We study voting rules for participatory budgeting, where a group of voters collectively decides which projects should be funded using a common budget. We allow the projects to have arbitrary costs, and the voters to have arbitrary additive valuations over the projects. We formulate two axioms that guarantee proportional representation to groups of voters with common interests. To the best of our knowledge, all known rules for participatory budgeting do not satisfy either of the two axioms; in addition we show that the most prominent proportional rules for committee elections (such as Proportional Approval Voting) cannot be adapted to arbitrary costs nor to additive valuations so that they would satisfy our axioms of proportionality. We construct a simple and attractive voting rule that satisfies one of our axioms (for arbitrary costs and arbitrary additive valuations), and that can be evaluated in polynomial time. We prove that our other stronger axiom is also satisfiable, though by a computationally more expensive and less natural voting rule.

中文翻译：

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Cardinal Utilities 的比例参与式预算

我们研究参与式预算的投票规则，其中一组选民共同决定应使用共同预算资助哪些项目。我们允许项目具有任意成本，并且选民可以对项目进行任意附加估值。我们制定了两个公理，以保证具有共同利益的选民群体的比例代表。据我们所知，所有已知的参与式预算规则都不满足这两个公理中的任何一个；此外，我们表明委员会选举最突出的比例规则（例如比例批准投票）不能适应任意成本或附加估值，因此它们将满足我们的比例公理。我们构建了一个简单而有吸引力的投票规则，它满足我们的一个公理（对于任意成本和任意附加估值），并且可以在多项式时间内进行评估。我们证明了我们的另一个更强的公理也是可满足的，尽管通过计算上更昂贵且更不自然的投票规则。