Method of Equal Shares

The Method of Equal Shares ^[1] (sometimes referred to as MES) is a proportional method of counting ballots that applies to participatory budgeting and to committee elections.^[2]^[3] MES uses rated ballots; the Expanding Approvals Rule allows for ranked ballots to be used instead, though expanding approvals works better when using cardinal ballots.

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Method of Equal Shares

Motivation

The method is an alternative to knapsack voting which is used by most cities even though it is a disproportional method. For example, if 51% of the population support 10 red projects and 49% support 10 blue projects, and the money suffices only for 10 projects, the knapsack budgeting will choose the 10 red supported by the 51%, and ignore the 49% altogether.^[4] In contrast, the method of equal shares would pick 5 blue and 5 red projects.

The method guarantees proportional representation: it satisfies the strongest known variant of the justified representation axiom that is known to be satisfiable in participatory budgeting.

Intuitive explanation

In the context of participatory budgeting the method assumes that the municipal budget is initially evenly distributed among the voters. Each time a project is selected its cost is divided among those voters who supported the project and who still have money. The savings of these voters are decreased accordingly. If the voters vote via approval ballots, then the cost of a selected project is distributed equally among the voters; if they vote via cardinal ballots, then the cost is distributed proportionally to the utilities the voters enjoy from the project. The rule selects the projects which can be paid this way, starting with those that minimise the voters' marginal costs per utility.

Example 1

The following example with 100 voters and 9 projects illustrates how the rule works. In this example the total budget equals $1000, that is it allows to select five from the nine available projects. See the animated diagram below, which illustrates the behaviour of the rule.

There are 9 projects. For example, the third group of 11 voters voted for D and G. The total budget of $1000 is divided equally among 100 voters. Each voter is given 10. Click on the arrow above the image in order to see the next steps of the method.
Project D obtained most votes. If we divided the cost of D equally among its supporters, each voter would pay $3.03. D is the project that minimises the maximal voter payment and so it is selected.
Project A obtained 60 votes. Analogously to the previous step: if we divided the cost of A equally among its supporters, each voter would pay at most $3.33. Project A minimises the maximal voter payment and so is selected.
Project C obtained 56 votes and is selected in the third round. Each supporter of C needs to pay $3.64, and this is the minimal possible payment. At this point the first 46 voters run out of their money.
In the fourth step project G is selected. Some voters do not have enough money to participate equally in the purchase, so they pay all money left. The maximal payment for this candidate equals $6.97.
In the last step, project H is selected. Now, only the fourth group of voters has money. They have enough money to afford the project they voted for. The maximal payment for the selected project is now $10.

The budget is first divided equally among the voters, thus each voters gets $10. Project $\mathrm {D}$ received most votes, and it is selected in the first round. If we divided the cost of $\mathrm {D}$ equally among the voters, who supported $\mathrm {D}$ , each of them would pay $\$200/66\approx \$3.03$ . In contrast, if we selected, e.g., $\mathrm {E}$ , then the cost per voter would be $\$200/46\approx \$4.34$ . The method selects first the project that minimises the price per voter.

Note that in the last step project $\mathrm {H}$ was selected even though there were projects which were supported by more voters, say $\mathrm {E}$ . This is because, the money that the supporters of $\mathrm {E}$ had the right to control, was used previously to justify the selection of $\mathrm {D}$ , $\mathrm {A}$ , and $\mathrm {C}$ . On the other hand, the voters who voted for $\mathrm {H}$ form 20% of the population, and so shall have right to decide about 20% of the budget. Those voters supported only $\mathrm {H}$ , and this is why this project was selected.

For a more detailed example including cardinal ballots see Example 2.

Definition

This section presents the definition of the rule for cardinal ballots. See discussion for a discussion on how to apply this definition to approval ballots and ranked ballots.

We have a set of projects $P=\{p_{1},p_{2},\ldots ,p_{m}\}$ , and a set of voters $N=\{1,2,\ldots ,n\}$ . For each project $p\in P$ let $\mathrm {cost} (p)$ denote its cost, and let $b$ denote the size of the available municipal budget. For each voter $i\in N$ and each project $p\in P$ let $u_{i}(p)$ denote the $i$ 's cardinal ballot on $p$ , that is the number that quantifies the level of appreciation of voter $i$ towards project $p$ .

