Title: Tight Bounds on the Distortion of Randomized and Deterministic Distributed Voting
Venue: NeurIPS 2025
Accepted: yes
Keywords: Distributed voting

Distortion

Metric

Randomization
Abstract: 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 Anshelevich \et~\cite{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.


Peer reviews:
comments: SUMMARY:
This paper presents a comprehensive study of metric distortion in distributed voting systems, where voters are grouped, and local winners are selected before a final aggregation. The authors consider both deterministic and randomized voting rules and analyze their performance across four natural cost objectives: avg-avg, avg-max, max-avg, and max-max. Significant theoretical advances are made, including tight or nearly tight bounds for all objectives in both rand-det and rand-rand settings. For example, the paper improves the deterministic upper bound for avg-max from 11 to 7, establishes a tight lower bound of 5 for max-avg, and proves distortion bounds of 3 for several randomized objectives. The analysis introduces novel tools, such as the Bias Tournament, and explores both general metric and Euclidean spaces. The results offer a near-complete characterization of distortion in this setting and highlight the benefits and limitations of randomization in distributed social choice. The paper is well-structured and clearly written, with detailed proofs and illustrative examples that support its contributions, making it a valuable reference for future research in metric voting theory and distributed decision-making.

STRENGTHS AND WEAKNESSES:
Strengths
The paper makes substantial progress in understanding the distortion of distributed voting mechanisms, closing gaps in existing bounds and introducing novel techniques like the Bias Tournament for lower bounds.

Weaknesses:
1.Authors may refine their descriptions, such as some proof sketches, in order to include a conclusion section.
2.Authors do not propose new mechanisms.

QUESTIONS:
Why didn't you consider the det-rand mechanism?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-20

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies distributed voting systems, where n voters are partitioned into k groups. Each group selects a local representative, and a final winner is chosen from the set of representatives (or all candidates). The key measure of interest is \"distortion,\" which quantifies how much worse the outcome chosen by a voting rule (using only ordinal preferences) can be compared to the optimal outcome (knowing all voters' actual numerical costs or utilities) in a metric space. Distributed voting was first considered by Anshelevich et al. '22, and since then, only deterministic voting rules have been explored. This paper is the first to explore randomized voting rules in the distributed setup, though it is worth noting that randomized voting mechanisms in the distortion framework (standard non-distributed setting) have already been explored and are known to provide a better distortion bound compared to deterministic ones.\n\nThe current paper mostly focuses on the randomized-deterministic and randomized-randomized mechanisms. For rand-det, the authors provide matching (upper and lower) bounds for all four (avg-avg, avg-max, max-avg, and max-max) cases, while on rand-rand, they provide matching 3-distortion bounds for max-avg and max-max, and near-matching bounds for the other two cases. Further, the authors also manage to strengthen the lower bound of det-det mechanisms in the max-avg case, provide a tight 5-distortion lower bound, and improve the upper bounds of avg-max (10 to 7) and max-max case (5 to 3). In the end, the authors focus on the Euclidean metric and provide a lower bound on distortion for both rand-rand and rand-det mechanisms in the avg-avg case.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper shows a host of new results on distributed voting mechanisms and shows many interesting tight bounds. One can say that a small weakness is that this paper does not introduce any new interesting techniques. Upper bound results, though interesting (some improve the known bounds, such as how to merge distortion bounds of outer and inner mechanisms to get a better final bound), are pretty simple. However, the lower bound constructions are clever, albeit with their simplicity.\n\nQUESTIONS:\nNothing specific\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-27"}

comments: SUMMARY:
This paper studies metric distortion in distributed voting systems, where voters are split into groups, each choosing a representative. The final winner is determined from these representatives. It focuses on four common cost objectives: avg-avg, avg-max, max-avg, and max-max, and provides tight upper and lower bounds for distortion with respect to deterministic and randomized mechanisms in general and line metric spaces. Some applications for this problem include certain elections, such as the US presidential election.

The authors use methods such as bias tournament constructions and shortest path distances in order to derive these bounds. Some of the existing bounds are improved, and most of the developed bounds are tight.

STRENGTHS AND WEAKNESSES:
Strengths:
1. The paper is well written and includes the necessary details, while leaving the details to the appendix.
2. The problem in question regarding studying distortion bounds is an important one, and the paper provides tight upper and lower bounds for an extensive number of deterministic and randomized mechanisms.
3. The methods used, such as bias tournament and shortest path distances, in the proofs are interesting.

Weakness:
1. There is a lack of empirical results, and real-world applications could be better explained.

QUESTIONS:
1. This is mostly a well-written paper; however, it would be better to explain how distortion can be used in the real world.
2. Distortion seems to be a worst-case measure of sorts. It might help to include experiments to show how much distortion occurs in practice.

LIMITATIONS:
Since this is a theoretical paper, it is not expected to consider the societal impact. However, is there a potential use case where the knowledge of distortion can be used to manipulate elections?
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies metric distortion in distributed voting, where (1) voters are divided into groups, (2) each group selects a local winner, and (3) the group winners then select a final winner. The authors consider four social cost objectives and present improved upper and lower bounds on the distortion of both deterministic and randomized voting rules.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- The paper studies voting rules quantitatively through the lens of metric distortion, which complements the traditional axiomatic approach. There have been many theoretical advances in this area recently.\n- It is natural to study the distortion of voting rules in the distributed setting, which could be of practical relevance in the long run. The presentation of the paper is clear.\n- The paper presents solid technical results. The authors close the gap in worst-case distortion for the max-avg and max-max social objective for deterministic rules (and reduce the gap for avg-max), establish tight bounds in 6 out of 8 randomized settings they consider (and give nearly tight bounds in the 2 remaining cases).\n\nWeaknesses\n- The paper has a wide range of results but can read like a Cartesian product of different settings: 4 objectives, 3 randomness models (where is det-rand? I am not advocating for adding it), and various metric spaces. This can seem tedious to readers unfamiliar with the area. The paper could be strengthened by emphasizing which results are most novel or challenging to prove, and what ideas the authors propose (e.g., Bias Tournament) that are missing from prior work. How much are the authors' proofs inspired by that in Kizilkaya-Kempe? \n- The authors could consider splitting Table 1 into two tables (one for deterministic and one for randomized) and/or adding the avg-avg row in det-det to provide a complete overview. (Are the upper and lower bounds of the avg-avg deterministic case still $7$ and $11$?). The paper should define $n^*$ in Table 1 (in addition to Theorem 4.4). The phrase \"tight bounds\" in the title may be a bit of an overstatement as some bounds are not tight, perhaps \"tighter bounds\"?\n\nTypos\n- Line 78: $\\alpha$ is not defined yet, and should it be $\\alpha+2$ instead of $5$?\n- Line 330 and 955: \"an unanimous\"\n- Line 347: \"can not\"\n- Line 951: \"these information\"\n- Line 960: \"Observation A.11\" missing brackets in the long equation\n\nQUESTIONS:\nThe authors are encouraged but not required to answer the questions mentioned in “Strengths & Weaknesses” above.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-16"}