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How to please voters. The Method of Equal Shares

Grafika ilustracyjna. Graficzne postaci ludzkie skupione obok siebie w taki sposób, że tworzą kształty dwóch przeciwstawnych strzałek.

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How to please voters. The Method of Equal Shares

​​​​​​​For small projects to have a chance at being carried out from civic funds, their budget does not need to increase. It is enough to change the method of selecting the winners.

Decisions to spend a few hundred thousand zlotys can be taken hastily although they should not. All the more so if it is civic budget spending that is intended to best serve the entire community. Researchers from several academic centres around the world, including the AGH University, have developed a vote-counting system that maximises the impact of each individual vote on the final outcome. The Method of Equal Shares was recently used during the participatory budget vote in Wieliczka. The researchers' analysis of the results shows that all the assumptions worked in practice and that more voters are satisfied with the Method of Equal Shares than with the previous vote-counting methods.

“Cities generally use a greedy method, which is the majority method, i.e. if someone collects a lot of votes they will get their project, which means, by definition as it were, that small projects, created by small groups of local people, stand no chance. The Method of Equal Shares is a proportional method, which means that a project's chances actually depend not just on how many people vote for it, but rather on how many people vote for it relative to its cost. So if someone in a small housing estate comes up with a small, cheap project and convinces their neighbours of it, they have a chance in the Method of Equal Shares,” claims Prof. Dr hab. Piotr Faliszewski, an employee of the Faculty of Computer Science, Electronics, and Telecommunications who cooperates with its originators, emphasising the advantages of the method developed. “In fact, all data analysis confirms that the Method of Equal Shares guarantees that there are far, far fewer people who casted their votes in the civic budget and none of their projects were funded. With this method, there are significantly less excluded people who took part and may be disappointed or not motivated to vote the following year.”

The winner does not take it all

Under the simplest and most commonly used voting formula, the greedy majority system, those projects that win the most votes and have the budget to cover their cost are implemented. These are basically the only two conditions that make it possible for funds to be distributed by one, not necessarily large, group of people. If the most popular project was very expensive, there may not be enough funds to implement the others, in which case a significant proportion of voters are left without any influence on the outcome of the election. If they do not vote in line with the majority, their votes are simply not reflected in the final outcome.

The Method of Equal Shares works differently. First of all, it allows everyone to influence the final outcome equally – it starts by dividing the available funds equally among all voters. Of course, the division is purely symbolic – no voter gets any money, but we can think of it in the way that casting a vote is a declaration to allocate part of the divided money to a particular project. Unlike the traditional greedy method, the most popular project does not win everything – only the amount available to those who voted for it can be used to fund it. Leaving aside the actual amounts of participatory budgets, let us assume that it is PLN 100, and among the projects the most popular was the renovation of a swing on the playground amounting to PLN 100, and only nine out of ten voters voted for it, the project will not be able to be implemented as those supporting this project have only PLN 90 at their disposal (everyone is entitled to PLN 10, as the whole budget [100] was divided equally among the voters). It is therefore necessary to find a project that is supported by enough voters to cover its cost, e.g. planting a tree by the playground at a cost of PLN 50, for which seven people voted.

Later, we consider more projects – is their cost proportional to the number of people who voted for it? And do they still have the finances to cover this cost? After all, we have to take into account that the people who have already cast their vote for the most popular of the possible projects (planting a tree) have already spent some of their money. If 7 people are willing to "chip in" for a tree for 50 zlotys, each has spent just over 7 zlotys. So they still have 2.85 zlotys to spare each, and only to this extent can they participate in further collections. Their vote also counts proportionally, which only confirms that all votes are equal. With further approved projects, on which some or all of the voters have already spent their money, deciding on the viability of further projects becomes somewhat more complicated. The overriding principle for the next steps, however, is that each voter should spend the smallest possible amount on a project in order to optimise the equality of the distribution of costs between all those voting for that project.

“Let's say that there is a project which is approved by many people,” explains Prof. Piotr Faliszewski, “and it turns out that if we divide it among these people, everyone has to give virtual two hundred zlotys. But some still have four hundred zlotys on their account, and some only one hundred. It may turn out that those who have four hundred zlotys, however, will have to pay not two hundred, but three hundred, and those who have only one hundred will pay one hundred, because they have no more. Then, the Method of Equal Shares works in such a way that those who pay three hundred count as the whole vote, but those who pay one hundred will be counted as one third of the vote. So that is how we will calculate the support for that project, and it might then turn out that somewhere next door there is a project where the drop was 250 zlotys, but exactly all those people who would have paid that 250 have that much. And it may be that their weighted power is greater and it is their project that will be included in the round of the Method of Equal Share. So the method takes into account the fact that at different stages voters will be able to pay, virtually of course, different amounts.”

In the photo, the creators of the Method, from left: Dr Jarosław Flis, Prof. UJ; Prof. Piotr Faliszewski; Dr Piotr Skowron; Grzegorz Pierczyński; Stanisław Szufa. Photo from the AGH University archives.

Na zdjęciu stoi obok siebie pięciu mężczyzn. Uśmiechają się.

