带并列的线性排序问题的Condorcet型性质
Condorcet-type properties of the linear ordering problem with ties
- Hosei University（法政大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对带并列的线性排序问题，引入非严格强Condorcet准则（NSCC）并证明其对任意实例的最优解成立，同时给出两候选人在所有最优解中必须并列的互补结构性质。
AI中文摘要：
Kemeny规则将多个严格排序聚合为单个严格排序，使得该排序与输入排序之间的距离之和最小。由此产生的优化问题称为Kemeny问题（\ exttt{KP}），它是线性排序问题（\ exttt{LOP}）的一个特例。Kemeny规则满足社会选择理论中的若干理想性质，包括扩展Condorcet准则（\ exttt{XCC}）。Ando等人通过引入强Condorcet准则（\ exttt{SCC}）并证明该准则对任意\ exttt{LOP}实例的每个最优解都成立，强化了这一结果。Yoo和Escobedo将Kemeny规则扩展到带并列的排序，并证明了由此产生的规则满足非严格扩展Condorcet准则（\ exttt{NXCC}）。该准则给出了一个条件，在此条件下，某个候选人在每个最优解中都必须严格排在另一个候选人之上。在本文中，我们引入了非严格强Condorcet准则（\ exttt{NSCC}），它是针对带并列排序的\ exttt{SCC}的对应物，并证明该准则对任意带并列的线性排序问题（\ exttt{LOPT}）实例的每个最优解都成立。我们还建立了一个互补的结构性质，该性质给出了在任意\ exttt{LOPT}实例的每个最优解中两个候选人必须并列的条件。
英文摘要：
The Kemeny rule aggregates multiple strict rankings into a single strict ranking that minimizes the sum of its distances from the input rankings. The resulting optimization problem, called the Kemeny problem (\texttt{KP}), is a special case of the linear ordering problem (\texttt{LOP}). The Kemeny rule satisfies several desirable properties in social choice theory, including the extended Condorcet criterion (\texttt{XCC}). Ando et al. strengthened this result by introducing the strong Condorcet criterion (\texttt{SCC}) and showing that it holds for every optimal solution to an arbitrary \texttt{LOP} instance. Yoo and Escobedo extended the Kemeny rule to rankings with ties and showed that the resulting rule satisfies the non-strict extended Condorcet criterion (\texttt{NXCC}). This criterion gives a condition under which one candidate must be ranked strictly above another in every optimal solution. In this paper, we introduce the non-strict strong Condorcet criterion (\texttt{NSCC}), a counterpart of the \texttt{SCC} for rankings with ties, and show that it holds for every optimal solution to an arbitrary instance of the linear ordering problem with ties (\texttt{LOPT}). We also establish a complementary structural property that gives conditions under which two candidates must be tied in every optimal solution to an arbitrary \texttt{LOPT} instance.
