In recent years, hot debates have emerged regarding electoral system reforms in the United States, as many localities substituted the traditional plurality voting rule with Ranked Choice Voting (RCV)~\citep{RCVfairvote,RCVWP,RCVIndiana,RCVUtah} or approval voting~\citep{AVStLouis} and the fifth president who lost the popular vote but won the election emerges~\citep{indp2016}.
Supporters of the reform argue that new voting rules can give more opportunities to independent candidates and underrepresented groups, while the opponents claim that the chance that a different winner would be elected is minimal under the two-party system. This dispute inspires our question: {\bf how likely is the outcome of different voting rules being (drastically) different?}

The impact of the question is double-sided. A confirmative answer supports the reforms and drives researchers and localities to discover, design, and deploy different voting rules to fit diverse scenarios. On the other hand, a dissenting answer justifies the legitimacy of the current plurality rule and eliminates the possibility of undermining democracy by campaigns on changing voting rules. 

% The impact of such events is double-edged. On the positive side, the distinctions between voting rules allow us to apply different rules to fit different scenarios. On the negative side, however, the existence of a different outcome from another rule undermines the legitimacy of the current rule. Moreover, the possibility of changing the outcome by switching the voting rule may incentivize political parties to project more power on changing the voting rule rather than fulfilling their political promises, posing a threat to the operation of the democratic system. 

The significance of the different outcome problems gives rise to a spectrum of theoretical work on studying its existence and likelihood, which can be dated back to Borda's paradox \citep{borda1781memoire}. When a (strong) Borda's paradox occurs, a plurality winner is a Condorcet loser. 
An even stronger version, named {\em strict} Borda's paradox, additionally requires the Condorcet winner to be in the last place under the plurality rule. 

\begin{example}[Borda's Paradox,~\citep{gehrlein2010probability}]\label{exmp:borda}
    Suppose that there are three candidates $A$, $B$, and $C$. The election contains one vote for $[A\succ B\succ C]$, seven votes for $[A\succ C\succ B]$, seven votes for $[B\succ C\succ A]$, and six votes for $[C\succ B\succ A]$. Under the plurality rule that counts the top ranks, $A$ becomes the winner, while $C$ is in the last place. However, the Condorcet winner/loser is elected by pairwise comparisons. Therefore, $C$ becomes the Condorcet winner while $A$ is the loser. Therefore, both strong Borda's paradox and strict Borda's paradox occur in the vote.
\end{example}

% Such phenomenon occurs not only between plurality and Condorcet but also among a wide range of common voting rules, including general positional scoring rules, ranked-choice voting, and Copeland. In real-world elections, the controversy between the US Electoral College and the popular vote has always been a disputed topic~\cite{nyt2016,wp2016,pew2023} and became unprecedentedly heated after Donald Trump became the fifth president who lost the popular vote but won the election~\cite{indp2016}.

% The significance of Borda's paradox gives rise to a spectrum of theoretical work on studying its existence and likelihood. 

% \cite{fishburn1976borda} show that Borda's rule is the only positional scoring rule that immunes strong and strict Borda's paradox. 

There is a large literature~\citep{tataru1997relationship,cervone2005scoring,gehrlein2010probability,diss2012borda} on the likelihood for strong and strict Borda's paradox in a three-candidate election under the standard assumption of impartial culture (IC) or impartial anonymous culture (IAC). Nevertheless, previous work faces three aspects of challenges.

Firstly, most papers are restricted to the comparison between positional scoring rules and the Condorcet winner/loser. On the other hand, (drastically) different outcomes occur among a wide range of common voting rules, including general positional scoring rules, ranked-choice voting, Copeland, Ranked Pairs, and Single Transferable vote (STV).  

Secondly, the assumption of IC and IAC has widely been criticized as unrealistic (see, for example, \citep{aki2007unreal,nurmi1999voting}) and fails to reflect real-world scenarios. 

Finally, few previous analyses on the likelihood can be extended to settings with more than three candidates due to the restriction on the techniques or overburden calculations. This largely undermines their application to real-world scenarios.

Therefore, our question on how likely the outcome of different voting rules is (drastically) different remains largely unanswered. 
% Therefore, the following research question remains largely unanswered: 
% \noindent\textbf{how likely are the outcomes of different voting rules being (drastically) different?}