Minimal and Maximal Representation of Degressively Proportional Allocation

Abstract

Boundary conditions are a significant element of a process developing the degressive proportionality principles. With regard to voting issues, this is equivalent to determining a minimum and a maximum representation, and a size of the whole assembly. In case of a practical problem, the choice of these numbers is evidently constrained. The political conditions as well as the necessity of ensuring efficient functioning of the elected body significantly restrict a set of all possible alternatives. The paper analyzes the feasibility of boundary conditions under a given minimum and maximum representation.

Keywords: Elections, fair division, European Parliament

1. Introduction

The Treaty of Lisbon (Laslier, 2012) introduced a principle of degressive proportionality of goods

and burdens as a legal norm for the first time. The principle was adopted as a rule of distributing

mandates to the European Parliament among the member states. The respective provision (The Treaty

of Lisbon, article 9A) reads as follows (Treaty, 2010): “

representatives of the Union’s citizens. They shall not exceed seven hundred and fifty in number, plus

the President. Representation of citizens shall be degressively proportional, with a minimum threshold

of six members per Member State. No Member State shall be allocated more than ninety-six seats”.

Additional explanations that interpret the notion of degressive proportionality can be found in the

resolution titled “Proposal to amend the Treaty provisions concerning the composition of the European

Parliament” (Lamassoure, & Severin, 2007). The two statements contained there, i.e. “the larger the

population of a country, the greater its entitlement to a large number of seats” and “the larger the

population of a country, the more inhabitants are represented by each of its Members of the European

Parliament”, allow to rigorously define this principle in the language of mathematics: a positive

sequence1,2, ..., is degressively proportional with respect to 0 <1≤2≤ ... ≤ if and only if

integers that denote the numbers of allocated mandates, whereas the terms of the sequence1,2, ...,denote the numbers of populations in respective member countries of the European Union.

An essential part of the quoted article 9A are so-called boundary conditions that define a minimum

and a maximum number of representatives from a given country, and a total size of an assembly. The

two former numbers are given by inequalities, however, due to a large number of possible solutions

(Łyko, & Rudek, 2013), they are adopted as indicated, especially because the resolution (Lamassoure,

& Severin, 2007) explicitly reads that “the minimum and maximum numbers set by the Treaty must be

fully utilised to ensure that the allocation of seats in the European Parliament reflects as closely as

possible the range of populations of the Member States”, thus furthermore confirming such

interpretation. As a result, the problem of allocating seats in the European Parliament can be

considered as a degressively proportional distribution problem subject to boundary conditions: = 6,

= 96 and = 751, where denotes the number of representatives from the least populous state of

the European Union, – the number of representatives from the most populous country, and – the

number of all members of the European Parliament (Dniestrzański, 2014; Dniestrzański, & Łyko 2014;

Serafini, 2012; Felgado-Marquez, Kaeding, & Palomares, 2013; Grimmett et al., 2011).

2.Boundary conditions of a degressively proportional distribution

Interestingly enough, the boundary conditions significantly influence both the likelihood of finding

a distribution as well as the number of feasible solutions, when the problem is not inconsistent (Łyko,

2013; Dniestrzański, & Łyko, 2014). Therefore, a feasibility analysis of specific boundary conditions

should precede any political discussions, and consequently, legal regulations. For that reason, an

answer to a question which triples of natural numbers (,,) can produce a system of boundary

conditions for a degressively proportional distribution becomes important. In such a case we often say

that a triple (,,) is not an inconsistent system of boundary conditions. Unfortunately, such

reasoning must not ignore the elements of the sequence1,2, ...,. One cannot expect just one universal answer. Feasibility or inconsistency of a given system of boundary conditions (,,) depends on the sequence1,2, ..., that determines the allocation.

It is easy to find such numbers1,2, ..., whose sole degressively proportional distribution is the one with =, so the elements of the sequence1,2, ..., are constant. An obvious trivial example is the sequence1 =2 = ... =. However, this is not the only case. More such sequences can be

condition, also – 1 is such a number. By assumption, we have >min, and as a consequence, for some there exists at least one degressively proportional distribution1,2, ..., subject to boundary conditions (,,), that for some,+1 = and < hold. If is the largest number among 1, 2, … ,–1 with this property, then a sequence1,2, …,,11,21,1 is degressively proportional with respect to1,2, ...,, satisfying the boundary conditions (, – 1,ʹ′), where

ʹ′1,2, …,1,1,11,1.

