# Explanation of the Method of Equal Shares

On this page, we explain how the Method of Equal Shares works. The method can be applied with two different input types:

1. Approval voting, where each voter votes for some of the projects, but votes for each of them with the same "strength". For instance, voters can select up to 5 projects. See this Wikipedia page for more details: Approval Voting.
2. Utilities, where each voter assigns a utility number to each project (0 or higher). For instance, voters can distribute 10 points across the projects. See this Wikipedia page for more details: Cardinal Voting.

Approval voting is a special case of utilities, where all projects are assigned the utility value 0 or 1. On this page, we explain the method for approval voting, with more information about utilities coming soon.

tip

## Main mechanism​

We start by giving the main principles of the method, and then we explain the technical details.

1. The overall budget is divided equally among the voters.
Example

Suppose the overall budget is 1 000 000 (1 million), and there are 100 000 voters. Then each voter gets assigned a share of 10.

1. We remove all projects from consideration that cost more than the combined share of all voters that voted for the project. (For utilities, the combined share of all voters who assigned a utility value of more than 0 to the project.)
Example

Suppose a project costs 10 000, and it received 500 votes. Because every voter was assigned a share of 10, these 500 votes together have a share of 5 000. Since the project costs more than the combined share of all voters who voted for it, it cannot be funded. We remove the project from consideration.

1. If no projects remain, the computation of the method is finished.
2. If projects remain, we calculate the "effective vote count" of every project.
Explanation

For approval voting, the vote count is the number of voters who voted for the project. For utilities, the vote count is the sum of the utilities assigned to the project. The effective vote count is smaller than the vote count, because we do not count voters who do not have enough money to pay an equal share for the project.

1. We select the project with the highest effective vote count. We split the cost of this project as equally as possible among the voters who voted for the project. Each voter's share of the cost is subtracted from the voter's budget.
2. We repeat from Step 2 to determine the next project to be funded.
info

As written, the Method of Equal Shares may finish in Step 3 without having spent all the available budget. Therefore, in Step 1, we usually use an overall budget that is larger than the actually available budget. This is done to obtain a better outcome. See Completion for more information about this.

## How to split project costs equally​

When determining the winning projects of the Method of Equal Shares, a key ingredient is to divide the cost of a project as equally as possible. In this section, we will explain what this means.

To be concrete, let's say that the overall budget is 300, and there are 10 voters. Thus, in Step 1, each voter is allocated a budget share of 30. We can represent this in the following picture, where each voter is represented by a bar of height 30.

30
0

Suppose there is a project that costs 50, and it received 5 votes from the 5 voters on the left. Then we can spread the cost equally, with each supporter paying 10. We can show this in the picture as follows:

30
0
10
10
10
10
10
0
0
0
0
0

There is a slightly more complicated case, when some of the voters have already spent some of their budget share, and so it is not possible to spread the cost of a project perfectly equally between its supporters.

For example, consider again the project costing 50, which received 5 votes from the 5 voters on the left. But suppose that the left-most voter has already spent her entire budget share of 30 in prior rounds Then we can only spread the cost of the project between the remaining 4 voters, who each now have to pay 12.50 (because 4 · 12.50 = 50).

30
0
12.5
12.5
12.5
12.5
0
0
0
0
0

Or for another example, suppose that the two voters on the left have only 5 left to spend (because they each spent a total of 25 in prior rounds already). Then to spread the cost most equally, we charge those voters their entire remaining budget, which total 5 + 5 = 10. Then we split the remaining cost of 40 equally between the remaining 3 voters, who each have to pay 13.33 (because 3 · 13.33 = 40).

30
0
5
5
5
5
13.3
13.3
13.3
0
0
0
0
0

In general, the procedure for splitting the cost as equally as possible between supporters is as follows. We first try to split the cost equally (so we divide the project's cost by the number of the project's supporters). If some voters do not have enough budget left to cover what they would have to pay, we instead take as much money as possible from those voters (namely their entire remaining budget). Then, in the next step, we try to split the remaining cost equally between the remaining voters. If this is again not possible, we take as much money as possible from the voters who do not have enough, and so on. (An alternative mathematical way of describing this is that we split the cost of the projects between its supporters in such a way that the maximum payment of any voter is as small as possible.)

You can try out how this works on the following example, where the cost of a project (initially 30) is split between 5 voters who support it. These voters have a remaining budget of 10, 10, 10, 20, 30, respectively. You can change the cost of the project by dragging the slider, and see how the resulting cost would be split as equally as possible between the 5 voters.