Satisfaction visible in figures

The main principles of the Equal Share Method were developed by Dr Piotr Skowron of the University of Warsaw, a PhD student at the University of Warsaw, Grzegorz Pierczyński, and Dr Dominik Peters of the University of Paris-Dauphine. They were looking for a method that would allow the winners of elections to be determined proportionally. They later realised that the system they had developed should work well for voting on civic budgets. Stanisław Szufa, an assistant at the AGH University, a doctoral student at the Jagiellonian University, and a member of the Jagiellonian University's Center for Quantitative Research in Political Science, collected hundreds of copies of participatory budget documentation, which allowed researchers from the AGH University and the ETH Zurich to analyse voter satisfaction levels. The findings allowed them to conclude that, in practice, the method would deliver all the anticipated benefits.

Wieliczka was the first town to hold a vote based on the Method of Equal Shares. In the spring of this year, its residents were able to cast their votes for the distribution of funds in the Green Million project, an innovative programme dedicated to the implementation of green solutions. Professor Piotr Faliszewski took part in the analysis of the results, which confirmed that the method has many advantages that the solutions used so far lack. As the scientist points out, the method works very well in practice and clearly reduces the percentage of residents’ exclusion, i.e. limits the number of voters whose votes did not translate into the implementation of the projects chosen by them. More people than in the method used so far could be satisfied because at least one of the projects they voted for was eventually funded.

“We analysed the results from Wieliczka and found them to be very stable. It might have seemed that in some cases the project was just short of receiving funding, so to dispel doubts, we generated a large number of explanations as to why the result was as it was. Subsequent statistical analysis showed that this was no coincidence. If, for example, small random changes had been made to the votes, the results would still have been the same with high probability. Incidentally, this is not obvious, because there are cases of civic budgets and we have also seen such data from the past, where the results are unstable and this is a natural thing, because sometimes a project really is "on the verge"."

The sensitivity of the Method of Equal Shares to low volatility may at first glance be seen as a disadvantage, but in a way it is due to its foundation. As Professor Piotr Faliszewski explains:

“Proportionality, by its very nature, is a little less immune to small changes; if a little more people vote for this project, that means it should have a slightly better chance, rather than just the majority deciding. If we want to be proportional, it could be said that the lack of sensitivity to smaller movements of voters is even a certain flaw in the majority method.”

Consult a specialist before use

Now the researchers hope that more towns will soon be convinced to use the Method of Equal Shares, which was recently applied in the Swiss town of Aarau. The authors of the method are already in talks with Świecie, where a resolution to use the Method has just been passed for the next vote. Although verification procedures at higher levels are still ongoing, no one expects problems with the application of the Method.

The algorithms that allow votes to be recalculated according to the Method are available in a public repository free of charge, so anyone can use them.

“You can, without asking us for anything, use them and they will work, but we of course encourage municipalities and towns to contact us and we make efforts to make the process run smoothly. We always help the municipalities and towns to write the right resolution, to describe the Method, so that it is all done according to the law. At this point, we offer assistance in basically everything from writing the resolution to conducting the vote. We are happy to travel to such municipalities – we can do training, we can talk about the Method, make a presentation. There are videos on the internet describing the Method, but we want to make an even shorter video, for voters, for project developers. From the project developer's point of view, you need to know that it is a little bit different from the majority method: if I expect the support of three hundred people and I know how many people are estimated to vote, I also know approximately what my chances are where it comes to the cost. It is useful to know this so that you are aware that if someone has always had the most support so far, it does not mean that they can take the whole budget for themselves.”

While voters' detailed understanding of the Method may facilitate informed decision-making, from the voters' perspective, there need not be any change in the electoral process itself  – voters can mark their preferred project, or several of them, on the card. The organiser of the election can allow as many projects to be marked on the card as they wish, or limit this number to a few. It is also possible to conduct the election on a scale to take into account voters' preferences – in which case each voter should specify numerically (e.g. on a scale of 1 to 10) which project they support more and which they support less. However, the creators themselves do not recommend this method, as a high level of voter involvement is required for it to work in practice (they should know dozens of projects and assign appropriate weights to them). The tedious process of evaluating projects can discourage voters.

The size of budgets also does not impose a constraint, although some advantages of the Method may be apparent particularly in cases of voting on budgets that have significant amounts to distribute.

“I would say that like many of these methods, the larger the funds and the greater the choice, the greater the chance that it will all go well,” explains Professor Piotr Faliszewski. “You could say that the smaller the process, the more noise and the greater the risk that we end up with a very specific case. On the other hand, we have built a number of safety valves into the resolution that we always propose. In particular, one such safety valve is that at the very end, when we have calculated the results by the Method of Equal Shares, we also calculate the results by the majority method. We have assumed that the voter wants as many of their projects as possible, and therefore prefers one result to another if more of the projects they voted for have been adopted. So we compare whether more voters are satisfied with the Method of Equal Shares or with this majority method. And if it turns out that the majority method gives greater satisfaction, then we return to this method. This very rarely happens, but theoretically it is possible, especially in such very small civic budgets. But there is this safety valve, so there is no possibility of us messing things up, and practically it will certainly be better.”

A detailed description of how the procedure works along with illustrative videos can be found at