An analogous situation is when we arbitrarily take the value of, given the sequence1,2, ...,.

A constant sequence-1=…1 is always degressively proportional, hencemax = and

max =. A recursively defined sequence =,⎡⎤−1⎢⎥⎢⎥

for =,–1, …, 2, satisfies the

condition of degressive proportionality for the sequence1,2, ...,, and its elements are the minimum values among all possible degressively proportional sequences with=. If any element would be smaller, then the condition of degressive proportionality would be violated. The values ofmin andmin are determined, asmin=1 andmin =1.

Similarly we can also demonstrate that for any∈min ,max  there exists such and a sequence

1,2, ..., degressively proportional with respect to1,2, ...,, whose triple (,,) represents

boundary conditions. In this case, it suffices to prove that if∈min ,max  can define a boundary

condition, then also + 1 is such number. Indeed, < holds under the given assumptions, therefore

there exists at least one sequence1,2, ..., degressively proportional with respect to1,2, ...,, whose triple (,,) represents boundary conditions for some. Then for any sequence with this

property, we can find such that and1. Ifis the smallest number among 1, 2, …,–1 with this property, then the sequence1 + 1,2 + 1, …, + 1,1,2, …, is degressively

proportional with respect to1,2, ..., that satisfies the boundary conditions ( + 1,,ʹ′), whereʹ′is the sum of its elements.

In both cases however, we can find a sequence1,2, ..., with such∈min ,max  that a

system of boundary conditions is inconsistent, i.e. there does not exist a sequence1,2, ...,degressively proportional with respect to1,2, ...,, whose triple (,,) represents a system of

2 =3 = …=, > 2, Then1 =,2 =3 =…= =+1 = is a degressively proportional sequence with respect to1,2, ..., with–1. Yet, there does not exist a sequence1,2, ...,degressively proportional with respect to1,2, ...,, whose boundary conditions are represented by a triple (,, – 1), because the value of1 cannot be reduced by one, and the decrease of any element among1,2, ..., requires the decrease of all remaining, thus yielding a sum that is smaller than – 1.

Therefore boundary conditions cannot be specified arbitrarily. Firstly, a minimum and a maximum

representation, i.e. the values and are restricted as above mentioned, and secondly, even if these

constraints are fulfilled, the existence of a sequence1,2, ..., with the sum satisfying the

inequalitiesmin≤≤max is not ensured. For =min and =max the sequences1,2, ..., are of

course determined uniquely, yielding either distributions that are closest to proportional allocations or

equal distributions. Generally these are the only possible boundary conditions leading to unique

solutions. However, they seem unacceptable from a practical point of view. Therefore, we have to seek

specific distribution consenting an arbitrary selection or giving additional criteria that allow a unique

solution.

3.Distribution of mandates in the European Parliament

Table 1 presents populations of the member states of the European Union (as of 1 January 2012,

based on Eurostat data, column 2), percentage shares (column 3) and examples of distribution of seats

in the European Parliament. Column 4 presents the distribution during the 2014-2019 term, column 7 –

a maximum distribution, column 7 – a maximum distribution. Columns 5, 8 and 11 give the numbers of

citizens of a given country represented by one member of the EP under a given distribution, and

columns 6, 9 and 12 – the percentage shares of mandates allocated to a given country in total number

of mandates.

Comparing populations with numbers of mandates allows to examine whether the principle of

degressive proportionality is satisfied. The allocation of mandates to the European Parliament among

all member countries for the 2014-2019 term was proposed by the Committee on Constitutional Affairs

and does not meet the condition of degressive proportionality (see columns 4 and 5 in table 1). This is a

consequence of methodology chosen by the Committee. Having in mind previous, historical allocations

of seats in the past terms of the European Parliament and the accession of Croatia to the European

Union, a distribution of seats was adopted as binding so that no member state loses more than one seat

of those allocated in the 2009-2014 term and the distribution is close to a degressively proportional

one. However, the adopted report explicitly states that this solution is temporary and that efforts will be

made to establish “a durable and transparent system which, in future, before each election to the

European Parliament, will allow seats to be apportioned amongst the Member States in an objective

manner, based on the principle of degressive proportionality” (Gualtieri, & Trzaskowski 2013).

