Approximately fair and population consistent budget division via simple payment schemes

In approval-based budget division, a budget needs to be distributed to some candidates based on the voters' approval ballots over these candidates. In the pursuit of a simple, consistent, and approximately fair rule for this setting, we introduce the maximum payment rule (MP). Under this rule, each voter controls a part of the budget and, in each step, the corresponding voters allocate their entire budget to the candidate approved by the largest number of voters with non-zero budget. We show that MP meets our criteria as it satisfies monotonicity and a demanding population consistency condition and gives a 2-approximation to a fairness notion called average fair share (AFS). Moreover, we generalize MP to the class of sequential payment rules and prove that it is the most desirable rule within this class: nearly all other sequential payment rules fail monotonicity while offering only small improvements in the approximation ratio to AFS.