all degressively proportional sequences with respect to $\mathit{p}$ on plane $(s_1,s_n)$ can be naturally interpreted geometrically. Sequence $(s_1,s_n)$ is degressively proportional with respect to a sequence $(p_1,p_n)$ if and only if point $(s_1,s_n)$ lies in the region:

$$\mathrm{DP}\left(\mathit{p}\right)=\{(x,y)\in R^2:x\le y\le \mathrm{ax}\}.$$

This relationship is shown in Figure 1. The line denoted by $k_{\mathrm{prop}(\mathit{p})}$ has equation $y=\mathrm{ax}$ . Similarly, symbol $k_{\mathrm{equal}}$ denotes the line with equation $y=x$ .

Figure 1.

A projection of the set of degressively proportional allocations onto a plane $(s_1,s_n)$ .

Notice that for a fixed sequence $(p_1,p_n)$ , every nonzero sequence $(s_1,s_n)$ that is DP with respect to $(p_1,p_n)$ is a DP allocation for the problem $((p_1,p_n),s_1+s_n)$ . Special cases of DP allocations, that is, equal allocation and proportional allocation, can also be geometrically interpreted in a similarly natural way. For $(s_1,s_n)$ and $(p_1,p_n)$ the first sequence is a proportional allocation of $h=s_1+s_n$ goods between agents with entitlements $p_1,p_n$ if and only if point $(s_1,s_n)$ lies on line $k_{\mathrm{prop}}$ and an equal allocation of $h=s_1+s_n$ goods between agents with entitlements $p_1,p_n$ if and only if point $(s_1,s_n)$ lies on line $k_{\mathrm{equal}}$ .

In the following subsections we explain typical methods of goods allocation to the smallest and the greatest agent that can be regarded as methods determining boundary conditions of the allocation. Furthermore, we shall present approaches to control domination with respect to the equal pattern and proportional pattern in the context of each method.

Let ${\left\{{\mathrm{\phi }}_{\alpha }\right\}}_{\alpha \in X}$ be family of DP rules indexed by parameter $\alpha \in X$ (where $X$ is some fixed set). We know that rule ${\mathrm{\phi }}_{\alpha }(\mathit{p},h)$ is balanced for problem $(\mathit{p},h)$ , if it minimizes sum $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left({\mathrm{\phi }}_{\alpha }(\mathit{p},h)\right)+\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left({\mathrm{\phi }}_{\alpha }(\mathit{p},h)\right)$ . Based on Corollary 2, for each allocation s that results from a rule that satisfies the condition DP, $\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathit{s}\right)=\frac{s_n}{s_1}=b$ and $\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathit{s}\right)=\frac{s_1}{s_n}·\frac{p_n}{p_1}=\frac{1}{b}·a$ holds. Since DP rules satisfy condition $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }(\mathit{p},h)\right)=\kappa \mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }(\mathit{p},h)\right)$ if and only if $b=\sqrt{\frac{a}{\kappa }}$ , then all balanced rules from this family satisfy condition $\frac{\mathrm{\phi }(\mathit{p},h)\left(p_n\right)}{\mathrm{\phi }(\mathit{p},h)\left(p_1\right)}=b=\sqrt{\frac{a}{1}}=\sqrt{a}$ . Let $f_{\alpha }:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a function that assigns a value $\frac{{\mathrm{\phi }}_{\alpha }(\mathit{p},h)\left(p_n\right)}{{\mathrm{\phi }}_{\alpha }(\mathit{p},h)\left(p_1\right)}$ to each argument $a$ . From the above considerations it follows that finding the parameter $\alpha$ for which the rule ${\mathrm{\phi }}_{\alpha }\in {\left\{{\mathrm{\phi }}_{\alpha }\right\}}_{\alpha \in X}$ is balanced for the problem $(\mathit{p},h)$ requires solving the equation $f_{\alpha }\left(a\right)=\sqrt{a}$ . It should be noted that, although in some trivial cases its solution will be independent of $a$ (i.e., the appropriate rule will be balanced for each problem $(\mathit{p},h)$ ), in the case of more complex allocation rules the rule determined in this way will be balanced only for a certain class of problems $(\mathit{p},h)$ . The parameter $\alpha$ for which the rule ${\mathrm{\phi }}_{\alpha }\in {\left\{{\mathrm{\phi }}_{\alpha }\right\}}_{\alpha \in X}$ is balanced will be denoted by the symbol ${\alpha }_{\mathrm{balance}}$ . Parameterized families of allocation functions are used to solve fair distribution problems. In particular, in the problems of determining DP allocation, as mentioned, the family of power functions plays an important role, for which, according to the previous considerations, the generally defined ${\alpha }_{\mathrm{balance}}$ is equal to $\frac{1}{2}$ . The ${\alpha }_{\mathrm{balance}}$ coefficient therefore allows for a subjective assessment of the selected parameter used for a specific allocation function. Later in the article, we will determine the ${\alpha }_{\mathrm{balance}}$ values for selected non-power-law allocation rules.

