\appendix
\setcounter{lemma}{0}
\setcounter{theorem}{0}
\setcounter{proposition}{0}

\section{Missing Proofs in Section 3}
\label{appx: sec: 2way}
\begin{proposition}[\bf\boldmath $\beta$-Pareto Efficiency, Upper Bound]
    \label{appx-prop: tradeoff-DP-Pareto}
    There is no neutral rule $f\colon \LA^n\to \R(A)$ that satisfies $\epsilon$-DP and $\beta$-Pareto efficiency with $\beta> \erm^{\frac{n\epsilon}{m-1}}$.
\end{proposition}
\begin{proof}
    Let $f\colon \LA^n\to A$ be a voting rule satisfying $\epsilon$-DP and $\beta$-Pareto efficiency. Let $P$ be a profile, where $a_1\succ a_2\succ\cdots\succ a_m$, for all $i\in N$. Since $a_1$ Pareto dominates $a_2$, $a_2$ Pareto dominates $a_3$, etc., we have
    \begin{align*}
        \p[f(P)=a_1] & \geqslant \beta\cdot \p[f(P)=a_2]        \\
                     & \geqslant \beta^2\cdot \p[f(P)=a_3]      \\
                     & \cdots                                   \\
                     & \geqslant \beta^{m-1}\cdot \p[f(P)=a_m].
    \end{align*}
    Then we claim that for profile $P$,
    \begin{align*}
        \p[f(P)=a_1] & \leqslant \erm^{n\epsilon}\cdot \p[f(P)=a_m].
    \end{align*}
    Theorefore, we have $\beta^{m-1}\leqslant \erm^{n\epsilon}$, i.e., $\beta\leqslant \erm^{\frac{n\epsilon}{m-1}}$, as desired.

    Finally, we prove the claim above. In fact, for any voting rule $f\colon \LA^n\to \R(A)$ satisfying $\epsilon$-DP and neutrality, we have
    \begin{align}
        \label{appx-equ: n-eps-pareto-bound}
        \p[f(P)=a] & \leqslant \erm^{n\epsilon}\cdot \p[f(P)=b],\quad \text{for all }a,b\in A.
    \end{align}
    Now, we prove Equation (\ref{appx-equ: n-eps-pareto-bound}). For any profile $P,P'\in \LA^n$, let $\ell_0(\cdot,\cdot)$ represents the $\ell_0$-distance between them, i.e., $\ell_0(P,P')=\{j\in N: \succ_j\neq \succ_j'\}$. Then, by considering the following operation {\bf Op}, we can see that $P$ can be transferred to $P'$ through $k=\ell_0(P,P')$ times of operations.
    \begin{itemize}
        \item {\bf Op}: Choose a voter $i\in N$ that $\succ_i\neq \succ_i'$, let $\succ_i=\succ_i'$.
    \end{itemize}
    Letting $P_0,P_1,\ldots,P_k$ denote all of the profiles, we have the following diagram.
    \begin{align*}
        P=P_0\xrightarrow{\text{\bf Op}}P_1\xrightarrow{\text{\bf Op}}P_2\xrightarrow{\text{\bf Op}}\cdots\xrightarrow{\text{\bf Op}}P_k=P'.
    \end{align*}
    Notice that in each step, only one voter's preference is changed. Consequently, for each $i$, $P_i$ and $P_{i+1}$ are neighboring profiles. Since $f$ satisfies $\epsilon$-DP, we have
    \begin{align*}
        \p[f(P)=a] &\leqslant \erm^\epsilon \cdot \p[f(P_1)=a] \\
        &\leqslant \erm^{2\epsilon}\cdot \p[f(P_2)=a] \\
        &\leqslant\ldots \\
        &\leqslant \erm^{k\cdot\epsilon}\cdot\p[f(P')=a].
    \end{align*}
    Besides, for any given $P,P'\in\LA^n$, there are at most $n$ distinct voters $j\in N$ that $\succ_j\neq \succ_j'$. Therefore, for any profile $P,P'\in \LA^n$ and any $a\in A$, we have
    \begin{align*}
        \p[f(P)=a] \leqslant \erm^{n\epsilon}\cdot \p[f(P')=a].
    \end{align*}
    Now, for any profile $P\in\LA^n$ and an arbitrarily chosen paif of alternatives $a,b\in A$, let $P'$ be the profile transferred from $P$ by swapping $a$ and $b$ in each voter's preference. By the neutrality of $f$, we have
    \begin{align*}
        \p[f(P)=a] &= \p[f(P')=b], \\
        \p[f(P')=a] &= \p[f(P)=b].
    \end{align*}
    Then it follows that
    \begin{align*}
        \p[f(P)=a] \leqslant &\; \erm^{n\epsilon}\cdot \p[f(P')=a] \\
        =                    &\; \erm^{n\epsilon}\cdot \p[f(P)=b],
    \end{align*}
    which completes the proof. 
\end{proof}

\begin{proposition}[\bf\boldmath $\beta$-Pareto Efficiency, Lower Bound]
    \label{appx-prop: bordaExp-pareto}
    Given $\epsilon\in\mathbb{R}_+$, Mechanism 1 satisfies $\epsilon$-DP and $\erm^{\frac{n\epsilon}{2m-2}}$-Pareto efficiency.
\end{proposition}
\begin{proof}
    Let $\mathfrak{E}_{\borda}\colon\LA^n\to \R(A)$ denote the mapping introduced by BordaEXP. Then for any profile $P\in\LA^n$ and alternative $a\in A$, we have
    \begin{align*}
        \p[\mathfrak{E}_{\borda}(P)=a]=\frac{\erm^{\borda_P(a)\epsilon/(2m-2)}}{\sum\limits_{c\in A}\erm^{\borda_P(c)\epsilon /(2m-2)}}.
    \end{align*}
    First, we establish the bound for Pareto efficiency. Given profile $P\in\LA^n$, suppose $a,b\in A$ are a pair of alternatives that $a\succ_j b$ for all $j\in N$. It follows that, for each voter $j \in N$, the number of alternatives that are considered worse than $b$ according to her preference order $\succ_j$ is strictly less than the number of alternatives considered worse than $a$. Formally, we have
    \begin{align*}
        |\{c\in A: a\succ_j c\}| - |\{c\in A: b\succ_j c\}| \geqslant 1, \quad \text{for all }j\in N.
    \end{align*}
    By the definition of Borda score, we have
    \begin{align*}
        \borda_P(a) - \borda_P(b) \geqslant n.
    \end{align*}
    Then it follows that
    \begin{align*}
        \p[\mathfrak{E}_{\borda}(P)=a] & = \frac{\erm^{\borda_P(a)\cdot \epsilon / (2m-2)}}{\sum\limits_{c\in A} \erm^{\borda_P(c)\cdot \epsilon / (2m-2)}}                 \\
                                       & \geqslant \frac{\erm^{\borda_P(b)}\cdot \erm^{n\epsilon / (2m-2)}}{\sum\limits_{c\in A} \erm^{\borda_P(c)\cdot \epsilon / (2m-2)}} \\
                                       & = \erm^{n\epsilon / (2m-2)}\cdot \p[\mathfrak{E}_{\borda}(P)=b],
    \end{align*}
    which indicates that $\mathfrak{E}_{\borda}$ satisfies $\erm^{\frac{n\epsilon}{2m-2}}$-Pareto efficiency.

