Proportionality in Multiple Dimensions to Design Electoral Systems

How to elect the representatives in legislative bodies is a question that every modern democracy has to answer. This design task has to consider various elements so as to fulfill the citizens' expectations and contribute to the maintenance of a healthy democracy. The notion of proportionality, in that the support of a given idea in the house should be nearly proportional to its support in the general public, lies at the core of this design task. In the last decades, demographic aspects beyond political support have been incorporated by requiring that they are also fairly represented in the body, giving rise to a multidimensional version of the apportionment problem. In this work, we provide an axiomatic justification for a recently proposed notion of multidimensional proportionality and extend it to encompass two relevant constraints often used in electoral systems: a threshold on the number of votes that a list needs in order to be eligible and the election of the most-voted candidate in each district. We then build upon these results to design methods based on multidimensional proportionality. We use the Chilean Constitutional Convention election (May 15-16, 2021) results as a testing ground -- where the dimensions are given by political lists, districts, and genders -- and compare the apportionment obtained under each method according to three criteria: proportionality, representativeness, and voting power. While local and global methods exhibit a natural trade-off between local and global proportionality, including the election of most-voted candidates on top of methods based on 3-dimensional proportionality allows us to incorporate both notions while ensuring higher levels of representativeness and a balanced voting power.