We study the model of metric voting proposed by Feldman et al. [2020]. In this model, experts and candidates are located in a metric space, and each candidate possesses a quality that is independent of her location. An expert evaluates each candidate as the candidate's quality less a bias term--the distance between the candidate and the expert in the metric space. The expert then votes for her favorite candidate. The goal is to select a voting rule and a committee of experts to mitigate the bias. More specifically, given $m$ candidates, what is the minimum number of experts needed to ensure that the voting rule selects a candidate whose quality is at most $\varepsilon$ worse than the best one? Our first main result is a new way to select the committee using exponentially less experts compared to the method proposed in Feldman et al. [2020]. Our second main result is a novel construction that substantially improves the lower bound on the committee size. Indeed, our upper and lower bounds match in terms of $m$, the number of candidates, and $\varepsilon$, the desired accuracy, for general convex normed spaces, and differ by a multiplicative factor that only depends on the dimension of the underlying normed space but is independent of other parameters of the problem. We extend the nearly matching upper and lower bounds to the setting in which each expert returns a ranking of her top $k$ candidates and we wish to choose $\ell$ candidates with cumulative quality at most $\varepsilon$ worse than that of the best set of $\ell$ candidates, settling an open problem of Feldman et al. [2020]. Finally, we consider the setting where there are multiple rounds of voting. We show that by introducing another round of voting, the number of experts needed to guarantee the selection of an $\varepsilon$-optimal candidate becomes independent of the number of candidates.

翻译：我们研究Feldman等人[2020]提出的度量投票模型。在该模型中，专家与候选人位于度量空间中，每位候选人都具有与其位置无关的固有质量。专家对候选人的评分为其质量减去偏差项——即候选人与专家在度量空间中的距离。随后，专家投票给其最偏好的候选人。目标是选择一种投票规则和一个专家委员会以缓解偏差。具体而言，给定$m$名候选人，至少需要多少专家才能确保投票规则选出的候选人质量与最优候选人相差不超过$\varepsilon$？我们的第一个主要成果提出了一种新方法，其所需专家数量较Feldman等人[2020]的方法呈指数级减少。第二个主要成果是一种新型构造，显著改进了委员会规模的下界。实际上，在一般凸赋范空间中，我们的上下界在候选人数量$m$和期望精度$\varepsilon$上相匹配，仅相差一个取决于底层赋范空间维度的乘法因子，且该因子与问题其他参数无关。我们将近乎匹配的上下界扩展至以下场景：每位专家返回其前$k$名候选人的排名，且我们希望选出$\ell$名候选人，其累积质量与最优$\ell$名候选人集合相比相差不超过$\varepsilon$，从而解决了Feldman等人[2020]提出的开放问题。最后，我们考虑多轮投票场景。研究表明，通过引入新一轮投票，确保选出$\varepsilon$-最优候选人所需要的专家数量将独立于候选人数目。