Between proportionality and diversity: Balancing district sizes under the Chamberlin-Courant rule

The Monroe and Chamberlin-Courant (CC) multiwinner rules proc eed by partitioning the voters into virtual districts and assigning a unique committee member to each district, so that the voters are as satisfied with the assignment as possible. The difference between Monroe and CC is that the former creates equal-sized districts, while the latter has no constraints. We generalize these rules by requiring that the largest district can be at most X times larger than the smallest one (where X is a parameter). We show that our new rules inherit worst-case computational properties from their ancestors; evaluate the rules experimentally (in particular, we provide their visualizations, analyze actual district sizes and voter satisfaction); and analyze their approximability.