30
0
10
6
10
6
10
6
20
6
6
Cost: 30

In this example, if the project cost is 50 or less, then we can split the cost perfectly equally. Between 50 and 80, the three voters with a remaining budget of 10 cannot pay an equal fraction of the cost, so the two other voters need to pay more. Note that if you drag the slider so that the project costs more than 80, then the voters do not have enough budget left to pay for the project, and so the project cannot be funded.

by the way

Our logo is inspired by these pictures, and is meant to invoke the picture of a city skyline.

## Computing the effective vote count​

When determining the winning projects of the Method of Equal Shares, we need to compute something called the effective vote count. In this section, we will explain what this means.

The most important rule is that we do not count voters if they have already spent their entire budget share. The reasoning behind this rule is that if a voter has already spent her entire budget share, then she has already been satisfied by the projects that were selected, and so it is more important to fund projects that other voters like. As an example, consider a project costing 30 that has received 5 votes, but 2 of the voters who voted for the project have already spent their entire budget. Then the effective vote count of the project is 3. The following picture displays this situation.

30
0
10
1
10
1
10
1
0
0
3

In the orange boxes below the bars, we can see that the 3 voters with budget remaining each contribute 1 to the project's vote count, while the 2 voters without remaining budget contribute 0. This adds up to an effective vote count of 3.

The second rule is that voters who still have money left, but not enough money to pay for the project when its cost is equally divided, will count as a fraction. For example, if voters who have lots of budget left each pay 10 for a project, and another voter only has 5 left to contribute to the project, then that latter voter counts for only half a vote. We can see this effect in the following example, concerning a project costing 25.

30
0
10
1
10
1
5
5
0.5
0
0
2.5

We can see that in this example, the third voter from the left counts as only 0.5 votes, leading to an effective vote count of 2.5 in total.

In the following example, you can move the slider to adjust the cost of the project, and see how the effective vote count changes. You can notice that the effective vote count goes down as the cost increases, because with high costs some voters do not have enough budget left to contribute an equal share.

30
0
10
6
1
10
6
1
10
6
1
20
6
1
6
1
5
Cost: 30

## Example 1​

You can also watch a video on YouTube [5:14 min] explaining this example.

Let us illustrate how the method works using a small example. Suppose the overall budget is 1100, and there are 11 voters.

There are 5 projects on the ballot, and we use approval voting. The following table lists the projects and their costs. For each voter, there is a column that indicates all the projects that the voter voted for. For example, the left-most voter selected the bike path and the outdoor gym.

ProjectCosts👤👤👤👤👤👤👤👤👤👤👤vote count
🚲 bike path 7007
🏋️ outdoor gym 4006
🌳 new park 2505
🛝 new playground 2004
📚 library for kids1003

Most cities use a simple method to select the winners of the participatory budgets, where they go through the projects in order of the number of votes. In this case, they would select the 🚲 bike path (with 7 votes), and then the 🏋️ outdoor gym (with 6 votes), and then stop because the budget is exhausted. Note that these two "sports" projects are both voted on by roughly the same voters. On the other hand, the "kids" projects (🛝 new playground, 📚 library for kids) are voted on by a different set of voters, who remain unrepresented.

Spoiler

As we will see, the Method of Equal Shares will select the 🚲 bike path, then the 🛝 new playground, and then the 📚 library for kids.

We begin computing the outcome by dividing the budget equally among the voters. Thus, each voter gets assigned a share of 100.

100
0

We need to determine the so-called effective vote count for the project with the most votes, which is the 🚲 bike path with 7 votes.

100
0
100
1
100
1
100
1
100
1
100
1
100
1
0
0
0
0
0
0
0
0
100
1
7

In this case, the effective vote count of the 🚲 bike path is 7. All the other projects have a lower effective vote count (because the effective vote count can only be lower than the vote count), and the next-most popular project (the 🏋️ outdoor gym) has a vote count of only 6.

Thus, the Method of Equal Shares selects the 🚲 bike path as the first project to be funded, and divides its cost equally between its supporters. Thus, the 7 voters who voted for the 🚲 bike path each pay 100, and thus do not have any budget left anymore.

100
0

Note that none of the people who voted for the 🏋️ outdoor gym have any budget left, so its effective vote count has dropped to 0. Similarly, 3 of the 5 people who voted for the 🌳 new park have no budget left, so its effective vote count has dropped to 2.