    Then we prove the DP-bound. For all neighboring profiles $P,P'\in\LA^n$,
    \begin{align*}
        &\frac{\p[\mathfrak{E}_{\borda}(P)=a]}{\p[\mathfrak{E}_{\borda}(P')=a]} \\
        =&\; \frac{\erm^{\borda_P(a)\cdot \frac{\epsilon}{2m-2}}}{\erm^{\borda_{P'}(a)\cdot \frac{\epsilon}{2m-2}}} \cdot \frac{\sum\limits_{c\in A} \erm^{\borda_{P'}(c)\cdot \frac{\epsilon}{2m-2}}}{\sum\limits_{c\in A}\erm^{\borda_P(c)\cdot \frac{\epsilon}{2m-2}}} \\
                                                                               \leqslant&\; \erm^{\epsilon/2}\cdot \sup\limits_{P\in\LA^n} \frac{\sum\limits_{c\in A} \erm^{\borda_{P'}(c)\cdot \frac{\epsilon}{2m-2}}}{\sum\limits_{c\in A}\erm^{\borda_P(c)\cdot \frac{\epsilon}{2m-2}}}                                                       \\
                                                                               \leqslant&\; \erm^{\epsilon},
    \end{align*}
    which completes the proof.
\end{proof}

\begin{lemma}
    \label{appx-lem: proof-gamma-SD-eff}
    Given $\gamma>0$, a voting rule $f$ satisfies $\gamma$-SD-efficiency if and only if
    \begin{align*}
        \frac{1}{\gamma} \geqslant \sup\limits_{P\in \LA^n, \xi\in\R(A)}\inf\limits_{j\in N, y\in A}\frac{\sum\limits_{x:x\succ_j y} \p[\xi=x]}{\sum\limits_{x:x\succ_j y} \p[f(P)=x]}.
    \end{align*}
\end{lemma}
\begin{proof}
    Suppose that $f$ is not $\gamma$-SD-efficient. Then there must be some profile $P\in\LA^n$ that $f(P)$ is $\gamma$-SD-dominated by some $\xi\in\R(A)$, i.e., for all $y\in A$ and $\succ_j \in P$,
    \begin{align*}
        \sum\limits_{x:x\succ_j y} \p[\xi=x] \geqslant \frac{1}{\gamma}\cdot \sum\limits_{x:x\succ_j y}\p[f(P)=x],
    \end{align*}
    which is equivalent to
    \begin{align*}
        \frac{1}{\gamma} \leqslant \inf\limits_{j\in N, y\in A}\frac{\sum\limits_{x:x\succ_j y}\p[\xi=x]}{\sum\limits_{x:x\succ_j y}\p[f(P)=x]}.
    \end{align*}
    Therefore, $f$ is $\gamma$-SD-efficient if and only if for each $P\in\LA^n$, there does not exist such $\xi$, i.e., for all $y\in A$ and $P\in\LA^n$,
    \begin{align*}
        \frac{1}{\gamma} \geqslant \inf\limits_{j\in N, y\in A}\frac{\sum\limits_{x:x\succ_j y}\p[\xi=x]}{\sum\limits_{x:x\succ_j y}\p[f(P)=x]},
    \end{align*}
    which is equivalent to
    \begin{align*}
        \frac{1}{\gamma} \geqslant \sup\limits_{P\in\LA^n, \xi\in\R(A)}\inf\limits_{j\in A, y\in A}\frac{\sum\limits_{x:x\succ_j y}\p[\xi=x]}{\sum\limits_{x:x\succ_j y}\p[f(P)=x]}.
    \end{align*}
    That completes the proof.
\end{proof}


\begin{proposition}[\bf\boldmath $\gamma$-SD-Efficiency, Upper Bound]
    \label{appx-prop: tradeoff-DP-SD-eff}
    Given $\gamma \in\mathbb{R}_+$, there is no neutral voting rule $f\colon \LA^n\to \R(A)$ satisfying $\epsilon$-DP and $\gamma$-SD-efficiency with $\gamma > \frac{(m-1)\erm^{n\epsilon}}{(m-1)\erm^{n\epsilon}+1}$.
\end{proposition}

\begin{proof}
    Consider two profiles, $P_1$ and $P_2$, where all voters in $P_1$ share the same preference order $a_1\succ a_2\succ \cdots \succ a_m$. In contrast, in $P_2$, the voters' preferences are $a_m\succ' a_2\succ' a_3 \succ' \cdots\succ' a_{m-1} \succ' a_1$. Then the unique SD-efficient lottery for $P_1$ and $P_2$ should be $\mathbb{I}_{a_1}$ and $\mathbb{I}_{a_m}$, respectively. Here, $\mathbb{I}_{a_1}$ and $\mathbb{I}_{a_m}$ represent the indicator functions of $a_1$ and $a_m$, formally defined as follows.
    \begin{align*}
        \p[\mathbb{I}_{a_1}=a] &= \begin{cases}
                                      1 & a = a_1          \\
                                      0 & \text{otherwise}
                                  \end{cases},\\
                                  \p[\mathbb{I}_{a_m}=a] &= \begin{cases}
                                                                                  1 & a = a_m          \\
                                                                                  0 & \text{otherwise}
                                                                              \end{cases}.
    \end{align*}
    Let $f$ be any neutral voting rule satisfying $\epsilon$-DP. By Equation (\ref{appx-equ: n-eps-pareto-bound}), we have
    \begin{align*}
        \p[f(P_1) = a_1] & \leqslant \erm^{n\epsilon}\cdot \p[f(P_2) = a_1] \tag*{(by $\epsilon$-DP)} \\
                         & = \erm^{n\epsilon}\cdot \p[f(P_1)=a_m]. \tag*{(by neutrality)}
    \end{align*}
    By symmetry, for any $a\neq a_m$, $\p[f(P_1)=a] \leqslant \erm^{n\epsilon}\cdot \p[f(P_1)=a_m]$. Therefore,
    \begin{align*}
        1 &= \sum\limits_{a\in A} \p[f(P_1)=a] \\
        &\leqslant ((m-1)\erm^{n\epsilon}+1)\cdot \p[f(P_1)=a_m],
    \end{align*}
    i.e., $\p[f(P_1)=a_m] \geqslant \frac{1}{(m-1)\erm^{n\epsilon}+1}$. If there exists some $\gamma$ that $f$ satisfies $\gamma$-SD-efficiency, there does not exist any $\xi\in\R(A)$, such that
    \begin{align*}
        \sum\limits_{x:x\succ y} \p[\xi=x]\geqslant \frac{1}{\gamma}\cdot\sum\limits_{x:x\succ y} \p[f(P_1)=x], ~~\text{for all } y\in A.
    \end{align*}
    Therefore, we have
    \begin{align*}
        \frac{1}{\gamma} & \geqslant \sup\limits_{P\in \LA^n}\sup\limits_{\xi\in\R(A)}\inf\limits_{y\in A} \frac{\sum\limits_{x: x\succ y} \p[\xi=x] }{\sum\limits_{x: x\succ y} \p[f(P)=x] } \\
                         & \geqslant \sup\limits_{\xi\in\R(A)}\inf\limits_{y\in A} \frac{\sum\limits_{x: x\succ y} \p[\xi=x] }{\sum\limits_{x: x\succ y} \p[f(P_1)=x] }                       \\
                         & \geqslant \inf\limits_{y\in A} \frac{\sum\limits_{x: x\succ y} \p[\mathbb{I}_{a_1}=x] }{\sum\limits_{x: x\succ y} \p[f(P_1)=x] }                                  \\
                         & = \frac{1}{\max\limits_{y\in A} \sum\limits_{x: x\succ y} \p[f(P_1)=x]}                                                                                            \\
                         & = \frac{1}{1-\p[f(P_1)=a_m]}                                                                                                                                       \\
                         & \geqslant \frac{(m-1)\erm^{n\epsilon}+1}{(m-1)\erm^{n\epsilon}}.
    \end{align*}
    In other words, we have $\gamma\leqslant \frac{(m-1)\erm^{n\epsilon}}{(m-1)\erm^{n\epsilon}+1}$, as desired.
\end{proof}

\begin{proposition}[\bf\boldmath $\gamma$-SD-Efficiency, Lower Bound]
    \label{appx-prop: anti-exp-SDeff}
    Mechanism 2 satisfies $\epsilon$-DP and $\frac{(m-1)\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1}$-SD-efficiency.
\end{proposition}
\begin{proof}
    Letting $\mathfrak{E}_{\mathsf{Anti}}\colon \LA^n\to \R(A)$ denote the mapping introduced by Mechanism 2.

