\subsection{Drastically Difference between Different Positional Rules}

\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 $\pi_{uni}\in CH(\Pi)$, then 
    \[
        \widetilde{\DI}^{\max}_{r_{\Vsi},r_{\Vsii}}(n)=\Theta(1).
    \]
\end{theorem}

\noindent\textbf{Proof Sketch.} We construct a sub-event $Y_a$ and check it satisfies the conditions of Lemma \ref{lem:Theta1}, and then the $\Theta(1)$ result can be derived by the lemma. Given alternative $a$, we define the sub-event $Y_a$ by ``$a$ is the winner under $r_{\Vsi}$ and the loser under $r_{\Vsii}$ in $P$''. $\mathcal{H}^a$ will satisfy the conditions whenever it is not empty, and hence we construct a profile $P$ such that $\Hist(P)\in \mathcal{H}^a$. Finally we shown 
\[\widetilde{\DI{}}^{\max}_{r_{\Vsi},r_{\Vsii}}(n)=\Theta(1).\]

The full proof is in Appendix~\ref{apx:diffposi1}.

%\begin{proof}
    %\noindent\textbf{Step 1: Sub-Event and Polyhedron $\mathcal{H}^{a}$} 

    %Given alternative $a$, we define the sub-event $Y_a$ by ``$a$ is the winner under $r_{\Vsi}$ and the loser under $r_{\Vsii}$ in $P$''. Let $A^{\Vsi,a}$ be the matrix whose row vectors are $\{Score_{k,a}^{\Vsi}:k\in[m], k\neq a\}$ and $A^{\Vsii,a}$ be the matrix 
    %with row vectors $\{Score_{a,k}^{\Vsii}:k\in[m], k\neq a\}$. Let $A^{a}=\begin{pmatrix}
    %    A^{\Vsi,a} \\
    %    A^{\Vsii,a}
    %\end{pmatrix}$ and $\vec b=-\vec{1}$. Then $\mathcal{H}^{a}:=\{\vec x\in\mathbb{R}^{m!}:A^{a}\vec x\leq \vec %b\}$. Then it's not hard to verify that $\mathcal{H}^{a}$ represents sub-event $Y_a$. 

    %\noindent\textbf{Step 2: Prove the conditions on $\mathcal{H}^a$} 

    %It suffices to show that $\mathcal{H}^a$ is non-empty since the rest of the reasoning is similar to the proof of Theorem \ref{thm:SgBorda}. We then construct a profile $P$ with $\Hist(P)\in\mathcal{H}^a$. 

    %We consider a series of profiles $\{P_i\}_{i\in [m]}$ defined as follows. Reindex the set of alternatives by $\mathcal{A}=\{a, c_1,\cdots, c_{m-1}\}$. Given $i\in [m]$, let 
    %\[
    %    P_i=\{v_j: (v_j)_{-i}=(c_j, \cdots, c_{m-1}, \cdots, c_{j-1}), (v_j)_i=a\}, 
    %\]
    %be the profile of $m-1$ votes. 
    %Let $\Vsi=(s_1^1, s_2^1,\cdots,s_m^1)$, $\Vsii=(s_1^2, s_2^2, \cdots, s_m^2)$, and $S_k:=\sum_{j=1}^m s_k^j$ where $k\in\{1,2\}$. We have 
    %\[
    %    Score_{a, c_j}^{\Vsk}(P_i)= ms_i^k-S_k, k\in\{1,2\},
    %\]
    %for any  $j\in\{1,\cdots,m-1\}$, since the score of $c_j$ are all the same.
    
