We study the budget aggregation problem in which a set of strategic voters must split a finite divisible resource (such as money or time) among a set of competing projects. Our goal is twofold: We seek truthful mechanisms that provide fairness guarantees to the projects. For the first objective, we focus on the class of moving phantom mechanisms [Freeman et al., 2021], which are -- to this day -- essentially the only known truthful mechanisms in this setting. For project fairness, we consider the mean division as a fair baseline, and bound the maximum difference between the funding received by any project and this baseline. We propose a novel and simple moving phantom mechanism that provides optimal project fairness guarantees. As a corollary of our results, we show that our new mechanism minimizes the $\ell_1$ distance to the mean (a measure suggested by Caragiannis et al. [2022]) for three projects and gives the first non-trivial bounds on this quantity for more than three projects.

翻译：我们研究预算聚合问题，其中一组策略性投票者必须在多个竞争项目之间分配有限的可分割资源（如资金或时间）。我们的目标有两方面：寻求在项目间提供公平保障的真实机制。针对第一个目标，我们聚焦于移动幻影机制类[Freeman等人，2021]——迄今为止，这本质上仍是该设置下唯一已知的真实机制。在项目公平性方面，我们将平均分配作为公平基线，并约束任意项目获得的资金与该基线之间的最大差异。我们提出了一种新颖且简洁的移动幻影机制，该机制能提供最优的项目公平保障。作为我们结果的推论，我们证明新机制在三个项目时最小化了与均值的$\ell_1$距离（由Caragiannis等人[2022]提出的度量指标），并首次在项目数量超过三个时给出了该量的非平凡界。