    For any neighboring profiles $P,P'\in\LA^n$ that $P_{-j}=P'_{-j}$ and $\succ_j\neq \succ_j'$, suppose that the chosen ballot in the mechanism is $\succ_i$. Then
    \begin{align*}
        \p[\mathfrak{E}_{\mathsf{Anti}}(P)=a~|~i\neq j] = \p[\mathfrak{E}_{\mathsf{Anti}}(P')=a~|~i\neq j] \tag*{(use $C$ to denote them)}
    \end{align*}
    Further, for any $a\in A$,
    \begin{align*}
        &\;\frac{\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a]}{\p[\mathfrak{E}_{\mathsf{Anti}}(P')=a]} \\
        =&\; \frac{\p[i = j \wedge \mathfrak{E}_{\mathsf{Anti}}(P)=a] + \p[i\neq j\wedge \mathfrak{E}_{\mathsf{Anti}}(P)=a]}{\p[i = j \wedge \mathfrak{E}_{\mathsf{Anti}}(P')=a] + \p[i\neq j\wedge \mathfrak{E}_{\mathsf{Anti}}(P')=a]}                       \\
                                                                                             =&\; \frac{\frac{1}{n}\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a~|~i=j]+\frac{n-1}{n}\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a~|~i\neq j]}{\frac{1}{n}\p[\mathfrak{E}_{\mathsf{Anti}}(P')=a~|~i=j]+\frac{n-1}{n}\p[\mathfrak{E}_{\mathsf{Anti}}(P')=a~|~i\neq j]} \\
                                                                                             =&\; \frac{\frac{1}{n}\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a~|~i=j]+\frac{n-1}{n}\cdot C}{\frac{1}{n}\p[\mathfrak{E}_{\mathsf{Anti}}(P')=a~|~i=j]+\frac{n-1}{n}\cdot C} \tag*{($P_{-j}=P'_{-j}$)}                                                        \\
                                                                                             \leqslant&\; \frac{\erm^{\epsilon}\cdot \frac{1}{n}\p[\mathfrak{E}_{\mathsf{Anti}}(P')=a~|~i=j]+\frac{n-1}{n}\cdot C}{\frac{1}{n}\p[\mathfrak{E}_{\mathsf{Anti}}(P')=a~|~i=j]+\frac{n-1}{n}\cdot C}                                                    \\
                                                                                             \leqslant&\; \erm^{\epsilon},
    \end{align*}
    which indicates that $\mathfrak{E}_{\mathsf{Anti}}$ satisfies $\epsilon$-DP. On the other hand, given profile $P$, suppose the top-ranked and the last-ranked alternative of $\succ_i$ are $a_{\top}$ and $a_{\bot}$, respectively. Then, for any $\xi\in\R(A)$, we have
    \begin{align*}
        \frac{\sum\limits_{x:x\succ_i y} \p[\xi=x]}{\sum\limits_{x:x\succ_i y} \p[\mathfrak{E}_{\mathsf{Anti}}(P)=x]} \leqslant&\; \frac{1}{\sum\limits_{x:x\succ_i y} \p[\mathfrak{E}_{\mathsf{Anti}}(P)=x]} \\
        =&\; \frac{\sum\limits_{x:x\succ_i y} \p[\mathbb{I}_{a_{\top}}=x]}{\sum\limits_{x:x\succ_i y} \p[\mathfrak{E}_{\mathsf{Anti}}(P)=x]}.
    \end{align*}
    Theorefore,
    \begin{align*}
        &\;\sup\limits_{\xi\in\R(A)}\inf\limits_{y\in A} \frac{\sum\limits_{x:x\succ_i y} \p[\xi=x]}{\sum\limits_{x:x\succ_i y} \p[\mathfrak{E}_{\mathsf{Anti}}(P)=x]} \\
        =&\; \inf\limits_{y\in A} \frac{1}{\sum\limits_{x:x\succ_i y} \p[\mathfrak{E}_{\mathsf{Anti}}(P)=x]} \\
                                                                                                                                                                    =&\; \frac{1}{1-\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a_{\bot}]}                                        \\
                                                                                                                                                                    =&\; \frac{(m-1)\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1}.
    \end{align*}
    By Lemma \ref{appx-lem: proof-gamma-SD-eff}, $\mathfrak{E}_{\mathsf{Anti}}$ satisfies $\frac{(m-1)\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1}$-SD-efficiency, which completes the proof.
\end{proof}

\begin{lemma}
    \label{appx-prop: PC->SD}
    Given $\gamma\leqslant 1$, $\gamma$-PC-efficiency implies $\gamma$-SD-efficiency.
\end{lemma}

\begin{proof}
    In fact, we only need to prove that for any $\xi,\zeta\in\R(A)$, $\xi\succeq^{\gamma-SD} \zeta$ implies $\xi\succeq^{\gamma-PC} \zeta$. Let $\xi$ and $\zeta$ be two lotteries satisfying $\xi\succeq^{\gamma-SD} \zeta$, i.e.,
    \begin{align*}
        \sum\limits_{x\succ y} \p[\xi=x] \geqslant \frac{1}{\gamma}\cdot \sum\limits_{x\succ y} \p[\zeta=y],~\quad \text{ for any }y\in A.
    \end{align*}
    Then, on the one hand, we have
    \begin{align*}
        &\;\sum\limits_{x,y\in A\wedge x\succ y} \p[\xi=x]\cdot \p[\zeta=y] \\
        =&\; \sum\limits_{y\in A} \p[\zeta=y]\cdot \sum\limits_{x\succ y} \p[\xi=x]                          \\
                                                                         \geqslant&\; \sum\limits_{y\in A} \p[\zeta=y]\cdot \frac{1}{\gamma}\sum\limits_{x\succ y}\p[\zeta=y] \\
                                                                         =&\; \frac{1}{\gamma}\cdot \sum\limits_{x,y\in A\wedge x\succ y} \p[\zeta=y]\cdot \p[\zeta=y].
    \end{align*}
    On the other hand, we have
    \begin{align*}
        &\;\frac{1}{\gamma}\cdot \sum\limits_{x,y\in A\wedge x\succ y} \p[\zeta=y]\p[\xi=y] \\
        =&\; \sum\limits_{x\in A} \p[\zeta=y]\cdot \sum\limits_{y\prec x} \p[\xi=y]                                                                                           \\
         =&\; \frac{1}{\gamma}\cdot \sum\limits_{x\in A} \p[\zeta=y]\cdot \sum\limits_{y\prec x} \left(1-\sum\limits_{x\succeq y} \p[\xi=x] \right)                            \\
         \leqslant&\; \frac{1}{\gamma}\cdot \sum\limits_{x\in A} \p[\zeta=y]\cdot \sum\limits_{y\prec x} \left(1-\frac{1}{\gamma} \sum\limits_{x\succeq y} \p[\zeta=y] \right) \\
         \leqslant&\; \frac{1}{\gamma}\cdot \sum\limits_{x\in A} \p[\zeta=y]\cdot \sum\limits_{y\prec x} \left(1-\sum\limits_{x\succeq y} \p[\zeta=y] \right)                  \\
         & = \frac{1}{\gamma}\cdot \sum\limits_{x,y\in A\wedge x\succ y} \p[\zeta=y]\cdot \p[\zeta=y].
    \end{align*}
    Then it follows that
    \begin{align*}
        &\;\sum\limits_{x,y\in A\wedge x\succ y} \p[\xi=x]\cdot \p[\zeta=y] \\
        \geqslant&\; \frac{1}{\gamma}\cdot \sum\limits_{x,y\in A\wedge x\succ y} \p[\zeta=x]\cdot \p[\xi=y],
    \end{align*}
    which completes the proof.
\end{proof}

\begin{proposition}[\bf\boldmath $\kappa$-PC-Efficiency, Upper Bound]
    \label{appx-prop: tradeoff-DP-PCeff}
    Given any $\kappa,\epsilon\in\mathbb{R}_+$, there is no voting rule $f\colon \LA^n\to\R(A)$ satisfying $\epsilon$-DP and $\kappa$-PC-efficiency.
\end{proposition}

\begin{proof}
    Consider the profile $P$, where all voters share the same prefereoce
    \begin{align*}
        a_1 \succ a_2 \succ \cdots \succ a_m.
    \end{align*}
    Then the unique PC-efficient distribution on $A$ is $\mathbb{I}_{a_1}$. Further, we have
    \begin{align*}
        &\;\sum\limits_{x,y:~x\succ_i y} \p[\mathbb{I}_{a_1}=x]\cdot \p[f(P)=y] \\
        =&\; \sum\limits_{y:~a_1\succ_i y} \p[f(P)=y] = 1-\p[f(P)=a_1].
    \end{align*}
    However,
    \begin{align*}
        &\;\sum\limits_{x,y:~x\succ_i y} \p[f(P)=x]\cdot \p[\mathbb{I}_{a_1}=y] \\
        =&\; \sum\limits_{x\in A} \p[f(P)=x]\cdot \sum\limits_{y:~x\succ_i y} \p[\mathbb{I}_{a_1}=y] = 0.
    \end{align*}
    In other words, for all $\kappa\in\mathbb{R}_+$, the lottery $\mathbb{I}_{a_1}$ can $\kappa$-PC-dominate any $f(P)$, which completes the proof.
\end{proof}


\begin{proposition}[\bf\boldmath $\alpha$-Condorcet Criterion, Lower Bound]
    \label{appx-prop: CW-DP-lower}
    Mechanism 3 satisfies $\erm^\epsilon$-Condorcet criterion and $\epsilon$-DP.
\end{proposition}
\begin{proof}
    Let $\mathfrak{R}_{\CW}\colon\LA^n\to \R(A)$ denote the mapping introduced by CWRR. Then for any profile $P\in\LA^n$ and alternative $a\in A$, we have
    \begin{align*}
        \p[\mathfrak{R}_{\CW}(P)=a]=\begin{cases}
                                        \frac{\erm^{\epsilon}}{\erm^{\epsilon} + m-1}, & a = \CW(P)       \\
                                        \frac{1}{\erm^{\epsilon} + m-1},               & \text{otherwise}
                                    \end{cases},
    \end{align*}
    for all $P\in\LA^n$ that $\CW(P)$ exists.

