Equi-Percentage Scheme in Participatory Budgeting

• Equi-Percentage Scheme is a family of budgeting mechanisms that assigns equal virtual endowments to voters and allocates funds via percentage-based splits.
• It encompasses methods such as MES, EES, AMES, and KES, where supporters share project costs equally, ensuring transparency and proportionality.
• The approach balances strict equal-sharing with adaptive strategies to achieve verifiable extended justified representation and computational efficiency.

Searching arXiv for papers relevant to “Equi-Percentage Scheme,” especially where the term is absent but closely related equal-shares or percentage-based allocation frameworks appear. “Equi-Percentage Scheme” is not a standard rule name in the arXiv literature cited here. In participatory budgeting and related participatory allocation work, the nearest formal objects are the Method of Equal Shares (MES), Adaptive Method of Equal Shares (AMES), Exact Equal Shares (EES), Komitee Equal Shares (KES), Fractional Equal Shares (FrES), and BOS Equal Shares. Across these formulations, the common structure is that participants receive virtual budget entitlements, projects are funded only when the relevant supporters can cover their costs, and proportionality is interpreted through equal endowments, equal per-supporter payments, or explicit percentage splits of the total budget rather than through a rule literally named “Equi-Percentage Scheme” (Kraiczy et al., 2023, Kraiczy et al., 17 Feb 2025, Yang et al., 2 Oct 2025, Papasotiropoulos et al., 2024).

1. Terminology and conceptual scope

In the participatory-budgeting literature, the phrase “Equi-Percentage Scheme” does not appear as a canonical label. The closest concept is an equal-shares or equal-endowment rule: each voter begins with the same virtual budget, typically $b/n$, and projects are financed by supporters out of these equal initial entitlements. Under this interpretation, “percentage” is implicit in the equal fraction of the global budget assigned to each voter, while “equal shares” is the operative term used by the literature (Kraiczy et al., 2023).

A second, distinct interpretation appears in hybrid allocation frameworks that begin from an explicit percentage split of the total budget across decision modes. KES is the clearest example: a parameter $r$ assigns $rB$ to evaluator mode and $(1-r)B$ to individual-voter mode, after which the mechanism distributes the individual part equally and the evaluator part proportionally to impact-field weights. If “Equi-Percentage Scheme” is understood as a scheme that first partitions the budget by explicit percentages and then allocates within those shares through virtual budgets, KES is the closest formal instance in the cited literature (Yang et al., 2 Oct 2025).

A common misconception is that an equal-share or equi-percentage interpretation requires every voter to pay the same percentage toward every selected project. The cited PB rules do not work that way. They are support-based financing rules: only supporters of a project pay for it, while equality is imposed on initial endowments, on per-supporter prices, or on within-project cost splits among the participating supporters (Kraiczy et al., 2023, Kraiczy et al., 17 Feb 2025).

2. Core formalism of equal-shares allocation

The standard approval-based PB model is an election

$E=(N,P,(A(i))_{i\in N},b,c),$

where $N=[n]$ is the set of voters, $P=\{p_1,\dots,p_m\}$ is the set of projects, $A(i)\subseteq P$ is voter $i$’s approval set, $b\in\mathbb{Q}_{>0}$ is the total budget, and $r$0 gives project costs. For $r$1, total cost is

$r$2

and $r$3 is feasible if $r$4 (Kraiczy et al., 2023).

Equal-shares formulations represent payments by a load distribution or price system. In the load-distribution notation, a matrix $r$5 is priceable if no voter pays more than her endowment: $r$6 where $r$7. It is equal-shares if it is priceable and, for every selected project $r$8, all voters who pay for $r$9 pay the same amount. Thus, if $rB$0 is paid for by $rB$1 voters, each pays

$rB$2

This is the exact formal sense in which equal shares implements an equal per-supporter contribution rule (Kraiczy et al., 2023).

MES starts by giving each voter virtual budget

$rB$3

For each not-yet-selected project $rB$4, it computes

$rB$5

MES then selects a project with minimum $rB$6 and charges each approving voter

$rB$7

The method stops when all remaining projects have $rB$8. Structurally, MES minimizes a per-head price, but it does not always produce exact equal splitting because some supporters may hit their budget cap and contribute less than the common target (Kraiczy et al., 2023).

EES is a stricter equal-splitting variant. It repeatedly searches for a project $rB$9 and a subset $(1-r)B$0 such that every voter in $(1-r)B$1 can pay

$(1-r)B$2

Among feasible pairs $(1-r)B$3, it selects one maximizing

$(1-r)B$4

Here the utilities are uniform, with cardinal utilities $(1-r)B$5 and cost utilities $(1-r)B$6 as important special cases. EES is therefore an exact equal-shares rule in which contributors to a funded project must pay exactly equally (Kraiczy et al., 17 Feb 2025).

