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入正题：

A, B, C: 30%

A, C, B: 20%

If we had four players, we would need to go down another level, eliminating the second-place finisher from the game, repeating the process for players C and D, then multiplying together all the probabilities we obtained (first place, second place, third place) to get an overall probability for that finish sequence.

Repeating the process for B and C finishing first yields:

A, B, C: 30%

A, C, B: 20%

B, A, C: 21.43%

B, C, A: 8.57%

C, A, B: 12.5%

C, B, A: 7.5%

All these possibilities should sum to 100%, since they represent all the possible orders in which the players can finish.

Supposing we want to determine the overall equity for player A. We need to multiply the chance that A finishes first, second and third by the prizes he receives for each placing (expressed as a fraction of the prize pool).

Equity(A) = 0.5 * (0.3 0.2) 0.3 * (0.2143 0.125) 0.2 * (0.0857 0.075)

Equity(A) = 38.39%

A’s stack of 10,000, according to the Independent Chip Model, is worth 38.39% of the tournament prize pool.

ICM tools

At the time of writing, two commercial tools are available which calculate the correct play in all-in or fold decisions, based on ICM equities. The programs require you to enter hand ranges for your opponent(s).

These programs are:

SNGPT – http://sitngo-analyzer.com/

SitNGo Wizard – http://sngwiz.com/

A free online ICM calculator, which calculates ICM equities based on inputted stack sizes, is available at http://www.bol.ucla.edu/~sharnett/ICM/ICM.html.

Consequences of the ICM – the Bias Against Confrontation

Earlier we discussed the fact that in tournaments, a chip earned is always worth less than a chip already in your stack. One consequence of this is that there is a natural bias against putting any chips in the pot. In a cash game, to call an allin raise, a player only has to be the barest favourite to justify a call. In an SNG, we are going to need to be a more substantial favourite. Thanks to the ICM, we can calculate exactly how much of a favourite we need to be.

Suppose you are playing a SNG where everyone starts with 2,000 chips and the blinds are 10-20. You’re in the big blind the first hand and everyone folds to the small blind, who moves allin. He then accidentally exposes his cards to you. He has the AcKd. You have the 2h2d. Should you call?

If you fold, you’ll have just lost the blind and will have 1980 chips, almost your starting stack. The ICM values this at 9.91%, very slightly less than the 10.00% you started with.

If you call and lose, your equity is pretty obvious: zero. You’ll be out of the tournament.

If you call and win, you’ll have 4000 chips, while your remaining 8 opponents will each have 2000. The ICM values this stack at 18.44%.

Suppose that we wanted to know what our probability needed to be of winning the hand before it would be a breakeven call. Breakeven means that the equity of calling would equal the equity of folding.

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