Approximately Fair and Population Consistent Budget Division via Simple Payment Schemes

In approval-based budget division, a budget needs to be distributed to 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 rule and prove that it is the most desirable rule in this class: all sequential payment rules but MP and one other rule fail monotonicity while only allowing for a small improvement in the approximation ratio to AFS.