Let $x$ and $y$ be two correlated random variable (say, standard normal) with correlation coefficient $\rho>0$. Let $X= \{x_1, x_2, ..., x_L\}$ and $Y= \{y_1, y_2, .. y_L\}$ be samples of size $L$ from $x$ and $y$ respectively.
What is the probability that $\mbox{argmax}\ X = \mbox{argmax}\ Y$.
Alternatively, suppose that $x_1$ is maximal element, what is the probability that $y_1$ is maximal too.
Any references pointing to the solution of either question would also be appreciated.
The question can apparently be clarified as follows:
Let $(X_1,Y_1),\dots,(X_n,Y_n)$ be iid pairs of real-valued random variables with a common continuous joint cumulative distribution function (cdf) $F$, so that $F(x,y)=P(X_i<x,Y_i<y)$ for all $i$ and all real $x,y$. What is then the probability \begin{equation} p:=P(\exists i\ \forall j\ne i\ X_i>X_j, Y_i>Y_j)? \end{equation}
To answer this question, note first that the random pairs $(X_1,Y_1),\dots,(X_n,Y_n)$ are exchangeable and also, because of the continuity of $F$, $P(X_i=X_j)=0=P(Y_i=Y_j)$ for all distinct $i$ and $j$. So,
\begin{equation}
p=np_1,\quad p_1:=P(\forall j\ne 1\ X_1>X_j, Y_1>Y_j).
\end{equation}
Next,
\begin{equation}
p_1=Eg(X_1,Y_1),\quad g(X_1,Y_1):=P(\forall j\ne 1\ X_1>X_j, Y_1>Y_j|X_1,Y_1).
\end{equation}
Further, for all real $x,y$
\begin{equation}
g(x,y)=P(\forall j\ne 1\ x>X_j, y>Y_j)=F(x,y)^{n-1}.
\end{equation}
Thus,
\begin{equation}
p_1=Eg(X_1,Y_1)=\int_{\mathbb R^2}F(x,y)^{n-1}\,dF(x,y)
\end{equation}
and
\begin{equation}
p=n\int_{\mathbb R^2}F(x,y)^{n-1}\,dF(x,y).
\end{equation}
In particular, if $X_1$ and $Y_1$ are independent, so that $F(x,y)=G(x)H(y)$ for all real $x,y$ (where $G$ and $H$ are the cdf's of $X_1$ and $Y_1$), then \begin{multline} p=n\int_{\mathbb R^2}G(x)^{n-1}H(y)^{n-1}\,dG(x)\,dH(y) \\ =n\int_{\mathbb R}G(x)^{n-1}\,dG(x)\; \int_{\mathbb R}H(y)^{n-1}\,dH(y) \\ =n\int_0^1 u^{n-1}\,du\;\int_0^1v^{n-1}\,dv =\frac1n, \end{multline} which also follows immediately because the random pairs $(X_1,Y_1),\dots,(X_n,Y_n)$ are exchangeable.
