The (Computational) Social Choice Take on Indivisible Participatory Budgeting

TL;DR

This survey analyzes the computational social choice perspective on indivisible participatory budgeting, framing PB as the problem of aggregating voter opinions under a budget constraint. It surveys ballot designs (cardinal, approval, ordinal, and their enriched variants) and a wide array of PB rules (welfare-maximising, sequential Phragmén, maximin support, MES, and more), detailing fairness notions such as extended justified representation, the core, and priceability, together with their axiomatic and algorithmic properties. Key contributions include formal definitions, comparative results (e.g., distortion analyses, complexity of welfare maximisation, and existence/computability of core-like outcomes), and empirical insights from real-life PB experiments. The work highlights both established results and important open problems—such as core non-emptiness with cardinal ballots, polynomial-time EJR satisfaction, and broader applicability of fairness axioms to diverse PB settings—laying a foundation for theoretically grounded, practically applicable PB design and analysis.

Abstract

In this survey, we review the literature investigating participatory budgeting as a social choice problem. Participatory Budgeting (PB) is a democratic process in which citizens are asked to vote on how to allocate a given amount of public money to a set of projects. From a social choice perspective, it corresponds then to the problem of aggregating opinions about which projects should be funded, into a budget allocation satisfying a budget constraint. This problem has received substantial attention in recent years and the literature is growing at a fast pace. In this survey, we present the most important research directions from the literature, each time presenting a large set of representative results. We only focus on the indivisible case, that is, PB problems in which projects can either be fully funded or not at all. The aim of the survey is to present a comprehensive overview of the state of the research on PB. We aim at providing both a general overview of the main research questions that are being investigated, and formal and unified definitions of the most important technical concepts from the literature.

Figures (3)

• Figure 1: Some of the experimental findings of FBG23 comparing different ballot formats. The voting time column indicates the time in seconds it took participants to submit their opinion for each ballot format. The reported ease of use and expressiveness columns represents the average value reported by the participants about the ease of use and the expressiveness of each ballot format, on a scale from 1 to 5 (the higher the better). The figures have been reproduced with the authorisation of the authors, using the data available in the GitHub repository https://github.com/rfire01/Participatory-Budgeting-Experiment.
• Figure 2: Taxonomy of the proportionality requirements for PB with cardinal ballots. An arrow between two concepts means that any budget allocation satisfying one also satisfies the other. All missing arrows are known to be missing. Most of this picture is based on LCG22, who showed: the absence of an arrow between either the core, EJR or PJR and priceability; the link between laminar proportionality and priceability (only for laminar instances); the absence of arrows between laminar proportionality and either PJR, EJR or the core. The link between FJR and EJR is due to PPS21. For the satisfiability of the concepts, see Table \ref{['tab:Summary_FairnessRulesPreCriteria_CardinalBallots']}.
• Figure 3: Taxonomy of the proportionality requirements for PB with approval ballots where $\mathit{sat}$ is an arbitrary satisfaction function. An arrow between two concepts means that any budget allocation satisfying one also satisfies the other. Some arrows are only valid for some satisfaction functions; the conditions are indicated on the arrows. All missing arrows are known to be missing. The links between EJR, PJR, Local-BPJR-L and priceability concepts are due to BFLMP23. The link from Strong-BPJR-L and BPJR-L is due to ALT18. The link between CPSC and PJR[$\mathit{sat}^{\mathit{cost}}$] is due to AzLe21. The one between IPSC and PJR-X[$\mathit{sat}^{\mathit{cost}}$] has never been published. The absence of arrows between the core, EJR and priceability is due to LCG22. The link between FJR and EJR is due to PPS21. All links including forms of EJR+ are due to BP23. For the satisfiability of the concepts, see Table \ref{['tab:Summary_FairnessRulesPreCriteria_ApprovalBallots']}.

Theorems & Definitions (68)

• Definition 1: Approval-Based Satisfaction Functions
• Definition 2: Greedy Scheme
• Definition 3: Ordinal Greedy Scheme
• Definition 4: Sequential Phragmén, Continuous Formulation
• Definition 5: Sequential Phragmén, Discrete Formulation
• Definition 6: Maximin Support Rule
• Definition 7: Method of Equal Shares for Cardinal Ballots
• Definition 8: Method of Equal Shares for Approval Ballots
• Definition 9: $(\alpha, P)$-Cohesive Groups
• Definition 10: $(\alpha, P)$-Cohesive Groups