We study deliberative social choice, where voters engage in small-group discussions to output collective preferences that are then aggregated by a social choice rule. We introduce a simple deliberation-via-matching protocol. In this protocol, for each pair of candidates, we form a maximum matching among voters who disagree on that pair, and have each matched pair deliberate. We then aggregate the resulting individual and deliberative preferences using the weighted uncovered set tournament rule. We show that this protocol has a tight distortion bound of $3$ within the metric distortion framework. In the absence of deliberation, general deterministic social choice rules can achieve this distortion, whereas deterministic tournament rules face a strictly larger lower bound of $3.11$. Our result closes this gap: Pairwise deliberation allows a tournament-based rule to attain distortion $3$. Conceptually, this shows that tournament rules can match the power of general deterministic social choice rules once they are given the minimal added power of pairwise deliberations. We prove this bound 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. This characterization therefore provides a new analytical tool for studying the distortion of deliberative protocols, and may be of independent interest. Finally, although our analysis is for the full protocol, we show that this mechanism also admits a lightweight sampling-based implementation, yielding a high-probability approximation to the deterministic guarantee with arbitrary accuracy and low per-voter complexity.

翻译：我们研究了商议性社会选择，其中选民参与小组讨论以输出集体偏好，这些偏好随后由社会选择规则汇总。我们引入了一个简单的“经由匹配的商议”协议。在该协议中，对于每一对候选者，我们在对该对候选者持不同意见的选民之间构建一个最大匹配，并让每一对匹配的选民进行商议。然后，我们使用加权未覆盖集锦标赛规则对由此产生的个体偏好与商议后偏好进行汇总。我们证明，在度量失真框架下，该协议具有严格的失真界限 $3$。在没有商议的情况下，一般的确定性社会选择规则可以达到这一失真度，而确定性锦标赛规则则面临严格更大的下界 $3.11$。我们的结果弥合了这一差距：成对商议使得基于锦标赛的规则能够实现失真度 $3$。从概念上讲，这表明，一旦锦标赛规则被赋予成对商议这一最小程度的额外能力，就能与一般确定性社会选择规则的能力相匹配。我们通过一种新颖的双线性松弛方法来证明这一界限，该方法针对捕获最优失真的非线性规划，其顶点可以被显式枚举，从而得出解析证明。粗略地说，我们的关键技术见解是：失真目标函数，作为到任意三个候选者之间度量距离的函数，同时具有超模性和凸性。因此，这一特性为研究商议协议中的失真提供了一种新的分析工具，并可能具有独立的意义。最后，尽管我们的分析针对完整的协议，但我们展示了该机制也可通过轻量级的基于采样的实现方式，以任意精度和低选民复杂度，高概率地逼近确定性保证。