A canonical problem in social choice is how to aggregate ranked votes: given $n$ voters' rankings over $m$ candidates, what voting rule $f$ should we use to aggregate these votes into a single winner? One standard method for comparing voting rules is by their satisfaction of axioms - properties that we want a "reasonable" rule to satisfy. Unfortunately, this approach leads to several impossibilities: no voting rule can simultaneously satisfy all the properties we want, at least in the worst case over all possible inputs. Motivated by this, we consider a relaxation of these worst case requirements. We do so using a "smoothed" model of social choice, where votes are perturbed with small amounts of noise. If, no matter which input profile we start with, the probability (post-noise) of an axiom being satisfied is large, we will consider the axiom as good as satisfied - called "smoothed-satisfied" - even if it may be violated in the worst case. Our model is a mild restriction of Lirong Xia's, and corresponds closely to that in Spielman and Teng's original work on smoothed analysis. Much work has been done so far in several papers by Xia on axiom satisfaction under such noise. In our paper, we aim to give a more cohesive overview on when smoothed analysis of social choice is useful. Within our model, we give simple sufficient conditions for smoothed-satisfaction or smoothed-violation of several previously-unstudied axioms and paradoxes, plus many of those studied by Xia. We then observe that, in a practically important subclass of noise models, although convergence eventually occurs, known rates may require an extremely large number of voters. Motivated by this, we prove bounds specifically within a canonical noise model from this subclass - the Mallows model. Here, we present a more nuanced picture on exactly when smoothed analysis can help.

翻译：社会选择中的一个经典问题是如何聚合排名投票：给定n个投票者对m个候选人的排名，应使用何种投票规则f来将这些投票聚合为单一获胜者？比较投票规则的标准方法之一是考察其对公理的满足程度——即我们期望"合理"规则应具备的性质。遗憾的是，这种方法会导致若干不可能性：没有投票规则能同时满足所有期望性质，至少在面向所有可能输入的最坏情况下如此。受此启发，我们考虑对这些最坏情况要求进行放松。我们采用社会选择的"平滑"模型来实现，其中投票被添加少量噪声干扰。如果无论初始输入概貌如何，公理被满足的概率（加噪后）都很高，我们就称该公理被"平滑满足"——即使它可能在最坏情况下被违反。我们的模型是对Xia Lirong模型的温和限制，与Spielman和Teng在平滑分析原始工作提出的模型高度对应。Xia已在多篇论文中对这种噪声下的公理满足性开展了广泛研究。本文旨在对平滑分析何时有助于社会选择给出更统一的概述。在我们的模型框架内，我们为多个先前未研究的公理与悖论（以及Xia研究过的许多公理）的平滑满足或平滑违反给出了简洁充分条件。随后发现，在实践重要的噪声模型子类中，尽管最终会收敛，但已知收敛速率可能需要极大数量的投票者。受此启发，我们专门针对该子类中的经典噪声模型——Mallows模型——证明了界值。在此，我们呈现了一个关于平滑分析确切何时能发挥作用的更为精细的图景。