[thylacine]

In the starting post to this thread Bozeman posted a link to "Ferguson method". (By the way, how do you do this, and what is the address of the link?) From this you get file gamblersruin.pdf, unpublished article by Thomas Ferguson.

I regard this as an exact solution (for the infinitesimal random walk, i.e. Brownian motion) for 3 players, although the solution is somewhat obscure and indirect. It finds the probability of a player coming third, and since the probability of a player coming first is easy to find, it gives prob of each position for each player for any given chip distribution.

I don't know if you'd call it "closed form".

It does not do n>3, and in any case for n>3, knowing P(first) and P(last) for each player would not tell you prob of other positions.

This general case is random walk on a regular n-vertex simplex. (n=2 line segment, n=3 triangle, n=4 tetrahedron etc)

For general n Ferguson does an outline for random walk in positive octant of n-dimensional space. This is equivalent to n+1 players, where one player has an infinite stack, so the totals for the others will vary but they will eventually all bust, and the question is, in what order? But it appears that this method would only give P(last) for each player and would not tell you prob of other positions.