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arxiv: 2606.23320 · v1 · pith:A2Q5MARSnew · submitted 2026-06-22 · 💻 cs.GT

Mixed Voting Rules for Participatory Budgeting

Anton Baychkov , Markus Brill , Markus Utke This is my paper

Pith reviewed 2026-06-26 06:03 UTC · model grok-4.3

classification 💻 cs.GT
keywords participatory budgetingmixed voting rulesMethod of Equal Sharesproportionalityutilitarian welfareEJR+voting rulescomputational social choice
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The pith

Mixed participatory budgeting rules that sequence welfare and adjusted proportional rules achieve improved trade-offs on both goals.

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

The paper introduces mixed voting rules for participatory budgeting elections as sequences of existing rules, each allocated a fraction of the total budget to select projects. It adapts the Method of Equal Shares to follow an initial rule such as Greedy by adjusting each voter's remaining budget according to their satisfaction from already selected projects. One specific adjustment method is shown to improve on a natural proportionality baseline for axioms such as EJR+, with the result holding for general additive satisfaction functions. Extensive experiments on real PB elections indicate that the resulting mixed rules can simultaneously deliver high utilitarian welfare and strong proportionality, outperforming pure applications of either component rule. The framework is also applied to rules other than Greedy and MES, identifying refinements that further improve the observed trade-offs.

Core claim

Sequencing a utilitarian rule such as Greedy with a satisfaction-adjusted version of the Method of Equal Shares allows the second rule to correct representation imbalances created by the first, producing outcomes that improve upon a natural proportionality baseline while retaining favorable utilitarian welfare; the improvement holds for general additive satisfaction functions and is confirmed empirically across real-world participatory budgeting elections.

What carries the argument

Mixed voting rules formed by sequential composition, in which each rule spends a predetermined fraction of the total budget and the Method of Equal Shares is adapted by reducing each voter's budget in proportion to their satisfaction from projects already chosen.

If this is right

  • One adjustment method for MES voter budgets improves upon the natural proportionality baseline for EJR+.
  • The improvement extends from unit-demand to general additive satisfaction functions.
  • Mixed rules achieve better welfare-proportionality trade-offs than pure rules on real PB election data.
  • Several refinements of the mixed framework further improve empirical performance.
  • The same sequential composition approach applies to PB rules other than Greedy and MES.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Organizers could tune the welfare-proportionality balance simply by varying the budget fractions allocated to each component rule.
  • The sequential correction idea might extend to online or multi-round participatory budgeting settings where satisfaction accumulates over time.
  • Similar mixing could be tested in related domains such as multi-winner voting to balance score-based and proportional selection.

Load-bearing premise

That adjusting voter budgets in the second rule according to prior satisfaction corrects representation shortfalls without violating the overall budget limit or introducing new proportionality failures.

What would settle it

A concrete election instance in which the proposed mixed rule with the improved adjustment either violates EJR+ or produces a strictly worse welfare-proportionality pair than both pure Greedy and pure MES.

Figures

Figures reproduced from arXiv: 2606.23320 by Anton Baychkov, Markus Brill, Markus Utke.

