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\begin{document}
\section{Definitions of Paradoxes}
\begin{itemize}
    \item Different Winner: Two voting rules $r_1$ and $r_2$, are said to have different winners if $r_1(P)\neq r_2(P)$, meaning that at least one winner of rule $r_1$ is not a winner of rule $r_2$. We use $\EI (r_1, r_2, P)$ to describe this event.
    \item Different Winner (Enhanced): Defined as $r_1(P)\cap r_2(P)=\emptyset$, which implies that none of the winners of rule $r_1$ are among the winners of $r_2$. We use $\EI E(r_1, r_2, O)$ to describe this event. 
    \item Drastically different: Occurs when $r_1(P)\cap \hat{r}_2(P)\neq \emptyset$, i.e. one of the winner in $r_1$ is actually a loser of $r_2$. We use $\DI (r_1, r_2, P)$ to describe this event.
    \item Drastically different (enhanced): Defined by $r_1(P)= \hat{r}_2(P)$, i.e. all winners of $r_1$ are losers of $r_2$. We use $\DI E(r_1, r_2, P)$ to describe this event
    
    .
\end{itemize}
\subsection{Relation of these four paradoxes}
First, it is easy to see $\EI E\implies \EI$ and $\DI E\implies \DI$. Also, when $r_1(P)\cap \hat r_1(P)=\emptyset$ (the only case where $r_1(P)\cap \hat r_1(P)\neq\emptyset$ is when all alternatives are tied), we have $\DI\implies\EI$ and $\DI E\implies\EI E$. Hence we have the diagram below. 

\begin{tikzcd}[arrows=Rightarrow]
& \text{$\EI E$} \arrow{r} & \text{$\EI$} \\
& \text{$\DI E$} \arrow{r} \arrow{u} & \text{$\DI$} \arrow{u}
\end{tikzcd}

Hence, $\DI E$ is the sub-event of any other paradox settings. If we can show $\DI E$ has an $\Theta(1)$ likelihood, the rest of events automatically has $\Theta(1)$ likelihood.
\section{Strong Borda's Paradox} 
\begin{theorem}\label{thm:SgBorda}
    Suppose the set of distributions $\Pi$ be strictly positive, closed and $\pi_{uni}\in CH(\Pi)$ then
    \[\widetilde{\DI E}^{\max}_{r_{\vec s},r_C}(n)=\begin{cases}
        0 & \text{if}\ r_{\vec s}\ \text{is the Borda's rule} \\
        \Theta(1) & otherwise 
    \end{cases}
    \]
\end{theorem}

\begin{theorem}\label{thm:diffposi1}
    For fixed $m\geq 3$ and any two linear independent score vectors $\Vsi$ and $\Vsii$. Suppose the set of distributions $\Pi$ is strictly positive and closed, and $\vec \pi_{uni}\in CH(\Pi)$, then 
    \[
        \widetilde{\DI E}^{\max}_{r_{\Vsi},r_{\Vsii}}(n)=\Theta(1).
    \]
\end{theorem}

\begin{theorem}\label{thm:CdandPosi}
    Let ${\mathrm{Cd}_\alpha}$ denotes the copeland rule with parameter $\alpha$.
    %and score vector $\vec s$.
    For fixed $m\geq 3$, $\alpha\in[0,1]$, and $\vec s$ which is an integer score vector, 
    suppose the set of distributions $\Pi$ be strictly positive, closed, and $\vec \pi_{uni}\in CH(\Pi)$. Then 
    \[
        \widetilde{\DI E}^{\max}_{r_{\vec s},{\mathrm{Cd}_\alpha}}(n)=\Theta(1).
    \]
\end{theorem}

\begin{theorem}\label{thm:stv_plur}
    Given $m\geq 3$, 
    \[
    \widetilde{\EI E}^{max}_{STV, plurality}(n)=\Theta(1).
    \]
\end{theorem}

\begin{corollary}\label{coro:iidSgBorda}

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