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

Participatory Budgeting With Multiple Degrees of Projects And Ranged Approval Votes