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 simple, well-behaved, and approximately fair rules for this setting, we introduce the class of sequential payment rules, where each voter controls a part of the budget and repeatedly spends his share on his approved candidates to determine the final distribution. We show that all sequential payment rules satisfy a demanding population consistency notion and we identify two particularly appealing rules within this class called the maximum payment rule (MP) and the $\frac{1}{3}$-multiplicative sequential payment rule ($\frac{1}{3}$-MP). More specifically, we prove that (i) MP is, apart from one other rule, the only monotonic sequential payment rule and gives a $2$-approximation to a fairness notion called average fair share, and (ii) $\frac{1}{3}$-MP gives a $\frac{3}{2}$-approximation to average fair share, which is optimal among sequential payment rules.

翻译：在基于批准的预算分配中，需要根据选民对候选人的批准投票将预算分配给部分候选人。为寻求适用于该场景的简单、行为良好且近似公平的规则，我们引入顺序支付规则类别，其中每位选民控制部分预算，并反复将其份额用于其批准的候选人以确定最终分配。我们证明所有顺序支付规则均满足严格的人口一致性要求，并在此类别中识别出两种特别具有吸引力的规则：最大支付规则（MP）与$\frac{1}{3}$乘性顺序支付规则（$\frac{1}{3}$-MP）。具体而言，我们证明：（i）除另一规则外，MP是唯一具有单调性的顺序支付规则，并对平均公平份额这一公平性概念提供$2$倍近似；（ii）$\frac{1}{3}$-MP对平均公平份额提供$\frac{3}{2}$倍近似，这在顺序支付规则中达到最优。