Re: ICM problems
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I recommended the use of the ICM as a settlement for stopped tournaments to a major poker server. They seemed interested until I mentioned that the exact calculation is complicated, and they may need to use an approximation if there are hundreds of players. Does anyone have a good approximation?
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I think I have found a fast way to calculate the probabilities of placing in each position when the stacks are small multiples of a common chip size. This may be good enough in practice. While the results are complicated if computed exactly, this method should be numerically stable. Errors in approximations of intermediate steps should not blow up.
Start with just the target player's stack. Add the other players' stacks one by one, maintaining a joint distribution of the location of the player's highest chip and the player's rank. At the end, ignore the extra information about the location of the player's highest chip, i.e., sum over the possible locations.
I think this basic idea can be used to prove the convexity conjecture, too, by adding the player's and opponent's stacks last.
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