    By definition, it is not hard to see that CWRR satisfies $\erm^\epsilon$-Condorcet criterion. Thus, we only need to prove that CWRR satisfies $\epsilon$-DP. In fact, for any neighboring profiles $P,P'\in \LA^n$ and $a\in A$,
    \begin{align*}
        \frac{\p[\mathfrak{R}_{\CW}(P)=a]}{\p[\mathfrak{R}_{\CW}(P')=a]} & \leqslant \frac{\max_{a\in A} \p[\mathfrak{R}_{\CW}(P)=a]}{\max_{a\in A} \p[\mathfrak{R}_{\CW}(P')=a]} \\
                                                                         & \leqslant \frac{\erm^{\epsilon}}{\erm^\epsilon+m-1} / \frac{1}{\erm^\epsilon+m-1}                      \\
                                                                         & = \erm^\epsilon,
    \end{align*}
    which completes the proof.
\end{proof}

\begin{proposition}[\bf\boldmath $\eta$-Condorcet Loser Criterion, Upper Bound]
    \label{appx-prop: CL-DP-upper}
    There is no voting rule satisfying $\epsilon$-DP and $\eta$-Condorcet loser criterion with $\eta>\erm^{\epsilon}$.
\end{proposition}

\begin{proof}
    Suppose $f\colon\LA^n\to A$ be a voting rule satisfying $\epsilon$-DP and $\eta$-Condorcet loser criterion. Consider the profile $P$ ($n=2k+1$):
    \begin{itemize}
        \item $k+1$ voters: $a_1\succ a_2\succ \cdots\succ a_m$,
        \item $k$ voters: $a_m\succ a_{m-1}\succ\cdots\succ a_1$.
    \end{itemize}
    By definition, we have $w_P[a_m,a_i]=-1$, for all $a_i\in A\backslash\{a_m\}$, i.e., $a_m$ is the Condorcet loser. Now, letting one voter change her preferece from $a_1\succ a_2\succ\cdots\succ a_m$ to $a_m\succ a_{m-1}\succ \cdots\succ a_1$, we can obtain another profile $P'$:
    \begin{itemize}
        \item $k$ voters: $a_1\succ' a_2\succ' \cdots\succ' a_m$,
        \item $k+1$ voters: $a_m\succ' a_{m-1} \succ \cdots\succ' a_1$.
    \end{itemize}
    Now we have $w_{P'}[a_1,a_i]=-1$, for all $a_i\in A\backslash\{a\}$, i.e., $a_1$ is the Condorcet loser for $P'$. Then
    \begin{align*}
        \p[f(P)=a_1] & \geqslant \eta\cdot \p[f(P)=a_m] \tag*{(By $\eta$-Condorcet loser)}                          \\
                     & \geqslant \erm^{-\epsilon}\cdot \eta\cdot \p[f(P')=a_m] \tag*{(By $\epsilon$-DP)}            \\
                     & \geqslant \erm^{-\epsilon}\cdot \eta^2\cdot \p[f(P')=a_1] \tag*{(By $\eta$-Condorcet loser)} \\
                     & \geqslant \erm^{-2\epsilon}\cdot \eta^2\cdot \p[f(P)=a_1] \tag*{($\epsilon$-DP)},
    \end{align*}
    which indicates that $\erm^{-2\epsilon}\cdot \eta^2\leqslant 1$, i.e., $\eta\leqslant \erm^{\epsilon}$. That completes the proof.
\end{proof}

\begin{proposition}[\bf\boldmath $\eta$-Condorcet Loser Criterion, Lower Bound]
    \label{appx-prop: CL-DP-lower}
    Mechanism 4 satisfies $\erm^\epsilon$-Condorcet loser criterion and $\epsilon$-DP.
\end{proposition}
\begin{proof}
    Let $\mathfrak{R}_{\CL}\colon \LA^n\to \R(A)$ denote the mapping introduced by CLRR, we have
    \begin{align*}
        \p[\mathfrak{R}_{\CL}(P)=a] = \begin{cases}
                                          \frac{1}{(m-1)\erm^{\epsilon}+1},               & a=\CL(P)         \\
                                          \frac{\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1}, & \text{otherwise}
                                      \end{cases},
    \end{align*}
    for all $P\in\LA^n$ that $\CL(P)$ exists.

    By definition, it is not hard to see that CLRR satisfies $\erm^\epsilon$-Condorcet criterion. Thus, we only need to prove that CLRR satisfies $\epsilon$-DP. In fact, for any neighboring profiles $P,P'\in \LA^n$ and $a\in A$,
    \begin{align*}
        \frac{\p[\mathfrak{R}_{\CL}(P)=a]}{\p[\mathfrak{R}_{\CL}(P')=a]} & \leqslant \frac{\max_{a\in A} \p[\mathfrak{R}_{\CL}(P)=a]}{\max_{a\in A} \p[\mathfrak{R}_{\CL}(P')=a]} \\
                                                                         & \leqslant \frac{\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1} / \frac{1}{(m-1)\erm^{\epsilon}+1}            \\
                                                                         & = \erm^\epsilon,
    \end{align*}
    which completes the proof.
\end{proof}

% \newpage
\section{Missing Proofs in Section 4}

\label{appx: sec: 3way}

\subsection{Results in Table \ref{tab: 3way-tradeoff-lower} and their proofs}

\label{appx: subsec: 3way-lower}

% \begin{proposition}
%     Given $\epsilon\in\mathbb{R}_+$, BordaEXP satisfies $\erm^{\left(\lfloor\frac{n}{2}\rfloor + 1\right)\cdot m-n\cdot (m-1)}$-Condorcet criterion, $\left(1-\frac{\erm^{\frac{n\epsilon}{m-1}+1} -\erm}{\erm^{\frac{mn\epsilon}{m-1}} -1}\right)$-SD-efficiency, and $\erm^{n-\left(\lceil \frac{n}{2} \rceil - 1\right)\cdot m}$-Condorcet loser criterion.
% \end{proposition}

