[thylacine]

But I don't think you have to do this integral. For n players you want prob P( x ,i j) the prob that player i comes in position j given starting stack distribution x=(x_1,x_2,...,x_n). These will be the solutions of some differential equation for these functions of x in an n-vertex simplex, with boundary conditions coming by induction from the (n-1)-player case.