Method of Equal Shares for Fair Budgeting

Updated 5 December 2025

Method of Equal Shares is a participatory budgeting mechanism that allocates equal voter budgets to projects using an iterative, price-based selection process.
It guarantees Extended Justified Representation (EJR) by charging supporters proportionally based on declared utilities and project costs.
MES and its variants, such as EES and BOS, optimize the trade-off between fairness and efficiency while enabling explainable and robust allocation outcomes.

The Method of Equal Shares (MES) is a class of iterative selection rules developed for participatory budgeting under arbitrary project costs and additive voter utilities. MES provides proportionality guarantees aligned with the Extended Justified Representation (EJR) axiom, ensures efficient computability, and underpins a family of robust, priceable, and explainable allocation mechanisms for diverse collective decision-making settings.

1. Formal Model and Definition

Consider a participatory budgeting instance with:

A set of voters $N = \{1, \ldots, n\}$ ,
A set $C$ of projects, each with cost function $\mathrm{cost}(c) > 0$ for $c \in C$ ,
Total available budget $b \in \mathbb{Q}_{>0}$ ,
Additive utilities: Each voter $i$ specifies $u_i: C \to \mathbb{Q}_{\geq 0}$ .

An outcome $W \subseteq C$ is feasible if $\sum_{c \in W} \mathrm{cost}(c) \leq b$ . MES aims to select a feasible set $W$ respecting proportionality constraints.

The distinguishing feature of MES is the equal initial allocation of the budget: each voter $i$ receives $b_i \gets b/n$ in an internal “bank account.” Projects are iteratively selected and funded via “price per unit utility” charged to their supporters, subject to supporters’ remaining balances.

Formal Rule

At each iteration:

For every $c \notin W$ and scalar price $p > 0$ , compute per-voter payment $p_i(c) = \min\{b_i,\, p \cdot u_i(c)\}$ .
Project $c$ is $p$ -affordable if $\sum_{i} p_i(c) = \mathrm{cost}(c)$ .
For each $c \notin W$ , let $p(c)$ be the minimum such $p$ .
Select $c^* \in \arg\min_{c \notin W} p(c)$ ( $c^*$ is the project with the lowest “price per unit utility” at this stage).
Update all $i$ : $b_i \gets b_i - \min\{b_i,\, p(c^*) \cdot u_i(c^*)\}$ .
Add $c^*$ to $W$ . Repeat until no project is $p$ -affordable.

This rule ensures no voter expends more than their allotted budget and that payments for each project are proportional to declared utility, subject to individual limits (Peters et al., 2020).

2. Pseudocode and Computational Properties

A prototypical polynomial-time implementation (for $m$ projects and $n$ voters) executes as follows:

Initialize: For all i, set b_i := b/n, W := ∅
While ∃ c ∉ W p-affordable:
    For each c ∉ W:
        Let S_c := {i | u_i(c) > 0}
        If ∑_{i∈S_c} b_i < cost(c): set p(c) = ∞
        Else:
            Sort S_c by increasing b_i/u_i(c)
            For s in 1..|S_c|:
                Compute minimal p so that
                   sum_{j < s} b_{i_j} + p * sum_{j=s}^t u_{i_j}(c) = cost(c)
                If p * u_{i_s}(c) ≤ b_{i_s}, set p(c) := p; break; update sum
    Pick c* := argmin_{c ∉ W} p(c)
    If p(c*) = ∞, halt and return W
    For each i: Δ_i := min(b_i, p(c*) * u_i(c*)); b_i := b_i - Δ_i
    Add c* to W

MES is computable in strongly polynomial time $O(m^2 n \log n)$ given unit-cost arithmetic (Peters et al., 2020). Each step fully pays for the selected project, and the running sequence of $p(c^*)$ is non-decreasing.

3. Proportionality Guarantees and Axiomatic Foundations

The axiomatic foundation of MES is the Extended Justified Representation (EJR) property. For additive or approval utilities with arbitrary project costs, MES guarantees for any $(a,T)$ -cohesive group $S \subseteq N$ (i.e., $|S|/n > \mathrm{cost}(T)/b,\, \forall i \in S,\, \forall c \in T: u_i(c) \geq a(c)$ ), there exists $i^* \in S$ such that $u_{i^*}(W) \geq \sum_{c \in T} a(c)$ , or at worst, $u_{i^*}(W \cup \{c^*\}) > \sum_{c \in T} a(c)$ for some $c^* \in T$ .

In approval-cost settings (i.e., $u_i(c) \in \{0, \mathrm{cost}(c)\}$ ), MES achieves exact EJR. In the general additive case, the guarantee is EJR up to one project. Checking EJR with arbitrary utilities is NP-hard (would solve a knapsack), but MES's construction assures the EJR-up-to-one-project guarantee by design (Peters et al., 2020).

A stronger axiom, Full Justified Representation (FJR), can be achieved by a computationally demanding greedy core rule, but not by MES in general.

MES has been empirically validated to outperform cost-greedy rules on Gini-based fairness metrics and happiness (fraction of non-empty-handed voters), even when trading off some utilitarian welfare (Nelissen, 2023).