ProjectCosts👤👤👤👤👤👤👤👤👤👤👤effective vote count
🏋️ outdoor gym 4000
🌳 new park 2502
🛝 new playground 2004
📚 library for kids1003

Thus, the 🛝 new playground has the highest vote count with 4 votes. We can spread its cost of 200 equally among its supporters, and thus each of the 4 voters who voted for the 🛝 new playground pays 50:

100
0
50
50
50
50

We select the 🛝 new playground as a winning project, and subtract the payments. Thus, the 4 supporting voters have 50 left:

100
0
50
50
50
50

The 📚 library for kids costs 100. The 3 voters who voted for it together have 150 left, so this project is still affordable. They can each pay 33.3.

100
0
50
33.3
50
0
50
33.3
50
33.3

We select the 📚 library for kids as a winning project, and subtract the payments. The remaining budgets are:

100
0
16.7
50
16.7
16.7

None of the remaining projects can be afforded using the budgets of their supporters, so we have finished computing the election winners.

Final result: The winning projects are the 🚲 bike path, the 🛝 new playground, and the 📚 library for kids.

## Example 2 (more complicated)​

The computation turns out to be a bit more complicated if the right-most voter votes for the 🏋️ outdoor gym instead of the 🚲 bike path:

ProjectCosts👤👤👤👤👤👤👤👤👤👤👤vote count
🚲 bike path 7006
🏋️ outdoor gym 4007
🌳 new park 2505
🛝 new playground 2004
📚 library for kids1003

Going through the details will be instructive.

As before, the available budget is divided equally among the voters, and each is allocated a share of 100:

100
0

We take the project with the highest vote count, which this time is the 🏋️ outdoor gym with 7 votes, and divide its cost equally between its supporters, with each voter contributing 400/7 = 57.14.

100
0
57
57
57
57
57
57
0
0
0
0
57

The 🏋️ outdoor gym is selected, and we subtract the payments. Now the remaining budgets are as follows:

100
0
43
43
43
43
43
43
43

The project with the next-highest vote count is the 🚲 bike path with 6 votes. The 6 voters supporting it have 6 · 43 = 258 left, which is not enough to pay for the 🚲 bike path which costs 700. Thus, the bike path is not affordable, and we denote this by giving it an effective vote count of 0.

Thus, we move on to the project with the next-highest vote count, which is the 🌳 new park with 5 votes. If we divide its cost of 250 equally among its supporters, each of the 5 voters supporting it would need to pay 50. But some of the supporters of the 🌳 new park only have a budget share of 43 left, so they cannot afford to pay 50. Thus (as described above), we divide the cost so that the three supporters with a remaining budget share of 43 each pay 43, and the two supporters with a budget share of 100 divide the remaining 250 − 3 · 43 = 121 equally, so that each pays 60.5:

100
0
43
0
0
43
43
0.71
43
0
0
43
43
0.71
43
43
0.71
43
0
0
60.5
1
0
0
0
0
60.5
1
43
0
0
4.13

Because we were not able to split the cost of the 🌳 new park completely equally, we have to compute the effective vote count of the 🌳 new park as follows: The two voters able to pay 60.5 each count as a full vote, while the three voters able to pay 43 each count as only a fraction of the vote, namely 0.71 each (because 43/60.5 ≈ 0.71). Thus we get an effective vote count of 2 + 3 · 0.71 = 4.13, which we can write in the table:

ProjectCosts👤👤👤👤👤👤👤👤👤👤👤effective vote count
🚲 bike path 7000
🌳 new park 2504.13
🛝 new playground 2004
📚 library for kids1003

(The effective vote count for the 🛝 new playground and 📚 library for kids is the same as before, because their costs can be equally split between its supporters.)

Despite the reduction in its effective vote count, the 🌳 new park is still the project with the highest effective vote count, so we select it. We subtract the payments. Now the remaining budgets are as follows:

100
0
43
43
43
39.5
39.5
43

In the next round, the 🛝 new playground is selected, obtaining an effective vote count of 3.31 (which is the highest):

100
0
43
0
0
0
43
0
0
0
0
43
0
0
39.5
39.5
0.65
60.5
1
60.5
1
39.5
39.5
0.65
43
0
0
3.31

After subtracting the payments, the remaining budgets are as follows:

100
0
43
43
43
39.5
39.5
43

At this point, none of the remaining projects (in particular the 📚 library for kids) can be afforded using the remaining budgets of their supporters, so the computation of the method ends.

Final result: The winning projects are the 🏋️ outdoor gym, the 🌳 new park, and the 🛝 new playground.