We analyze properties of apportionment functions in context of the problem of allocating seats in the European Parliament. Necessary and sufficient conditions for apportionment functions are investigated. Some exemplary families of apportionment functions are specified and the corresponding partitions of the seats in the European Parliament among the Member States of the European Union are presented. Although the choice of the allocation functions is theoretically unlimited, we show that the constraints are so strong that the acceptable functions lead to rather similar solutions.
One of the major mathematical approaches to the problem of allocating seats in the European Parliament can be described by the following general scheme. First, one has to choose a concrete characterization of the size of a given Member State i by a number p i (for example, equal to the total number of its inhabitants, citizens, or voters 1 ) we call here population, and precisely define by which means these data should be collected and how often they should be updated. Then, one needs to transform these numbers by an allocation (or apportionment) function A belonging to a given family indexed (usually monotonically and continuously) by some parameter d, whose range of variability is determined by the requirement that the function fulfills constraints imposed by the treaties: is non-decreasing and degressively proportional.
Additionally, the apportionment function satisfies certain boundary conditions, A (p) = m and A (P) = M, where the population of the smallest and the largest state equals, respectively, p and P, and the smallest and the largest number of seats are predetermined as, respectively, m and M. (In the case of the European Parliament these quantities are explicitly bounded by the treaty, m ≥ M min = 6 and M ≤ M max = 96.) To obtain integer numbers of seats in the Parliament one has to round the values of the allocation function, e.g., using one of three standard rounding methods (upward, downward or to the nearest integer). Finally, one has to choose the parameter d in such a way that the sum of the seat numbers of all Member States equals the given Parliament size S , solving (if possible) in d the equation
where N stands for the number of Member States, p i for the population of the i-th state (i = 1, . . . , N), and [•] denotes the rounded number. Though usually there is a whole interval of parameters satisfying this requirement, nonetheless, in a generic case, the distribution of seats established in this way is unique. Thus, this technique bears a resemblance to divisor methods in the proportional apportionment problem applied first by Thomas Jefferson in 1792 (Balinski & Young, 1978, Toplak, 2009). The crucial role in this apportionment scheme plays the notion of degressive proportionality. The principle of degressive proportionality enshrined in the Lisbon Treaty was probably borrowed from the discussions on the taxation rules, where the term has appeared already in the nineteenth century, when many countries introduced income tax for the first time in their history (Young, 1994). It was already included in the debate on the apportionment in the Parliament in late 1980s, but at first, it was a rather vague idea that gradually evolved into a formal legal (and mathematical) term in the report Lamassoure & Severin (2007) adopted by the European Parliament. There were also suggestions to apply this general principle to other parliamentary or quasi-parliamentary bodies like the projected Parliamentary Assembly of the United Nations (Bummel, 2010).
In fact the entire problem of apportionment of seats in the Parliament is mathematically similar (not counting rounding) to the taxation problem, what is illustrated in the table below.
In consequence, the similar mathematical tools can be used to solve both of them; see for instance Young (1987), Thomson (2003), Kaminski (2006), Hougaard (2009), Ju &Moreno-Ternero (2011), andMoreno-Ternero (2011), where the authors use the above presented scheme to consider possible parametric solutions of the taxation problem or the dual profit-sharing problem. Of course, the analogy has clear limitations since income and post-tax income are calculated in the same units, whereas population and seats are not. Moreover, money is (at least theoretically) infinitely divisible, while seats are indivisible.
Although quite a novelty in politics, nevertheless, the concept of degressive proportionality is not entirely new in mathematics. It was already analysed in late 1940s under the name of ‘quasi-homogeneity’ by Rosenbaum (1950, Definition 1.4.1), see also Kuczma (2009, p. 480), and since then studied also under the name of ‘subhomogeneity’, see e.g. Burai & Száz (2005). Moreover, an increasing function such that its inverse is degressively proportional (and so it is an allocation function) is called ‘star-shaped’ (with respect to the origin) in the mathematical literature. In other words, the function is degressively proportional if and only if the lines joining points lying below its graph with the origin do not cross the graph. Star-shaped functions were introduced in Bruckner & Ostrow (1962), and since then have been studied in many areas of pure and applied mathematics, see e.g. Ding & Wolfstetter (2009), Dahm (2010). Thus, the results concerning this class of functions can be applied, mutatis mutandis, to degressively proportional functions.
Note that in the original definition of the degressive proportionality formulated in Lamassoure & Severin (2007) it wa
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