\begin{proposition}
    Given $\epsilon\in\mathbb{R}_+$, BordaEXP satisfies
    \begin{enumerate}[label={(\arabic*)}]
        \item $\frac{\erm^{\frac{n}{2}}+(m-2)\cdot \erm^{\frac{n(m-2)}{4m-4}}}{\erm^{\frac{n}{2}}+(m-1)\cdot \erm^{\frac{n(m-2)}{4m-4}}}$-SD-efficiency,
        \item $\erm^{\left(\lfloor \frac{n}{2} \rfloor + 1\right)\cdot \frac{m}{2m-2}-\frac{n}{2}}$-Condorcet criterion,
        \item $\erm^{\frac{n}{2m-2}-\left( \lceil \frac{n}{2} \rceil - 1 \right)\frac{m}{2m-2}}$-Condorcet loser criterion.
    \end{enumerate}
\end{proposition}
\begin{proof}
    Let $\mathfrak{E}_{\borda}$ denote the voting rule introduced by BordaEXP. First, we prove (1). In fact,
    \begin{align*}
        &\;\sup\limits_{P, \xi}\inf\limits_{j, y}\frac{\sum\limits_{x\succ_j y}\p[\xi=x]}{\sum\limits_{x\succ_j y}\p[\mathfrak{E}_{\borda}(P)=x]} \\
        \leqslant &\; \sup\limits_{P}\inf\limits_{j, y}\frac{1}{\sum\limits_{x\succ_j y}\p[\mathfrak{E}_{\borda}(P)=x]} \\
        = &\; \sup\limits_{P}\inf\limits_{j}\frac{1}{1-\p[\mathfrak{E}_{\borda}(P)=a^j_{\bot}]} \\
        \leqslant &\; \frac{1}{1-\sup\limits_P \inf\limits_{j} \p[\mathfrak{E}_{\borda}(P)=a^j_{\bot}]}.
    \end{align*}
    where $a^j_{\bot}$ denote the last-ranked alternative in $\succ_j$. By symmetry, we have
    \begin{align*}
        \sup\limits_P \inf\limits_{j}\p[\mathfrak{E}_{\borda}(P)=a^j_{\bot}] = \frac{\erm^{\frac{n(m-2)}{4m-4}}}{\erm^{\frac{n}{2}}+(m-1)\cdot \erm^{\frac{n(m-2)}{4m-4}}}.
    \end{align*}
    Then BordaEXP satisfies $\frac{\erm^{\frac{n}{2}}+(m-2)\cdot \erm^{\frac{n(m-2)}{4m-4}}}{\erm^{\frac{n}{2}}+(m-1)\cdot \erm^{\frac{n(m-2)}{4m-4}}}$. Second, we prove (2). By definition, for any profile $P$ that $\CW(P)$ exists, $\CW(P)$ must defeat each alternative $a \neq \{\CW(P)\}$ in at least half of the votes, i.e., $\borda_P(\CW(P)) \geqslant (m-1)\left(\lfloor\frac{n}{2}\rfloor + 1\right)$. And for each $a\neq \CW(P)$, $\borda_P(a) \leqslant (m-1)n -(\lfloor\frac{n}{2}\rfloor + 1)$. Therefore,
    \begin{align*}
        &\;\frac{\p[\mathfrak{E}_{\borda}(P)=\CW(P)]}{\p[\mathfrak{E}_{\borda}(P)=a]} \\
        \geqslant &\; \erm^{\frac{(m-1)\left(\lfloor\frac{n}{2}\rfloor + 1\right)}{2m-2}-\frac{(m-1)n -(\lfloor\frac{n}{2}\rfloor + 1)}{2m-2}} \\
        =&\; \erm^{\left(\lfloor \frac{n}{2} \rfloor + 1\right)\cdot \frac{m}{2m-2}-\frac{n}{2}},
    \end{align*}
    which indicates that BordaEXP satisfies $\erm^{\left(\lfloor \frac{n}{2} \rfloor + 1\right)\cdot \frac{m}{2m-2}-\frac{n}{2}}$-Condorcet criterion. Finally, we prove (3). By definition, for any profile $P$ that $\CL(P)$ exists, $a\neq \CL(P)$ must be ranked than $\CL(P)$ in at least a half of votes, i.e., $\borda_P(\CL(P)) \leqslant (m-1)\left(\lfloor \frac{n}{2} \rfloor - 1\right)$. And for each $a\neq \CL(P)$, $\borda_P(a)\geqslant n-\lceil \frac{n}{2}\rceil + 1$. Therefore,
    \begin{align*}
        \frac{\p[\mathfrak{E}_{\borda}(P)=a]}{\p[\mathfrak{E}_{\borda}(P)=\CL(P)]} \geqslant &\; \erm^{\frac{n-\lceil \frac{n}{2}\rceil + 1}{2m-2}-\frac{(m-1)\left(\lfloor \frac{n}{2} \rfloor - 1\right)}{2m-2}} \\
        =&\; \erm^{\frac{n}{2m-2}-\left( \lceil \frac{n}{2} \rceil - 1 \right)\frac{m}{2m-2}},
    \end{align*}
    which indicates that the BordaEXP mechanism satisfies $\erm^{\frac{n}{2m-2}-\left( \lceil \frac{n}{2} \rceil - 1 \right)\frac{m}{2m-2}}$-Condorcet loser criterion.
\end{proof}

% \begin{proposition}
%     Given $\epsilon\in\mathbb{R}_+$, RD-Anti satisfies $\frac{\left(\lfloor\frac{n}{2}\rfloor-1\right)\erm^{\epsilon}+\lceil\frac{n}{2}\rceil+1}{n\erm^{\epsilon}}$-Condorcet criterion, $1$-Pareto efficiency, and $\frac{\left(\lfloor\frac{n}{2}\rfloor-1\right)\erm^{\epsilon}+\lceil\frac{n}{2}\rceil+1}{n\erm^{\epsilon}}$-Condorcet loser criterion.
% \end{proposition}

\begin{proposition}
    Given $\epsilon\in\mathbb{R}_+$, RD-Anti satisfies
    \begin{enumerate}[label={(\arabic*)}]
        \item $1$-Pareto efficiency,
        \item $\frac{\left(\lfloor\frac{n}{2}\rfloor-1\right)\erm^{\epsilon}+\lceil\frac{n}{2}\rceil+1}{n\erm^{\epsilon}}$-Condorcet criterion,
        \item $\frac{\left(\lfloor\frac{n}{2}\rfloor-1\right)\erm^{\epsilon}+\lceil\frac{n}{2}\rceil+1}{n\erm^{\epsilon}}$-Condorcet loser criterion.
    \end{enumerate}
\end{proposition}
\begin{proof}
    First, given profile $P$, for any $a,b\in A$, $a$ Pareto dominates $b$ means that $a\succ_j b$ for all $j\in N$. Then $a$, the Pareto dominator, is never ranked last in any $\succ_j$. Therefore, $\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a] \geqslant \p[\mathfrak{E}_{\mathsf{Anti}}(P)=b]$, which completes the proof of (1).
    Second, we prove (2). For any profile $P\in\LA^n$,
    \begin{align*}
        |\{j\in N: a^j_{\bot}=\CW(P)\}| \leqslant \lceil \frac{n}{2}\rceil - 1,
    \end{align*}
    otherwise $\CW(P)$ will be the Condorcet loser. Therefore,
    \begin{align*}
        &\;\p[\mathfrak{E}_{\mathsf{Anti}}(P)=\CW(P)] \\
        \geqslant&\; \frac{\lceil\frac{n}{2}\rceil - 1}{n}\cdot \frac{1}{(m-1)\erm^{\epsilon}+1}+\frac{\lfloor\frac{n}{2}\rfloor + 1}{n}\cdot \frac{\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1}.
    \end{align*}
    For any $a\neq \CW(P)$,
    \begin{align*}
        \p[\mathfrak{E}_{\mathsf{Anti}}(P)=a] \leqslant \frac{\erm^{\epsilon}}{(m-1)\erm^{\epsilon} + 1}.
    \end{align*}
    Hence, we have
    \begin{align*}
        \frac{\p[\mathfrak{E}_{\mathsf{Anti}}(P)=\CW(P)]}{\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a]} \geqslant \frac{\left(\lfloor\frac{n}{2}\rfloor-1\right)\erm^{\epsilon}+\lceil\frac{n}{2}\rceil+1}{n\erm^{\epsilon}},
    \end{align*}
    which completes the proof. Finally, we prove (3). Given a profile $P$ that $\CL(P)$ exists,
    \begin{align*}
        |\{j\in N: a^j_{\bot}=a\}| \leqslant \lceil \frac{n}{2}\rceil - 1,
    \end{align*}
    otherwise $a$ will be the Condorcet loser. Therefore,
    \begin{align*}
        &\;\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a] \\
        \geqslant&\; \frac{\lceil\frac{n}{2}\rceil - 1}{n}\cdot \frac{1}{(m-1)\erm^{\epsilon}+1}+\frac{\lfloor\frac{n}{2}\rfloor + 1}{n}\cdot \frac{\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1}.
    \end{align*}
    However, 
    \begin{align*}
        \p[\mathfrak{E}_{\mathsf{Anti}}(P)=\CL(P)] \leqslant \frac{\erm^{\epsilon}}{(m-1)\erm^{\epsilon} + 1}.
    \end{align*}
    Hence we have
    \begin{align*}
        \frac{\p[\mathfrak{E}_{\mathsf{Anti}}(P)=a]}{\p[\mathfrak{E}_{\mathsf{Anti}}(P)=\CL(P)]} \geqslant \frac{\left(\lfloor\frac{n}{2}\rfloor-1\right)\erm^{\epsilon}+\lceil\frac{n}{2}\rceil+1}{n\erm^{\epsilon}},
    \end{align*}
    which completes the proof of (3).
\end{proof}

\begin{proposition}
    Given $\epsilon\in\mathbb{R}_+$, CWRR satisfies $1$-Pareto efficiency, $\frac{m-1}{m}$-SD-efficiency, and $1$-Condorcet loser criterion.
\end{proposition}

