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Method of equal shares
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Method of equal shares
The method of equal shares is a proportional method of counting ballots that applies to participatory budgeting, to committee elections, and to simultaneous public decisions. It can be used when the voters vote via approval ballots, ranked ballots or cardinal ballots. It works by dividing the available budget into equal parts that are assigned to each voter. The method is only allowed to use the budget share of a voter to implement projects that the voter voted for. It then repeatedly finds projects that can be afforded using the budget shares of the supporting voters. In contexts other than participatory budgeting, the method works by equally dividing an abstract budget of "voting power".
In 2023, the method of equal shares was being used in a participatory budgeting program in the Polish city of Wieliczka. The program, known as Green Million (Zielony Milion), was set to distribute 1 million złoty to ecological projects proposed by residents of the city. It was also used in a participatory budgeting program in the Swiss city of Aarau in 2023 (Stadtidee).
The method of equal shares was first discussed in the context of committee elections in 2019, initially under the name "Rule X". From 2022, the literature has referred to the rule as the method of equal shares, particularly when referring to it in the context of participatory budgeting algorithms. The method can be described as a member of a class of voting methods called expanding approvals rules introduced earlier in 2019 by Aziz and Lee for ordinal preferences (that include approval ballots).
The method is an alternative to the knapsack algorithm which is used by most cities even though it is a disproportional method. For example, if 51 percent of the population support 10 red projects and 49 percent support 10 blue projects, and the money suffices only for 10 projects, the knapsack budgeting will choose the 10 red supported by the 51 percent, and ignore the 49 percent altogether. In contrast, the method of equal shares would pick 5 blue and 5 red projects.
The method guarantees proportional representation: it satisfies a strong variant of the justified representation axiom adapted to participatory budgeting. This says that a group of X percent of the population will have X percent of the budget spent on projects supported by the group (assuming that all members of the group have voted the same or at least similarly).
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.
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.
The budget is first divided equally among the voters; thus, each voter gets $10. Project received most votes, and it is selected in the first round. If we divided the cost of equally among the voters, who supported , each of them would pay . In contrast, if we selected, e.g., , then the cost per voter would be . The method selects first the project that minimises the price per voter.
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Method of equal shares
The method of equal shares is a proportional method of counting ballots that applies to participatory budgeting, to committee elections, and to simultaneous public decisions. It can be used when the voters vote via approval ballots, ranked ballots or cardinal ballots. It works by dividing the available budget into equal parts that are assigned to each voter. The method is only allowed to use the budget share of a voter to implement projects that the voter voted for. It then repeatedly finds projects that can be afforded using the budget shares of the supporting voters. In contexts other than participatory budgeting, the method works by equally dividing an abstract budget of "voting power".
In 2023, the method of equal shares was being used in a participatory budgeting program in the Polish city of Wieliczka. The program, known as Green Million (Zielony Milion), was set to distribute 1 million złoty to ecological projects proposed by residents of the city. It was also used in a participatory budgeting program in the Swiss city of Aarau in 2023 (Stadtidee).
The method of equal shares was first discussed in the context of committee elections in 2019, initially under the name "Rule X". From 2022, the literature has referred to the rule as the method of equal shares, particularly when referring to it in the context of participatory budgeting algorithms. The method can be described as a member of a class of voting methods called expanding approvals rules introduced earlier in 2019 by Aziz and Lee for ordinal preferences (that include approval ballots).
The method is an alternative to the knapsack algorithm which is used by most cities even though it is a disproportional method. For example, if 51 percent of the population support 10 red projects and 49 percent support 10 blue projects, and the money suffices only for 10 projects, the knapsack budgeting will choose the 10 red supported by the 51 percent, and ignore the 49 percent altogether. In contrast, the method of equal shares would pick 5 blue and 5 red projects.
The method guarantees proportional representation: it satisfies a strong variant of the justified representation axiom adapted to participatory budgeting. This says that a group of X percent of the population will have X percent of the budget spent on projects supported by the group (assuming that all members of the group have voted the same or at least similarly).
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.
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.
The budget is first divided equally among the voters; thus, each voter gets $10. Project received most votes, and it is selected in the first round. If we divided the cost of equally among the voters, who supported , each of them would pay . In contrast, if we selected, e.g., , then the cost per voter would be . The method selects first the project that minimises the price per voter.