Condorcet-type properties of the linear ordering problem with ties

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.