\begin{proof}
    The bounds of  Pareto efficiency and Condorcet loser criterion are evident, since for any profile $P$, neither a Pareto dominated alternative nor the Condorcet loser can be the Condorcet winner. Then we only need to prove the bound of SD-efficiency. Given profile $P$, we have
    \begin{align*}
        &\;\sup\limits_{P, \xi}\inf\limits_{j, y}\frac{\sum\limits_{x\succ_j y}\p[\xi=x]}{\sum\limits_{x\succ_j y}\p[\mathfrak{R}_{\CW}(P)=x]} \\
        \leqslant &\; \frac{1}{1-\sup\limits_P \inf\limits_{j} \p[\mathfrak{R}_{\CW}(P)=a^j_{\bot}]}.
    \end{align*}
    Then there are two possible cases for the profile, discussed as follows
    \begin{enumerate}
        \item If $\CW(P)$ exists, then there must exist some $j$ that $a^j_{\bot}\neq \CW(P)$. Therefore
        \begin{align*}
            \inf\limits_{j} \p[\mathfrak{R}_{\CW}(P)=a^j_{\bot}]=\frac{1}{\erm^{\epsilon}+m-1}.
        \end{align*}
        \item If $\CW(P)$ does not exist, then
        \begin{align*}
            \inf\limits_{j} \p[\mathfrak{R}_{\CW}(P)=a^j_{\bot}]=\frac{1}{m} \geqslant \frac{1}{\erm^{\epsilon+m-1}}.
        \end{align*}
    \end{enumerate}
    In other words, we have
    \begin{align*}
        \sup\limits_P \inf\limits_{j} \p[\mathfrak{R}_{\CW}(P)=a^j_{\bot}]=\frac{1}{m},
    \end{align*}
    which indicates that CWRR satisfies $\frac{m-1}{m}$-SD-efficiency. That completes the proof.
\end{proof}

\begin{proposition}
    Given $\epsilon\in\mathbb{R}_+$, CLRR satisfies $1$-Pareto efficiency, $\frac{(m-2)\erm^{\epsilon}+1}{(m-1)\erm^{\epsilon}+1}$-SD-efficiency, and $1$-Condorcet criterion.
\end{proposition}

\begin{proof}
    The bounds of  Pareto efficiency and Condorcet criterion are evident, since for any profile $P$, neither a Pareto dominator nor the Condorcet winner can be the Condorcet loser. Then we only need to prove the bound of SD-efficiency. Given profile $P$, we have
    \begin{align*}
        &\;\sup\limits_{P, \xi}\inf\limits_{j, y}\frac{\sum\limits_{x\succ_j y}\p[\xi=x]}{\sum\limits_{x\succ_j y}\p[\mathfrak{R}_{\CL}(P)=x]} \\
        \leqslant &\; \frac{1}{1-\sup\limits_P \inf\limits_{j} \p[\mathfrak{R}_{\CL}(P)=a^j_{\bot}]}.
    \end{align*}
    Then there are two possible cases for the profile, discussed as follows
    \begin{enumerate}
        \item If $\CL(P)$ exists, considering the profile $P$, where each $a^j_{\bot}\neq \CL(P)$ for each $j\in N$, we have
        \begin{align*}
            \inf\limits_{j} \p[\mathfrak{R}_{\CL}(P)=a^j_{\bot}]=\frac{\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1}.
        \end{align*}
        \item If $\CL(P)$ does not exist, then
        \begin{align*}
            \inf\limits_{j} \p[\mathfrak{R}_{\CL}(P)=a^j_{\bot}]=\frac{1}{m} \leqslant \frac{\erm^{\epsilon}}{(m-1)\erm^{\epsilon}+1}.
        \end{align*}
    \end{enumerate}
    In other words, we have
    \begin{align*}
        \sup\limits_P \inf\limits_{j} \p[\mathfrak{R}_{\CL}(P)=a^j_{\bot}]=\frac{1}{m},
    \end{align*}
    i.e., CWRR satisfies $\frac{(m-2)\erm^{\epsilon}+1}{(m-1)\erm^{\epsilon}+1}$-SD-efficiency. That completes the proof.
\end{proof}

\subsection{Proofs of Theorems 1-6}
\label{appx: subsec: 3way-upper}

\begin{theorem}
    \label{appx-thm: Condorcet-winner-loser}
    There is no voting rule satisfying $\epsilon$-DP, $\alpha$-Condorcet criterion and $\eta$-Condorcet loser criterion with $\alpha\cdot \eta>\erm^{\epsilon}$.
\end{theorem}
\begin{proof}
    Consider the profile $P$ ($n=2k+1$):
    \begin{itemize}
        \item $k+1$ voters: $a_1\succ a_2\succ \cdots\succ a_m$,
        \item $k$ voters: $a_m\succ a_{m-1}\succ\cdots\succ a_1$.
    \end{itemize}
    By definition, we have $\CW(P)=a_1$, since $w_P[a_1,a_i]=1$, for all $a_i\neq a_1$. Now consider another profile $P'$ with the same number of voters:
    \begin{itemize}
        \item $k$ voters: $a_1\succ' a_2\succ' \cdots\succ' a_m$,
        \item $k+1$ voters: $a_m\succ a_{m-1}\succ\cdots\succ a_1$.
    \end{itemize}
    Then $w_P[a_1,a_i]=-1$, for all $a_i\in A\backslash\{a\}$, i.e., $a_1$ is a Condorcet loser. Since there is only one voter changes her preference from $P$ to $P'$, we have
    \begin{align*}
        \p[f(P)=a_1] & \geqslant \alpha\cdot \p[f(P)=a_2] \tag*{($\alpha$-Condorcet criterion)}                                           \\
                     & \geqslant \alpha\eta \cdot \p[f(P)=a_m] \tag*{($\eta$-Condorcet loser criterion)}                                  \\
                     & \geqslant \erm^{-\epsilon}\cdot \alpha\eta \cdot \p[f(P')=a_m] \tag*{($\epsilon$-DP)}                              \\
                     & \geqslant \erm^{-\epsilon}\cdot \alpha^2\cdot \eta \cdot \p[f(P')=a_2] \tag*{($\alpha$-Condorcet criterion)}       \\
                     & \geqslant \erm^{-\epsilon}\cdot \alpha^2\cdot \eta^2 \cdot \p[f(P')=a_1] \tag*{($\eta$-Condorcet loser criterion)} \\
                     & \geqslant \erm^{-2\epsilon}\cdot \alpha^2\cdot \eta^2 \cdot \p[f(P')=a_1] \tag*{($\epsilon$-DP)},
    \end{align*}
    which indicates that $\erm^{-2\epsilon} \alpha^2 \eta^2\leqslant 1$, i.e., $\alpha \eta\leqslant \erm^{\epsilon}$. That completes the proof.
\end{proof}

\begin{theorem}
    \label{appx-thm: tradeoff-DP-Pareto-Condorcet}
    If a neutral voting rule $f\colon \LA^n\to A$ satisfies $\epsilon$-DP, $\beta$-Pareto efficiency, and $\alpha$-Condorcet criterion, then $\alpha\beta^{m-2}\leqslant \erm^{n\epsilon}$.
\end{theorem}
\begin{proof}
    Consider the following profile $P$ ($n=2k+1$):
    \begin{itemize}
        \item $k+1$ voters: $a_1\succ a_2\succ\cdots\succ a_m$;
        \item $k$ voters: $a_2\succ \cdots\succ a_m\succ a_1$.
    \end{itemize}
    By definition, we have $w_P[a_1,a_i]=1$, for all $a_i\in A\backslash \{a_1\}$, which indicates that $\CW(P)=a_1$. Also notice that $w_P[a_i,a_j]=n$ for all $i<j$. Thus, $a_i$ Pareto dominates $a_j$ for all $i<j$. The relations among all alternatives are shown in the following graph.
    \begin{align}
        \label{appx-equ: Condorcet->Pareto}
        a_1\xrightarrow{\bf CW} a_2\xrightarrow{\bf Pareto} a_3\xrightarrow{\bf Pareto} \cdots \xrightarrow{\bf Pareto} a_m.
    \end{align}
    Since $f$ satisfies $\alpha$-Condorcet criterion and $\beta$-Pareto efficiency, we have
    \begin{align*}
        \p[f(P)=a_1] & \geqslant \alpha\cdot \p[f(P)=a_2]             \\
                     & \geqslant \alpha\beta\cdot \p[f(P)=a_3]        \\
                     & \geqslant \cdots                               \\
                     & \geqslant \alpha\beta^{m-2}\cdot \p[f(P)=a_m].
    \end{align*}
    Now, consider another profile $P'$, where all voters' preferences are exactly the same:
    \begin{align*}
        a_m \succ a_{m-1} \succ \cdots \succ a_1.
    \end{align*}
    Then we have the following graph.
    \begin{align*}
        a_m\xrightarrow{\bf CW} a_{m-1}\xrightarrow{\bf Pareto} a_{m-2}\xrightarrow{\bf Pareto} \cdots \xrightarrow{\bf Pareto} a_1.
    \end{align*}
    Similarly, we have
    \begin{align*}
        \p[f(P')=a_m] \geqslant \alpha\beta^{m-2} \cdot \p[f(P')=a_1].
    \end{align*}
    Notice that $|\{j\in N: \succ_j\neq \succ_j'\}|=n$. Therefore,
    \begin{align*}
        \p[f(P)=a_1] & \geqslant \alpha\beta^{m-2}\cdot \p[f(P)=a_m]                                                       \\
                     & \geqslant \erm^{-n\epsilon}\cdot \alpha\beta^{m-2} \cdot \p[f(P')=a_m] \tag*{($\epsilon$-DP)}       \\
                     & \geqslant \erm^{-n\epsilon}\cdot \alpha^{2}\beta^{2m-4} \cdot \p[f(P')=a_1]                         \\
                     & \geqslant \erm^{-2n\epsilon}\cdot \alpha^{2}\beta^{2m-4} \cdot \p[f(P)=a_1] \tag*{($\epsilon$-DP)}.
    \end{align*}
    Then $\erm^{-2n\epsilon}\alpha^{2}\beta^{2m-4}\leqslant 1$, i.e., $\alpha\beta^{m-2}\leqslant \erm^{n\epsilon}$, which completes the proof.
\end{proof}


