# Completion of the Method of Equal Shares

Summary: There are a few different variants of the Method of Equal Shares. In its simplest variant, the method may not spend all the available budget (but only about 70% of it). This is usually not desirable, so the method needs to be "completed". We describe the standard way to do this, and discuss some alternatives. Most users should use the standard method, which is also implemented in software packages.

In the explanation of the Method of Equal Shares, Step 1 says:

The overall budget is divided equally among the voters.

For example, if the overall budget is 100 000, and there are 1000 voters, then each voter would be allocated a share of 100. However, when we compute the Method of Equal Shares with this initial division of the budget, we will typically find that the election outcome does not use up the entire budget. This is because for some of the voters, the method cannot productively use the budget share assigned to them. For example, if a voter only votes for 1 project, and this project does not receive enough votes to win, then the method cannot spend the 100 assigned to this voter.

In simulations using voting data from Warsaw, the Method of Equal Shares spends approximately 60-70% of the available budget. Of course, election officials would usually prefer to spend close to 100% of the budget. Therefore, the Method of Equal Shares needs to be completed, which means that its outcome needs to be modified so that almost the entire budget is used.

Chart: If all voters vote for many projects, completion is not as crucial.

The root cause of the Method of Equal Shares often only spending 60-70% of the available budget is that some voters only vote for a small number of projects. We can see this effect by running a simulation based on data from Warsaw (2020-22).

In Warsaw, voters are allowed to vote for up to 15 projects in their district. We can simulate what would have happened if Warsaw had a minimum number of votes in place by counting only those votes in the dataset that have voted for sufficiently many projects. This way, we can simulate a minimum vote count of 2, 3, 4, ..., 15. For each minimum, we can check what percent of the budget is used by the Method of Equal Shares when only run on the subset of votes that have voted for at least the minimum number of projects.

The results suggest that once voters vote for at least 8 projects, more than 90% of the budget is used.

The basic idea is to give voters a larger initial budget share. In the example, instead of giving each voter 100, we could give them 130. More precisely, we go through each possible value 100, 101, 102, and so on. For each value, we compute the outcome of the Method of Equal Shares. We stop in one of two cases:

• The outcome is exhaustive, which means that the budget limit would be exceeded if we added any additional project to the outcome.
• If we were to run the Method of Equal Shares with the next higher value, then the resulting outcome would exceed the budget limit.

Simulation results suggest that the Method of Equal Shares, completed in this way, retains its desirable fairness properties.

Note that in general, it is not a good sign if we have to increase the voters' initial budget share too much in the completion process. In this case, there will be much more virtual money distributed to voters than is actually spent, and thus some voters will have been left with a lot of unspent virtual money. This means that some voters got to spend much more money than other voters on their preferred projects. This leads to an inequality of the amount of influence that voters have on the outcome. A way to avoid having to increase the initial budget share too much is to ask voters to vote for several projects. For example, some cities require voters to vote for at least 3 projects. This helps because a major cause of unspent virtual money is because some voters only vote for a single project, which does not receive enough votes to win. In this case, it is impossible for the Method of Equal Shares to spend the virtual money assigned to the voter.

## Other completion methods​

There are also other ways of using the part of the budget that is not spent by the Method of Equal Shares:

• Save the money for next year. The remaining funds can remain unspent and be used to increase the available budget in the following year.
• Fund projects with the highest vote counts. The remaining funds can be used to fund projects with high vote counts that were skipped by the Method of Equal Shares. However, this will lead to an outcome that is less fair.
• Spend the money on a flexible back-up solution. For example, in Chicago, a flexible part of the budget was used to fund street repairs of potholes.