In approval-based budget division, the task is to allocate a divisible resource to the candidates based on the voters' approval preferences over the candidates. For this setting, Brandl et al. [2021] have shown that no distribution rule can be strategyproof, efficient, and fair at the same time. In this paper, we aim to circumvent this impossibility theorem by focusing on approximate strategyproofness. To this end, we analyze the incentive ratio of distribution rules, which quantifies the maximum multiplicative utility gain of a voter by manipulating. While it turns out that several classical rules have a large incentive ratio, we prove that the Nash product rule ($\mathsf{NASH}$) has an incentive ratio of $2$, thereby demonstrating that we can bypass the impossibility of Brandl et al. by relaxing strategyproofness. Moreover, we show that an incentive ratio of $2$ is optimal subject to some of the fairness and efficiency properties of $\mathsf{NASH}$, and that the positive result for the Nash product rule even holds when voters may report arbitrary concave utility functions. Finally, we complement our results with an experimental analysis.

翻译：在基于审批的预算分配中，任务是根据选民对候选人的审批偏好，将可分割的资源分配给候选人。对于这一设定，Brandl 等人 [2021] 已证明，没有任何分配规则能同时满足防策略性、效率性和公平性。本文旨在通过关注近似防策略性来规避这一不可能性定理。为此，我们分析了分配规则激励比率——该比率量化了选民通过操纵行为所能获得的最大乘性效用增益。尽管结果表明若干经典规则具有较大的激励比率，但我们证明纳什乘积规则（$\mathsf{NASH}$）的激励比率为 $2$，从而表明我们可通过放宽防策略性来绕过 Brandl 等人的不可能性结果。此外，我们证明在满足 $\mathsf{NASH}$ 的某些公平性和效率性属性约束下，激励比率 $2$ 是最优的，并且该关于纳什乘积规则的正面结论甚至在选民可能报告任意凹效用函数时仍然成立。最后，我们通过实验分析对结果进行了补充。