Method of Equal Shares (MES)

Updated 27 November 2025

Method of Equal Shares (MES) is a polynomial-time allocation rule for participatory budgeting that divides the budget equally among voters to achieve strong proportionality.
It satisfies Extended Justified Representation and offers transparent priceability, though it often results in leftover budget due to non-exhaustiveness.
Variants such as fractional MES, bounded overspending, and adaptive MES improve budget utilization, scalability, and efficiency while preserving fairness.

The Method of Equal Shares (MES) is a polynomial-time allocation rule for participatory budgeting (PB) that achieves strong proportionality guarantees by dividing the total budget equally among all voters and iteratively allocating funds to projects in a manner that charges supporters of each project as uniformly as possible. Originating from the work of Peters and Pierczyński, MES has established itself as a central algorithm in the computational social choice literature and has seen real-world deployments in municipal budgeting processes across Europe. MES satisfies Extended Justified Representation (EJR), a demanding proportionality axiom, and offers priceability—a property that makes the cost distribution transparent and verifiable to participants. However, MES is not exhaustive: it typically finishes with leftover budget even though further projects could be feasibly funded. Recent research has focused on adaptive, fractional, and heuristic-completion variants of MES that improve exhaustion, scalability, and efficiency, while retaining its core fairness properties.

1. Formal Framework and Algorithmic Definition

MES operates in a participatory budgeting instance where $N = \{1,\ldots,n\}$ is the set of voters, $P = \{p_1, ..., p_m\}$ is the set of projects with costs $c(p) > 0$ , and the total public budget is $b > 0$ . Each voter $i$ submits an approval ballot $A(i) \subseteq P$ . An outcome is a subset $W \subseteq P$ such that $\sum_{p \in W} c(p) \leq b$ .

MES assigns each voter an initial virtual budget $b_i = b/n$ and iteratively selects projects according to the following steps (Kraiczy et al., 2023, Nelissen, 2023, Kraiczy et al., 17 Feb 2025):

For each project $p \notin W$ , compute the minimal $\rho(p)$ such that:

$\sum_{i: p \in A(i)} \min(\rho(p), \mathrm{rem}_i) \geq c(p)$

where $\mathrm{rem}_i$ is the current remaining budget of voter $i$ .

Identify $p^* = \arg\min_p \rho(p)$ among unaffordable projects. If $\rho(p^*) = \infty$ , halt.
Add $p^*$ to $W$ . Each supporter $i$ contributes $\min(\rho(p^*), \mathrm{rem}_i)$ , and $\mathrm{rem}_i$ is reduced accordingly.
Repeat until no further project can be afforded by its supporters.

This iterative process results in a “priceable” solution, with each project funded by a coalition of its approvers, each paying an equal or maximal possible share, subject to individual budget constraints.

A formal pseudocode sketch appears in multiple sources and can be summarized as follows:

def MES(N, P, A, cost, b):
    rem = {i: b/n for i in N}
    W = set()
    x = { (i,p): 0 for i in N for p in P }
    while True:
        rho = {}
        for p in P - W:
            # Find smallest rho such that sum_{i: p in A(i)} min(rho, rem[i]) >= cost(p)
            /* details omitted for brevity */
            rho[p] = compute_affordability(p, rem)
        if all(rho[p] == float('inf') for p in rho):
            break
        p_star = argmin_p(rho[p])
        for i in N:
            if p_star in A(i):
                contr = min(rho[p_star], rem[i])
                x[i, p_star] = contr
                rem[i] -= contr
        W.add(p_star)
    return W, x

(Kraiczy et al., 2023, Kraiczy et al., 17 Feb 2025)

2. Proportionality and Extended Justified Representation

MES is foundationally motivated by proportionality, specifically the Extended Justified Representation (EJR) axiom. Intuitively, EJR ensures that any group of voters large and cohesive enough to “deserve” a set of projects (according to their approval overlap and aggregate virtual budget) will see at least one member’s justified share of those projects funded.

Formally, a group $S \subseteq N$ is $T$ -cohesive for $T \subseteq P$ if $\frac{|S|}{n} \geq \frac{\sum_{p \in T} c(p)}{b}$ and $T \subseteq \cap_{i \in S} A(i)$ . MES provides EJR if for every $T$ and $T$ -cohesive $S$ , at least one member $i \in S$ satisfies $|A(i)\cap W| \geq |T|$ (Kraiczy et al., 2023, Nelissen, 2023).

MES was established as the first polynomial-time participatory budgeting rule (for approval ballots) to satisfy EJR comprehensively, advancing beyond prior approaches that could not efficiently guarantee such strong proportional protections (Kraiczy et al., 2023, Kraiczy et al., 17 Feb 2025).

3. Complexity, Non-exhaustiveness, and Practical Heuristics

The computational complexity of MES is $O(m^2 n)$ , with each iteration examining $O(mn)$ project-voter pairs and at most $m$ iterations (Kraiczy et al., 2023, Kraiczy et al., 17 Feb 2025). MES’s key limitation is non-exhaustiveness: it may stop before fully allocating the available budget, as some projects cannot be equitably funded by their remaining supporters due to per-head constraints.

To address non-exhaustiveness, practical implementations frequently employ “completion heuristics.” The most common is the “add-one” heuristic, which incrementally increases the per-voter virtual budget and reruns MES until the resulting outcome is exhaustive for the original budget—a process that is computationally expensive, especially for fine-grained increments or large elections (Kraiczy et al., 17 Feb 2025, Kraiczy et al., 2023).

