Approximate Single-Peakedness in Large Elections

Zhihuai Chen, Q. Li, Xiaoming Sun, Lirong Xia, Jialin Zhang
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Abstract

Single-peaked preferences are a natural way to avoid paradoxes and impossibility theorems in social choice and have recently been involved in the study of various computational aspects of social choice. Since strict single-peakedness is hard to achieve in practice, approximate single-peakedness appears more appropriate and is gaining popularity. In this paper, we study approximate single-peakedness of large, randomly-generated profiles. We focused on characterizing the asymptotically optimal social axis, which is asymptotically consistent with most agents’ preferences generated from a statistical model. We characterize all asymptotically optimal social axes under the Mallows model for two case: the case where the dispersion parameter $\varphi$ is close to 0, and the case where $\varphi$ is close to 1. We also design an algorithm to help characterize all asymptotically optimal social axes for all $\varphi$ when the number of alternative is no more than 10. These results help us understand the structure of approximate single-peakedness in large elections.
大型选举中的近似单峰性
单峰偏好是避免社会选择中悖论和不可能定理的一种自然方式,最近被用于社会选择的各种计算方面的研究。由于严格的单峰在实践中很难实现,近似单峰显得更为合适,并且越来越受欢迎。在本文中,我们研究了随机生成的大型剖面的近似单峰性。我们专注于描述渐近最优社会轴,它与统计模型生成的大多数代理的偏好渐近一致。我们在两种情况下描述了Mallows模型下的所有渐近最优社会轴:离散参数$\varphi$接近于0的情况,以及$\varphi$接近于1的情况。我们还设计了一种算法来帮助描述所有$\varphi$的渐近最优社会轴,当选项的数量不超过10时。这些结果有助于我们理解大型选举中近似单峰的结构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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