Democratic Budget Aggregation

Updated 7 November 2025

Democratic budget aggregation is a method that combines individual votes into a vector allocation of public funds using formal algorithms and utility metrics.
It distinguishes between divisible and indivisible allotments while integrating diverse inputs like approvals, rankings, and cumulative votes.
Mechanisms such as moving phantoms and MES ensure fairness, proportionality, and computational efficiency in resource allocation.

Democratic budget aggregation refers to the formal and algorithmic processes by which individual preferences, proposals, or ballots regarding resource allocation are combined into a collective decision over how public funds or budgets are distributed. The field encompasses models and methods for participatory budgeting, allocation games, multi-winner voting, and priceable mechanisms, with particular focus on incentive, fairness, and computational guarantees. It draws on techniques from computational social choice, mechanism design, operations research, and welfare economics.

1. Formal Foundations and Models

Democratic budget aggregation generalizes social choice by considering outcomes that are not simply selections from a set, but vectors (or multisets) corresponding to resource shares among multiple projects, issues, or alternatives. Two primary modes are distinguished:

Divisible budget aggregation: Each alternative/project can receive any fraction from zero to full budget, resulting in outcomes in the $m$ -dimensional simplex $\Delta_{m-1}$ .
Indivisible (discrete) budget aggregation: Each project is either funded in whole, in pre-specified degrees, or not at all, subject to overall resource constraints.

The individual input takes various forms—ideal point proposals, approval ballots, cumulative votes, rankings, or ordinal preferences—and aggregation rules must interpret and integrate these into a single feasible allocation.

Underlying models assume objective functions or utility metrics:

$\ell_1$ -disutility (linear distance from proposal, (Freeman et al., 2019))
Submodular or non-additive utility to capture complex interactions or diminishing returns (Yuan et al., 19 Jun 2024)
Ordinal and approval axioms (PR, EJR, PJR, PSC, etc.) (Aziz et al., 2017, Aziz et al., 2019)
Game-theoretic utility functions (linear, Leontief, etc.) for strategic or Nash equilibrium-based settings (Becker et al., 10 Sep 2025)

2. Proportionality and Representation: Axiomatic Developments

A principal axis of research is the formalization and achievement of proportional (fair) representation, required to prevent domination by majorities and guarantee minorities' influence commensurate with their size.

Key Axioms

Axiom Name	Description
Justified Representation (JR)	Large cohesive groups entitled to some budget/project from their consensus (Aziz et al., 2017)
Proportional Justified (PJR)	Strengthens JR: groups receive (in sum) no less than what their share can afford
Extended JR (EJR)	Some member of each group gets the full bundle they could win by pooling their share
Local-BPJR	Weaker/feasible proportionality for indivisible goods, ensures some guarantee locally
Inclusion PSC (IPSC)	New inclusion-based axiom for ordinal PB, ensuring group-fundable candidates are not excluded

Axioms are generalized to accommodate arbitrary project costs (instead of uniform costs), variable group sizes, and flexible budget constraints.

Algorithmic and Complexity Insights

For divisible allocations, proportionality is algorithmically compatible with truthfulness under certain norms, but generally cannot be combined with welfare maximization (Freeman et al., 2019).
In indivisible PB, strong forms of proportionality may be NP-hard to check or enforce, and some (e.g., Strong-BPJR) may not always be feasible (Aziz et al., 2017).
Polynomial-time algorithms (e.g., Generalized Phragmén, PB Expanding Approvals (Aziz et al., 2019)) guarantee weaker but meaningful proportionality.

3. Aggregation Mechanisms and Rules

Truthful Mechanisms

Moving Phantom Mechanisms: Under $\ell_1$ disutility, these are the unique class guaranteeing incentive compatibility, anonymity, neutrality, and continuity. The mechanism computes the median (possibly with "phantom" proposals) for each coordinate, subject to normalization (Freeman et al., 2019, Freeman et al., 2023).
Ladder Mechanism: A simple moving phantom mechanism with optimal project fairness ( $\ell_\infty$ proximity to mean) for three or more projects (Freeman et al., 2023).
Beyond Moving-Phantoms: Mechanisms like CutoffGreedyMax exist, but do not improve worst-case fairness beyond moving-phantoms (Berg et al., 30 May 2024).

Impossibility Results

No mechanism can achieve both strict proportionality (e.g., full JR) and truthfulness for indivisible allocations (Schmidt-Kraepelin et al., 9 May 2025).
For fractional-input, integral-output, only dictatorial mechanisms can be truthful and onto (Schmidt-Kraepelin et al., 9 May 2025).
Even extending the class of mechanisms does not overcome mean-approximation bounds for truthfulness and fairness (Berg et al., 30 May 2024).

