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%\headline={\rm  \ifodd\pageno  \RightHead  \else  \LeftHead  \fi}
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\centerline
{
\bf
The Experimental Mathematics of Voting
}
\bigskip

\centerline
{\it By  Shalosh B. EKHAD and Doron ZEILBERGER}

By using the Maple package {http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/LinDiophantus} one derives the following theorem (in less than 20 seconds,
see \hfill\break {\tt http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/oLinDiophantus2}),  the following theorem.

{\bf Theorem 1} The weighted generating function of The Condorcet scenario $1 \rightarrow 2 \rightarrow 3 \rightarrow 1$ (with three candidates $1,2,3$ and
according to the weight 
$$
x123^{NumberOf123} \cdots  x321^{NumberOf321} t^NumberOfVotes
$$
is
$$
{\frac { \left( 1 \, + \, {t}^{3}{\it x123}\,{\it x231}\,{\it x312} \right) {t}^{3}{\it x312}\,{\it x123}\,{\it x231}}
{ \left( 1- {t}^{2}{\it x123}\,{\it x321} \right)  
\left( 1- {t}^{2}{\it x213}\,{\it x312} \right)  
\left( 1- {t}^{2}{\it x312}\,{\it x123} \right)  
\left( 1- {t}^{2}{\it x132}\,{\it x231} \right) 
 \left( 1- {t}^{2}{\it x123}\,{\it x231} \right)  
\left( 1- {t}^{2}{\it x231}\,{\it x312} \right) }} \quad .
$$

By applying the `umbra' 
$$
x123^{a123} \cdots x321^{a321} \rightarrow (a123+ \dots +a321)!/((a123! \cdots a321!) \cdot 6^{a123+ \dots + a321})
$$
to the above one generating functions, one gets a differential equation (via WZ-theory) that translates into
a recurrence that translates (for the odd part) to the following theorem.

{\bf Theorem 2}:
Suppose three candidates, $1$,$2$, and $3$, are running for office, and 
there are    $2n-1$ voters, each of them choosing one of the 3!=6
possible ranking with the same probability $(1/6)$. 
The votes are counted and  it turns out that in the vote of

                                 1 vs. 2, 1 won \quad 

                                 2 vs. 3, 2 won \quad 

                                 1 vs. 3, 3 won \quad 


In other words, there is a cycle $1 \rightarrow 2 \rightarrow 3 \rightarrow 1$.

Let a(n) be the probability of that happening times $6^{2n-1}$ (i.e. the numerator)

The integer sequence, $a(n)$, satisfies the following linear recurrence equation with polynomial coefficients
$$
-1296\,{\frac {n \left( 2\,n+3 \right)  \left( 2\,n+1 \right) a \left( n \right) }{ \left( n+1 \right)  \left( n+2 \right) ^{2}}}
+36\,{\frac { \left( 2\,n+3 \right) \left( 22\,{n}^{2}+33\,n+12 \right) a \left( n+1 \right) }{ \left( n+1 \right)  \left( n+2 \right) ^{2}}}
$$
$$
-4\,{\frac { \left( 19\,{n}^{2}+57\,n+45
 \right) a \left( n+2 \right) }{ \left( n+2 \right) ^{2}}}+a \left( n+3 \right) =0 \quad .
$$

The asymptotic expression for $a(n)$ is $0.043869914022955- 0.021101164\,{n}^{-1} +O(n^{-2})$  .

By symmetry, this is also the probability of the scenario $1 \rightarrow 3 \rightarrow 2 \rightarrow 1$. 
So doubling the asympotic expression for the Condorcet scenario is
$$
 0.087739828045910- 0.042202328\,{n}^{-1} \quad .
$$

{\bf Corollary}: 
The probability for a strict Condorcet scenario with three candidates and $2n-1$ voters tends as $n \rightarrow \infty$ to
a constant, that is approximately equal to  $0.087739828045910\dots$, and that should be named the
Condorcet constant.

{\bf Comment}
This number was first derived by cosnsiderable human effort by W.V. Gehrlein and P.C. Fishburn,
{\it The probability of paradox of voting: a computable solution}, J. of Economic Theory {\bf 13}  (1976), 14-25
[bottom of page 17, left column, where the complementary probability,  $0.91226$ (only to five digits) is given]


\bigskip
\hrule
\bigskip
Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA; \hfill \break
DoronZeil at gmail dot com \quad ;  \quad {\tt http://www.math.rutgers.edu/\~{}zeilberg/} \quad .
\bigskip
\hrule
\bigskip
Exclusively published in The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger  \hfill \break
({\tt http://www.math.rutgers.edu/\~{}zeilberg/pj.html}) \quad .
\bigskip
\hrule
{\bf Nov. 3, 2016}

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