大型选举的可能性有多大？
How Likely Are Large Elections Tied?
摘要：
了解选举被捆绑的可能性是许多学科的经典主题，包括社会选择，博弈论，政治学和公共选择。尽管有大量的文献和普遍的信念，领带是罕见的，很少有人知道如何罕见的关系，在大型选举中，除了一些简单的位置评分规则下的i.i.d.选票的均匀分配，被称为社会选择中的公平文化（IC）。特别是，在2001年Marchant明确提出在IC下k路关系的可能性作为一个开放的问题之后，几乎没有取得进展。我们在比i.i.d.更通用和更现实的模型下，对广泛的常用研究投票规则的开放问题给出了渐进答案模型（特别是IC）--夏的平滑社会选择框架，受到斯皮尔曼和滕的著名平滑复杂性分析的启发。我们证明二分法定理的平滑可能性的关系下的位置评分规则，边顺序为基础的规则，和一些多轮评分为基础的消除规则，其中包括通常研究的投票规则，如复数，Borda，否决权，最大，Copeland，排名对，舒尔茨，STV，和Coombs作为特例。我们还通过在Preflib上对合成数据和真实世界的排名数据进行实验来补充理论结果。我们的主要技术工具是一个改进的二分表征的平滑似然泊松多项式变量是在一个多面体，这是证明通过探索之间的相互作用的V-表示和多面体的矩阵表示，可能是独立的兴趣。
英文摘要：
Understanding the likelihood for an election to be tied is a classical topic in many disciplines including social choice, game theory, political science, and public choice. Despite a large body of literature and the common belief that ties are rare, little is known about how rare ties are in large elections except for a few simple positional scoring rules under the i.i.d. uniform distribution over the votes, known as the Impartial Culture (IC) in social choice. In particular, little progress was made after Marchant explicitly posed the likelihood of k-way ties under IC as an open question in 2001. We give an asymptotic answer to the open question for a wide range of commonly studied voting rules under a model that is much more general and realistic than i.i.d. models (especially IC) -- the smoothed social choice framework by Xia that was inspired by the celebrated smoothed complexity analysis by Spielman and Teng. We prove dichotomy theorems on the smoothed likelihood of ties under positional scoring rules, edge-order-based rules, and some multi-round score-based elimination rules, which include commonly studied voting rules such as plurality, Borda, veto, maximin, Copeland, ranked pairs, Schulze, STV, and Coombs as special cases. We also complement the theoretical results by experiments on synthetic data and real-world rank data on Preflib. Our main technical tool is an improved dichotomous characterization on the smoothed likelihood for a Poisson multinomial variable to be in a polyhedron, which is proved by exploring the interplay between the V-representation and the matrix representation of polyhedra and might be of independent interest.
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