4. Welfare Guarantees and Efficiency–Proportionality Trade-offs

MES’s selection for proportionality can, in principle, sacrifice global utilitarian welfare. For classes of weakly-decreasing normalized satisfaction (DNS) functions (which includes cost-satisfaction and cardinality-satisfaction), MES admits the following tight lower bound (Baychkov et al., 25 Nov 2025):

Let $c_{\min} = \min_{p} \mathrm{cost}(p)$ , $c_{\max} = \max_{p} \mathrm{cost}(p)$ , and $k_{\min} = b/c_{\max}$ , $k_{\max} = b/c_{\min}$ . Then

$\frac{\operatorname{UW}_s(\mathrm{MES}_s(I))}{\operatorname{UW}_s(P^*)} \geq 2\sqrt{\frac{c_{\min}}{b}-\frac{c_{\min}+c_{\max}}{b}}$

where $P^*$ is the utilitarian-optimal bundle and $s$ is the satisfaction function. This bound is asymptotically tight among all EJR-1 rules. In unit-cost settings, this reproduces classical multiwinner lower bounds.

Empirical data confirm that while MES may reduce average cost satisfaction slightly, it yields more equal distribution (reducing the satisfaction Gini index) and increases the fraction of satisfied voters (Nelissen, 2023).

5. Exhaustiveness, Variants, and Implementation Considerations

A characteristic limitation of MES is its non-exhaustiveness: after termination, some budget may remain unused with no affordable projects left under current balancing constraints (Kraiczy et al., 17 Feb 2025). In practical deployments, this is commonly addressed by "add-one" heuristics (increasing the virtual budget in unit steps until an exhaustive bundle is reached), but this results in high computational overhead.

The Exact Equal Shares (EES) variant restricts payments to exact equal shares per project and supports a more efficient completion via the add-opt heuristic, which computes, in $O(mn)$ (cardinal) or $O(m^2 n)$ (uniform), the minimal budget increment that changes the EES outcome. Empirical analyses show EES+add-opt matches or exceeds MES+add-one in spending efficiency but with dramatically fewer rule invocations (Kraiczy et al., 17 Feb 2025).

The Adaptive Method of Equal Shares (AMES) incrementally builds on existing MES output to quickly adapt to increased budgets, leveraging invariants in the per-voter price structure. Verifiable certificates for EJR can be efficiently constructed after each update (Kraiczy et al., 2023).

6. Extensions: Hybrid, Fractional, and Bounded-Overspending MES

The Komitee Equal Shares (KES) framework generalizes MES by introducing parallel "voter" and "impact-field agent" modes, allowing budget partitions between individuals and co-designed collective value proxies. KES retains priceability (a ledger of who pays for what) and enables fine-grained, explainable allocations with per-agent "voting receipts" (Yang et al., 2 Oct 2025).

The Method of Equal Shares with Bounded Overspending (BOS) permits controlled budget overruns for individual voters, relaxing strict EJR in favor of higher budget utilization and efficiency. BOS provides approximate proportionality: for any cohesive group $S$ , some group member secures utility at least $\text{cost}(T) - \frac{n-|S|}{2|S|}c^*$ , where $c^* = \max_c \text{cost}(c)$ . BOS and its fractional version FrES close the budget utilization gap seen in MES while maintaining practical proportionality (Papasotiropoulos et al., 23 Sep 2024).

These frameworks offer trade-offs between provable fairness, welfare maximization, and practical computational performance. Empirical studies across real-world participatory budgeting instances have demonstrated scalability, rapid execution, and favorable fairness-efficiency profiles for MES and its variants (Kraiczy et al., 17 Feb 2025, Papasotiropoulos et al., 23 Sep 2024, Nelissen, 2023).

7. Comparison with Proportional Approval Voting (PAV)

MES generalizes the proportional reasoning underpinning Proportional Approval Voting (PAV) from the unit-cost, approval committee context to the full participatory budgeting model with arbitrary costs and additive utilities. Crucially, PAV’s cost ignorance makes it incompatible with cost-sensitive proportionality when projects differ in costs: it can systematically underrepresent majorities or minorities in district-structured settings purely due to its fixed objective function (Peters et al., 2020). MES, in contrast, ensures that proportionality accounts for both support and cost, enforcing district-level (or group) fairness automatically.

The following table summarizes key differences:

Rule	Cost-Aware	Proportionality	Computability
MES	Yes	EJR (EJR-1)	Strongly polynomial
PAV	No	EJR (unit-cost)	Strongly polynomial

MES is the unique (up to minor variations) polynomial-time rule with EJR guarantees under arbitrary utilities and costs, and, unlike PAV, is robust to cost heterogeneity (Peters et al., 2020, Baychkov et al., 25 Nov 2025).

In summary, the Method of Equal Shares framework provides a theoretically-sound, computationally efficient, and widely adopted approach for proportional public goods selection in participatory budgeting and related allocation problems. Its formal and empirical properties have motivated a suite of extensions balancing proportionality, efficiency, and practical explainability, making it central to the design of contemporary fair allocation mechanisms.