The utilitarian distortion framework evaluates voting rules by their worst-case efficiency loss when voters have cardinal utilities but express only ordinal rankings. Under the classical model, a longstanding tension exists: Plurality, which suffers from the spoiler effect, achieves optimal $Θ(m^2)$ distortion among deterministic rules, while normatively superior rules like Copeland and Borda have unbounded distortion. We resolve this tension under probabilistic voting with the Plackett-Luce model, where rankings are noisy reflections of utilities governed by an inverse temperature parameter $β$. Copeland and Borda both achieve at most $β\frac{1+e^{-β}}{1-e^{-β}}$ distortion, independent of the number of candidates $m$, and within a factor of 2 of the lower bound for randomized rules satisfying the probabilistic Condorcet loser criterion known from prior work. This improves upon the prior $O(β^2)$ bound for Borda. These upper bounds are nearly tight: prior work establishes a $(1-o(1))β$ lower bound for Borda, and we prove a $(1-ε)β$ lower bound for Copeland for any constant $ε>0$. In contrast, rules that rely only on top-choice information fare worse: Plurality has distortion $Ω(\min(e^β, m))$ and Random Dictator has distortion $Θ(m)$. Additional `veto' information is also insufficient to remove the dependence on $m$; Plurality Veto and Pruned Plurality Veto have distortion $Ω(β\ln m)$. We also prove a lower bound of $(\frac{5}{8}-ε)β$ (for any constant $ε>0$) for all deterministic finite-precision tournament-based rules, a class that includes Copeland and any rule based on pairwise comparison margins rounded to fixed precision. Our results show that the distortion framework aligns with normative intuitions once the probabilistic nature of real-world voting is taken into account.

翻译：功利主义扭曲框架通过评估投票规则在最坏情况下的效率损失来衡量其性能，该损失发生在选民具有基数效用但仅表达序数排名时。在经典模型下，长期存在一种张力：易受搅局效应影响的多数决（Plurality）在确定性规则中实现了最优的 $Θ(m^2)$ 扭曲，而规范性上更优的规则如 Copeland 和 Borda 则具有无界扭曲。我们在概率投票的 Plackett-Luce 模型下解决了这一张力，其中排名是由逆温度参数 $β$ 控制的效用的噪声反映。Copeland 和 Borda 均实现至多 $β\frac{1+e^{-β}}{1-e^{-β}}$ 的扭曲，与候选人数量 $m$ 无关，并且处于先前工作中已知的满足概率性孔多塞输家准则的随机化规则下界的 2 倍因子内。这改进了先前 Borda 规则的 $O(β^2)$ 上界。这些上界几乎是紧的：先前工作为 Borda 建立了 $(1-o(1))β$ 下界，我们为 Copeland 证明了对于任意常数 $ε>0$，存在 $(1-ε)β$ 下界。相比之下，仅依赖首选信息的规则表现更差：多数决（Plurality）具有 $Ω(\min(e^β, m))$ 扭曲，随机独裁者（Random Dictator）具有 $Θ(m)$ 扭曲。额外的“否决”信息也不足以消除对 $m$ 的依赖；多数否决（Plurality Veto）和修剪多数否决（Pruned Plurality Veto）具有 $Ω(β\ln m)$ 扭曲。我们还证明了所有基于锦标赛的确定性有限精度规则（包括 Copeland 以及任何基于固定精度舍入的成对比较边际的规则）具有 $(\frac{5}{8}-ε)β$（对于任意常数 $ε>0$）的下界。我们的结果表明，一旦考虑到现实世界投票的概率性本质，扭曲框架便与规范性直觉相一致。