Under a maximum representation (columns 7-9), it is assumed that the smallest country by

population is allocated 6 mandates, then each larger country, in an increasing order, is allocated the

largest possible number of mandates, so that the principle of degressive proportionality remains

satisfied. The model of maximum representation does not set the limits of mandates allocated to a

country or the total number of mandates. Given current populations, the largest country could be

allocated almost tenfold the current limit, i.e.max = 902 mandates. As known, this distribution is close to a proportional allocation that can be seen when we compare the percentage shares of seats allocated

to a country in the total number of seats in the EP with the percentage share of its population in the

total population of all member states of the EU (columns 3 and 9). This is also confirmed when we

compare the extreme values of citizens from a given country represented by one member of the EP

(column 8). Under a maximum representation, the difference between these values for the largest and

the smallest country by population is smallest among all possible distributions satisfying the principle

of degressive proportionality. The total number of seats allocated under this model ismax = 5666 (withmin = 6⋅ 28 = 168), considerably more than the adopted limit of 751. Modifying this distribution so

that the countries which should be allocated more than 96 mandates, get 96 seats, also yields a

distribution that satisfies the condition of degressive proportionality, even if most of countries (Austria

and states larger by population than Austria) will be allocated the same number of seats. With this

modification the size of the European Parliament would bemax = 1949.

Under a minimum representation (columns 10–12), the largest country by population is allocated 96

seats, then smaller countries, in a decreasing order, are allocated the smallest possible numbers of

mandates, so that the principle of degressive proportionality remains satisfied. Analogously, as before,

no limit to the smallest possible number of mandates is introduced. Under this model, the smallest

feasible number of mandates ismin = 666 (withmax = 96⋅ 28 = 2688).

This distribution clearly reveals that such countries as France, the UK or Spain have less mandates

in the 2014-2019 term of the European Parliament than required by the principle of degressive

proportionality. It is also helpful in case when additional constraints are introduced, such as that the

least populous country has to be allocated 6 mandates, and the total number of seats must not exceed

751. It suffices to modify the distribution in such a way that countries which have less than 6 seats gain

additional mandates, i.e. we have to allocate 15 additional seats. Thenʹ′min = 681 (see table 2). If the

total size is = 751, then it suffices to allocate additional 70 seats, so that the principle of degressive

proportionality is satisfied.

It is worth mentioning here that, on the one hand, the ‘surplus’ of 70 mandates results in numerous

degressively proportional distributions that satisfy the conditions (,,) = (6,96751), but on the

other hand, this exposes the fact that after the accession to the EU of large countries by population,

such as Ukraine or Turkey that should be allocated more than 70 mandates, either the size of the

European Parliament will exceed 751 or the upper limit will have to be much lower than 96.

In order to allocate the additional mandates, one can employ a sequence that recursively determines

the minimum distribution. For example, the second largest country by population (i.e. France) is

allocated a greater number of seats, also increasing the numbers of seats for other countries, according

⎡⎤to an algorithm based on this sequence28 = 96,27 =ʹ′, whereʹ′ > 77,−1⎢⎥⎢⎥

, for

i = 28, 27, …, 2, however, the total size must not exceed 751. If we take28 = 96,27 = 86, then the total will be 733 (see table 2, column 5). The procedure can go on, trying to increase the number of seats for subsequent, smaller countries (columns 6 and 7).

This approach is not able however to resolve the known problem when the selection of a

distribution is not unique (Cegiełka at al. 2010; Dniestrzański 2013; Słomczyński, & Życzkowski). For

instance, instead of allocating 86 seats to France, as in our example, France might be allocated 80 seats,

or any number between 77 and 86, and smaller countries might be allocated more. In any case, using

this algorithm guarantees that the principle of degressive proportionality is satisfied.

