To calculate the probability of coming in last for 3 players, you can solve Laplace's eqaution on a triangle with boundary conditions 1 on one edge and 0 on the others. I think the eigenfunctions for the triangle can be found
here. They might be helpful. It is much easier to solve Laplace's equation on a sixty degree wedge of a circle with the same boundry conditions but I don't think there are any conformal maps from the wedge to the equilateral triangle. Instead, you could just use numerical analysis to get approximate solutions to the PDE. That might also work for the higher dimensional cases.