A voting rule decides on a probability distribution over a set of m alternatives, based on rankings of those alternatives provided by agents. We assume that agents have cardinal utility functions over the alternatives, but voting rules have access to only the rankings induced by these utilities. We evaluate how well voting rules do on measures of social welfare and of proportional fairness, computed based on the hidden utility functions. In particular, we study the distortion of voting rules, which is a worst-case measure. It is an approximation ratio comparing the utilitarian social welfare of the optimum outcome to the social welfare produced by the outcome selected by the voting rule, in the worst case over possible input profiles and utility functions that are consistent with the input. The previous literature has studied distortion with unit-sum utility functions (which are normalized to sum to 1), and left a small asymptotic gap in the best possible distortion. Using tools from the theory of fair multi-winner elections, we propose the first voting rule which achieves the optimal distortion $\Theta(\sqrt{m})$ for unit-sum utilities. Our voting rule also achieves optimum $\Theta(\sqrt{m})$ distortion for a larger class of utilities, including unit-range and approval (0/1) utilities. We then take a worst-case approach to a quantitative measure of the fairness of a voting rule, called proportional fairness. Informally, it measures whether the influence of cohesive groups of agents on the voting outcome is proportional to the group size. We show that there is a voting rule which, without knowledge of the utilities, can achieve a $\Theta(\log m)$-approximation to proportional fairness, and thus also to Nash welfare and to the core, making it interesting for applications in participatory budgeting. For all three approximations, we show that $\Theta(\log m)$ is the best possible.

翻译：投票规则基于代理人提供的备选方案排序，决定一组m个备选方案上的概率分布。我们假设代理人对备选方案具有基数效用函数，但投票规则仅能访问这些效用所诱导的排序。我们评估投票规则在基于隐藏效用函数计算的社会福利和比例公平性指标上的表现。特别地，我们研究了投票规则的扭曲度（一种最坏情况度量）。该指标是一种近似比，比较最优结果的功利主义社会福利与投票规则所选结果产生的社会福利，在最坏情况下考虑可能的输入配置和与输入一致的效用函数。已有文献研究了单位总和效用函数（归一化至总和为1）下的扭曲度，并在可能的最优扭曲度上留下了一个小的渐近差距。利用公平多赢家选举理论中的工具，我们提出了首个在单位总和效用下达到最优扭曲度Θ(√m)的投票规则。我们的投票规则对更广泛的效用函数类别（包括单位区间和批准（0/1）效用）也实现了最优Θ(√m)扭曲度。随后，我们采用最坏情况方法研究投票规则的定量公平性指标——比例公平性。非正式地说，该指标衡量凝聚性代理人群体对投票结果的影响是否与群体规模成比例。我们证明存在一种投票规则，在无需知晓效用函数的情况下，可实现Θ(log m)近似比例公平性，进而也近似纳什福利和核心解，这使得该规则在参与式预算等应用中具有重要价值。对于所有三种近似，我们证明Θ(log m)是最优可能的结果。