Proportional and Welfare-Optimal Mechanisms

Method of Equal Shares (MES), Rule X, Substitute Rule X (SRX): Allocate virtual spending power (budget share) to agents, funding projects as soon as their supporters can collectively pay the cost (with extensions for project interactions/substitutes) (Fairstein et al., 2021, Yang et al., 2 Oct 2025).
GREEDY, Phragmén, Monroe/Chamberlin–Courant Generalizations: Greedy or sequential rules that select projects iteratively per support, cost, or representation; variously adapted for approvals, rankings, or costs (Page et al., 9 Oct 2024, Aziz et al., 2019).
Minimal Transfers over Costs (MTC): Proportional, CSTV-inspired method for cumulative votes; provides strong PR and robust minority protection (Skowron et al., 2020).
Nash Equilibrium Approaches: Allocate virtual power to each agent; Nash equilibria in these games match individual fair share and can now be computed in polynomial time for Leontief and related utilities (Becker et al., 10 Sep 2025).

4. Handling Project Interactions and Realism

Classical models assume additive utilities, but emerging research incorporates more realistic interaction structures:

Project Substitutes and Complements: Expressive ballots allow voters to encode substitution (e.g., need only one of group) and complementarity (value only if all are funded) dependencies. Aggregation mechanisms such as SRX refine proportional guarantees and dynamically update marginal utility per round (Fairstein et al., 2021, Goyal et al., 2023).
Submodular Utilities: Voters' satisfaction may obey diminishing returns, modeled as monotone submodular utility over funded sets. Aggregation with rankings, threshold approvals, or values can outperform additive-focused rules, with rigorously analyzed "distortion" guarantees (Yuan et al., 19 Jun 2024).
Multiple Degrees and Ranged Approvals: Each project may have several permissible funding levels, and voters approve ranges per project. The model unifies and extends knapsack and approval frameworks, with dedicated utility/disutility rules and tractable algorithms (Sreedurga, 2023).
Metric Aggregation: Treats all possible budgets as a metric space—each voter submits an ideal point, and aggregation minimizes total or worst-case distance (e.g., $L_p$ medians, Condorcet points) (Bulteau et al., 2018).

5. Computational and Implementation Considerations

Mechanism Type	Complexity	Notable Results
Moving phantom (fractional)	Poly-time	Unique in incentive compatibility.
Indivisible Priceable	Poly-time/FPT/CNP-hard	MES, Generalized Phragmén are tractable; strong BPJR-L is NP-hard (Aziz et al., 2017).
Project Interactions	FPT in group size (Goyal et al., 2023)	Social welfare maximization NP-hard; tractable by dynamic programming for bounded group size.
Nash equilibrium (Leontief)	Poly-time (Becker et al., 10 Sep 2025)	Polynomial algorithm for a class previously only characterizable, not computable efficiently.
Submodular PB	Poly-time for approval-based rule	$O(\log m)$ distortion for threshold approvals (improves previous $O(\log^2 m)$ ) (Yuan et al., 19 Jun 2024).

Performance guarantees often depend on the input size, utility class, and parameterization by group size, project cost variance, or number of approval levels.

6. Empirical and Behavioral Foundations

Standard theoretical assumptions (e.g., $\ell_1$ , $\ell_2$ , Leontief) align poorly with observed human behavior. Empirical studies reveal that:

Most participants' preferences are "star-shaped" (single-peaked about their ideal) but frequently asymmetric by project and direction.
Peak-linear and weighted models fit data better than symmetric distance metrics.
Mechanism design should be empirically grounded, possibly leveraging direct polling and flexible utility forms rather than elegant but unrealistically symmetric axioms (Amster et al., 28 Oct 2025).

7. Integration of Deliberation, Evaluation, and Process Design

Recent frameworks emphasize that democratic aggregation should support not only preference collection but also co-evaluation and deliberation:

Hybrid Aggregation: Algorithms such as Komitee Equal Shares (KES) extend MES to aggregate both individual votes and collective group evaluation over impact fields, unifying subjective and deliberative signals in a priceable, auditable manner (Yang et al., 2 Oct 2025).
Transparency and Receipts: Voting receipts detail who (which voters or impact fields) funded which projects, improving explainability and procedural legitimacy.
Reality-aware Process: Algorithms can incorporate last-year’s budget for continuity and provide participants with tools for ranked, partial, or modification-oriented input (Shapiro et al., 2017).

8. Ongoing Challenges and Research Directions

Formal impossibility theorems delineate the limits of joint truthfulness, proportionality, and efficiency, leaving scope for context- or goal-specific tradeoffs (Berg et al., 30 May 2024, Schmidt-Kraepelin et al., 9 May 2025).
Strengthening empirical foundations for axioms and utility assumptions remains critical to robust, democratically legitimate mechanism design (Amster et al., 28 Oct 2025).
Integrating expressivity, computational tractability, and platform usability is an active area, especially as real-world participatory budgeting scales up in city and national contexts.

In summary, democratic budget aggregation is a multi-faceted domain where technical, axiomatic, and behavioral considerations intersect. Rigorous models and algorithms provide a spectrum of possible mechanisms, balancing truthfulness, proportionality, welfare, and practicability, while empirical and process design work identifies the requirements and constraints of real democratic deliberation and allocation.