Participatory budgeting with cumulative votes

The idea of participatory budgeting (PB) originated in Brazil in the 1980s, when political reformers sought ways to move beyond the legacy of the military dictatorship (1964–1985), which had been marked by exclusion and corruption (Wampler, 2012). Their aim was to make local decision-making more transparent and to strengthen social justice and democracy (Cabannes, 2004). In this context, PB emerged as a natural innovation.

In the initial stage of PB, governments and civil society organizations define a set of goals based on certain principles, without yet referring to the budget. For instance, many PB programs in Brazil rely on the Quality of Life Index, first developed by the local government in Belo Horizonte. These goals are then translated into specific projects to be put forward for public selection. In some European cities, for example, in Poland, citizens themselves can also propose projects (Walczak & Rutkowska, 2017). The municipality subsequently reviews these proposals, filtering out inadmissible ones and assigning costs to those that are approved. Several researchers (e.g., Sintomer et al. (2012)) emphasise importance of deliberation but this rarely happens. The second stage is actual voting in which citizens express their views on the relative importance of the projects on offer. Thus, Participatory Budgeting is indeed a direct-democracy approach to budgeting.

The most prominent applications of PB have been at the municipal level, where residents collectively decide how to allocate a fraction of the city’s budget. Beyond this setting, PB-inspired methods have also been applied in diverse contexts, such as: (1) an airline company choosing which movies to include in its in-flight entertainment system (Skowron et al., 2016), where licensing costs vary across films; (2) a fan-owned soccer club deciding which athletes to sign; (3) generating a concise collection of statements that proportionally reflects societal viewpoints (Fish et al., 2024; Boehmer et al., 2025); and (4) selecting validators in proof-of-stake blockchain protocols (Cevallos & Stewart, 2020; Burdges et al., 2020).

Recently, PB has attracted considerable attention, and an increasing share of public funds is now allocated through this process.¹ This growing interest is also reflected in the scientific literature. Most research has focused on procedures based on approval ballots, where each voter selects a subset of projects they consider worth funding (sometimes subject to the constraint that the chosen projects do not exceed the available budget or that a certain number of projects needs to be approved). To a lesser extent, scholars have examined ordinal ballots, where voters rank (a subset of) projects from most to least desirable (Aziz & Shah, 2020; Rey & Maly, 2023). Several works have examined the use of cardinal utilities, where voters assign numerical scores to projects (Fain et al., 2018; Fluschnik et al., 2019; Benade et al., 2017; Peters et al., 2021).

In this paper, we study a model in which each of the n voters is endowed with virtual coins worth L/n, where L denotes the budget limit. Each voter distributes these coins across the projects, thereby indicating the intensity of their preferences. By analogy with voting theory (see, e.g., (Cole Jr, 1949)), we refer to such ballots as cumulative votes. Cumulative votes are closely related to cardinal ballots, with the key distinction that cardinal ballots are not required to sum to a fixed predefined value. This difference is significant, as it influences both the formulation of axioms and the design of voting rules. Finally, we note that even when voters cast approval ballots, it is often reasonable to interpret or convert them into cardinal or cumulative ones. For example, Faliszewski et al. (2023) argue that larger and more expensive projects tend to be more valuable from a voter’s perspective, and therefore propose interpreting a voter’s utility for a project as proportional to its cost.

We begin our study of aggregation methods for PB with cumulative votes by examining three greedy methods, analogous to top-k multiwinner voting rules (Faliszewski et al., 2017). We refer to these rules as (1) greedy by support, (2) greedy by excess, and (3) greedy by support over cost.

These methods are intuitive, easy to explain, and computationally efficient. However, they also have important shortcomings. In particular, as we will show, they fail to satisfy several desirable axiomatic properties and can, in some cases, completely ignore the preferences of minority groups.

To address this limitation, we propose several adaptations of the Single Transferable Vote (STV) rule to the PB setting. STV is a well-known ordinal-based multiwinner election rule, widely used in contexts where proportional representation is desired, such as parliamentary elections. Our adaptations differ in several key details; here we informally describe one of them. Given a budget limit L, each of the n voters is endowed with a bag of L/n coins, which they can distribute among the available projects as they wish. We then consider the coin stash accumulated by each project. In each iteration, if there exists a project whose stash meets or exceeds its cost, that project is funded, and any surplus coins are redistributed to other projects according to the preferences of the voters who contributed to it. If no project can be funded, we instead eliminate the project with the smallest stash, redistribute its coins to other projects, and continue iteratively.

Since cumulative votes generalize both approval and ordinal ballots,² our methods apply to these models as well. In practice, this means that voters need not use the full expressive power of cumulative ballots: they may simply provide approval or ordinal preferences, which can then be translated, by the user interface or by the aggregation algorithm, into cumulative form. At the same time, voters who wish to express their preferences in a more detailed manner retain the option to do so.

