Proportionality in multiple dimensions to design electoral systems

Abstract Proportionality is a foundational principle in the election of representatives for legislative bodies. In this work, we provide an axiomatic justification for a recently proposed notion of multidimensional proportionality and extend it to incorporate two common electoral constraints: vote threshold for list eligibility and the election of the most-voted candidate in each district. Building on these results, we design methods based on multidimensional proportionality and apply them to the 2021 Chilean Constitutional Convention election, considering political lists, districts, and genders as dimensions. We evaluate the resulting apportionments according to three criteria: proportionality, representativeness, and voting power. Whereas local and global methods exhibit a natural trade-off between local and global proportionality, incorporating the election of district-level winners into methods based on 3-dimensional proportionality enables a balance between both notions while ensuring higher representativeness and a more equitable voting power. These findings are further supported by experiments on the 2023 Finnish Parliamentary Election.