Tight Bounds on the Distortion of Randomized and Deterministic Distributed Voting

We study metric distortion in distributed voting, where $n$ voters are partitioned into $k$ groups, each selecting a local representative, and a final winner is chosen from these representatives (or from the entire set of candidates). This setting models systems like U.S. presidential elections, where state-level decisions determine the national outcome. We focus on four cost objectives from \citep{anshelevich2022distortion}: $\avgavg$, $\avgmax$, $\maxavg$, and $\maxmax$. We present improved distortion bounds for both deterministic and randomized mechanisms, offering a near-complete characterization of distortion in this model. For deterministic mechanisms, we reduce the upper bound for $\avgmax$ from $11$ to $7$, establish a tight lower bound of $5$ for $\maxavg$ (improving on $2+\sqrt{5}$), and tighten the upper bound for $\maxmax$ from $5$ to $3$. For randomized mechanisms, we consider two settings: (i) only the second stage is randomized, and (ii) both stages may be randomized. In case (i), we prove tight bounds: $5\!-\!2/k$ for $\avgavg$, $3$ for $\avgmax$ and $\maxmax$, and $5$ for $\maxavg$. In case (ii), we show tight bounds of $3$ for $\maxavg$ and $\maxmax$, and nearly tight bounds for $\avgavg$ and $\avgmax$ within $[3\!-\!2/n,\ 3\!-\!2/(kn^*)]$ and $[3\!-\!2/n,\ 3]$, respectively, where $n^*$ denotes the largest group size.