    %We then show that there exists $(t_1, t_2,\cdots, t_m)\in \mathbb{Z}^m$ such that $P=\sum_{i=1}^m t_iP_i$ satisfies $\Hist(P)\in \mathcal{H}^a$. Given profile $P$, $-P$ is defined as the profile that reverses each vote's rank in $P$, and an easy observation is $Score_{a,b}^{\vec s}(P)=-Score_{a,b}^{\vec s}(-P)$ for any alternatives $a,b$ and score vector $\vec s$. Hence, it suffices to find an integer vector $\vec t=(t_1,\cdots,t_m)$, such that 
    %\[
    %    \sum_{i=1}^m t_i(ms_i^1-S_1)=\vec t\cdot (m\Vsi-S_1\vec{1})>0, 
    %\]
    %\[
    %    \sum_{i=1}^m t_i(ms_i^2-S_2)=\vec t\cdot (m\Vsii-S_2\vec{1})<0.
    %\]

    %Since $\Vsi$ and $\Vsii$ are linear independent, we claim that $m\Vsi-S_1\vec{1}$ and $m\Vsii-S_2\vec{1}$ are linear independent. If not so, there exists $\lambda\in\mathbb{R}$ such that for all $i \in [m]$, $ms_i^1-S_1=\lambda(ms_i^2-S_2)$. Since $s_m^1=s_m^2=0$, we have $S_1=\lambda S_2$, hence $ms_i^1=ms_i^2$ for all $i\in[m]$, which contradicts to the fact that $\Vsi$ and $\Vsii$ are independent. Then there exists $\vec {t'}\in\mathbb{R}^m$ satisfies the inequalities, since $m\Vsi-S_1\vec{1}$ and $m\Vsii-S_2\vec{1}$ are independent. Suppose $\vec t'\cdot (m\Vsi-S_1\vec{1})=x_1$ and $\vec t'\cdot (m\Vsii-S_2\vec{1})=-x_2$ where $x_1,x_2>0$. Let $x:=\min\{x_1, x_2\}$ and $M:=\max \{\sum_{i=1}^m |m s_i^1-S_1|, \sum_{i=1}^m |m s_i^2-S_2| \}$. There exists $\vec{t^*}\in\mathbb{Q}^m$ such that $\sum_{i=1}^m |t'_i-t^*_i|<\frac{x}{2M}$ since $\mathbb{Q}^m$ are dense in $\mathbb{R}^m$. It can be verified that $t^*$ also satisfies these inequalities and the integer vector $\vec t$ is obtained by multiplying an appropriate integer to $t^*$. Hence we have constructed a profile $P=\sum_{i=1}^m t_iP_i$ where $\Hist(P)\in\mathcal{H}^a$.

    %\noindent\textbf{Step 3: Apply Lemma \ref{lem:Theta1}.}
    %Finally, we apply Lemma 1 and get 
    %\[        \widetilde{\DI{}}^{\max}_{r_{\Vsi},r_{\Vsii}}(n)=\Theta(1).
    %\] 
    %\qed
%\end{proof}

%\begin{conjecture}\ziqi{update 20240604: Conjecture of min-likelihood, positional rules.}
    %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. If for any $i\in [m]$ there exists a corresponding fractional profile $\vec\pi_i\in CH(\Pi)$ such that the score of $i$ is not the highest in $r_{\Vsi}$ and the lowest in $r_{\Vsii}$ (tie permitted), then
    %\[
    %    \widetilde{\DI}^{\min}_{r_{\Vsi},r_{\Vsii}}(n)=\exp(-\Theta(n)),
    %\]
    %otherwise 
    %\[
    %    \widetilde{\DI}^{\min}_{r_{\Vsi},r_{\Vsii}}(n)=\Theta(1).
    %\]
%\end{conjecture}
Similar to the Section~\ref{sec:SgBorda}, we have the following corollary.
\begin{corollary} \label{coro:iid}
    Suppose the distribution set $\Pi=\{\pi\}$ where $\pi$ is strictly positive. Let
    \[
    \DI{}_{r_{\Vsi}, r_{\Vsii}}(n):=\mathrm{Pr}_{P\sim  \pi^n}(\DI(r_{\Vsi}, r_{\Vsii},P)=1),
    \]
    where $\Vsi, \Vsii$ are two linearly independent score vectors. Consider $\pi$ as a fractional profile, we have 
    \[
    \DI{}_{r_{\Vsi}, r_{\Vsii}}(n)=
    \begin{cases}
        \Theta(1) & r_{\Vsi}(\pi)\cap \hat{r}_{\Vsii}(\pi)\neq \emptyset, \\
        \exp(-\Theta(n)) & \text{otherwise.}
    \end{cases}
    \]
\end{corollary}