Figure 1
Figure 1. Figure 1: Illustration of a mixed voting rule R as a sequence of rules with inputs and outputs. This process is illustrated in Fig￾ure 1. When the instance 𝐼 is clear from the context, we often drop it from the notation and write 𝑃 ∗ = R ( [𝐵𝑘 ], 𝑃0). We can think of a mixed rule R ( [𝐵𝑘 ]𝑘, 𝑃0) = [𝑅𝑘 ]𝑘 ( [𝐵𝑘 ]𝑘, 𝑃0) as splitting up the instance budget among the voting rules it contains, giving rule 𝑅𝑘 a budget of … view at source ↗
Figure 2
Figure 2. Figure 2: PB instance and pre-allocation outcomes for [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experimental results for mixing Greedy with MES (without budget increase, with Greedy completion) for different pre-allocation methods and greedy budget shares 𝛼𝐺 ∈ {0, 0.1, 0.2, . . . , 1}. Metrics are averaged over all instances with at least 20 projects. utilitarian welfare, further supports this observation. The average values of 𝛼 for which 𝛼-budget EJR+X is satisfied range between 1.5 and 2.9, well a… view at source ↗
Figure 4
Figure 4. Figure 4: Proportionality and welfare of mixing Greedy and MES with and without budget increase [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mixing [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relationships between PB proportionality notions, for [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Minimum voter budget share 𝛼 M in practice for different pre-allocation methods and Greedy bud￾get shares 𝛼G, averaged over all instances with at least 20 projects [PITH_FULL_IMAGE:figures/full_fig_p037_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Budget spending of each individual rule in the mixed rules [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Results for R 𝑀+ ( [𝛼G𝐵, 𝐵, 𝐵]) for 𝑀 ∈ {Null, MES-Style, Equal-Split, Value-Based} and 𝛼𝐺 ∈ {0, 0.1, 0.2, . . . , 1.0}. Metrics are averaged over all instances with at least 20 projects. Greedy MES Weak baseline Strong baseline Utilitarian welfare Proportionality [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Overlap between the outcomes of MES (with budget increase and greedy completion) and the mixed rule R Value-Based+ ( [𝛼G𝐵, 𝐵, 𝐵]), averaged over all instances with at least 20 projects. Splitting the Budget Between Exhaustive Rules. Splitting the budget between multiple rules can sometimes have undesirable consequences — for example, a project costing more than 50 % of the budget is unlikely to ever be se… view at source ↗
Figure 14
Figure 14. Figure 14: Performance of mixing Greedy with Greedy for different (first) Greedy budget shares 𝛼G from 0 to 1. (a) Proportionality over utilitarian ratio. (b) Budget spending of Greedy [PITH_FULL_IMAGE:figures/full_fig_p040_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Experimental results of mixing Greedy with and without early stopping with MESValue-Based+ , averaged over all instances with fewer than 20 projects. next project to be added is not affordable. We call this variant “Greedy with early stopping” and presented its effect on the experiments in Section 6.3. Curiously, if the first instance of Greedy in R G = [Greedy, Greedy] is implemented with early stopping,… view at source ↗
Figure 16
Figure 16. Figure 16: Comparison of experimental results when mixing [PITH_FULL_IMAGE:figures/full_fig_p041_16.png] view at source ↗
read the original abstract

Designing and analyzing voting rules for Participatory Budgeting (PB) elections is an active research area in computational social choice. Many PB voting rules aim to optimize a specific objective. For instance, the ubiquitous Greedy rule attempts to maximize utilitarian welfare, while the Method of Equal Shares (MES) aims to achieve proportional representation. However, it is often desirable to achieve good outcomes on multiple objectives rather than a close-to-perfect outcome for one. Inspired by mixed-member systems for parliamentary elections, we introduce mixed voting rules for PB. These are composed of a sequence of two or more rules that can each spend some fraction of the overall budget in order to add projects to the set selected by earlier rules. We develop a theoretical framework for formulating and analyzing mixed PB voting rules, and explore how existing rules can be adapted to this framework. We particularly focus on MES and its potential to address imbalances in representation created by earlier rules. We propose different ways to adjust MES voter budgets based on how satisfied voters are with previously chosen projects, and examine how well the resulting rules approximate well-known proportionality axioms such as EJR+. In particular, we show that one of these methods improves upon a natural proportionality baseline. We also extend our main positive result to general additive satisfaction functions. We complement our theoretical results with an extensive empirical analysis of real-world PB elections. Our experiments show that mixed rules can achieve favorable trade-offs between utilitarian welfare and proportionality. We identify several refinements that further improve their performance, and apply our framework to PB rules beyond Greedy and MES.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces a framework for mixed voting rules in participatory budgeting, where rules such as Greedy and MES are applied sequentially with allocated budget fractions. It proposes several satisfaction-based adjustments to MES voter budgets to correct representation imbalances from prior rules, proves that one such adjustment improves upon a natural proportionality baseline (with an extension to general additive satisfaction functions), and presents empirical results on real-world PB elections showing favorable trade-offs between utilitarian welfare and proportionality, along with refinements and extensions to other rules.

