Alpha Proportionality and Penrose Square Root Law

Abstract

Problem concerns the distribution of mandates in collegial bodies of communities of states. The number of delegates representing each member of community depends on the number of its population. The paper analyzes the fairness of different kinds of relationship involving the number of representatives in collegial bodies of communities of states, in search of the fairest relationship. The research aims at comparing different ideas of how the numbers of representatives are determined reflecting the population of member states. Literature review, elementary probabilistic tools and mathematical analysis of degressive proportionality are employed. The main result of the paper demonstrates that the two fundamentally different approaches to the problem of distribution of mandates in collegial bodies may lead to similar solutions. A probabilistic approach (based on the voting power) and the concept of degressive proportionality (without additional assumptions) do not exclude each other, but, quite the contrary, may complement each other. The Penrose law and the rule of degressive proportionality, when suitably interpreted, are two sides of the coin. Degressive proportionality, as written in the Lisbon Treaty, is a vague idea.

Relative Entropy in Determining Degressive Proportional Allocations

“…The analysis of power functions in this context is not a new approach. For example, in papers [ 3 , 19 , 20 ], the exponent is proposed for the use in allocation. Interestingly enough, the arguments in favor of the fairness of such a solution are not the same.…”

“…Interestingly enough, the arguments in favor of the fairness of such a solution are not the same. Penrose [ 3 ] employs an assumption of voters’ independent decisions, Theil and Schrage [ 19 ] minimize the average number of citizens represented by the Parliament members from each country, whereas Dniestrzański [ 20 ] examines the conditions to be satisfied by the derivative of the allocation function in degressively proportional division.…”

Asymmetrization of a Set of Degressively Proportional Allocations with Respect to Lexicographic Order. An Algorithmic Approach

“…The increased interest in degressively proportional allocation, and thereby the growth of research concerning it, has been occurring since 2007, when the Treaty of Lisbon introduced the principle as a basis of allocating mandates in the European Parliament among the member states [7][8][9][10][11][12][13]. In it, the populations of the respective countries are assumed as their values or entitlements.…”

Application of a Power Function to Allocating Seats in the European Parliament Before and After Brexit

“…In this case, subject to boundary conditions for , , there still exist many sequences , which satisfy them, and a problem arises how to select a concrete allocation. In consequence, an area of both political as well as scientific discourse [3,4,5,6,7,8,9,10] is opened.…”