The method of equal shares works in rounds. At the beginning it puts an equal part of the budget, in each voter's virtual bank account, $b_{i}=b/n$ . In each round the method selects one project according to the following procedure.

For each not-yet-selected project $p\in P$ the method tries to spread the cost of the project proportionally to the cardinal ballots submitted by the voters, taking into account the fact that some voters might have already run out of money. Formally, for $\rho \geq 0$ , we say that a not-yet-selected project $p$ is $\rho$ -affordable if
${\begin{aligned}\sum _{i\in N}\min \left(b_{i},u_{i}(p)\cdot \rho \right)=\mathrm {cost} (p){\text{.}}\end{aligned}}$
Intuitively, if a project $p$ is $\rho$ -affordable then, the cost of the project can be spread among the voters in a way that each voter pays the price-per-utility of at most $\rho$ .
If there are no $\rho$ -affordable projects then the method of equal shares finishes. This happens when for each not-yet selected project $p$ the remaining amount of money in the private accounts of those voters who submitted a positive ballot on $p$ is lower than the cost of $p$ : $\textstyle \sum _{i\in N\colon u_{i}(p)>0}b_{i}<\mathrm {cost} (p){\text{.}}$
It might happen that when the method finishes, there is still some money left that would allow to fund a few more projects. This money can be spent using the simple greedy procedure that select the remaining projects $p$ starting from those with the lowest ratio $\mathrm {cost} (p)/\textstyle \sum _{i\in N}u_{i}(p)$ , until the budget is exhausted. Yet, the method of equal shares keeps most of its properties independently of how the remaining budget is spent.
If there is at least one not-yet-selected $\rho$ -affordable project, the method selects the project $p$ that is $\rho$ -affordable for the lowest value of $\rho$ (the project that minimises the price-per-utility that the voters need to pay). The voters' budgets are updated accordingly: for each $i\in N$ the method sets $b_{i}:=\max(0,b_{i}-u_{i}(p)\cdot \rho )$ .

Example 2

The following diagram illustrates the behaviour of the method.

There are 8 available projects and 250 voters. For example, the first 65 voters assign value 30 to project B and value 10 to projects E and G. The total budget of $2500 is divided equally among 250 voters. Each voter is given $10. Click on the arrow above the image in order to see the next steps of the method.
Project B is selected first, and its cost is divided proportionally to the values that the voters assigned to the project. In this case, this means it is divided equally among the voters from the first and the second group. Each such voter pays $2, and for those $2 they get the utility of 30. Thus the maximal payment-per-utility equals $2/30\approx 0.066$ . If a different project was selected, the maximal payment-per-utility would be higher.
Consider project G and the payments presented in the picture. The payments are not equal, but they are still proportional to the values that the voters' assigned to G. The maximal voter's payment-per-utility for project G equals $1/10=4/40=10/100=0.1$ and this value is minimal across all projects. Consequently, G is selected. After this round the voters from the fourth group have run out of money.
In the third round project F is selected. Every supporter of F pays an equal part of the price - except for voters from the fourth group, who have no money. If they had any, they would also need to participate. Nevertheless, the maximal payment-per-utility for project F is minimal (it equals 0.2), hence F is elected.
In the fourth round, project E is elected. Consider the payments presented in the picture, and note that the voters from the third group have too little money to participate in paying proportionally to their utilities, yet they all still have $4 dollars left. In such case, they pay all money they still have. The maximal payment-per-utility (paid by the voters from the first group) is minimal, and equals circa 0.54.
In the last step, project C is elected. The voters from the second and the sixth group have too little money to participate in paying proportionally to their utilities, hence they pay as much as possible. The maximal payment-per-utility is paid for the fifth group of voters and equals 0.7.
The rule spent $2380 out of $2500 in the budget. While the voters from the first and the fifth group have positive savings, no project can be afforded by their supporters. Hence the algorithm stops. The outcome can be further completed. According to the utilitarian strategy project H would be selected as its cost per utility equals $110/(2\cdot 50+1\cdot 10+1\cdot 55)$ and is maximal across the projects that would fit within the budget constraint.

Discussion

This section provides a discussion on other variants of the method of equal shares.

Other types of ballots

The method of equal shares can be used with other types of voters ballots.

