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.
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Cite this article as:
Łyko, J., & Maciuk, A. (2016). Minimal and Maximal Representation of Degressively Proportional Allocation. In Z. Bekirogullari, M. Y. Minas, & R. X. Thambusamy (Eds.), Political Science, International Relations and Sociology - ic-PSIRS 2016, vol 10. European Proceedings of Social and Behavioural Sciences (pp. 10-18). Future Academy. https://doi.org/10.15405/epsbs.2016.05.03.2