Borda's paradox~\citep{borda1784paper}, a well-known paradox in the social choice theory, refers to the event that the voting outcome of the plurality rule would be drastically different from the Condorcet winner. When (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]\label{exmp:borda}
    Suppose 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 that different voting rules have (drastically) different outcomes occurs not only between plurality and Condorcet but also among a wide range of common voting rules, including general positional scoring rules, Maximin, and Copeland. In real-world elections, the controversy between the US Electoral College and the popular vote has always been a disputed topic~\citep{nyt2016,wp2016,pew2023} and became unprecedentedly heated after Donald Trump became the fifth president who lost the popular vote but won the election~\citep{indp2016}.

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 Borda's paradox gives rise to a spectrum of theoretical work on studying its existence and likelihood. \citet{fishburn1976borda} show that Borda's rule is the only positional scoring rule that immunes strong and strict Borda's paradox. And 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 and do not characterize a wider range of voting rules. 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. Therefore, the following research question remains largely unanswered: 
\noindent\textbf{how likely are the outcomes of different voting rules being (drastically) different?}