3. Adaptivity, stability, and verifiable proportionality

A major structural problem of MES is non-exhaustiveness. MES may stop even though additional projects could still be added without violating the global budget constraint. In practice, this motivates a virtual-budget search: given real budget $(1-r)B$7, one increases the virtual budget to $(1-r)B$8 and reruns the rule until the outcome becomes maximal feasible for the original budget (Kraiczy et al., 2023).

AMES was introduced to avoid recomputing from scratch when the virtual budget increases. It starts from an equal-shares solution $(1-r)B$9 and performs greedy update steps until the solution becomes stable. For an equal-shares solution, the per-voter price of project $E=(N,P,(A(i))_{i\in N},b,c),$0 is

$E=(N,P,(A(i))_{i\in N},b,c),$1

for $E=(N,P,(A(i))_{i\in N},b,c),$2, and $E=(N,P,(A(i))_{i\in N},b,c),$3 otherwise. Capacities are defined by

$E=(N,P,(A(i))_{i\in N},b,c),$4

where $E=(N,P,(A(i))_{i\in N},b,c),$5 is voter $E=(N,P,(A(i))_{i\in N},b,c),$6’s largest current project payment and $E=(N,P,(A(i))_{i\in N},b,c),$7 is the minimum gap between distinct per-voter prices in the finite set

$E=(N,P,(A(i))_{i\in N},b,c),$8

The first term captures possible reallocation away from the voter’s most expensive supported project; the second is remaining budget (Kraiczy et al., 2023).

A solution is unstable if there exists a project $E=(N,P,(A(i))_{i\in N},b,c),$9 and an integer $N=[n]$0 such that at least $N=[n]$1 approving voters have capacity at least $N=[n]$2. If instability exists, AMES adds or reinforces the project with the best new per-voter price, possibly removing more expensive projects from affected voters to free budget. Each update preserves equal-shares structure, improves the solution in a lexicographic order over sorted per-voter-price vectors, and greedy updates imply fast convergence (Kraiczy et al., 2023).

The main normative result is that stable solutions satisfy Extended Justified Representation (EJR) for approval utilities. This makes AMES “certifiably proportional”: given the output $N=[n]$3, one can verify stability, and thus EJR, in time $N=[n]$4; with auxiliary sorted capacities $N=[n]$5, the bound becomes $N=[n]$6. The paper also proves that the minimum next budget $N=[n]$7 at which a stable solution becomes unstable can be computed in time

$N=[n]$8

This combines adaptivity with an efficiently checkable proportionality certificate (Kraiczy et al., 2023).

4. Completion, critical budgets, and streamlined search

The computational bottleneck created by non-exhaustiveness is especially clear in completion heuristics. The standard “add-one” heuristic increases each voter’s budget by one unit, equivalently increasing total budget by $N=[n]$9 at each step, and reruns MES or EES until the next outcome would overspend the true budget. The cited analysis identifies several drawbacks: many consecutive budget increments may produce the same outcome, the unit step depends on currency denomination, the first overspending virtual budget need not be best, and relevant critical budgets can be fractional even when project costs are integers (Kraiczy et al., 17 Feb 2025).

“Streamlining Equal Shares” replaces this search-by-probing with direct threshold computation. The problem is: given the current EES solution for budget $P=\{p_1,\dots,p_m\}$0, find the smallest per-voter increment $P=\{p_1,\dots,p_m\}$1 such that the EES outcome changes for budget $P=\{p_1,\dots,p_m\}$2. In the cardinal-utility case, the paper derives an $P=\{p_1,\dots,p_m\}$3 algorithm, \textsc{add-opt}, that computes exactly this next critical budget. For uniform utilities, the corresponding bound is $P=\{p_1,\dots,p_m\}$4 (Kraiczy et al., 17 Feb 2025).

The key structural device is stability. EES outcomes are characterized as stable equal-shares solutions, and the next budget threshold is the minimum increase at which some project certifies instability. In the cardinal case this is implemented by the subroutine GreedyProjectChange, which scans possible buyer sets for a project and computes the minimum subsidy needed to make one of them viable. The full \textsc{add-opt} heuristic applies this projectwise and jumps directly to the next budget at which the outcome changes (Kraiczy et al., 17 Feb 2025).

The paper also introduces \textsc{add-opt-skip}, a faster heuristic that runs GreedyProjectChange only on projects not currently selected and repeats until every project has been considered for inclusion at least once. This sacrifices the guarantee of visiting every payer-set modification on already selected projects, but it sharply reduces the number of rule executions in practice. The work leaves open the corresponding optimal virtual-budget problem for MES itself and shows that, for cost utilities, the number of distinct EES outcomes reachable as the budget varies can be exponential (Kraiczy et al., 17 Feb 2025).

