循环序投票、群论与选票
Voting on Cyclic Orders, Group Theory, and Ballots
AI总结：
本文利用对称群表示论分析循环序投票的多种选票类型及基于点数的程序，对n=4给出完整刻画，对n=5给出关键观察，并强调对称性（中立性）要求。
摘要：
循环序可以非正式地理解为一种安排人们围坐在桌子旁的方式，例如用于赌博游戏或晚餐。给定一组智能体如$\{A,B,C\}$，我们可以通过将循环序定义为该有限集合上的一个置换或线性序，在等价关系下将$A\succ B\succ C$与$B\succ C\succ A$及$C\succ A\succ B$视为相同，从而形式化这一概念。与其他具有某种结构的集合族一样，我们可能希望聚合一组（可能不同的）投票者对于选择循环序的可能方式的偏好。\n然而，鉴于循环序的完整排序数量存在组合爆炸，人们可能不希望使用通常的投票机制。这引出了什么样的选票可能合适的问题：单个循环序、一组循环序，还是其他类型的选票？此外，置换群在智能体集合上存在自然作用。一个合理的选举程序要求是尊重这种对称性（相当于普通投票理论中的中立性）。\n在本文中，我们将利用对称群的表示论来分析投票于循环序的几种自然选票类型，以及使用这些选票的基于点数的程序。我们针对$n=4$的两种相当不同的选票类型提供了此类程序的完整刻画，并给出了$n=5$的最重要观察结果。
英文摘要：
A cyclic order may be thought of informally as a way to seat people around a table, perhaps for a game of chance or for dinner. Given a set of agents such as $\{A,B,C\}$, we can formalize this by defining a cyclic order as a permutation or linear order on this finite set, under the equivalence relation where $A\succ B\succ C$ is identified with both $B\succ C\succ A$ and $C\succ A\succ B$. As with other collections of sets with some structure, we might want to aggregate preferences of a (possibly different) set of voters on the set of possible ways to choose a cyclic order. However, given the combinatorial explosion of the number of full rankings of cyclic orders, one may not wish to use the usual voting machinery. This raises the question of what sort of ballots may be appropriate; a single cyclic order, a set of them, or some other ballot type? Further, there is a natural action of the group of permutations on the set of agents. A reasonable requirement for a choice procedure would be to respect this symmetry (the equivalent of neutrality in normal voting theory). In this paper we will exploit the representation theory of the symmetric group to analyze several natural types of ballots for voting on cyclic orders, and points-based procedures using such ballots. We provide a full characterization of such procedures for two quite different ballot types for $n=4$, along with the most important observations for $n=5$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