Geometric Interpretation of DP Allocation

In this section the two geometric interpretations of DP allocations of goods among agents with the smallest and the greatest entitlements will be presented. Both use the idea of balance or the equal distance from equal and proportional allocation. In the first case one considers the points lying on bisectors of line segments representing the $\mathrm{DP}$ allocations of a given amount $h$ of goods. Therefore, by definition of angle bisector, these points are equidistant from the endpoints of given segments which represent equal allocation and proportional allocation. In the other case, symmetry is comprehended as the location on the bisector of an acute angle between the lines $k_{\mathrm{equal}}$ and $k_{\mathrm{prop}}$ , which represent equal and proportional allocations, respectively. According to the definition of angle bisector, it is a set of points equidistant from angle’s sides, that is, in this case from lines $k_{\mathrm{equal}}$ and $k_{\mathrm{prop}}$ .

Convex Combination of Equality and Proportionality

Let a sequence of entitlements $(p_1,p_n)=(p_1,a{p}_1),a\ge 1$ be given. Therefore, for a fixed $h=s_1+s_n$ we have: ${\mathit{s}}_{\mathrm{equal}}((p_1,a{p}_1),h)=(\frac{h}{2},\frac{h}{2})=\frac{h}{2}(1,1)$ , ${\mathit{s}}_{\mathrm{prop}}((p_1,a{p}_1),h)=(\frac{h}{1+a},\frac{\mathrm{ah}}{1+a})=\frac{h}{1+a}(1,a).$ Obviously, lies on straight line $y=x$ and ${\mathit{s}}_{\mathrm{prop}}((p_1,a{p}_1),h)$ lies on straight line $y=\mathrm{ax}.$ Let $A={\mathit{s}}_{\mathrm{equal}}((p_1,a{p}_1),h)$ and $B={\mathit{s}}_{\mathrm{prop}}((p_1,a{p}_1),h)$ . Then allocation $(s_1,s_n)=(s_1,b{s}_1)$ , such that $s_1+b{s}_1=h$ is DP if and only if it lies in a line segment with endpoints $A$ and $B$ . Each such allocation can therefore be written as a convex combination of allocations $A$ and $B$ :

$$C(a,\alpha )=(1-\alpha )A+\alpha B,\alpha \in [0,1],$$

thus,

$$C(a,\alpha )=\frac{h}{2a+1}(2\alpha +(1-\alpha )(a+1),2\alpha a+(1-\alpha )(a+1)).$$

Taking a particular value of the coefficient $\alpha \in [0,1]$ , a respective allocation rule is determined. For $\alpha =0$ we have an equal allocation, for $\alpha =1$ we have a proportional allocation, and for $\alpha =\frac{1}{2}$ we have an allocation which can be considered as equidistant both from equal allocation as well as from proportional allocation. It results from the elementary geometric properties that point $C$ is the intersection of the segment $\mathrm{AB}$ with its bisector (see Figure 2).

Figure 2.

Convex combination of equal and proportional allocations among agents with the greatest and the smallest entitlements.

