A General Theory of Proportionality with Additive Utilities

We consider a model where a subset of candidates must be selected based on voter preferences, subject to general constraints that specify which subsets are feasible. This model generalizes committee elections with diversity constraints, participatory budgeting (including constraints specifying how funds must be allocated to projects from different pools), and public decision-making. Axioms of proportionality have recently been defined for this general model, but the proposed rules apply only to approval ballots, where each voter submits a subset of candidates she finds acceptable. We propose proportional rules for cardinal ballots, where each voter assigns a numerical value to each candidate corresponding to her utility if that candidate is selected. In developing these rules, we also introduce methods that produce proportional rankings, ensuring that every prefix of the ranking satisfies proportionality.