Method of equal shares

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results The method of equal shares[1][2][3][4] is a proportional method of counting ballots that applies to participatory budgeting,[2] to committee elections,[3] and to simultaneous public decisions.

It works by dividing the available budget into equal parts that are assigned to each voter.

It then repeatedly finds projects that can be afforded using the budget shares of the supporting voters.

[1] In 2023, the method of equal shares was being used in a participatory budgeting program in the Polish city of Wieliczka.

It was also used in a participatory budgeting program in the Swiss city of Aarau in 2023 (Stadtidee).

[7] The method of equal shares was first discussed in the context of committee elections in 2019, initially under the name "Rule X".

[2][8] 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).

[10] In contrast, the method of equal shares would pick 5 blue and 5 red projects.

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.

The method selects first the project that minimises the price per voter.

This section presents the definition of the rule for cardinal ballots.

At the beginning it puts an equal part of the budget, in each voter's virtual bank account,

In each round the method selects one project according to the following procedure.

This section provides a discussion on other variants of the method of equal shares.

The method of equal shares can be used with other types of voters ballots.

The method can be applied in two ways to the setting where the voters vote by marking the projects they like (see Example 1): The method applies to the model where the voters vote by ranking the projects from the most to the least preferred one.

Each voter is initially assigned an equal part of the budget

The method of equal shares can return a set of projects that does not exhaust the whole budget.

There are multiple ways to use the unspent budget: In the context of committee elections the method is often compared to Proportional Approval Voting (PAV), since both methods are proportional (they satisfy the axiom of Extended Justified Representation (EJR)).

[13][14] The methods compare as follows: MES with adjusting initial budget, PAV and Phragmen's voting rules can all be viewed as extensions of the D'Hondt method to the setting where the voters can vote for individual candidates rather than for political parties.

[2] Below there is a Python implementation of the method that applies to participatory budgeting.

For the model of committee elections, the rules is implemented as a part of the Python package abcvoting.

Fairstein, Meir and Gal[16] extend MES to a setting in which some projects may be substitute goods.

Fairstein, Benade and Gal[17] compare MES to greedy aggregation methods.

They find that greedy aggregation leads to outcomes that are highly sensitive to the input format used, and the fraction of the population that participates.

In contrast, MES leads to outcomes that are not sensitive to the type of voting format used.

These outcomes are stable even when only 25 to 50 percent of the population participates in the election.

Fairstein, Meir, Vilenchik and Gal[18] study variants of MES both on real and synthetic datasets.