The value of quotient $\frac{s_n}{s_1}=b$ of the amount of goods for the greatest agent and the amount of goods for the smallest agent is

$$f_{\alpha }\left(a\right)=\frac{a+1+\alpha (a-1)}{a+1-\alpha (a-1)}.$$

For equal and proportional allocations we have: $f_0(a)=1$ , $f_1(a)=a$ , respectively. The upper limit of this coefficient with respect to $a$ is

$$\underset{a\to \infty }{lim}{f}_{\alpha }\left(a\right)=\frac{1+\alpha }{1-\alpha }.$$

This constraint does not depend on the disparity of entitlements and is finite for an arbitrary parameter $\alpha \in [0,1)$ , irrespective of the degree of domination of the greater agent’s entitlements over the smaller one, which means that for every $\alpha \in [0,1)$ , $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)<\infty$ and, obviously, $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\infty$ . For example, $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)$ of the allocation which is the midpoint of the segment with endpoints $A$ and $B$ , that is, allocation $C(a,\frac{1}{2})=\frac{1}{2}(A+B)$ , is equal to 3.

In the case of the rule presented in this section the solution of this equation is ${\alpha }_{\mathrm{balance}}=\frac{a+1}{{(\sqrt{a}+1)}^2}$ .

Angle Bisection

Let a sequence of entitlements be given in the form $(p_1,a{p}_1),a\ge 1$ . Proportional allocations with respect to this sequence lie on the line: $y=\mathrm{ax}$ . Let $m_{\alpha }$ denote the line passing through the origin of the coordinate system and forming angle $\frac{\pi }{4}+\alpha \gamma$ , $\alpha \in [0,1]$ , with the $\mathrm{OX}$ axis. Evidently, the choice of coefficient $\alpha$ determines the allocation rule. Allocation $(s_1,b{s}_1)$ is given by point $D(a,\alpha )$ lying in the intersection of line $m_{\alpha }$ and segment $\mathrm{AB}$ (Figure 3). Its value depends clearly on parameters $\alpha$ and $\gamma$ , that is, indirectly on $a$ , because $\gamma$ depends on $a$ . Elementary geometric properties yield: $\frac{\mathrm{AD}}{\mathrm{OA}}=\tan (\alpha \gamma )$ and $\frac{\mathrm{AB}}{\mathrm{OA}}=\tan (\gamma )$ , that is, $\frac{\mathrm{AD}}{\mathrm{AB}}=\frac{\tan \left(\alpha \gamma \right)}{\tan \left(\gamma \right)}$ , and therefore fixing a coefficient $\alpha$ does not determine the proportion of division of segment $\mathrm{AB}$ (see point 3.1). These proportions change depending on $\gamma$ and hence on $a$ , that is, depending on the ratio of agents’ entitlements. For $\alpha =0$ the allocation is equal, while for $\alpha =1$ the allocation is proportional. Line $m_{1/2}$ is a bisector of angle $\gamma$ . Consequently, one may regard allocation $D(a,1/2)$ , in a sense, as a symmetrization of the allocation problem. A bisector divides equally the angle between equal allocation and proportional allocation, therefore it is another approach to interpret the location between equality and proportionality. The definition of a bisector implies that it is a set of points equidistant from the angle’s sides, that is, in this case from lines $y=x$ and $y=\mathrm{ax}.$

Figure 3.

Division of angle between equal and proportional allocations among agents with the greatest and the smallest entitlements.

