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.

翻译：我们考虑一个模型，其中必须根据选民偏好从候选人集合中选出一个子集，并受限于指定哪些子集可行的通用约束条件。该模型推广了具有多样性约束的委员会选举、参与式预算（包括规定资金如何从不同资金池分配给项目的约束）以及公共决策制定。最近已为该通用模型定义了比例性公理，但所提出的规则仅适用于认可投票制——即每位选民提交其认为可接受的候选人子集。我们为基数投票制提出了比例性规则，其中每位选民为每位候选人分配一个数值，对应若该候选人被选中时她所获得的效用。在制定这些规则的过程中，我们还引入了能够生成比例性排序的方法，确保排序的每个前缀均满足比例性要求。