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：我们研究了参与式预算（PB）中的低样本复杂度机制，其中每位选民投票支持一个偏好分配，即在满足项目成本与总支出约束下，将资金分配给各个项目。我们分析了PB机制相对于期望最小社会成本结果的失真度。针对该问题的随机独裁者机制实现了2的失真度。在每位选民仅投票支持一个项目的特殊情形中，[Fain等 '17]获得了4/3的失真度。我们证明，当PB结果由两位选民的投票的任意凸组合决定时，失真度为2。当使用三个均匀随机抽样的投票时，我们提出了一种失真度至多为1.66的PB机制，从而以最小可能的样本复杂度打破了2的阈值。我们给出一种随机纳什议价方案，其中两位均匀随机选择的选民进行议价，以一位均匀随机选择的选民的投票作为分歧点。该机制的失真度至多为1.66。我们为该方案提供了1.38的失真度下界。进一步，我们证明，输出三位均匀随机选择的选民的投票中位数的PB机制，其失真度至多为1.80。