# Higher and more equal voter satisfaction

**Summary**:
Full voting data from PB elections in Warsaw is publicly available, and we can use these to simulate the election outcome under the Method of Equal Shares.
We analyze how many projects an average voter approves in the election outcome, and find that this number is significantly higher under the Method of Equal Shares compared to the actual outcome.
We also find that under the Method of Equal Shares, a larger fraction of the voters approves at least one of the winning projects.
On the other hand, an average voter sees slightly less funding going to the projects that the voter voted for.

When simulating the election outcomes in the participatory budgeting elections of several major cities, we found that voter satisfaction would increase when switching to the Method of Equal Shares. More importantly, we find that the satisfaction is more equally distributed among the voters, and the least satisfied voters see significantly more spending on their preferred projects under the Method of Equal Shares.

On this page, we present results from a simulation on election data from PB elections in Warsaw, taken from the Pabulib library. In the simulations, we compute what the election outcome would have been if Warsaw had used the Method of Equal Shares. We compare this outcome to the actual election outcome, which is obtained by just selecting the projects with the highest number of votes until the budget runs out.

Our conclusions are as follows:

**Voters approve more winning projects on average.**In Warsaw, voters are allowed to vote for up to 10 projects. In the actual outcome, voters approved 4.5 of the winning projects on average, while under the Method of Equal Shares, voters approve 6 of the winning projects on average. [more]**More voters have voted for at least one of the winning projects.**In the actual outcome, 87% of the voters voted for at least one of the winning projects, while under the Method of Equal Shares, 93% of the voters voted for at least one of the winning projects. [more]**For an average voter, slightly less money is spent on approved projects.**In the actual outcome in 2022, the average voter approved projects that received 27% of the budget in total, while under the Method of Equal Shares, the average voter approved projects that received 24% of the budget in total. Thus, on this metric, the Method of Equal Shares does slightly worse than the actual outcome. [more]**Voter satisfaction is more equally distributed.**When using the standard voting method, some voters are very satisfied, and others get 0 approved projects in the outcome. In contrast, the Method of Equal Shares gives more voters an average satisfaction level where they are happy with between 10% and 40% of the spending. [more]

## Voters approve more winning projects on average

A natural way to measure how satisfied voters are with an election outcome is to count how many projects in the outcome an average voter voted for. In Warsaw, each voter votes in 1 district, and can approve up to 10 projects. Averaging over all districts and over the years 2020-22, we find that the average voter approves about 4.5 of the winning projects in the actual outcome. Under the Method of Equal Shares, the average voter approves 6 of the winning projects, a substantial increase of 33%.

There is not a single district where the average number is lower under the Method of Equal Shares. On the other hand, there are several examples of districts where the average number of approved projects almost tripled. An underlying reason for this is that the Method of Equal Shares tends to select more projects overall, whereas the standard method in current use in Warsaw selects only a few but very expensive projects.

In the maps below, we show for each district the average number of approved projects for the election in that district. The third map shows the percent difference. For example, +85% means that the average number of approved projects under the Method of Equal Shares is 1.85 times the number under the actual outcome.

- 2022
- 2021
- 2020

## More voters have voted for at least one of the winning projects

One aim of participatory budgeting is to give a broad segment of the population an influence over budgeting decisions. Therefore, it is desirable to use a voting method where a large fraction of the population is represented in the outcome. A minimal requirement would be that for a large fraction of the voters, there is at least one winning project that the voter voted for. This is also desirable to obtain high participation: if a voter votes for many projects but none of them win, the voter might be discouraged and feel that their vote was wasted.

In our simulation, we find that under the Method of Equal Shares, a larger fraction of the voters get at least one winning project from the projects they voted for. On average (over the years and districts), 93% of the voters get at least one winning project under the Method of Equal Shares, compared to just 86% under the actual outcome. In some districts, the difference is very large: for example in Bielany in 2022, 94% of voters get at least one winning project under the Method of Equal Shares, compared to just 80% under the actual outcome. In a small number of cases, the fraction of voters with at least one winning project is lower under the Method of Equal Shares, but by at most 1 percentage point.

In the maps below, we show for each district the fraction of voters who get at least one winning project from the projects they voted for (as a percentage). In the "difference" map, we show the difference as a number of percentage points. A positive number indicates a higher fraction under the Method of Equal Shares, and a negative number indicates a lower fraction.

- 2022
- 2021
- 2020

## For an average voter, slightly less money is spent on approved projects

Above, we saw that under the Method of Equal Shares, voters on average approve a higher number of projects in the outcome. An alternative way of measuring voter satisfaction is to look at the total amount of money spent on approved projects. These two measurements can be different, because a voter could get just 1 approved project but a very large and expensive one, or many small and cheap projects.

In this section, we calculate the percent of the overall budget that is spent on approved projects for an average voter. We find that under the Method of Equal Shares, slightly less money is spent on approved projects for an average voter, usually 3-4 percentage points less. On average over the years and districts, voters see 27% of the budget spent on approved projects under the actual outcome, compared to 24% under the Method of Equal Shares.

The reason that the Method of Equal Shares does not perform as well as the actual outcome is that the standard voting method can be seen as an optimization algorithm that selects an outcome that maximizes this measure. Therefore, any other outcome must be worse than the actual outcome. Encouragingly, this difference is not very large, and the Method of Equal Shares arguably still performs well.

- 2022
- 2021
- 2020

## Voter satisfaction is more equally distributed

Above, we saw that in the actual election outcome, an average voter will get slightly more spending on approved projects compared to the Method of Equal Shares. However, this is not true for all voters: in the actual outcome some voters have much more satisfaction and others much less. By this, we mean that there are more voters with very little spending on their approved projects, and more voters with very high spending on their approved projects. In contrast, under the Method of Equal Shares, the distribution of spending on approved projects is more equal, and in particular there are more voters who get a moderate amount of spending on their approved projects.

We can see this phenomenon in the following chart. For example, the chart shows that in the actual outcome, about 7 000 voters see 20% of the budget going to approved projects. Under the Method of Equal Shares, this number is about 9 000 voters. For the entire range between 5% and 40% of the budget going to approved projects, the Method of Equal Shares has more voters at that level than does the actual outcome.