4.Summary

Due to the lack of unambiguous indications regarding the methods of degressively proportional

distribution of goods, there emerge various interpretations of provisions of the Lisbon Treaty. As a

result, it becomes significant to find the acceptable solutions. This leads to an analysis of boundary

conditions of a degressively proportional distribution. It turns out that defining a minimum and a

maximum representation is subject to the smallest constraints. For every sequence, and associated

degressively proportional allocation, one can determine a minimum or maximum allocation, assuming

either of them. In addition, any number from resulting intervals can be considered a boundary

condition. However, this is not valid as regards the size of the assembly. One can define the minimum

and maximum values subject to a minimum or maximum representation, but some numbers from the

respective intervals cannot be elements of boundary conditions. What is more, determining such

numbers seems a computationally complex problem.

There are no indications concerning the number of feasible solutions. Apart from trivial cases when

allocation is unique, it is difficult to find this number. It is known however, that for large, under a

certain system of boundary conditions, the set of feasible solutions cannot be searched in a manageable

time. Yet, some simulations are possible if restrictions on boundary conditions are known. Such

simulations can lead to establishing an additional rule that points towards a unique solution.

Acknowledgements

The results presented in this paper have been supported by the Polish National Science Centre under grant no. 2013/09/B/HS4/02702.

References

• Cegiełka, K., Dniestrzański, P., Łyko, J., Misztal, A. (2010). Division of seats in the European

• Parliament. Journal for Perspectives of Economic Political and Social Integration. Journal for Mental Changes, XVI (1-2), 39-55.

• Delgado-Márquez, B., Kaeding, M., Palomares, A. (2013). A more balanced composition of the European Parliament with degressive proportionality. European Union Politics,14, 458-471.

• Dniestrzański, P. (2013). Degressive proportionality – notes on the ambiguity of the concept. Mathematical Economics,9(13), 21-28.

• Dniestrzański, P. (2014). The Proposal of Allocation of Seats in the European Parliament – The Shifted Root. Procedia – Social and Behavioral Sciences,124, 536-543.

• Dniestrzański, P., Łyko, J. (2014). Base Sequence of Degressively Proportional Divisions. Procedia – Social and Behavioral Sciences, 124, 381-387.

• Dniestrzański, P., Łyko J., (2014). Influence of boundary conditions of digressively proportional division on the potential application of proportional rules. Procedia – Social and Behavioral Sciences, 109, 722-729.

• Grimmett, G., Laslier, J. F., Pukelsheim, F., Ramirez-González, V., Rose, R., Słomczyński, W., Zachariasen, M., Życzkowski, K. (2011). The allocation between the UE Member States of seats in the European Parliament. European Parliament Studies, PE 432.760.

• Gualtieri, R., Trzaskowski, R. (2013). European Parliament Resolution of 13 March 2013 on the composition of the European Parliament with a view to the 2014 elections. (INI/2012/2309).

• Lamassoure, A., Severin A. (2007). European Parliament Resolution on “Proposal to amend the Treaty Provisions Concerning the composition of the European Parliament” adopted on 11 October 2007 (INI/2007/2169).

• Laslier, J-F. (2012). Why not Proportional? Mathematical Social Sciences, 63(2), 90-93.

• Łyko, J. (2012). The boundary conditions of degressive proportionality. Procedia – Social and Behavioral Sciences, 65, 76-82.

• Łyko, J. (2013). Boundary, Internal and Global Measure of the Degression of Distribution. Procedia Economics and Finance, 5, 519-523.

• Łyko, J., Rudek, R. (2013). A fast exact algorithm for the allocation of seats for the EU Parliament.

• Expert Systems with Applications, 40 (13), 5284-5291. DOI: 10.1016/j.eswa.2013.03.035.

• Serafini, P. (2012). Allocation of the EU Parliament seats via integer linear programming and revised quotas. Mathematical Social Sciences, 63(2), 107-113.

• Słomczyński ,W., Życzkowski K. (2012). Mathematical aspects of degressive proportionality.

• Mathematical Social Sciences, 63(2), 94-101.

• Treaty (2010). The Treaty of Lisbon. Available at: http://europa.eu/lisbon_treaty/full_text/index_en.htm.