Proportional Participatory Budgeting with Additive Utilities

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 an axiom (Extended Justified Representation, EJR) that guarantees proportional representation to groups of voters with common interests. We propose a simple and attractive voting rule called the Method of Equal Shares that satisfies this axiom for arbitrary costs and approval utilities, and that satisfies the axiom up to one project for arbitrary additive valuations. This method can be computed in polynomial time. In contrast, we show that the standard method for achieving proportionality in committee elections, Proportional Approval Voting (PAV), cannot be extended to work with arbitrary costs. Finally, we introduce a strengthened axiom (Full Justified Representation, FJR) and show that it is also satisfiable, though by a computationally more expensive and less natural voting rule.