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Learning Aggregation Rules in Participatory Budgeting: A Data-Driven Approach

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper shows that a neural network can learn participatory budgeting aggregation rules from example votes and outcomes, reproducing standard rules and recovering Proportional Approval Voting as a 30/70 compromise between welfare and…

desk verdict A genuinely useful extension of Set Transformer learning to participatory budgeting, with a PAV-resemblance claim that overstates what average ratios can support. read the letter →

arxiv 2412.01864 v1 pith:FYZ6FFCJ submitted 2024-12-01 cs.LG cs.AIcs.CYcs.GT

Roy Fairstein , Dan Vilenchik , Kobi Gal This is my paper · ORCID
classification cs.LGcs.AIcs.CYcs.GT MSC 68T0791B1291B14
keywords participatorybudgetingaggregationrulesneuralnetworksSetTransformerproportionalapprovalvotingsocialwelfarerepresentationcompromise
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that participatory budgeting rules do not have to be hand-crafted formulas: a neural network can learn the rule from examples of votes, project costs, and chosen outcomes. Trained on small synthetic instances, the network reproduces the behavior of Approval Voting, Chamberlin–Courant, and Proportional Approval Voting, and it keeps that accuracy when applied to real Warsaw municipal budgets with thousands of voters. When the training data blends 30 percent AV outcomes with 70 percent CC outcomes, the learned rule lands on exactly the welfare-versus-representation trade-off that Proportional Approval Voting achieves, which the authors offer as a data-driven rationale for PAV's compromise position.

What carries the argument

The load-bearing component is the ST+PMA architecture: a Set Transformer in which a pooling-by-multihead-attention (PMA) layer is moved from the decoder to before the encoder, compressing the voter-by-project approval matrix into a project-dimensional latent space so that the number of voters no longer affects the network's input size. Project costs, normalized so the budget is 1, are appended to this latent representation, and the network outputs a sigmoid score per project from which a feasible bundle is selected greedily. Training runs in two modes: implicit, where the cross-entropy loss compares the network's bundle to an optimal bundle from the target rule, and explicit, where the loss directly optimizes a smooth approximation of the welfare or representation score—the explicit mode being designed for datasets with many tied optima.

What would settle it

Run the network trained on synthetic data on published participatory budgeting elections from several cities other than Warsaw, comparing the welfare and representation ratios of its chosen bundles with those of the rule it was trained to imitate; if the ratios fall well short on average, or if the 30/70 AV/CC rule's average ratios on a fresh, diverse set of real instances diverge from Proportional Approval Voting's by more than the variance reported in the paper, the transfer and compromise claims would be falsified.

Watch

Extended reading notes

Core claim

The central discovery is that an aggregation rule for participatory budgeting can be embedded directly into a neural network's weights and then decoded on new, larger instances. Using a Set Transformer with a pooling-by-multihead-attention layer placed before the encoder, the network's input size depends only on the number of projects, not the number of voters, allowing training on instances with 10–900 voters and transfer to instances with up to 50,000. On the UNIQUE datasets, the learned rule attains total RMSE between 0.013 and 0.076 in the welfare-representation space, well below the 0.168–0.242 of sequential baselines, and it runs on the real-world Warsaw instances where the plain architecture exhausts GPU memory. The new observation is that a rule trained on a blend of 30% AV outcomes and 70% CC outcomes reproduces PAV's average welfare and representation ratios, giving a data-driven and closed-form rationale for PAV's balance of welfare and representation.

Load-bearing premise

The learned rules are trained on synthetic instances from four stylized generative models, so the approach rests on these models mimicking the approval patterns, project costs, and project interactions of real participatory budgeting settings where the rule would be deployed.

Editorial extensions

If this is right

  • PB organizers could define a desired rule by supplying examples of outcomes, without knowing or stating the rule's formula.
  • Rules learned on small synthetic instances transfer to large real elections, making the approach practical for municipal budgeting with tens of thousands of voters.
  • The same trained network can be fine-tuned as objectives shift, so a community's aggregation rule can evolve without redesigning a rule from scratch.
  • A compromise between competing objectives can be produced by mixing examples from each objective; the 30/70 AV/CC mix specifically recovers PAV's welfare-representation trade-off.
  • Because inference with the network is fast, the learned rule is usable in real-time settings such as multi-winner voting in games or sensor-network decisions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the approach generalizes beyond Warsaw, it suggests a paradigm in which aggregation rules are validated empirically by behavior rather than derived axiomatically; one could test whether other known rules, such as the Method of Equal Shares, are also recovered by particular AV/CC mixture levels.
  • The paper's own caution that the network is a black box with no worst-case guarantees implies the safest deployment is as a decision-support tool for ordinary PB, with human veto, rather than as the final authority on public funds.
  • The p=0.3 coincidence invites a direct experiment: train networks on AV/CC mixtures with p ranging from 0 to 1 in fine increments and check whether the welfare-representation curve is exactly the PAV curve at any other p, or whether 0.3 is special.
  • A natural extension the paper leaves implicit is to train on mixtures of PAV and another proportional rule to see whether the learned compromise outperforms both parents on proportionality axioms such as justified representation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a neural-network framework for learning participatory budgeting (PB) aggregation rules from labeled examples. The authors adapt the Set Transformer architecture by moving the pooling-by-multihead-attention (PMA) layer before the encoder, so that the network input is a fixed-size project representation regardless of the number of voters, and they introduce implicit (cross-entropy to an optimal bundle) and explicit (surrogate losses for welfare and representation) training modes. They evaluate the approach on synthetic datasets (Euclidean, One-vs-Many, Trade-Off AV, Trade-Off CC) and on real Warsaw PB data (2017–2021), reporting that the ST+PMA model learns AV, CC, and PAV rules on 'unique' datasets with low RMSE and generalizes to larger instances. The paper also learns mixture rules AV-CC-p and claims that a 30%-70% blend of AV and CC examples produces a rule that 'closely resembles' PAV, presented as a data-driven and closed-form rationale for PAV's welfare-representation balance.

