Deliberation via Matching

TL;DR

This work introduces deliberation via matching, a minimal two-person dyadic-deliberation protocol for social choice under a metric distortion model. By pairing voters who disagree on each candidate pair and weighting deliberation outcomes within a lambda-weighted uncovered set tournament, the authors establish a tight distortion bound of 3, matching the deterministic optimum for non-deliberative rules and surpassing prior tournament-only bounds. A key technical advance is a bilinear reformulation of the distortion objective, leveraging supermodularity and convexity to enable analytic proofs and vertex enumeration, with a warm-up showing distortion 2 for two candidates. The results imply that tournament-based rules, when equipped with minimal pairwise deliberation, are as powerful as general social choice rules in preserving welfare under ordinal information. The paper also provides lower bounds and open questions about extending these insights to other rules and larger deliberation groups, highlighting practical and theoretical avenues for leveraging small-group deliberation in collective decision-making.

Abstract

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.

Figures (6)

• Figure 1: Counter-monotonic coupling of $X$ (in blue) and $Y$ (in red). Note the graphs partition $[0,1]$ into $AC/CA$ (by $X$) and $BC/CB$ (by $Y$).
• Figure 2: The prefix-suffix structure of $A\succ C$ pairs.
• Figure 3: A visualization of continuous tie-handling on $X$. Left (a): At time step $t'$, both $P_{t'}$ and $S_{t'}$ have positive mass. Middle (b): We handle $P_{t'}$ by continuously allocating increasing subsets $P'\subseteq P_{t'}$ to $C\succ A$ pairs. Right (c): We then handle $S_{t'}$ by continuously allocating increasing subsets $S'\subseteq S_{t'}$ to $CA$ and argue that the change in $W_{AC}(t')$ is also continuous.
• Figure 4: The lines go from $0$ to $1$, capturing cumulative voter mass. The top line represents $X$ values in decreasing order and the bottom line represents $Y$ values in increasing order. A voter appears at the same position in both lines. The pink masses $p_1$ and $p_5$ represent a set of $A \succ C$ matching pairs. This means $p_1 = p_5$. The same holds for the purple masses $p_2$ and $p_4$. The masses $p_1, p_2, p_3$ correspond to non-negative $X$ values, hence together capture $\lvert AC\rvert$. The masses $p_1, \ldots, p_4$ have non-positive $Y$ values and together capture $\lvert BC\rvert$. The $C \succ B$ pairs are captured by the pairs of masses $(p_2, p_9)$, $(p_3, p_8)$ and $(p_4,p_7)$.
• Figure 5: Interval split for Case 2. The interpretation of this figure is similar to \ref{['fig:ac_small_LP']}. Note that analogous to that case, we have $p_2 = p_4$.

Theorems & Definitions (34)

• Theorem 1.1: Main Theorem, proved in \ref{['sec:general']}
• Theorem 4.1
• proof
• Corollary 1
• Theorem 4.2
• proof
• Corollary 2
• Theorem 4.3
• proof
• Definition 1