This is a common problem in the analysis of poker tournaments, particularly at the final table. Because there are prizes for second and lower places, to calculate your expected prize money, you would like a reasonable function from the distribution of chips to your place distribution.
Of course, mathematically, this problem is underdetermined. So, you need to make more assumptions.
The usual model used by professional poker players who specialize in single table tournaments, where this is most important, is the Independent Chip Model (ICM). The ICM is commonly described in a few different ways. One is that the chips of the players are independently removed from the table one by one, and a players is eliminated when his last chip is eliminated. Another is that the winner is chosen with probability in proportion with his share of the chips, then his chips are removed, and you recursively determine the second place finisher, etc. It may not be obvious that these are equivalent, since the latter can easily be extended to non-integer chip counts and makes it obvious that doubling the number of chips for everyone does not change the finishing probabilities. However, they are equivalent since you can shuffle the chips and rank players by their highest chips. Revealing the shuffle from the bottom gives you the first interpretation. Revealing the shuffle from the top gives you the second.
For example, with stacks of 5, 2, 2, and 1, the ICM says that the chip leader finishes
- 1st: 0.5000
- 2nd: 0.3056
- 3rd: 0.1508
- 4th: 0.0437
You can download a free advanced ICM calculator I wrote called ICM Explorer, which performs the calculations for tournaments of up to 10 players, and tells you how much equity you need to call, bluff, or semibluff. For example, you might be considering risking 3000 chips to gain 4000, but the equity you might need against your opponent's range might be 55%, since you might lose more equity when you lose than you get when you win.
An alternative model considered by Thomas Ferguson is diffusion. This might be a more realistic model under some circumstances, as chips move from player to player instead of being removed from the table. It may be a better model for poker tournaments with a limit betting structure instead of no limit, since in the latter it is possible for the second stack to be eliminated in one hand. However, this model has not found favor among poker players because it's much harder to calculate. Nevertheless, there have been some efforts to compare the ICM, diffusion, and empirical data.
A drawback of the ICM is that players might predictably gain equity or lose equity in the course of a hand, while tournament equity (and all place probabilities) ought to be a martingale. You can see this according to a Nash equilibrium calculator for push/fold play. With stacks of 3000, 2000, and 1000, and a 50-30-20 prize structure, if you average over all permutations, the player with 3000 chips averages a slight gain in equity, and in general the chip leader gains slightly since players are the most risk-averse against the chip leader. There are at least three groups of people trying to come up with a better model. In the mean time, the ICM with slight adjustments is what serious single table tournament players use.
Poker servers have expressed interest in using the Independent Chip Model to settle poker tournaments interrupted by crashes. It's not easy to calculate the ICM for several thousand players. The naive algorithms for n players use about n! calculations, although faster computations are possible. As far as I know, poker servers only use it to suggest possible deals at the final table.
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