Significance. If the results hold, the paper makes a useful contribution to data-driven social choice: it demonstrates that a reasonably simple neural architecture can learn standard PB rules from example optimal outcomes and can scale to the voter counts of real PB instances, which prior work on learning voting rules did not achieve for the PB setting. The architectural change (PMA before the encoder) is a practical engineering contribution, and the mixture-learning experiment, if properly supported, would offer an interesting perspective on the relationship between AV, CC, and PAV. The paper is honest about two limitations (black-box explainability and lack of worst-case guarantees), and the experimental setup is described in enough detail (dataset generators, hyperparameters, tie analysis) to be replicable. However, the central quantitative claims need strengthening: the headline RMSE tables lack uncertainty quantification, one of the four described model variants is omitted from results, and the PAV-resemblance claim is supported only by aggregate welfare/representation ratios rather than outcome-level comparisons.

major comments (4)
  1. [Section 7, Figure 2; Contribution (4)] The claim that AV-CC-0.3 'closely resembles' PAV and offers a 'data-driven rationale' for PAV is supported only by average welfare and representation ratios. The manuscript itself states that this 'does not mean that those two rules give the same outcome as PAV,' and average ratios can be matched by rules that select very different bundles. To make the resemblance claim load-bearing, the paper should report outcome-level similarity metrics (e.g., Jaccard overlap or rank correlation between funded bundles of AV-CC-0.3 and PAV) with confidence intervals, or explicitly weaken the claim to state that the average welfare/representation tradeoff of PAV is reproduced by the mixture on the evaluated datasets. In addition, the phrase 'closed-formula rationale' is misleading: the orange points in Figure 2 come from the ILP with objective p·welfare+(1−p)·representation, which is not the PAV score; the coincidence at p=0.3 is an empirical observation on these datasets, not a derivation.
  2. [Section 6, Tables 2 and 3; Appendix Tables 4 and 5] The RMSE values in Tables 2 and 3 are computed from dataset-level average welfare and representation ratios, with no error bars, confidence intervals, or repeated-seed information. The appendix tables show high per-instance variance for several cells, e.g., Table 5 reports OVM small AV implicit representation ratio 0.824 ± 0.249 and TOCC small CC implicit welfare ratio 0.095 ± 0.103; these large standard deviations mean that the aggregate RMSE may be dominated by a few instances or runs. The paper should report bootstrap intervals over instances and multiple training seeds, or otherwise quantify the uncertainty of the RMSE claims.
  3. [Section 6, first paragraph] The paper states that 'for brevity' it does not report the ST model with explicit learning because 'its performance was like that of Random.' This is a negative result for one of the four model variants the paper claims to evaluate, and its omission weakens the comparison, especially because the explicit method shows different behavior on TIED datasets. Please include the ST-explicit results (at least in the appendix) or provide a concrete summary of its performance, so readers can verify the claim and understand why this variant fails.
  4. [Section 5 and Appendix 1.A; Section 6.2] The generalization claim ('able to generalize from small-scale synthetic PB examples to large, real-world PB instances') rests on a single real-world collection (Warsaw) and on synthetic generators whose parameters are not varied in a sensitivity analysis. The paper should add cross-generator transfer experiments (e.g., train on EUC and test on OVM or TOCC, and vice versa) and, if feasible, evaluate on additional Pabulib datasets, to assess how sensitive the learned rules are to distribution shift. Without such experiments, a reader cannot tell whether the success on Warsaw reflects general representativeness of the synthetic benchmark or an accidental match to that particular city's instances.
minor comments (5)
  1. [Figure 2 caption] The caption says 'The first four points from the left correspond to p = 0, 0.01, 0.04, 0.8'; given the discussion of p ≤ 0.1, '0.8' is likely a typo for '0.08'. Please correct.
  2. [Section 7] The text 'Alongside AC-CC-p' should be 'Alongside AV-CC-p' to match the notation used everywhere else.
  3. [Section 4.2] The explicit representation loss uses a shifted sigmoid with parameters a=3 and b=6 that were chosen 'after testing a few candidates.' Please clarify how these values were selected and whether the results are robust to their choice, since the loss is an approximation of the true representation score.
  4. [Section 5 and Appendix 1.A] There are several typos and spacing inconsistencies, e.g., 'T rade-Off A V', 'TOA V', 'optimal soultions', 'mult-winner', and 'it possible possible to use it' in Section 8. A careful proofreading pass is needed.
  5. [Section 1.1] The notation introduces [k] but does not use it consistently; please either use it in the definitions of the scores or remove it.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the supervised learning setup and the explicitly qualified PAV analogy keep the derivation self-contained.

