2. What is ICM

ICM stands for independent chip model and describes a model for giving monetary values to chip stacks during tournaments, that is, to calculate the $EV from the EV. The money value of your own stack ($EV) is dependent on the stack ratios of the other players.

How is the $EV calculated under the ICM?

To calculate the $EV of stacks, 3 simplifying assumptions are made:

• All players are equal in skill

• The current position is irrelevant

• The table image of the player is irrelevant

Now, for each player, the probability can be calculated of his taking the first, second, and third places.
This is modeled as a lottery where each chip is a ticket. All tickets are put in a drum and we draw for first place. So with a quarter of all chips, you will be first 25% of the time. After this draw, all of the tickets belonging to the winner are removed and we draw for second place. This is repeated for third place. If this procedure is repeated many times, we can get the probability of each player obtaining a particular place by keeping track of how many times they win it. Once we have these, it's easy to calculate how much each player will win on average by using the payoff structure.
So if Klaus has 3000 chips in a 10+1$ SnG and takes places 1/2/3 with probabilities 15/17/20%, then his payoff is

$EV = 0.15*$50 + 0.17*$30 + 0.20*$20 = $16.6

You can see that the size of the stack doesn't appear in this calculation explicitly. His and the other stack sizes get sucked into the placement probabilities, and are, therefore, included implicitly.
Mathematically, there is a closed-form solution for obtaining the probabilities of placement using conditional probabilities. But I think this metaphor is sufficient for our explanation.

2.1 Using the ICM

Since we can now calculate our $EV from the stacks, we can now decide which action has the greater $EV given a push-or-fold or call-or-fold decision. We then make this decision with the knowledge that it is correct.
I'll now explain the calculation of the $EV with a push or fold example:

$10+$1 PartyPoker SnG, Blinds 300/600
CO 8000
BU 2000
SB 6000
BB 4000 (Hero)
CO folds, BU folds, SB pushes All-In 6000, Hero A9 ??


We believe that the SB will often put us under pressure, pushing his hand 85% of the time. It would be +EV to call against this handrange with A9o. But is it also +$EV?

There are 3 scenarios for us:

• Fold

• Call & Win

• Call & Lose

We use an ICM calculator to get the $EV for all 3:

$EV(Fold): 21.50$ with 3400 Chips
$EV(Call&Win): 34.40$ with 8000 Chips
$EV(Call&Lose): 0$ with 0 Chips

To get $EV(Call), we must weight the last two cases with the probability of winning: (we will win 60% of the time against his range)

$EV(Call) = 0.60*$EV(Call&Win)+0.40*$EV(Call&Lose) = 0.60*$34.4 = ~$20.6

Now we compare $EV(fold) with $EV(Call) and see that it would be better to fold here, even if we will have fewer chips on average. The reason is that we'll bust on the bubble 40% of the time with a call even though there's another short stack at the table. He would profit greatly from our call. For the SB, a call from us would be -$EV (and also -EV). In this example, a fold would even be correct if the SB pushed with any two cards.

Let's look at the same example from the SB's side, supposing we hold 23o.
Here there are two alternatives and 4 scenarios:

• Fold (F)

• Push & Fold (P,F)

• Push & Call & Win (P,C,W)

• Push & Call & Lose (P,C,L)

The decisive factor is how we estimate the calling range of the BB. Harry, a TAG who uses the ICM, will clearly fold A9o and has a call range of around 10% (66+, A9s+, ATo+, KJs+).
Hermine, a normal low-limit player that we all know and love, has a call range of perhaps 26% (22+,A2+,KT+,K7s+,QTs+,JTs+).

$EV(F) = $28.7
$EV(P,F) = $31
$EV(P,C,W) = $38.9
$EV(P,C,L) = $15.6

Our $EV(Push) is calculated by weighting these values according to their probabilities.
We will win 28.4% of the time against Hermine.

Our $EV(Push) = 0.74*$EV(P,F) + 0.26*[$EV(P,C,W)*0.284+$EV(P,C,L)*0.716] = 0.74*$31 + 0.26*($11+$11) = ~$23+$5.7 = $28.7

Here we have an interesting case where the $EV is the same for push and fold. For this reason, we should fold since we are hopefully better than our opponent and can then use our edge in later situations instead of entering a break even gamble here.

Without showing the calculations, against Harry our $EV(Push) = $30 meaning we have a clear push. We only win 25.3% of the time against his range, but we'll get the blinds without a fight in 90% of cases. You can see that estimating the opponent's handrange is important, as is which player is holding the gun to the other's chest. This is especially true of good players.

If that was too much math for you, just know that there are programs to do it for you. The decisive skill remains the estimation of the opposition's push or calling range.

2.2 ICM Programs

The two most widespread ICM analysis programs are SNG Wizard and SNG Power Tools. Both of them are expensive at $99 and $79 respectively and have similar functions whose description is beyond the scope of this article. I would recommend trying the 30 day test version of SNG Wizard. SNG Wizard also includes the so-called bubble trainer. This feature generates random situations given parameters like position, stack size, etc. and lets you decide which action to take. Afterward, you can evaluate your decision. This is a good way to automate ICM decisions and to find errors in your own estimation.

2.3 The ICM Strategy

The obvious strategy resulting from the ICM is simply to always choose the decision with the highest $EV. This is called playing "according to the ICM" and is particularly important with 3 to 6 players. To do this you must be able to estimate the handranges of the opposition and calculate the $EV. With time, you'll do this automatically. You should analyze difficult decisions with the tools above and post them in the forums for discussion. The strategy basically results in many first-in pushes to put the opposition under pressure, but seldom calling an all-in yourself. Furthermore, your game is heavily dependent on stack size. The goal is to duck out of confrontations with big stacks and putting maximum pressure on the smaller but not destitute stacks. As a middle stack on the bubble with short stacks around, you take fewer risks while you play more aggressively when stacks are the same. As a big stack you play very aggressively, the same as for a single short stack.

2.4 Limitations of the ICM

The publishers of ICM software and many players who play strictly according to the $EV as calculated by the ICM will tell you that it is a perfect and indeed the only correct mode of play. This is not so. As we said in the beginning, the ICM neglects player skill, table image, and position. Even if player skill is included in the ICM program with an edge input, it still does not capture all the various skills of the opposition. Suppose a decision raises the chance that I will go heads up against a complete fish. This could then be a deciding factor. Or if I'm sitting UTG with 6 players then my 5 BB stack is clearly worth less than it would be on the button. The optimal mode of play is therefore different from the ICM when the decision is very hard. However, strict ICM strategy is very close to the optimal mode and is therefore a winning strategy on almost all limits. Keep in mind, however, that there is more than just the ICM. This becomes important on particularly high buy-ins and will be addressed in further articles. Furthermore, very small stacks (less than 1BB) and very high blinds can cause errors in the ICM calculations. In these cases, trust your own intuition rather than the calculated value.