Quotient $\frac{s_n}{s_1}=b$ of the amount of goods for the greater agent to the amount of goods for the smaller agent can be expressed by the formula:

$$f_{\alpha }(a)=\tan (\frac{\pi }{4}+\alpha (\arctan \left(a\right)-\frac{\pi }{4})).$$

It is easily seen that when $a\to \infty$ , then $\gamma \to \frac{\pi }{4}$ and thus $f_{\alpha }\left(a\right)\to \tan \left(\frac{(1+\alpha )\pi }{4}\right)$ . Likewise in point 4.1.1 this value does not depend on disparity of entitlements, therefore irrespective of how much the entitlements of the greater agent dominate over the entitlements of the smaller agent, the quotient of the amount of their goods has got an upper limit, which depends on parameter $\alpha$ . This means that for every $\alpha \in [0,1)$ , $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)<\infty$ and, obviously, $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\infty$ . Now it holds: $\underset{a\to \infty }{lim}{f}_{1/2}(a)=\tan \left(\frac{3}{8}\pi \right)=1+\sqrt{2}$ . As a result, aside from the domination of entitlements of the greater agent over the smaller agent the quotient of goods allocated to them in this manner will always be limited by $1+\sqrt{2}$ .

It is also worth noticing that $f_{1/2}\left(3\right)=(1+\sqrt{5})/2$ , that is, when the entitlements of the greater agent are threefold greater than the entitlements of the smaller agent, then the quotient of the amounts of goods they are allocated is expressed as the golden number. What’s more, ${\alpha }_{\mathrm{balance}}=\frac{4\mathrm{arccot}\left(\sqrt{a}\right)-\pi }{\pi -4\arctan (a)}$ .

Index Approach to Balance of DP Allocation

Measures of inequality or disproportionality play a significant role in welfare economics (see e.g., Cowell, 2011). They help to measure income inequality and wealth inequality as well as inequality of goods distribution. In the latter case they indicate the deviation of a given allocation from equal or proportional allocation. These indexes have various forms and different properties. They can be normed or may take arbitrary values.

The so-called Pigou-Dalton transfer principle (Dalton, 1920; Pigou, 1912) is one of the frequently assumed properties of inequality measures. This inequality index is said to satisfy this condition if each transfer of a good from an agent with smaller entitlements to an agent with greater entitlements does not reduce the value of the index.

This property is significant in the analysis of DP allocations. In the case of two agents participating in a division, a measure satisfying the Pigou-Dalton transfer principle assumes the minimum value for equal allocation, because any other DP allocation appears from equal allocation by means of transferring a good to an agent with greater entitlements. Similarly, the maximum value is reached by proportional allocation, because any other DP allocation occurs from proportional allocation by transferring a good to an agent with smaller entitlements. As a result, one can easily regard symmetric allocation, in the sense of a given inequality measure, as a mean value, for instance the arithmetic mean of equal and proportional allocations. Many indexes are equal to zero for an equal allocation, therefore an allocation whose value of inequality index is equal to half its value computed for a proportional allocation can be regarded as a symmetric allocation.

Gini Index

The Gini index (Gini, 1921) is one of the best known inequality measures in the literature dealing with inequality measurement. For a two-element sequence $(p_1,a{p}_1)$ , where $a\ge 1$ , this index can be expressed in the form: $\mathrm{Gini}((p_1,a{p}_1),)=\frac{a{p}_1-p_1}{2(p_1+a{p}_1)}=\frac{a-1}{2(a+1)}$ . In the case of equal allocation $\mathrm{Gini}({\mathit{s}}_{\mathrm{equal}}(p_1,a{p}_1))=0$ , and because the Gini index has the Pigou-Dalton property, it suffices to solve equation $G(s_1,a{s}_1)=\alpha \mathrm{Gini}({\mathit{s}}_{\mathrm{prop}}(p_1,a{p}_1))$ , where $\mathrm{Gini}({\mathit{s}}_{\mathrm{prop}}(p_1,a{p}_1))=\mathrm{Gini}((p_1,a{p}_1))$ in order to analyze the problem of domination. It is reduced to finding parameter $b$ from equation $\frac{b-1}{2(b+1)}=\alpha \frac{a-1}{2(a+1)}$ . A simple reworking yields $b=\frac{a+1+\alpha (a-1)}{a+1-\alpha (a-1)}$ , which means that the quotient $\frac{s_n}{s_1}$ of goods for the greater agent to goods for the smaller agent equals

$$f_{\alpha }(a)=\frac{a+1+\alpha (a-1)}{a+1-\alpha (a-1)}.$$

Therefore, the solution is the same as in the case of a convex combination of equality and proportionality. As a result, all properties, including limited ones, are identical to those obtained in section “Convex Combination of Equality and Proportionality.” In particular, $\underset{a\to \infty }{lim}{f}_{\alpha }(a)=\frac{1+\alpha }{1-\alpha }$ , and ${\alpha }_{\mathrm{balance}}$ for the balanced rule equals $\frac{a+1}{{(\sqrt{a}+1)}^2}$ .