5. Percentage-split and relaxed variants

KES extends equal-shares logic to a hybrid participatory budgeting and grant-allocation setting with two agent classes: individuals $P=\{p_1,\dots,p_m\}$5 and impact fields $P=\{p_1,\dots,p_m\}$6. Utilities are normalized to

$P=\{p_1,\dots,p_m\}$7

A parameter $P=\{p_1,\dots,p_m\}$8 divides the total budget $P=\{p_1,\dots,p_m\}$9 into evaluator mode and individual-voter mode. Individual budgets are initialized as

$A(i)\subseteq P$0

while impact-field budgets are

$A(i)\subseteq P$1

For each unfunded project $A(i)\subseteq P$2, the pseudocode computes support

$A(i)\subseteq P$3

although the surrounding prose states that KES integrates both individuals and impact fields into one allocation. Projects are affordable when $A(i)\subseteq P$4 and $A(i)\subseteq P$5; among affordable projects, the rule selects a project with maximum support and assigns receipts proportionally: $A(i)\subseteq P$6 KES therefore begins from an explicit percentage split of the total budget and then implements a virtual-budget allocation with receipts as priceability witnesses (Yang et al., 2 Oct 2025).

FrES and BOS Equal Shares show a different line of generalization. FrES introduces $A(i)\subseteq P$7-affordability: $A(i)\subseteq P$8 so the rule can buy an $A(i)\subseteq P$9-fraction of a project. The cited theorem states that FrES satisfies Fractional EJR. BOS then converts this fractional structure into an integral rule. Among projects fitting in the remaining real budget, BOS chooses a $i$0-affordable candidate minimizing

$i$1

and updates virtual budgets as

$i$2

Some supporters can therefore overspend relative to their remaining virtual entitlement (Papasotiropoulos et al., 2024).

Under cost utilities, BOS satisfies EJR up to

$i$3

where $i$4, but the same paper also proves that BOS can fail EJR badly in committee elections. BOS is thus not a strict equi-percentage rule. It is better read as a bounded-overspending relaxation of equal-shares logic, designed to trade strict proportionality for higher efficiency (Papasotiropoulos et al., 2024).

6. Empirical behavior and analytical benchmarks

The empirical evidence in these papers supports the view that equal-shares outcomes often change slowly as virtual budgets move. On five real PB instances from Pabulib, AMES was tested by repeatedly topping up each voter’s budget by 1 and comparing consecutive outcomes over 50 runs. Small budget increases typically produced very small changes. In Warszawa Bielany, around 50 projects were selected on average in each run, but the average change between consecutive outcomes was just below 3. In Lodz, each iteration added over 25 projects on average, while the average difference between consecutive outcomes was less than 7 (Kraiczy et al., 2023).

The streamlining results are similarly concrete. On over 250 real-world PB instances from Pabulib, EES with add-one achieved spending efficiency at least as high as MES in over 77% of cost-utility cases and 85% of cardinal-utility cases. For add-opt, the average per-voter budget jump was 37.3 units for cost utilities and 35 for cardinal utilities, with medians 6.4 and 4.6 respectively. The practical heuristic add-opt-skip reduced average executions from 535 to 28 on cardinal datasets and from 466 to 12 on cost-utility datasets, while maintaining comparable spending efficiency (Kraiczy et al., 17 Feb 2025).

BOS Equal Shares was evaluated on 991 Pabulib instances using score satisfaction, cost satisfaction, exclusion ratio, and EJR+ violations. MES had 0 EJR+ violations, Utilitarian had average 0.933 violations per instance, BOS had average 0.046, and BOS+ had average 0.051. The same study reports that BOS and BOS+ were roughly an order of magnitude faster than MES with Add1U completion. KES was also deployed in Kultur Komitee Winterthur 2025, with a 50:50 budget split between individual votes and impact fields; the final allocation funded 43 projects and spent CHF 378,901 (Papasotiropoulos et al., 2024, Yang et al., 2 Oct 2025).

A broader analytical benchmark comes from the threshold framework for multiwinner elections. There, proportionality is expressed through thresholds $i$5, which ask how large a voter group must be, as a fraction of the electorate, to be guaranteed $i$6 seats under a given scenario $i$7. In that framework, the property that a group with vote fraction $i$8 always gets at least $i$9 seats is equivalent to

$b\in\mathbb{Q}_{>0}$0

and, in practical terms,

$b\in\mathbb{Q}_{>0}$1

This suggests that any “equi-percentage” claim should specify its benchmark carefully: a Hare-type share $b\in\mathbb{Q}_{>0}$2 is stronger than the Droop-type benchmark $b\in\mathbb{Q}_{>0}$3, and the distinction matters when proportionality claims are translated into formal guarantees (Janson, 2018).

The cumulative picture is therefore precise rather than terminologically uniform. “Equi-Percentage Scheme” is best understood not as a single named rule, but as a family resemblance among participatory allocation mechanisms that distribute influence through equal voter endowments, exact equal per-supporter payments, or explicit percentage splits of the total budget. Within that family, MES and EES provide the strict equal-shares core, AMES adds adaptivity and verifiable EJR, KES introduces explicit percentage partitioning across decision modes, and FrES and BOS explore fractional and bounded-overspending relaxations (Kraiczy et al., 2023, Kraiczy et al., 17 Feb 2025, Yang et al., 2 Oct 2025, Papasotiropoulos et al., 2024).