A neutral core of degressively proportional allocations under lexicographic preferences of agents

Abstract

One of the main problems of practical applications of degressively proportional allocations of goods and burdens is lack of uniqueness of this principle. Even under given boundary conditions of allocation, i.e. determined minimal and maximal amounts of a good that can be assigned in a given allocation, there are usually many feasible solutions. The lack of formal rules of allocation is the reason why the allocation is typically a result of negotiations among its agents. A number of allocations favor some of agents or their groups, therefore other agents cannot accept them. The aim of this paper is to indicate a way of reducing the set of all feasible solutions exclusively to those that are neutral to all agents. As a result of the term of lexicographic preference of allocation agents defined on the basis of the relation theory followed by a numerical analysis of sets of all feasible solutions, it is possible to determine a core of this set in the form of a subset of all feasible solutions that are acceptable by all agents. In addition, this subset can be further divided into smaller subsets with regard to the degree of acceptance of their elements. Theoretical analysis is complemented by case studies, one of which is application of this idea to the allocation of seats in the European Parliament among the member states of the European Union.

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).…”