The final results can be expressed by a bunch of integrals.
Assume that the random variables and the corresponding density functions are:
X₁---f₁(x₁),
X₂---f₂(x₂),
...
X_{n}---f_{n}(x_{n}).
Obviously, if n=6, you can write down the density functions as you know the means and the standard deviations.
Suppose that you want to calculate the probability that X₁ win this game. Then you want to calculate the probability of following events:
X₁-X₂ < 0,
X₁-X₃ < 0,
...
X₁-X_{n} < 0,
which are
P(X₁-X₂<0),
P(X₁-X₃<0),
...
P(X₁-X_{n}<0),
and then you multiply them together to get the probability you want:
P(X₁-X₂<0)P(X₁-X₃<0)...P(X₁-X_{n}<0).
Let's only calculate one of them. The others can be similarly obtained.
P(X₁-X₂<0)
= ∫∫_{x₁-x₂<0}f₁(x₁)f₂(x₂)dx₁dx₂
= ∫_{-∞}^{∞}f₂(x₂)dx₂∫_{-∞}^{x₂}f₁(x₁)dx₁.
Oh, well, this is all I can get to. I don't have a good idea to carry out this integral.