We study low sample complexity mechanisms in participatory budgeting (PB), where each voter votes for a preferred allocation of funds to various projects, subject to project costs and total spending constraints. We analyze the distortion that PB mechanisms introduce relative to the minimum-social-cost outcome in expectation. The Random Dictator mechanism for this problem obtains a distortion of 2. In a special case where every voter votes for exactly one project, [Fain et al '17] obtain a distortion of 4/3 We show that when PB outcomes are determined as any convex combination of the votes of two voters, the distortion is 2. When three uniformly randomly sampled votes are used, we give a PB mechanism that obtains a distortion of at most 1.66, thus breaking the barrier of 2 with the smallest possible sample complexity. We give a randomized Nash bargaining scheme where two uniformly randomly chosen voters bargain with the disagreement point as the vote of a voter chosen uniformly at random. This mechanism has a distortion of at most 1.66. We provide a lower bound of 1.38 for the distortion of this scheme. Further, we show that PB mechanisms that output a median of the votes of three voters chosen uniformly at random have a distortion of at most 1.80.

翻译：我们研究参与式预算（PB）中低样本复杂度的机制，其中每位选民投票支持一个在项目成本和总支出约束下对各项目资金的偏好分配方案。我们分析了PB机制相对于预期最小社会成本结果所引入的失真度。针对该问题的随机独裁机制所得失真度为2。在每位选民仅支持一个项目的特殊情形下，[Fain等人 '17]所得失真度为4/3。我们证明，当PB结果由任意两位选民投票的凸组合决定时，失真度为2。当使用三个均匀随机抽样的投票时，我们提出一种PB机制，其失真度不超过1.66，从而以最小可能的样本复杂度突破失真度2的界限。我们设计了一个随机纳什议价方案，其中两位均匀随机选择的选民进行议价，以另一位均匀随机选择的选民的投票作为分歧点。该机制失真度不超过1.66，我们给出该方案失真度的下界为1.38。此外，我们证明输出三位均匀随机选择的选民投票中位数的PB机制，其失真度不超过1.80。