Significance. If the central positive result holds, the work offers a flexible, axiomatically grounded approach to balancing multiple objectives in PB that builds directly on existing rules without requiring entirely new mechanisms. The extension to additive satisfaction functions and the empirical validation on real elections add practical value; the framework's modularity for composing rules is a clear strength.

major comments (2)
  1. [§4.3] §4.3, Theorem 2: the proof that the satisfaction-adjusted MES improves the EJR+ baseline assumes the first rule's output is fixed before the second rule runs; it is unclear whether the argument extends without additional conditions when the budget fraction allocated to the first rule varies continuously.
  2. [§5.2] §5.2, the empirical comparison of mixed rules: the reported improvements in the proportionality-utilitarian trade-off rely on post-hoc refinements whose selection criteria are not pre-specified; this raises the possibility that the favorable outcomes are partly due to data-driven tuning rather than the core mixed-rule construction.
minor comments (3)
  1. [§3.2] §3.2: the definition of the voter-budget adjustment function could include a short numerical example to illustrate how prior satisfaction is subtracted before the second rule is invoked.
  2. [Figure 3] Figure 3: axis labels and legend entries are too small for readability when printed; consider increasing font size or splitting into two panels.
  3. [§6] The paper mentions applicability beyond Greedy and MES but provides only brief sketches; a short additional subsection with one concrete example for another rule would strengthen the framework claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the recommendation of minor revision. We address each major comment below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [§4.3] §4.3, Theorem 2: the proof that the satisfaction-adjusted MES improves the EJR+ baseline assumes the first rule's output is fixed before the second rule runs; it is unclear whether the argument extends without additional conditions when the budget fraction allocated to the first rule varies continuously.

    Authors: The framework defines mixed rules for any fixed budget fractions allocated in advance; the first rule's output is therefore fixed once its share is chosen. Theorem 2 shows the improvement holds for every such fixed allocation. The proof does not claim to cover continuous variation of the fraction as a free parameter, which would require separate continuity arguments. We will add a short clarifying sentence noting that the result applies pointwise to any chosen fixed fractions. revision: partial

  2. Referee: [§5.2] §5.2, the empirical comparison of mixed rules: the reported improvements in the proportionality-utilitarian trade-off rely on post-hoc refinements whose selection criteria are not pre-specified; this raises the possibility that the favorable outcomes are partly due to data-driven tuning rather than the core mixed-rule construction.

    Authors: We accept the concern. The primary empirical claims in §5.2 concern the basic mixed rules; the refinements are presented as optional, post-hoc observations. In the revision we will (i) explicitly separate the core results from the refinements, (ii) state the refinement criteria upfront, and (iii) move the detailed refinement experiments to an appendix so that the main findings stand independently of them. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proposes a new sequential composition framework for mixed PB rules, defining budget fractions and satisfaction-based MES adjustments directly from the input rules (Greedy, MES) and axioms (EJR+). The central positive result—that one adjustment method improves a natural proportionality baseline, with extension to additive satisfaction—is stated as a theorem derived from the explicit construction, without reducing to fitted parameters, self-definitions, or load-bearing self-citations. Empirical trade-offs on real-world data are independent of the theoretical claims. No enumerated circularity pattern applies; the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; no fitted values or new postulated objects are described.

pith-pipeline@v0.9.1-grok · 5804 in / 1013 out tokens · 25060 ms · 2026-06-26T06:03:49.217736+00:00 · methodology

discussion (0)

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