\begin{theorem}
    \label{appx-thm: tradeoff-DP-Pareto-CondorcetLoser}
    If a neutral voting rule $f\colon \LA^n\to \R(A)$ satisfies $\epsilon$-DP, $\beta$-Pareto efficiency, and $\alpha$-Condorcet loser criterion, then $\alpha\beta^{m-2}\leqslant \erm^{n\epsilon}$.
\end{theorem}
\begin{proof}
    Consider the following profile $P$ ($n=2k+1$):
    \begin{itemize}
        \item $k+1$ voters: $a_1\succ a_2\succ\cdots\succ a_m$;
        \item $k$ voters: $a_2\succ \cdots\succ a_m\succ a_1$.
    \end{itemize}
    By definition, we have $w_P[a_1,a_i]=1$, for all $a_i\in A\backslash \{a_1\}$, which indicates that $\CL(P)=a_1$. Also notice that $w_P[a_i,a_j]=n$ for all $i<j$. Thus, $a_i$ Pareto dominates $a_j$ for all $i<j$. The relations among all alternatives are shown in the following graph.
    \begin{align*}
        a_1\xrightarrow{\bf Pareto} a_2\xrightarrow{\bf Pareto} \cdots \xrightarrow{\bf Pareto} a_{m-1} \xrightarrow{\bf CL} a_m.
    \end{align*}
    Since $f$ satisfies $\alpha$-Condorcet loser criterion and $\beta$-Pareto efficiency, we have
    \begin{align*}
        \p[f(P)=a_1] & \geqslant \beta\cdot \p[f(P)=a_2]               \\
                     & \geqslant \cdots                                \\
                     & \geqslant \beta^{m-2}\cdot \p[f(P)=a_{m-1}]     \\
                     & \geqslant \alpha\beta^{m-2} \cdot \p[f(P)=a_m].
    \end{align*}
    Now, consider another profile $P'$, where all voters' preferences are exactly the same:
    \begin{align*}
        a_m \succ a_{m-1} \succ \cdots \succ a_1.
    \end{align*}
    Then we have the following graph.
    \begin{align*}
        a_m\xrightarrow{\bf Pareto} a_{m-1}\xrightarrow{\bf Pareto} \cdots \xrightarrow{\bf Pareto} a_2 \xrightarrow{\bf CL} a_1.
    \end{align*}
    Similarly, we have
    \begin{align*}
        \p[f(P')=a_m] \geqslant \alpha\beta^{m-2} \cdot \p[f(P')=a_1].
    \end{align*}
    Notice that $|\{j\in N: \succ_j\neq \succ_j'\}|=n$. Therefore,
    \begin{align*}
        \p[f(P)=a_1] & \geqslant \alpha\beta^{m-2}\cdot \p[f(P)=a_m]                                                       \\
                     & \geqslant \erm^{-n\epsilon}\cdot \alpha\beta^{m-2} \cdot \p[f(P')=a_m] \tag*{($\epsilon$-DP)}       \\
                     & \geqslant \erm^{-n\epsilon}\cdot \alpha^{2}\beta^{2m-4} \cdot \p[f(P')=a_1]                         \\
                     & \geqslant \erm^{-2n\epsilon}\cdot \alpha^{2}\beta^{2m-4} \cdot \p[f(P)=a_1] \tag*{($\epsilon$-DP)}.
    \end{align*}
    Then $\erm^{-2n\epsilon}\alpha^{2}\beta^{2m-4}\leqslant 1$, i.e., $\alpha\beta^{m-2}\leqslant \erm^{n\epsilon}$, which completes the proof.
\end{proof}


\begin{proposition}
    Condorcet method satisfies SD-efficiency on $\mathcal{D}_C$.
\end{proposition}
\begin{proof}
    Let $P$ be an arbitrarily chosen profile in $\mathcal{D}_C$. Then we only need to proof that there does not exist $\xi\in\R(A)$ that SD-dominates $\mathsf{CM}(P)$.

    In fact, if there exists such a $\xi$, we can obtain by definition that for all $j\in N$ and $a\in A$,
    \begin{align*}
        \sum\limits_{b\succ_j a} \p[\xi=b] \geqslant \sum\limits_{b\succ_j a} \p[\mathsf{CM}(P)=b],
    \end{align*}
    Since for any $a\in A$ that $\CW(P)\succ_j a$, we have
    \begin{align*}
        \sum\limits_{b\succ_j a} \p[\mathsf{CM}(P)=b] = \p[\mathsf{CM}(P)= \CW(P)] =1,
    \end{align*}
    which indicates that
    \begin{align*}
        \sum\limits_{b\succ_j a} \p[\xi=b] \geqslant 1.
    \end{align*}
    Therefore, for any $a\in A$ that $\CW(P)\succ_j a$, $\p[\xi=a]=0$. However, according to the definition of $\CW(P)$, each $a\in A$ must be ranked behind $\CW(P)$ in some $\succ_j$. Hence we have $\xi=\mathsf{CM}(P)$, a contradiction.
\end{proof}

\begin{theorem}
    \label{appx-thm: tradeoff-DP-CW-SD}
    There is no neutral voting rule $f\colon \LA^n\to \R(A)$ satisfying $\epsilon$-DP, $\alpha$-Condorcet criterion, and $\gamma$-SD efficiency with $\gamma > \frac{\alpha+m-1-\alpha\erm^{-n\epsilon}}{\alpha+m-1}$.
\end{theorem}
\begin{proof}
    Consider the profile $P$, where all voters' vote are exactly the same, i.e.,
    \begin{align*}
        a_1\succ_j a_2\succ_j\cdots\succ_j a_m,\quad \text{for all } j\in N.
    \end{align*}
    It is not hard to see that $\CW(P)=a_1$. Since $f$ satisfies $\alpha$-Condorcet criterion, we have $\p[f(P)=a]\leqslant \p[f(P)=a_1]/\alpha$, for all $a\in A\backslash\{a_1\}$. Therefore,
    \begin{align*}
        1=&\; \p[f(P)=a_1]+\sum\limits_{a\in A\backslash\{a_1\}} p[f(P)=a] \\
        \leqslant&\; \left( 1+\frac{m-1}{\alpha}\right) \p[f(P)=a_1],
    \end{align*}
    i.e., $\p[f(P)=a_1]\geqslant \frac{\alpha}{\alpha+m-1}$. Further, by Equation (\ref{appx-equ: n-eps-pareto-bound}), we have
    \begin{align*}
        \p[f(P)=a_m] \geqslant \erm^{-n\epsilon}\cdot \p[f(P)=a_1] \geqslant \frac{\alpha\erm^{-n\epsilon}}{\alpha+m-1}.
    \end{align*}
    However, for profile $P$, the unique SD-efficient lottery is $\mathbb{I}_{a_1}$. In other words, all lotteries $\xi\in\R(A)$ that $\xi\neq \mathbb{I}_{a_1}$ are $\gamma$-SD-dominated by $\mathbb{I}_{a_1}$ with $\gamma > 1$. Further,
    \begin{align*}
        \frac{\sum\limits_{x\succ y}\p[\mathbb{I}_{a_1}=x]}{\sum\limits_{x\succ y}\p[f(P)=x]} \geqslant & \frac{\inf\limits_{y\in A} \sum\limits_{x\succ y}\p[\mathbb{I}_{a_1}=x]}{\sup\limits_{y\in A}\sum\limits_{x\succ y}\p[f(P)=x]} \\
        =                                                                                                & \frac{1}{1-\p[f(P)=a_m]}                                                                                                        \\
        \geqslant                                                                                        & \frac{1}{1-\frac{\alpha\erm^{-n\epsilon}}{\alpha+m-1}}                                                                          \\
        =                                                                                                & \frac{\alpha+m-1-\alpha\erm^{-n\epsilon}}{\alpha+m-1},
    \end{align*}
    i.e., $\mathbb{I}_{a_1}$ can $\frac{\alpha+m-1-\alpha\erm^{-n\epsilon}}{\alpha+m-1}$-dominates $f(P)$, which completes the proof.
\end{proof}


