Metric Distortion of Small-group Deliberation

TL;DR

This work shows that very small deliberation groups can substantially improve the metric distortion of social choice rules, beating long-standing deterministic and randomized lower bounds. By modeling group outputs via two mechanisms—deterministic averaging and probabilistic random choice—and aggregating outcomes with Copeland, the authors prove sharp distortion reductions: for k=3 (and k=4 in the randomized setting) the distortion falls below the classic 3 (and 2.11) bounds, and, crucially, distortion converges to 1 as group size grows, with k depending only on the target accuracy ε. The analysis hinges on a core small-deviation optimization problem, θ_k, solved via non-convex relaxations and Berry–Esseen-based asymptotics, yielding tight bounds such as θ_2=√2−1 and θ_3≈0.25–0.2522; for large k, distortion scales as 1+O(1/k) (averaging) or 1+O(√(log k / k)) (random choice). The results generalize to concave bias and opinion-change models, extend sample-complexity guarantees, and reinforce the practical potential of organizing deliberation into many tiny groups rather than a single large session.

Abstract

We consider models for social choice where voters rank a set of choices (or alternatives) by deliberating in small groups of size at most $k$, and these outcomes are aggregated by a social choice rule to find the winning alternative. We ground these models in the metric distortion framework, where the voters and alternatives are embedded in a latent metric space, with closer alternative being more desirable for a voter. We posit that the outcome of a small-group interaction optimally uses the voters' collective knowledge of the metric, either deterministically or probabilistically. We characterize the distortion of our deliberation models for small $k$, showing that groups of size $k=3$ suffice to drive the distortion bound below the deterministic metric distortion lower bound of $3$, and groups of size $4$ suffice to break the randomized lower bound of $2.11$. We also show nearly tight asymptotic distortion bounds in the group size, showing that for any constant $ε> 0$, achieving a distortion of $1+ε$ needs group size that only depends on $1/ε$, and not the number of alternatives. We obtain these results via formulating a basic optimization problem in small deviations of the sum of $i.i.d.$ random variables, which we solve to global optimality via non-convex optimization. The resulting bounds may be of independent interest in probability theory.

Figures (1)

• Figure 1: Distortion of the Random Choice model as a function of the group size $k$, for $k \in [2,30]$.

Theorems & Definitions (26)

• Example 1
• Theorem 1.1: Proved in \ref{['sec:avg']}.
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 3.1
• proof