\section{Experiment Results}

We conducted numerical experiments to estimate the probability of $\EI$ and $\DI$ for different voting rules using synthetic ranking data generated from common statistical models. 

\subsection{Experiment Setup}

We set the number of alternatives to $m=4$ and consider three settings. The first distribution set is $\Pi=\{\pi_{uni}\}$, which corresponds to our results under assumptions that IC lies within the convex hull and the $\Theta(1)$ case when $\Pi$ is single-element. The other two settings use the single-agent Mallows' Model with fixed dispersion rate $\varphi=0.8, 0.9$ (Example~\ref{ex:mallows}), which corresponds to our $\exp(-\Theta(n))$ results when the distribution set $\Pi$ is single-element.

For each setting, voter preferences are sampled independently and identically (i.i.d.) from the corresponding distribution. 
Each experiment consists of $10^5$ simulations, during which we compute the frequencies (probabilities) of $\EI$ and $\DI$ outcomes.
We vary the number of voters $n$ across the following values: $n = 100, 200, \dots, 1000$ to observe how the likelihoods converge as $n$ increases. 

The experiment involves six common voting rules/criteria: Borda, plurality, Condorcet winner/loser, ranked pairs, STV, and Copeland ($\alpha = 0.5$), as defined in the preliminaries. %\qishen{Are we stilling using PLU for STV, or we change the name? Also PLU and ''tie-breaking'' is controversial. Just saying we are using irresolute version as stated.}


\subsection{Results for single-element distribution}

\begin{figure}[!htb]
      \centering
      \subfigure{
           \includegraphics[scale=0.24]{Experiment/Plurality_vs._Borda_DW.pdf}
      }
      \subfigure{
           \includegraphics[scale=0.24]{Experiment/Plurality_vs._Borda_DD.pdf}
      }
      \subfigure{
           \includegraphics[scale=0.24]{Experiment/Plurality_vs._Condorcet_DW.pdf}
      }
      \subfigure{
           \includegraphics[scale=0.24]{Experiment/Plurality_vs._Condorcet_DD.pdf}
      }
      \caption{Two pairs of comparison. UD is the uniform distribution, and MM is the single-agent Mallows' model with fixed dispersion $\varphi$. }
      \label{fig:3comparisons}
\end{figure}

We show two pairs of comparison in Figure~\ref{fig:3comparisons}. The curve of uniform distribution converges to a value of $\Theta(1)$ while the single-agent Mallows' model curves converge to $0$ rapidly, as predicted in Corollary~\ref{coro:iidSgBorda} and Example~\ref{ex:mallowsexp}. 

%Since the Mallows model introduces a structured dependence in voter preferences, with rankings clustering more tightly around a central reference order as $\varphi$ decreases, the probability of disagreements ($\EI$ and $\DI$) to decrease exponentially with the number of voters, as the collective preferences increasingly reflect the central order, which aligns with our i.i.d results. The tighter clustering at $\varphi = 0.8$ compared to $\varphi = 0.9$ further amplifies this rate of convergence, resulting in steeper curves for $\EI$ and $\DI$. %\qishen{Give some explanation for this. Why it happens when you are proving some O(1) results?} 

\subsection{STV vs. Plurality rule}
Figure~\ref{fig:stv_plu} shows the change of $\EI$ and $\DI$ for STV and plurality rule as the number of voters increases. The curve of uniform distribution corroborates our result of $\Theta(1)$ for $\EI$ and  $\Theta(n^{-\frac{1}{2}})$ for $\DI$ as revealed in Theorem~\ref{thm:stv_plur} and Theorem~\ref{thm:stv_plur_dd}.
\begin{figure}[!htb]
      \centering
      \subfigure{
           \includegraphics[scale=0.24]{Experiment/STV_vs._Plurality_DW.pdf}
      }
      \subfigure{
           \includegraphics[scale=0.24]{Experiment/STV_vs._Plurality_DD.pdf}
      }
      \caption{The probability of $\EI$ and $\DI$ for the STV and Plurality under uniform distribution.}
      \label{fig:stv_plu}
\end{figure}

\subsection{Positional Rules vs. Ranked Pairs with three candidates}
When the number of candidates is $3$, Figure~\ref{fig:borda_rp_3} shows the change of $\DI$ as the number of voters increases. The curve of uniform distribution corroborates our result of $\Theta(n^{-\frac{3}{2}})$ for the Borda rule while the Plurality rule has a probability of $\Theta(1)$ to produce a drastically different outcome against Ranked Pairs.

\begin{figure}[!htb]
      \centering
      \subfigure{
           \includegraphics[scale=0.24]{Experiment/Borda_vs._RankedPairs_DD.pdf}
      }
      \subfigure{
           \includegraphics[scale=0.24]{Experiment/Plurality_vs._RankedPairs_3_DD.pdf}
      }
      \caption{The probability of $\DI$ under uniform distribution when $m=3$.}
      \label{fig:borda_rp_3}
\end{figure}


For additional experiments and results, see Appendix~\ref{Sim_UD}.
%\qishen{Add your computer setup for experiments.}