Probability of more than three normal distributions

Question

Let $x_i∼N(u_i,\sigma^2_i)$. For two normal distribution, I know that $P(x_1>x_2|u_1,u_2) = {\displaystyle \Phi }(\frac{u_1-u_2}{\sqrt{\sigma^2_1+\sigma^2_2}})$. But for more than two distributions, e.g. swimming, only 1 winner out of $n$ players, how to find $P(x_1>x_2,...,x_n)$ if the order of non-winners doesn't matter?