Let $N_1, N_2, N_3$ be three independent Gaussian random variables. Is there a closed form for $$ \mathbf P \big[\{N_1 \ge N_2\} \cap \{ N_1 \ge N_3\}\big]=\int_{\mathbb R} \Phi_2(y)\Phi_3(y) p_1(y)\,dy\,\,\,\,\,\,\,\,\,?$$ Here $p_n$ and $\Phi_n$ are the density and cdf of $N_n$.
If not, what is likely to be the fastest way to numerically evaluate this number (possibly considering the general case $\mathbf P [\cap_{n=2}^N \{ N_1 \ge N_n\}]$)?