We illustrate the behavior of our rules by analyzing several desirable properties that a PB procedure should satisfy. We identify one class of rules, Minimal Transfers, which behaves particularly well with respect to our axiomatic properties and which indeed produces proportional results.

In our model there is a set of projects \(P = \{p_1, \ldots , p_m\}\); the cost of a project \(p \in P\) is a natural number, denoted c(p). There is a set of n voters \(V = \{v_1, \ldots , v_n\}\), where voter \(v_j\) expresses her preferences over the projects by assigning a value \(v_j(p)\) to each \(p \in P\) such that \(v_j(p) \ge 0\) and \(\sum _{p \in P} v_j(p) = 1\) (notice that we refer to both the jth voter and her cumulative vote using the same symbol \(v_j\)); intuitively, the value of \(v_j(p)\) is understood as the fraction of the funds in disposal of voter \(v_j\) that the voter would like to assign to project p.³ We say that a voter \(v_i\) supports a project p if \(v_i(p) > 0\). The above notation naturally extends to sets. For each \(B \subseteq P\) and each \(v_j \in V\) we denote \(v_j(B) = \sum _{p \in B}v_j(p)\) and \(c(B) = \sum _{p \in B} c(p)\).

A budgeting scenario is a tuple \(E=(P, V, c, L)\), where P, V, and c are as defined above, and \(L \in \mathbb {N}\) is a budget limit. An aggregation method \(\mathcal {R}\) is a function that, given a budgeting scenario, selects a bundle of projects \(\mathcal {R}(E) \subseteq P\) such that \(c(\mathcal {R}(E)) \le L\).

We compare our methods against several axiomatic properties, focusing on three main categories: The first category addresses potential manipulation by the steering committee when setting the agenda. Monotonicity axioms capture the principle that collective outcomes should respond positively to increases in individual support. Proportionality axioms ensure fairness toward minority groups. Both monotonicity and proportionality are widely recognized as highly desirable in the context of participatory budgeting. Since our focus is on cumulative votes, we adapt these axioms to this setting in the following section.

In this section, we define and discuss our aggregation methods for PB with cumulative votes.

Our analysis is summarized in Table 1, with detailed proofs provided in the appendix. For most properties, our rules do not offer guarantees, and the corresponding proofs consist of small counterexamples. In contrast, the proofs showing that our rules satisfy specific proportionality axioms rely on an invariant: a designated group of voters consistently directs a fixed amount of money toward a corresponding group of projects, and this allocation is never redirected outside of this group.

None of the rules satisfies Support Monotonicity; this is not surprising, given how demanding monotonicity requirements are in voting procedures; indeed, even STV is not monotonic. Further, none of the proportional rules satisfy Resistance to Merging, and only MT satisfies Resistance to Splitting. We believe that weaker variants of these axioms merit further investigation. For example, one could study axioms that impose stricter conditions on how support is redistributed when a project is split, or on how support from other projects can be redirected toward a given one. It is possible that some impossibility theorems are at play here, though they remain to be discovered. As expected, the greedy rules perform poorly with respect to proportionality, whereas the CSTV variants satisfy Weak-PR and PR, with some (such as MT, MTS and MTC) even meeting Strong-PR.

To complement the axiomatic analysis provided in the section above, in this section we compare our eight aggregation methods through computer-based simulations. In particular, we report on simulations done on 1398 real-life PB elections, which come from the PabuLib library (Faliszewski et al., 2023).

In PabuLib, each voter v assigns to each project p a numeric score \(\text {sc}_v(p)\). For approval ballots, this value is 1 if v approves of p, and 0 otherwise. We convert these instances to the cumulative form using one of two approaches, following Faliszewski et al. (2023) and Papasotiropoulos et al. (2025):We consider the following statistics (all values are averages over instances). Given a winning bundle of projects B, we compute:We do not include metrics related to resilience against merging or splitting projects, nor to support monotonicity, as it is unclear how to compute such metrics efficiently. Investigating this remains an interesting open problem.

In Table 2, we report the data from elections with more than 30 projects; in Table 3, we provide the corresponding data for elections with fewer than 15 projects; and in Table 4, we present the data for elections with between 15 and 30 projects. This division allows us to compare how our rules behave separately in small, medium, and large elections. We additionally include in our comparison two known rules for participatory budgeting, the ADD1U variant of the method of equal shares (Peters & Skowron, 2020; Peters et al., 2021; Faliszewski et al., 2023), and the recently introduced method of equal shares with bounded overspending (BOS) (Papasotiropoulos et al., 2025).

