Relative Entropy in Determining Degressive Proportional Allocations

Abstract

The principle of degressively proportional apportionment of goods, being a compromise between equality and proportionality, facilitates the application of many different allocation rules. Agents with smaller entitlements are more interested in an allocation that is as close to equality as possible, while those with greater entitlements prefer an allocation as close to proportionality as possible. Using relative entropy to quantify the inequity of allocation, this paper indicates an allocation that neutralizes these two contradictory approaches by symmetrizing the inequities perceived by the smallest and largest agents participating in the apportionment. First, based on some selected properties, the set of potential allocation rules was reduced to those generated by power functions. Then, the existence of the power function whose exponent is determined so as to generate the allocation that symmetrizes the relative entropy with respect to equal and proportional allocations was shown. As a result, all agents of the apportionment are more inclined to accept the proposed allocation regardless of the size of their entitlements. The exponent found in this way shows the significant relationship between the problem under study and the well-known Theil indices of inequality. The problem may also be seen from this viewpoint.

The Implications of Possible Enlargements of the European Union for the Configuration of Power in the European Parliament

“…The literature on the subject includes both proposed methods dealing with division of seats in the European Parliament [for example: (Pukelsheim, 2007;Ramírez-González, 2007;Martínez-Aroza and Ramírez-González, 2008;Słomczyński and Życzkowski, 2010;Pukelsheim, 2010;Słomczyński and Życzkowski, 2010;Grimmett et al, 2011;Florek, 2012;Ramírez-González et al, 2012;Słomczyński and Życzkowski, 2012;Łyko and Rudek, 2013;Dniestrzański and Łyko, 2014;Łyko and Rudek, 2017;Cegiełka et al, 2019), as well as theoretical analyses to be applied regardless of what type of goods are divided (Słomczyński and Życzkowski, 2012;Cegiełka and Łyko, 2014;Haman, 2017;Dniestrzański and Łyko, 2020;Cegiełka et al, 2021a;Cegiełka et al, 2021b. The discussion around indicating a single rule to form the composition of the European Parliament has been ongoing for more than fifteen years. The best-known proposal is the so-called Cambridge Compromise (Grimmett et al, 2011).…”