Improving Randomized Metric Distortion to 2.1441

Metric distortion has primarily been studied for social choice functions, which select a single winner from ordinal preferences. We extend this framework to social welfare functions, which output a ranking of $m$ candidates. We associate each voter $v$ with a monotone weight vector $\mathbf{w}_v = (w_{v1},\ldots,w_{vm})$, specifying the importance of the $i$-th position for voter $v$, and define the cost of a ranking as the position-weighted sum of their distances to the ranked candidates. This model generalizes both single-winner voting and committee selection. We consider three information regimes. First, we study the setting where the positional weight vectors are known. A natural approach recursively applies a single-winner rule with distortion $β$ to construct the ranking one position at a time. We show that this yields distortion at most $3β$ in general. This gap is not merely an artifact of the analysis: we show that no analysis based solely on per-round guarantees can certify a bound better than $2β$. By exploiting structural properties specific to Fractional Veto of Kizilkaya and Kempe, we show that its recursive extension achieves the optimal distortion of 3. Second, when all voters share the same unknown weight vector, recursively applying any social choice function with distortion $β$ achieves distortion at most $1+(β-1)\text{range}(\mathbf{w})$, where $\text{range}(\mathbf{w})=(w_1-w_m)/w_1$ denotes the normalized range of the common weight vector $\mathbf{w}$. Finally, we study unknown heterogeneous weights. Without further assumptions, every rule has unbounded distortion. We therefore consider two natural normalizations: unit-sum, where each voter distributes one unit of value across the ranking, and unit-top, where every voter assigns unit value to the first position. Under both models, we show that the optimal distortion is $Θ(m)$.

We consider the metric distortion framework for voting, where $n$ voters and $m$ candidates are embedded in an unknown metric space. A voting rule selects a candidate using only voters' ordinal preferences, and its distortion is the worst-case ratio between the social cost of the selected candidate and the minimum social cost. We study how round complexity trades off against total communication in deterministic voting protocols with low distortion. Deterministic rules can achieve the optimal distortion of $3$ in one round by eliciting complete rankings using $Θ(nm\log m)$ bits, while a direct sequential implementation of PluralityVeto achieves the same distortion using only $O(n\log m)$ bits but requires $Θ(n)$ rounds. Our main result is TreePluralityVeto, a deterministic voting protocol that, for every $r\ge 2$, achieves distortion at most $2r-1$ using at most $r$ rounds and $O(n\log m+n s^{1/(r-1)}\log s)$ bits of total communication, where $s = \min\{n, m\}$. The protocol runs PluralityVeto on a hierarchical partition of the voters, collecting restricted rankings in parallel at each level. The branching factor controls the tradeoff between rounds and communication, and a matching-based analysis shows that each level increases the distortion bound by at most $2$.

Competitive selection processes, from scientific funding to admissions and hiring, use evaluations to score candidates, and eventually choose a subset of them based on those scores. Recently, many organizations have adopted partial lotteries, which randomize selection based on evaluation scores. However, existing lottery designs are inherently unstable, as a small change to a single candidate's score can cause large shifts in their selection probabilities. This instability undermines a key goal of lotteries: reducing the influence of fine-grained score distinctions near the decision boundary. We propose smoothness as a design principle for partial lotteries, formalizing it as a Lipschitz condition on the mapping from review scores over candidates to selection probabilities. We introduce the Clipped Linear Lottery, a simple mechanism in which selection probabilities scale linearly with estimated quality between an upper threshold, above which we always accept, and a lower threshold, below which we always reject. We prove that the Clipped Linear Lottery's worst-case regret matches a lower bound for any smooth selection rule up to a factor of $(1 - k/n)$, where $k/n$ is the acceptance rate. We compare smooth selection to other stability notions like Individual Fairness and Differential Privacy, showing that the Clipped Linear Lottery achieves a better smoothness-regret tradeoff than alternatives. Experiments on real peer review data from ICLR 2025, NeurIPS 2024, and the Swiss National Science Foundation demonstrate that existing lottery designs are highly unstable in practice even under perturbations to a single score. Our experiments also confirm the tightness of our theoretical analysis and show that our proposed Clipped Linear Lottery achieves a better smoothness-utility tradeoff than alternatives in practice.