Atkinson Index

We will focus on one of the most important formula of the Atkinson index, that is, the complement to $1$ of the ratio of geometric mean and arithmetic mean (Atkinson, 1970; Schneckenburger et al., 2017). For a given two-element sequence $(p_1,a{p}_1)$ , where $a\ge 1$ , it has the form

$$\mathrm{Atk}((p_1,a{p}_1),)=1-\frac{\sqrt{{\mathrm{ap}}_1^2}}{(p_1+a{p}_1)/2}=1-\frac{2\sqrt{a}}{a+1}.$$

In the case of equal allocation, $\mathrm{Atk}({\mathit{s}}_{\mathrm{equal}}(p_1,a{p}_1),)=0$ , and because the Atkinson index satisfies the Pigou-Dalton transfer principle, it suffices to solve the equation $\mathrm{Atk}\left(\mathit{s}\right)=\alpha \mathrm{Atk}({\mathit{s}}_{\mathrm{prop}}(p_1,a{p}_1),)$ , where $\mathrm{Atk}({\mathit{s}}_{\mathrm{prop}}(p_1,a{p}_1),)=\mathrm{Atk}((p_1,a{p}_1),)$ , similar to the case of the Gini index, in order to analyze the problem of domination. Hence it suffices to solve equation $1-\frac{2\sqrt{b}}{b+1}=\alpha (1-\frac{2\sqrt{a}}{a+1})$ in parameter $b$ , equivalent to equation $(1-\alpha +\frac{2\alpha \sqrt{a}}{1+a})b-2\sqrt{b}+1-\alpha +\frac{2\alpha \sqrt{a}}{1+a}=0$ . Substituting $t=\sqrt{b}$ and $k=1-\alpha +\frac{2\alpha \sqrt{a}}{1+a}$ yields a quadratic equation in $t$ , with two solutions: $t_1=\frac{1-\sqrt{1-k^2}}{k}$ and $t_2=\frac{1+\sqrt{1-k^2}}{k}$ . As a result, and in view of the fact that $t_1<1$ , the quotient $\frac{s_n}{s_1}=b$ of amounts of goods for the greater and the smaller agent is equal to

$$f_{\alpha }(a)={\left(\frac{1+\sqrt{1-k^2}}{k}\right)}^2.$$

If $a\to \infty$ , then $k\to 1-\alpha$ , and thus $\underset{a\to \infty }{lim}{f}_{\alpha }(a)={\left(\frac{1+\sqrt{1-{(1-\alpha )}^2}}{1-\alpha }\right)}^2$ for every $\alpha \in [0,1)$ . That means that for every $\alpha \in [0,1)$ , $\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)<\infty$ and, obviously, $\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\infty$ .

Finding coefficient $a_{\mathrm{balance}}$ in this case involves solving equation ${((1+\sqrt{1-{(1-\alpha +\frac{2\alpha \sqrt{a}}{1+a})}^2})/(1-\alpha +\frac{2\alpha \sqrt{a}}{1+a}))}^2=\sqrt{a}$ . Due to a strong nonlinearity of this equation, one can only indicate numerical approximations of its solution for a particular problem $(\mathit{p},h)$ . For example, in the case of the European Parliament of the ninth term, 2019 to 2024 $a_{\mathrm{balance}}\approx 0.58126$ .

