Participatory Budgeting With Multiple Degrees of Projects And Ranged Approval Votes

In an indivisible participatory budgeting (PB) framework, we have a limited budget that is to be distributed among a set of projects, by aggregating the preferences of voters for the projects. All the prior work on indivisible PB assumes that each project has only one possible cost. In this work, we let each project have a set of permissible costs, each reflecting a possible degree of sophistication of the project. Each voter approves a range of costs for each project, by giving an upper and lower bound on the cost that she thinks the project deserves. The outcome of a PB rule selects a subset of projects and also specifies their corresponding costs. We study different utility notions and prove that the existing positive results when every project has exactly one permissible cost can also be extended to our framework where a project has several permissible costs. We also analyze the fixed parameter tractability of the problem. Finally, we propose some important and intuitive axioms and analyze their satisfiability by different PB rules. We conclude by making some crucial remarks.