Axioms and Divisor Methods for a Generalized Apportionment Problem with Relative Equality

The allocation of seats in a legislative body to groups based on their size is a crucial issue in legal and political studies. However, recent findings suggest that an optimal allocation of seats may not be proportional to the size of the groups. For instance, the European Parliament (EP) utilizes a subproportional system known as degressive proportionality. Unfortunately, current apportionment methods for the EP lack a rigorous axiomatic analysis and fail to adequately address equality. Building upon recent research on equality in subproportional settings, this paper proposed a novel generalization of existing axioms and divisor methods for proportionality to encompass subproportionality with relative equality. Specifically, we consider a function f(p)=a+bp? on the standard number of seats for a group of size p, where a, b and ? are given non-negative constants, and a is an integer. This theory is exemplified through an empirical study focused on the EP.