Resolve distortion bounds for det-rand distributed mechanisms

Determine tight worst-case distortion bounds for det-rand distributed voting mechanisms in general metric spaces, where the in-group rule is randomized and the over-group rule is deterministic, within the metric social choice framework defined in the paper. Ascertain these bounds to complete the characterization alongside the analyzed det-det, rand-det, and rand-rand mechanisms.

Background

The paper provides tight or nearly tight distortion bounds for three classes of distributed mechanisms in general metric spaces: det-det (both stages deterministic), rand-det (randomized second stage only), and rand-rand (both stages randomized). These results cover all four cost objectives considered in the distributed metric setting.

However, the class where the first stage is randomized and the second stage is deterministic (det-rand) is explicitly identified as remaining open. The authors note that their primary lower-bound technique (Bias Tournament) is incompatible with randomization in the first stage, and thus current understanding for det-rand is limited to basic bounds inherited from other mechanism classes. Closing this gap would complete the distortion landscape for distributed voting under metric preferences.

References

A significant challenge about our work leaves open lies in analyzing the det-rand\ mechanisms, where random decisions in the first stage are followed by deterministic ones in the second.

— Tight Bounds On the Distortion of Randomized and Deterministic Distributed Voting  (2509.17134 - Abam et al., 21 Sep 2025) in Discussion and Open Problems

In general, we conjecture that the distortion should be upper bounded by the deterministic bound. For any sampling size the expected metric distortion is bounded by $\max (\frac{2-k}{k}, \frac{3-k}{1-k})$.

— A Constant Metric Distortion Protocol for Approval Voting Given Plurality Polls  (2608.28340 - Frank et al., 28 Aug 2026) in Conjecture following Theorem expectedTheorem, Section 4 (Robustness to Noise)