Our analysis indicates that certain variants of our rules perform particularly well across all metrics. Under the cost-utilities approach, both MTS and MTC consistently achieve strong results: MTC tends to select smaller projects than MTS and slightly improves the exclusion ratio, though at the expense of lower total cost-utility. Notably, neither rule ever violates EJR+ up to one, and in many respects they are comparable to Equal Shares and BOS. Compared with these two methods, MTS selects larger projects and attains higher cost-utility in medium and large elections, while in smaller elections it lies between Equal Shares and BOS. Among the elimination-with-transfers rules, EwTC provides the best balance between proportionality and total cost-utility. Under the score-utilities approach, MTS consistently outperforms MTC in terms of total cost-utility. Overall, we believe that MTS exhibit particularly good properties.

There is a large body of work on participatory budgeting (PB) and related topics in the social choice literature (see, e.g., Aziz and Shah (2020); Rey and Maly (2023)), which relies on different assumptions. A first point of divergence is the type of information solicited from voters. Since most PB aggregation rules adapt existing voting rules, the elicited input usually follows standard social choice formats: voters are asked either to approve projects (and thus disapprove others) or to rank projects by desirability. In approval ballots, voters are typically asked either to approve a fixed number of projects (say k) (Faliszewski & Talmon, 2019) or any subset of projects whose total cost does not exceed the budget (knapsack voting) (Goel et al., 2019). Goel et al. (2019) also considered pairwise comparisons, asking voters to compare projects by their value-to-cost ratio. These comparisons are then aggregated using variants of classic voting rules such as Borda and Kemeny. Klamler et al. (2012) and Lu and Boutilier (2011) proposed adaptations of multiwinner voting rules that, given project rankings, select k alternatives subject to their costs and the budget constraint. In these models, rankings are independent of costs. A number of works have instead solicited cardinal utilities for projects (Fain et al., 2018; Fluschnik et al., 2019; Peters et al., 2021; Papasotiropoulos et al., 2025), with voters specifying their utility for the implementation of each project. Benade et al. (2021) introduced a knapsack-style model in which voters support the bundle maximizing their utility. They also suggested other elicitation formats: Value voting (ranking by value), Value-for-money voting (ranking by value-to-cost), and Threshold voting (supporting all projects exceeding a given value threshold). Laruelle (2021) proposed that such utilities may be implicit and inferred from rankings. The question of proportional representation in PB naturally leads to fairness considerations. For instance, it should not happen that no project supported by a large minority is funded. The notion of justified representation for approval-based PB rules was introduced in Aziz and Lee (2018); Peters et al. (2021); Papasotiropoulos et al. (2025), while proportionality for solid coalitions in the ordinal setting was studied in Aziz and Lee (2019). Cumulative and cardinal ballots have also been explored in divisible models of PB, where voters allocate fractions of the budget across projects rather than selecting projects outright (Fain et al., 2016; Freeman et al., 2019). A broader literature on cumulative voting exists, but it mostly concerns single-winner elections and focuses on practical or legal aspects (Mills, 1968; Bhagat & Brickley, 1984; Cole Jr, 1949; Vengroff, 2003). There is also extensive research on the Single Transferable Vote (STV). It is widely regarded as a strong rule for both single-winner and multiwinner elections, especially when proportional representation is desired; see, e.g., Tideman and Richardson (2000); Elkind et al. (2017, 2017). This motivated us to adapt STV to PB via cumulative voting. Relatedly, Ford [Ford (2020), Section 3.4] proposed a cumulative version of STV for multiwinner elections. The work most closely related to ours is the Accurate Democracy project.⁶ It describes an iterative PB procedure reminiscent of STV. However, the description provided is informal, and the proposed rule differs from ours in several respects–for example, it combines ordinal and cumulative ballots. Moreover, no axiomatic analysis is offered.

We proposed the use of cumulative ballots for participatory budgeting and analyzed several aggregation methods within this framework. Our results–particularly the experimental findings presented in Tables 2, 3, 4–indicate that MTS and MTC deserve serious consideration in practical applications, for the following reasons:Looking forward, we see several promising directions for extending CSTV to more general PB settings. While in this paper we focused on the standard combinatorial PB model, cumulative ballots and the CSTV framework can be naturally adapted to richer ballot structures, enabling greater voter expressiveness and potentially yielding even better outcomes. In particular, we plan to investigate the following extensions:

PB with project interactions. Projects may interact, e.g., as in the model of Jain et al. (2020), where a substitution structure partitions the projects. Beyond cumulative ballots, voters could express preferences over substitutions and complementarities. Since CSTV redistributes support on behalf of voters, it could be extended to let voters explicitly direct how such redistribution should occur.