Recent work introduced more efficient approaches: the Exact Equal Shares (EES) rule and the “add-opt” heuristic. EES simplifies the sharing of project costs, strictly enforcing equal partition among supporters with sufficient remaining budget, while add-opt efficiently computes the minimum per-voter budget increment needed to change the EES outcome, reducing the number of full re-runs required for exhaustiveness (Kraiczy et al., 17 Feb 2025).

In Table 1, practical methods for MES outcome completion are outlined:

Method	Exhaustiveness	Complexity
Add-one	Yes (by increment)	$O(d_{\max}\cdot T)$
Add-opt (EES only)	Yes (efficiently)	$O(mn)$

This table distills completion methods and their scaling, as detailed in (Kraiczy et al., 17 Feb 2025).

4. Variants: Fractional, Bounded Overspending, and Adaptive MES

Strong MES variants have been proposed to overcome efficiency and fairness pathologies:

Fractional MES (FrES): Allows projects to be partially funded ( $W_c \in [0,1]$ ), leading to full budget exhaustion and satisfying full fractional EJR. Each round, the maximum affordable fraction of each project is purchased with remaining budgets, and the process continues until all budgets are expended or all projects are fully funded (Papasotiropoulos et al., 23 Sep 2024).
Bounded Overspending (BOS): Modifies MES to guarantee exhaustiveness by permitting controlled “overspending”: supporters may temporarily exceed their virtual budget to fund a project, with per-project overspend bounded so that proportionality and price-per-utility remain nearly optimal. BOS yields significantly higher budget utilization and retains quantitative approximate-EJR (Papasotiropoulos et al., 23 Sep 2024).
Adaptive Method of Equal Shares (AMES): AMES is an adaptive extension of MES that, given a MES outcome for some budget $b$ , efficiently computes the updated allocation for a new budget $b' > b$ by leveraging similarities between solutions, instead of recomputing from scratch. This yields substantial empirical speed-ups and enables dynamic, responsive participatory budgeting scenarios (Kraiczy et al., 2023).

5. Empirical Evaluation and Welfare Tradeoffs

Empirical studies demonstrate that MES and its variants deliver substantially improved fairness metrics against utilitarian or greedy cost-welfare rules, at a modest price in total voter satisfaction (Nelissen, 2023, Baychkov et al., 25 Nov 2025, Papasotiropoulos et al., 23 Sep 2024). Across hundreds of real-world PB instances, MES-based outcomes exhibit:

Lower Gini coefficients for both cost satisfaction and effort compared to greedy rules, indicating more equal distributions of funded project benefits.
Higher voter "happiness" (fraction approving at least one funded project), with increases from 94% (greedy) to 96–97% (MES) in real Amsterdam elections.
More, cheaper projects funded and greater category proportionality in most cases (Nelissen, 2023).

MES typically spends less of the available budget compared to exhaustive (utilitarian) rules. However, with effective completion heuristics or BOS, nearly full budget utilization can be achieved without material loss of proportional fairness (Papasotiropoulos et al., 23 Sep 2024, Kraiczy et al., 17 Feb 2025). The fundamental welfare tradeoff is that MES’s proportionality comes at a small but consistent reduction in aggregate voter satisfaction, which is asymptotically minimal for large budgets with well-distributed project costs (Baychkov et al., 25 Nov 2025).

Recent theoretical analysis establishes explicit lower bounds on MES’s utilitarian welfare for any strictly positive, weakly-decreasing normalized satisfaction function (DNS). The bound:

$UW_\mu(W) \geq \left( 2\sqrt{c_{\min}/b} - \frac{c_{\min}+c_{\max}}{b} \right)\cdot OPT$

is asymptotically tight for all proportional (EJR $_1$ ) rules (Baychkov et al., 25 Nov 2025).

6. Extensions: Virtual Budgets, Human-in-the-Loop, and Mixed Aggregation

Recent applied work has extended MES to support hybrid participatory processes and enhanced transparency:

Virtual Budgets and Dual-Role Aggregation: In Komitee Equal Shares, MES is extended to aggregate over both individual voters and deliberatively weighted “impact fields,” dividing the budget into virtual buckets allocated according to point-votes and group evaluations. Each participant’s and each field’s support is normalized before running MES, and funding splits (voting receipts) are provided to every agent (Yang et al., 2 Oct 2025).
Human-in-the-Loop MES: MES has been implemented with real-time digital feedback, slider-controlled delegation of budget to algorithmic allocation, and post-selection veto/discussion. This approach allows participants direct control over the degree of automation and interactive exploration of MES outcomes, enhancing trust and transparency (Yang et al., 7 Feb 2025).

These implementations underscore the modularity and explainability of MES: priceability facilitates the construction of participant-facing “receipts,” and the core algorithm adapts flexibly to a wide class of utility models, support structures, and hybrid aggregation schemes.

7. Summary Table: Main MES Families and Properties

Variant	Exhaustive	Proportionality (EJR)	Computation	Practical Notes
MES	No	Yes	$O(m^2 n)$	Baseline, widely implemented
EES	No	Yes	$O(m^2 n)$	Strict equal split (no partial pay)
FrES (fractional)	Yes	Yes (fractional EJR)	Poly	Partially funds projects
BOS	Yes	Approximate	$O(m^2 n)$	Controlled overspending
MES + Add-one	Yes (slow)	Yes	Up to $O(h m^2 n)$	Incremental completion
EES + Add-opt	Yes (fast)	Yes	$O(m n)$	Direct to next optimal budget
AMES	No/Partial	Yes	$O(n\log n+mn)$ per update	Adaptive budget extensions
Human-in-the-Loop	Configurable	Yes	Poly (interactive)	User-controlled, deliberative