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The Independent Chip Model or ICM as it's more commonly referred to is a mathematical model used in the late stages of tournaments to calculate your total equity in the prize pool, the money you stand to make in the tournament. This is calculated based on your chip stack in relation to the total amount of chips in play and the other players' chips. To help clarify the purpose of using ICM in a tournament let's use the example of a cash game situation
In a cash game every chip has a specific value to make up your total chip stack amount. Knowing this information allows you to quickly assess where you stand in the game in relation to the other players, but more importantly it helps you calculate the exact dollar amount of how much you stand to win or lose in the long run by making a particular call or fold, this is your expected value.
You're playing in a $1/$2 and your stack is $120
You have K♠ Q♠
You raise to $6 in the BB, everyone folds but the dealer calls.
The flop comes
9♠ 10♠ A♣
You bet out $10, and he raises all-in for $26.
The pot is now at $49
Now let's analyze the hand to see if you should call or fold.
You know the player on the button is a tight player so he is likely to have paired his Ace with a strong kicker, most likely AK.
On a board of 9♠ 10♠ A♣
K♠ Q♠ vs. A♥ K♥ = 45.15% vs. 53.94%
The pot odds were 4.9:1 ($10 to win $49), so let's calculate the EV
(-$10 * 53.49%) + ($49 * 45.15%) = $14.97
On average if you make the call you would make $14.97.
This is what we're trying to figure out by using the ICM, the dollar amount that you will win in the long run by making the call. In a tournament however you will not make money on a per hand basis so instead we're interested in knowing how calling or folding a hand will increase or decrease your odds of making it into the money.
Now that you have a better understanding of what the ICM is and how it can be used, let's take a look at the calculations behind it.
You are registered for a $20+$2 sit and go tournament with 6 players registered.
Each player started with 1,500 chips, for a total of 9,000 chips.
The sit and go has the following prize distribution:
First place: $80
Second place: $40
The tournament is down to the bubble with 3 players left with the following chip stacks
Player 1: 4,500
Player 2: 3,000
Player 3: 1,500
The first step is determining each players' equity. In order to do so we need to know the odds of each player finishing in the money. The ICM model stipulates that your odds of finishing in a certain position are based on your chips in relation to the total chip stacks. Using this logic we can determine:
Player 1 - 1st place 4,500/9,000 = 50%
Player 2 - 2nd place 3,000/4,500 = 66.67% *
Player 3 - 2nd place 1,500/4,500 = 33.33%
* Because the ICM assumes the chip leader will finish in 1st place, we need to omit these chips to determine the 2nd and 3rd places.
Now we need to repeat the process to determine the odds of players 2 and 3 of finishing in 1st place, In this case we'll need to omit the 2nd player's chips to determine the odds of the first player finishing 2nd and similarly with the 3rd player.
After the calculations we get the following results
Player 2 - 1st place 33.33%
Player 1 - 2nd place 75%
Player 3 - 2nd place 25%
Player 3 - 1st place 16.67%
Player 1 - 2nd place 60%
Player 2 - 2nd place 40%
Now let's calculate how often each player will finish in 2nd place.
Player 1 - 33.33% * 75% + 16.67% * 60% = 34.99%
Player 2 - 50% * 66.67% + 16.67% * 40% = 40%
Player 2 - 50% * 33.33% + 33.33% * 25% = 24.99%
Finally the odds of each player finishing in 3rd place is just a matter of calculating the odds of finishing in first place and second place, and subtracting this result from 100.
Player 1 - 15.01%
Player 2 - 26.67%
Player 3 - 58.34%
We're finally ready to calculate each player's equity. The equation is:
(Probability of being 1st * Payout for 1st) + (Probability of being 2nd * Payout for 2nd) + (Probability of being 3rd * Payout for 3rd) +
Crunch all those numbers and we get each player's equity:
Player 1 - $26.99
Player 2 - $21.33
Player 3 - $9.66
We're almost there. Now let's use these numbers in a practical application. Let's use the same example as we did with the cash game,
You're the chip leader with 4,500 chips in the BB and you have K♠ Q♠.
The blinds are at 400/800
You raise to 1,600
Player 3 folds
Player 2 moves all-in in the SB for an extra 1,000.
There's now 4,200 in the pot.
You put player 2 on A♥ K♥
If you remember from before we said that on a board of
9♠ 10♠ A♣
K♠ Q♠ vs. A♥ K♥ = 45.15% vs. 53.94%
If you call and win your odds of finishing first are: 7,100/9,000 = 78%
If you call and lose your odds of finishing first are: 1,900/9,000 = 21%
If you fold your odds of finishing first are: 2,900/9,000 = 32%
Let's calculate the expected value
(45.15% * 78%) + (53.94% * 21%) = 46.54%
$80 * (46.54% - 32%) = $11.63
According to the ICM, by calling the all-in in this situation you will make on average $11.63 in the long run.
Remember that the ICM is not without its flaws. Perhaps the most important of these is the fact that it fails to take into account a player's skill. The expected value above was calculated based on a speculative hand that you put your opponent on based on your observations of this player. Although we've all seen opponents push all-in wish live cards such as 9-10 in which case the EV would have been negative.
Doing the ICM calculations manually is very tedious and completely unpractical. There are several calculators available which can do all these calculations in under a second. The most notable of which is SNG Wizard which allows you to import your game history from the majority of online poker sites. From there it will analyze every hand you played and point out the ones that are questionable. You can then select these hands and analyze them further by assigning a range of hands for your opponents and running a simulation of what would have happened if you'd called/folded.
It's important to note that Poker Stars recently announced that it does not allow the use of ICM calculators while playing in a tournament on the basis that it gives players an unfair advantage.
These tools are a great way to improve your game, but they should be used to analyze your call / raise / fold decisions after the game instead of during. The use of ICM calculators during a game can be detrimental over time as you develop a dependency and your game becomes almost automated instead of using your own judgment.
Good luck at the tables!
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© Directory of Online Poker.
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