Under Proportionality

The above-presented rules have finite domination with respect to a proportional pattern. We shall now indicate a rule whose domination with respect to a proportional pattern is finite. Let a sequence of entitlements be given $(p_1,a{p}_1),a\ge 1$ . Let $\mathrm{\phi }((p_1,a{p}_1),,h)=(s_1,s_n)=\frac{h}{1+\alpha a}(1,\alpha a)$ , where $\alpha \in (\frac{1}{a},1)$ . . The rule $\mathrm{\phi }$ . is DP and $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)=\underset{a\to \infty }{lim}\frac{1}{\alpha a}a=\frac{1}{\alpha }<\infty$ . Obviously, by Proposition 3 we have $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=\infty$ . Moreover, ${\alpha }_{\mathrm{balance}}=\frac{1}{\sqrt{a}}$ .

Table 3 presents the summary of results obtained in section “Empirical Verification.”

Table 3.

Dominations With Respect to the Equal and Proportional Patterns and Coefficients $a_{\mathrm{balance}}$ .

$\mathrm{\phi }$	$f_{\alpha }(a)$	$\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)$	$\mathrm{domi}{\mathrm{n}}_{{\mathit{w}}_{\mathrm{prop}}}\left(\mathrm{\phi }\right)$	${\alpha }_{\mathrm{balance}}$
Conv, Gini	$\frac{a+1+\alpha (a-1)}{a+1-\alpha (a-1)}$	$\frac{1+\alpha }{1-\alpha }$	∞	$\frac{a+1}{{(\sqrt{a}+1)}^2}$
Angle	$\tan (\frac{\pi }{4}+\alpha (\arctan \left(a\right)-\frac{\pi }{4}))$	$\tan \left(\frac{(1+\alpha )\pi }{4}\right)$	$\infty$	$\frac{4\mathrm{arccot}\left(\sqrt{a}\right)-\pi }{\pi -4\arctan (a)}$
Atk	${\left(\frac{1+\sqrt{1-k^2}}{k}\right)}^2$ where $k=1-\alpha +\frac{2\alpha \sqrt{a}}{1+a}$	${\left(\frac{1+\sqrt{1-{(1-\alpha )}^2}}{1-\alpha }\right)}^2$	$\infty$	Only numerical solutions available
UProp	$\alpha a$	$\infty$	$\frac{1}{\alpha }$	$\frac{1}{\sqrt{a}}$

Conclusions

The principle of degressive proportionality in the context of apportionment problems emerged as an alternative to the principle of proportionality typically applied in such cases. Its introduction was primarily motivated by a desire to protect the interests of agents with small entitlements. In the case of European Parliament composition this principle was supposed to ensure an appropriate parliamentary representation of even the least populated member states in the European Union. Hence it was about excluding the excessive domination that is understood in this article to be the domination regarding an equal pattern allocation. This absence of domination was legally sanctioned in the Treaty of Lisbon by indicating the smallest and the greatest number of seats its member states are allowed, 6 and 96; in this particular case the domination was limited to 16.

If there are no boundary conditions one may consider the problem of domination in degressively proportional allocation, in the above-mentioned sense and more generally, with respect to any normed pattern allocation. In this context, there is also a proportional allocation in addition to an equal one among all normed pattern allocations. In particular, equal and proportional allocation represent two limits of degressively proportional allocation. Just like a small domination with respect to an equal pattern indicates the guaranteed protection of agents with small entitlements, a small domination with respect to a proportional pattern is supposed to guarantee the protection of interests of agents with greater entitlements against the excessive, absolute egalitarianism represented by an insufficiently diversified allocation.

The article basically concludes that no rule of degressively proportional allocation exists that would simultaneously protect against domination in both of the senses mentioned. This means that, by deciding to select the rule of finite domination with respect to a proportional pattern or finite domination with respect to an equal pattern, we undermine the protection of interests of agents with small entitlements or those with large entitlements. Consequently, fairer rules can be regarded as those whose domination is not finite with respect to both an equal as well to a proportional pattern, such as rules applying power functions which are often recommended as a convenient solution to an allocation problem in agreement with the principle of degressive proportionality (as in Dniestrzański, 2014; Martínez-Aroza & Ramírez-González, 2008; Słomczyński & Życzkowski, 2012). Among them, the fairest rules can be distinguished, that is, those whose ratio of these dominations is constant, for example rules based on a square root allocation function.