Approval ballots

The method can be applied in two ways to the setting where the voters vote by marking the projects they like (see Example 1):

Setting $u_{i}(p)=\mathrm {cost} (p)$ if project $p$ is approved by voter $i$ , and $u_{i}(p)=0$ otherwise. This assumes that the utility of a voter equals the total amount of money spent on the projects supported by the voter. This assumption is commonly used in other methods of counting approval ballots for participatory budgeting, for example in the knapsack algorithm, and typically results in selecting fewer more expensive projects.
Setting $u_{i}(p)=1$ if project $p$ is approved by voter $i$ , and $u_{i}(p)=0$ otherwise. This assumes that the utility of a voter equals the number of approved selected projects. This typically results in selecting more but less expensive projects.

Ranked ballots

The method applies to the model where the voters vote by ranking the projects from the most to the least preferred one. Assuming lexicographic preferences, one can use the convention that $u_{i}(p)$ depends on the position of project $p$ in the voter's $i$ ranking, and that $u_{i}(p)/u_{i}(p')\to \infty$ , whenever $i$ ranks $p$ as more preferred than $p'$ .

Formally, the method is defined as follows.

For each voter $i\in N$ let $\succ _{i}$ denote the ranking of voter $i$ over the projects. For example, $Y\succ _{i}X\succ _{i}Z$ means that $Y$ is the most preferred project from the perspective of voter $i$ , $X$ is the voter's second most preferred project and $Z$ is the least preferred project. In this example we say that project $Y$ is ranked in the first position and write $\mathrm {pos} _{i}(Y)=1$ , project $X$ is ranked in the second position ( $\mathrm {pos} _{i}(X)=2$ ), and $Z$ in the third position ( $\mathrm {pos} _{i}(Z)=3$ ).

Each voter is initially assigned an equal part of the budget $b_{i}=b/n$ . The rule proceeds in rounds, in each round:

For each not-yet-selected project $p\in P$ we say that $p$ is $\delta$ -affordable if the remaining budget of the voters who rank $p$ at position $\delta$ or better is greater than or equal to $\mathrm {cost} (p)$ :
${\begin{aligned}\sum _{i\in N\colon \mathrm {pos} _{i}(p)\leq \delta }b_{i}\geq \mathrm {cost} (p){\text{.}}\end{aligned}}$

If no project is affordable the rule stops. This happens when the total remaining budget of the voters $\textstyle \sum _{i\in N}b_{i}$ is lower than the cost of each not-yet-selected project.
If there are affordable projects, then the rule picks the not-yet-selected project $p$ that is $\delta$ -affordable for the lowest value of $\delta$ . The budgets of the voters are updated accordingly. First, the cost is equally spread among the voters who rank $p$ at the first position. If the budgets of these voters are insufficient to cover the cost of the project, the remaining part of the cost is further spread equally among the voters who rank $p$ at the second position, etc. Formally we start with $\delta :=1$ and $\mathrm {cost} :=\mathrm {cost} (p)$ , and proceed in the loop:
1. If $\textstyle \sum _{i\in N\colon \mathrm {pos} _{i}(p)=\delta }b_{i}\geq \mathrm {cost}$ then we find $\rho$ such that $\textstyle \sum _{i\in N\colon \mathrm {pos} _{i}(p)=\delta }\min(\rho ,b_{i})=\mathrm {cost}$ and for each voter $i\in N$ with $\mathrm {pos} _{i}(p)=\delta$ we set $b_{i}:=\max(0,b_{i}-\rho )$ .
2. Otherwise, we update the cost: $\mathrm {cost} :=\mathrm {cost} -\textstyle \sum _{i\in N\colon \mathrm {pos} _{i}(p)=\delta }b_{i}$ . We charge the voters: for each voter $i\in N$ with $\mathrm {pos} _{i}(p)=\delta$ we set $b_{i}:=0$ , and move to the next position $\delta :=\delta +1$ .

Committee elections

In the context of committee elections the projects are typically called candidates. It is assumed that cost of each candidate equals one; then, the budget $b$ can be interpreted as the number of candidates in the committee that should be selected.

Unspent budget

The method of equal shares can return a set of projects that does not exhaust the whole budget. There are multiple ways to use the unspent budget:

The utilitarian method (Bloc Score Voting): the projects $p$ are selected in the order of ${\frac {\sum _{i\in N}u_{i}(p)}{\mathrm {cost} (p)}}$ until no further project can be selected within the budget limit.
Adjusting initial budget: the initial budget can be adjusted to the highest possible value which makes the method select projects, whose total cost does not exceed the unadjusted budget.
Other Cardinal PR systems: The remaining winners can be found with the use of other Cardinal PR systems. Sequentially Spent Score would be a natural completion since it maintains the concept of spending.