full rationale

The paper's central pipeline is supervised learning: networks are trained on PB instances labeled by optimal outcomes of AV, CC, or PAV, and then evaluated on held-out instances, including larger synthetic instances and real Warsaw data, using welfare and representation ratios. This is a standard fit-and-generalization setup, not a circular derivation: the evaluation metric is not identical to the training objective, and generalization to unseen and larger instances provides independent evidence that the network has learned the target rule rather than memorized the training labels. The AV-CC-0.3 claim is an empirical post-hoc observation that the average welfare and representation ratios of a learned mixture lie close to PAV's aggregate ratios; the paper explicitly disclaims outcome-level equivalence ('this does not mean that those two rules give the same outcome as PA V'), so the resemblance claim is not forced by construction. The only self-citation is to Fairstein et al. [11] for the welfare-versus-representation trade-off datasets; those datasets are described in detail in the current paper and are not used to assume the paper's conclusions. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and the 'closed-formula' phrasing for PAV overstates what is an aggregate coincidence, but that is an interpretation/correctness concern rather than circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The method relies on supervised learning from synthetic data and a few hand-tuned numerical choices. No new physical or mathematical entities are introduced.

free parameters (1)
  • sigmoid parameters a and b in explicit representation loss = a=3, b=6
    Chosen after testing a few candidates (Section 4.2); these affect the explicit CC learning, though not the main implicit results.
assumptions (4)
  • domain assumption An aggregation rule can be approximated by a neural network trained on optimal outcomes of that rule.
    Core premise of the method; the paper assumes the network can learn a scoring function close to the rule from labeled examples.
  • domain assumption The optimal outcomes used as labels are computed correctly by an exact solver or rigorous analysis.
    Section 5.1 states they used an exact solver or rigorous analysis to compute ties; correctness of these labels is assumed.
  • domain assumption Welfare and representation ratios are appropriate metrics to compare rules.
    The paper uses RMSE in the welfare-representation space as the main evaluation; this assumes these two metrics capture the essential behavior of the rules.
  • ad hoc to paper The explicit representation loss with a shifted sigmoid is a valid differentiable surrogate.
    Section 4.2 introduces sigma(x) = 1/(1 + exp(a - bx)) with a=3, b=6 after testing; this is an ad hoc approximation to the non-differentiable representation score.

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Cite this review

Pith. "Pith review of Learning Aggregation Rules in Participatory Budgeting: A Data-Driven Approach." pith.science (2026). https://pith.science/paper/FYZ6FFCJ

@misc{pith2026241201864,
  author       = {Pith},
  title        = {Pith review of: Learning Aggregation Rules in Participatory Budgeting: A Data-Driven Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FYZ6FFCJ}},
  note         = {Machine review of arXiv:2412.01864}
}
read the original abstract

Participatory Budgeting (PB) offers a democratic process for communities to allocate public funds across various projects through voting. In practice, PB organizers face challenges in selecting aggregation rules either because they are not familiar with the literature and the exact details of every existing rule or because no existing rule echoes their expectations. This paper presents a novel data-driven approach utilizing machine learning to address this challenge. By training neural networks on PB instances, our approach learns aggregation rules that balance social welfare, representation, and other societal beneficial goals. It is able to generalize from small-scale synthetic PB examples to large, real-world PB instances. It is able to learn existing aggregation rules but also generate new rules that adapt to diverse objectives, providing a more nuanced, compromise-driven solution for PB processes. The effectiveness of our approach is demonstrated through extensive experiments with synthetic and real-world PB data, and can expand the use and deployment of PB solutions.

Figures

Figures reproduced from arXiv: 2412.01864 by the authors.

Figure 1
Figure 1. The network architecture. 4.1 Set Transformers (ST) Architecture The ST network was suggested as a permutation-invariant network, i.e., the network outputs the same no matter the input order. This property is important for our scenario as we would like the outcome to be independent of the order of the voters and projects. For this purpose, the network combines attention blocks and linear layers to create an encoder … view at source ↗
Figure 2
Figure 2. Welfare and rep. ratios on the UNIQUE datasets for the AV-CC-p models [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. The Jaccard similarity obtained for each of the aggregation methods: [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

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Cited by 1 Pith paper

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    In the instance creating, each voter approve the kv projects that are the closest to his location

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