\subsection{Positional Rules and Ranked Pairs are Drastically Different}\label{sect:RPPosi}
\begin{theorem}
\label{thm:psr_rp}
     Let $r_{\mathrm{RP}}$ denote the ranked pairs rule. 
     For fixed $m\geq 3$ and an integer score vector $\vec s$, suppose the set of distributions $\Pi$ be strictly positive, closed, and $\pi_{uni}\in CH(\Pi)$. Then 
    \[
         \widetilde{\DI}^{\max}_{r_{\vec s},r_{\mathrm{RP}}}(n)=
         \begin{cases}
             \Theta(n^{-\frac{3}{2}}) & m=3,\ r_{\vec s} \text{ is Borda},  \\
             \Theta(1)& otherwise.
         \end{cases}
    \]
\end{theorem}
\noindent\textbf{Proof Sketch.} On the one hand, when $r_{\vec s}$ is Borda and $m=3$, the desired event (DD) happens if and only if $WMG(P)$ is a directed cycle where the weight of each edge is the same, and hence it can be described by polyhedron and the result is derived by applying Theorem~\ref{thm:pmv1}. On the other hand, we follow a similar way of Theorem~\ref{thm:SgBorda}.
 The full proof is in Appendix~\ref{apx:psr_rp}.

We also consider the reverse version. Let $G$ and $G'$ be the acyclic graph constructed by the Ranked Pairs rule under profiles $P$ and $\bar P$, respectively. By the properties of the complement profile, the sink (source) of $G$ is exactly the source (sink) of $G'$. Hence, we can follow the same reasoning in Corollary~\ref{coro:reverse1} and state the following corollary. 
\begin{corollary}
\label{coro:reverse_rp}
     Suppose the set of distributions $\Pi$ be strictly positive, closed, and $\vec \pi_{uni}\in CH(\Pi)$. Then 
    \[
         \widetilde{\DI}^{\max}_{r_{\RP},r_{\vec s}}(n)=
         \begin{cases}
             \Theta(n^{-\frac{3}{2}}) & m=3,\ r_{\vec s} \text{ is Borda} \\
             \Theta(1)& otherwise
         \end{cases}
     \]
\end{corollary}

%We also state the i.i.d. case as follows. 

%\begin{corollary}
   %Suppose the distribution set $\Pi=\{\pi\}$ where $\pi$ is strictly positive. Let
    %\[
    %\DI{}_{r_{\vec s}, r_{RP}}(n):=\mathrm{Pr}_{P\sim \pi^n}(\DI(r_{\vec s}, r_{RP},P)=1),
    %\]
    %where $\vec s$ is a score vector. Consider $\pi$ as a fractional profile, we have 
    %\[
    %\DI{}_{r_{\vec s}, r_{RP}}(n)=
    %\begin{cases}
    %    \Theta(n^{-\frac{3}{2}}) & m=3, r_{\vec s}\text{ is Borda,}\ r_{\vec s}(\pi)\cap %\hat{r}_{RP}(\pi)\neq\emptyset\\
    %    \exp(-\Theta(n)) & r_{\vec s}(\pi)\cap \hat{r}_{RP}(\pi)=\emptyset \\
    %    \Theta(1) & o.w.
    %\end{cases}
    %\]
%\end{corollary}



%     %%REMOVE THESE MIN CONJECTURES?
% %\begin{conjecture}
%     %Let $r_{\mathrm{RP}}$ denote the ranked pairs rule. 
%     %For fixed $m\geq 4$ and an integer score vector $\vec s$, suppose the set of distributions %$\Pi$ be strictly positive and closed. If all fractional profiles $P$ in $CH(\Pi)$ satisfy
%     %\begin{itemize}
%         %\item $WMG(P_i)=G_0$ holds for all $P_i\in CH(\Pi)$, 
%         %\item The winner of positional rule is the loser of the ranked pair, given any $P\in %CH(\Pi)$,
%     %\end{itemize}
%     %then 
%     %\[
%     %    \widetilde{\DI}^{\min}_{r_{\vec s},r_{\mathrm{RP}}}(n)=\Theta(1).
%     %\]
% %\end{conjecture}