\begin{theorem}
    \label{appx-thm: tradeoff-DP-CL-SD}
    There is no neutral voting rule $f\colon \LA^n\to \R(A)$ satisfying $\epsilon$-DP, $\eta$-Condorcet loser criterion, and $\gamma$-SD-efficiency with $\gamma > \frac{\erm^{n\epsilon}-\eta}{\erm^{n\epsilon}}$.
\end{theorem}
\begin{proof}
    Let $m>2$. Consider the following profile $P$ with $k(m-2)$ voters $(k\geqslant 1)$.
    \begin{itemize}
        \item $k$ voters: $y\succ a_1\succ a_2\succ \cdots \succ x \succ a_{m-2}$,
        \item $k$ voters: $y\succ a_2 \succ a_3\succ \cdots \succ x \succ a_1$,
        \item $k$ voters: $y\succ a_3 \succ a_4\succ \cdots \succ x \succ a_2$,
        \item $k$ voters: $\cdots$,
        \item $k$ voters: $y \succ a_{m-2} \succ a_1\succ \cdots \succ x \succ a_{m-3}$.
    \end{itemize}
    Then it is quite evident that $\CL(P)=x$. Since $f$ satisfies $\eta$-Condorcet loser criterion, we have $\p[f(P)=a]\geqslant \eta\cdot \p[f(P)=x]$, for all $a\in A\backslash \{x\}$. By Equation (\ref{appx-equ: n-eps-pareto-bound}), we have $\p[f(P)=x] \geqslant \erm^{-n\epsilon}$. Since $f$ satisfies $\eta$-Condorcet loser criterion, for all $a\in A\backslash\{x\}$, we have
    \begin{align*}
        \p[f(P)=a] \geqslant \eta\cdot \p[f(P)=x] \geqslant \alpha\erm^{-n\epsilon}.
    \end{align*}
    However, the unique SD-efficient lottery of $P$ is $\mathbb{I}_y$, since $\mathbb{I}_y$ can SD-dominates any other lotteries on $A$. Further,
    \begin{align*}
        \frac{\sum\limits_{b\succ a}\p[\mathbb{I}_y=b]}{\sum\limits_{b\succ a}\p[f(P)=b]} \geqslant &\; \frac{\inf\limits_{a\in A}\sum\limits_{b\succ a}\p[\mathbb{I}_y=b]}{\sup\limits_{a\in A}\sum\limits_{b\succ a}\p[f(P)=b]} \\
        =                                                                                            &\; \frac{1}{1-\inf\min\limits_{1\leqslant i\leqslant m-2} \p[f(P)=a_i]}                                                       \\
        \geqslant                                                                                    &\; \frac{1}{1-\alpha\erm^{-n\epsilon}}.
    \end{align*}
    In other words, $\mathbb{I}_y$ can $\frac{\erm^{n\epsilon}-\eta}{\erm^{n\epsilon}}$-SD-dominates $f(P)$, which completes the proof.
\end{proof}

\begin{theorem}
    \label{appx-thm: SD-PE}
    There is no neutral voting rule $f\colon \LA^n\to \R(A)$ satisfying $\epsilon$-DP, $\gamma$-SD-efficiency, and $\beta$-Pareto efficiency with $\gamma > \frac{\erm^{n\epsilon}-\erm^{n\epsilon}\beta^{2-m}}{\erm^{n\epsilon}-\erm^{n\epsilon}\beta^{2-m}+\beta-1}$.
\end{theorem}
\begin{proof}
    Consider the profile $P$, where all voters' preferences are the same, i.e.,
    \begin{align*}
        a_1\succ_j a_2\succ_j \cdots\succ_j a_m, \quad \text{for all }j\in N.
    \end{align*}
    By definition, for all $i<j$, any $a_i$ Pareto dominates $a_j$ in profile $P$. In other words, we have the following diagram
    \begin{align*}
        a_1\xrightarrow{\bf Pareto} a_2 \xrightarrow{\bf Pareto} \cdots \xrightarrow{\bf Pareto} a_m.
    \end{align*}
    Since $f$ satisfies $\beta$-Pareto efficiency, $\p[f(P)=a_{i+1}]\leqslant \beta\cdot \p[f(P)=a_i]$ holds for any $i<m$. By Equation (\ref{appx-equ: n-eps-pareto-bound}), $\p[f(P)=a_m]\geqslant \erm^{-n\epsilon}\cdot \p[f(P)=a_1]$. Further,
    \begin{align*}
        \p[f(P)=a_1] \leqslant&\; \erm^{n\epsilon}\cdot \p[f(P)=a_m],                                                                           \\
        \p[f(P)=a_2] \leqslant&\; \frac{1}{\beta}\cdot \p[f(P)=a_1] \\
        \leqslant&\; \frac{\erm^{n\epsilon}}{\beta}\cdot \p[f(P)=a_m],                 \\
        \cdots                                                                                                                               \\
        \p[f(P)=a_{m-1}] \leqslant&\; \frac{1}{\beta^{m-2}}\cdot \p[f(P)=a_1] \\
        \leqslant&\; \frac{\erm^{n\epsilon}}{\beta^{m-2}}\cdot \p[f(P)=a_m].
    \end{align*}
    By summing up the above inequalities, we have
    \begin{align*}
        1 =&\; \sum\limits_{a\in A} \p[f(P)=a] \\
        \leqslant&\; \left(1+\left(1+\frac{1}{\beta}+\cdots+\frac{1}{\beta^{m-2}}\right)\erm^{n\epsilon}\right) \p[f(P)=a_m],
    \end{align*}
    i.e., $\p[f(P)=a_m] \geqslant \frac{\beta-1}{\erm^{n\epsilon}-\erm^{n\epsilon}\beta^{2-m}+\beta-1}$. However, the unique SD-efficient lottery of $P$ is $\mathbb{I}_{a_1}$, since it can SD-dominate any other lottery. Further, we have
    \begin{align*}
        \frac{\sum\limits_{b\succ a}\p[\mathbb{I}_y=b]}{\sum\limits_{b\succ a}\p[f(P)=b]} \geqslant &\; \frac{\inf\limits_{a\in A}\sum\limits_{b\succ a}\p[\mathbb{I}_y=b]}{\sup\limits_{a\in A}\sum\limits_{b\succ a}\p[f(P)=b]} \\
        =                                                                                            &\; \frac{1}{1-\p[f(P)=a_m]}                                                                                                   \\
        \geqslant                                                                                    &\; \frac{1}{1-\frac{\beta-1}{\erm^{n\epsilon}-\erm^{n\epsilon}\beta^{2-m}+\beta-1}}.
    \end{align*}
    In other words, $\mathbb{I}_{a_1}$ can $\frac{\erm^{n\epsilon}-\erm^{n\epsilon}\beta^{2-m}}{\erm^{n\epsilon}-\erm^{n\epsilon}\beta^{2-m}+\beta-1}$-SD-dominates $f(P)$, which completes the proof.
\end{proof}


% \newpage
\section{More Figures for the Tradeoff Curves}
\label{appx: sec: curves}

\begin{figure}[h]
    \centering
    \includegraphics[width=.8\linewidth]{CW-SDeff-1.pdf} \\
    \includegraphics[width=.8\linewidth]{CW-SDeff-2.pdf}
    \caption{Tradeoff curves between $\alpha$-Condorcet criterion and $\gamma$-SD-efficiency under $\epsilon$-DP (upper bounds). Above: $m=5,n=10$. Below: $m=5, n=20$.}
\end{figure}

\begin{figure}[h]
    \centering
    \includegraphics[width=.8\linewidth]{CL-SDeff-1.pdf} \\
    \includegraphics[width=.8\linewidth]{CL-SDeff-2.pdf}
    \caption{Tradeoff curves between $\eta$-Condorcet loser criterion and $\gamma$-SD-efficiency under $\epsilon$-DP (upper bounds). Above: $m=5,n=10$. Below: $m=5, n=20$.}
\end{figure}

The Python codes for drawing these curves are included in the supplementary materials.