Implementation

Below there is a Python implementation of the method.

import math

def leq(a, b):
    return a < b or math.isclose(a, b)

# N:     a list of voters.
# C:     a list of projects (candidates).
# cost:  a dictionary that assigns each project its cost.
# b:     the total available budget.
# u:     a dictionary; u[c][i] is the value that voter i assigns to candidate c.
#        an empty entry means that the corresponding value u[c][i] equals 0.

def complete_utilitarian(N, C, cost, u, b, W):
    util = {c: sum([u[c][i] for i in N]) for c in C}
    committee_cost = sum([cost[c] for c in W])
    while True:
        next_candidate = None
        highest_util = float("-inf")
        for c in C.difference(W):
            if leq(committee_cost + cost[c], b):
                if util[c] / cost[c] > highest_util:
                    next_candidate = c
                    highest_util = util[c] / cost[c]
        if next_candidate is None:
            break
        W.add(next_candidate)
        committee_cost += cost[next_candidate]
    return W

def method_of_equal_shares(N, C, cost, u, b):
    W = set()
    total_utility = {c: sum(u[c].values()) for c in C}
    supporters = {c: set([i for i in N if u[c][i] > 0]) for c in C}
    budget = {i: b / len(N) for i in N}
    while True:
        next_candidate = None
        lowest_rho = float("inf")
        for c in C.difference(W):
            if leq(cost[c], sum([budget[i] for i in supporters[c]])):
                supporters_sorted = sorted(supporters[c], key=lambda i: budget[i] / u[c][i])
                price = cost[c]
                util = total_utility[c]
                for i in supporters_sorted:
                    if leq(price * u[c][i], budget[i] * util):
                        break
                    price -= budget[i]
                    util -= u[c][i]
                rho = price / util \
                        if not math.isclose(util, 0) and not math.isclose(price, 0) \
                        else budget[supporters_sorted[-1]] / u[c][supporters_sorted[-1]]
                if rho < lowest_rho:
                    next_candidate = c
                    lowest_rho = rho
        if next_candidate is None:
            break
        W.add(next_candidate)
        for i in N:
            budget[i] -= min(budget[i], lowest_rho * u[next_candidate][i])
    return complete_utilitarian(N, C, cost, u, b, W)  # one of the possible completions

References

↑ In early papers the method has been also referred to as Rule X.
↑ Peters, Dominik; Skowron, Piotr (2020). "Proportionality and the Limits of Welfarism". Proceedings of the 21st ACM Conference on Economics and Computation. EC'20: 793–794. arXiv:1911.11747. doi:10.1145/3391403.3399465. ISBN 9781450379755.
↑ Pierczyński, Grzegorz; Peters, Dominik; Skowron, Piotr (2020). "Proportional Participatory Budgeting with Additive Utilities". Proceedings of the 2021 Conference on Neural Information Processing Systems. NeurIPS'21. arXiv:2008.13276.
↑ Fluschnik, Till; Skowron, Piotr; Triphaus, Mervin; Wilker, Kai (2019-07-17). "Fair Knapsack". Proceedings of the AAAI Conference on Artificial Intelligence. 33: 1941–1948. doi:10.1609/aaai.v33i01.33011941. ISSN 2374-3468.

[rulex-1] In early papers the method has been also referred to as Rule X.

[2] Peters, Dominik; Skowron, Piotr (2020). "Proportionality and the Limits of Welfarism". Proceedings of the 21st ACM Conference on Economics and Computation. EC'20: 793–794. arXiv:1911.11747. doi:10.1145/3391403.3399465. ISBN 9781450379755.

[3] Pierczyński, Grzegorz; Peters, Dominik; Skowron, Piotr (2020). "Proportional Participatory Budgeting with Additive Utilities". Proceedings of the 2021 Conference on Neural Information Processing Systems. NeurIPS'21. arXiv:2008.13276.

[:4-4] Fluschnik, Till; Skowron, Piotr; Triphaus, Mervin; Wilker, Kai (2019-07-17). "Fair Knapsack". Proceedings of the AAAI Conference on Artificial Intelligence. 33: 1941–1948. doi:10.1609/aaai.v33i01.33011941. ISSN 2374-3468.

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