We study deliberative social choice, where voters refine their preferences through small-group discussions before collective aggregation. We introduce a simple and easily implementable deliberation-via-matching protocol: for each pair of candidates, we form an arbitrary maximum matching among voters who disagree on that pair, and each matched pair deliberates. The resulting preferences (individual and deliberative) are then appropriately weighted and aggregated using the weighted uncovered set tournament rule. We show that our protocol has a tight distortion bound of $3$ within the metric distortion framework. This breaks the previous lower bound of $3.11$ for tournament rules without deliberation and matches the lower bound for deterministic social choice rules without deliberation. Our result conceptually shows that tournament rules are just as powerful as general social choice rules, when the former are given the minimal added power of pairwise deliberations. We prove our bounds via a novel bilinear relaxation of the non-linear program capturing optimal distortion, whose vertices we can explicitly enumerate, leading to an analytic proof. Loosely speaking, our key technical insight is that the distortion objective, as a function of metric distances to any three alternatives, is both supermodular and convex. We believe this characterization provides a general analytical framework for studying the distortion of other deliberative protocols, and may be of independent interest.

翻译：我们研究审议性社会选择，即选民在集体聚合前通过小组讨论完善其偏好。我们引入一种简单且易于实施的匹配审议协议：对于每对候选人，我们在对该对持不同意见的选民中形成任意最大匹配，每对匹配的选民进行审议。随后，使用加权未覆盖集锦标赛规则对产生的偏好（个体偏好与审议后偏好）进行适当加权与聚合。我们证明，在度量失真框架下，该协议具有紧致的失真上界$3$。这突破了无审议时锦标赛规则先前$3.11$的下界，并与无审议时确定性社会选择规则的下界相匹配。我们的结果在概念上表明，当锦标赛规则被赋予成对审议这一最小附加能力时，其效力与一般社会选择规则相当。我们通过一种新颖的双线性松弛方法证明该界，该方法处理刻画最优失真的非线性规划问题，其顶点可显式枚举，从而得到解析证明。粗略而言，我们的关键技术见解是：失真目标函数作为任意三个备选方案间度量距离的函数，既是超模的又是凸的。我们相信这一特征为研究其他审议协议的失真提供了通用分析框架，并可能具有独立的研究价值。