Also, the allocation rules of finite domination with respect to an equal pattern are important in practice (see Table 3). Such rules play a significant role in apportionment issues when analyzing the problem of a qualified majority, that is, in cases when decisions are not based on an ordinary or absolute majority of votes. In such cases the necessary majority is typically $2/3$ or $3/4$ . This is the equivalent of saying that the domination of a given rule with respect to an equal pattern is, for example, 2 or 3 respectively. In the case of a convex combination of equality and proportionality this means that $\alpha =1/3$ or $\alpha =1/2$ . Likewise, one may find $\alpha$ for an arbitrary qualified majority $r$ and a rule of allocation. It suffices to solve, if possible, the equation $\mathrm{domi}{n}_{{\mathit{w}}_{\mathrm{equal}}}\left(\mathrm{\phi }\right)=\frac{r}{1-r}$ with respect to $\alpha$ . This means that by an appropriate choice of parameter $\alpha$ one can develop methods which do not guarantee a fixed qualified majority even with an arbitrarily high domination of the sizes of the agents’ or their coalitions’ entitlements. Using a convex combination of equality and proportionality, and likewise the Gini index, one obtains relatively simple equations and intuitively interpreted results. In the remaining cases presented in this article, computations involve more complex equations and the interpretation of the results obtained is more difficult because of, for instance, the irrationality of solutions. Nevertheless, knowing even approximate values allows the control of an arbitrarily selected qualified majority.

Discussion

Domination and balance of an allocation rule depend only on minimal and maximal values of quotients $\frac{s_i}{w_i}$ . In the case of degressive proportional allocation, they are achieved for agents with the greatest and least entitlements. This naturally links our research with the area of determining the boundary conditions of degressively proportional allocations (Dniestrzański & Łyko, 2014; Łyko, 2012; Ramírez González et al., 2012; Słomczyński & Życzkowski, 2012) and that is why these issues will be further studied in this domain. As is also known, simply determining boundary conditions does not define the unambiguous allocation rule. There exist many rules of degressively proportional allocation of known, finite domination with respect to a given pattern. Thus a later, practical stream of research will deal with searching for allocations of fixed domination and the specific properties (such as consistency, homogeneity or monotonicity) which are desirable in order to solve a concrete problem.

Using the ${\alpha }_{\mathrm{balance}}$ coefficient defined in this article, it is also possible to objectively indicate one of many DP allocations defined using parameterized allocation rules. At this stage, this frees the considered problem from the subjective view of decision-makers, making it more transparent and acceptable.

The results presented in this article could also be used in other areas of scientific research. The domination of individual agents or groups of agents over others can be interpreted in terms of a power index. This allows for a new look at the issue of voting power, for example, in the problems of stakeholders or political groups in collegial bodies. The importance of the domination goes beyond the domain of electoral problems. An example could be applications in economics. The problem of the division of goods and burdens is analyzed in the literature in different ways. One of the theories dedicated to it is game theory. Problems of cooperation and competition are considered in mathematical economics. Among such issues is oligopoly and monopoly. The tool proposed in this work allows to easily detect and analyze potential advantages of some economic entities over others in terms of the existence of a monopoly or oligopoly, both on a single market and on many different markets at the same time. By determining the appropriate pattern for allocation of market shares between individual entities, corresponding to the absence of monopoly or oligopoly, it is possible, using the domination, to analyze market dominance in terms of the existence of an oligopoly or monopoly. It is known that an oligopoly (monopoly) takes place when there are several (one) entities dominating the others. The criterion for the existence of an oligopoly (monopoly) may be the existence of a group of entities (one entity) for which the coefficient of dominance over the set of all other entities (calculated in a similar way as in Example 2) is greater than a certain previously determined value greater than 1 or, in case of oligopoly, the existence of a group of entities for which the coefficient of dominance over the group of other entities exceeds fixed value.