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Chip Value (2): The Fundamentals of the Independent Chip Model (ICM)
Before reading this lesson, you should have previously read through:
As you saw in the previous lesson, multiplying your stack in SNGs doesn’t equally multiply its value, thus the number of chips in SNGs and their value are not directly correlated. That's why you should always think of your decisions in terms of monetary profitability ($EV), instead of profitability measured in chips (cEV).
In order to recognise the monetary profitability of your decisions, you should be aware of the real value of chips. But how can you assign a monetary value to a given chip stack? This is where the Independent Chip Model (ICM) comes into play. In this lesson, you will learn what the ICM is, how it works, and what its limitations are. In order to get a better understanding of the ICM, you will also see specific examples from the game.
The ICM – what it is & how it works
The ICM is the most popular way to ascertain the monetary value of chip stacks in a game. Its role is especially important in SNGs, but it applies for MTTs as well.
|The Independent Chip Model (ICM):
A model for assigning monetary value to chip stacks in tournaments.
The ICM approximates a given stack’s monetary value by calculating its equity in the prize pool. This equity is estimated by calculating the probability of finishing in particular places, based on given stack sizes. For the purposes of explanation, you can think of this model as a lottery wherein each chip is a ticket.
Imagine that you are playing in a 9-man SNG and you have a quarter of the total chips in play. All chips (tickets) are put in a drum and we draw for first place. After this draw, we have a winner; all of his tickets are removed. Now, we draw for second place and repeat for all remaining places.
If this procedure is repeated millions of times, we can see the probability of each player obtaining a particular place by seeing how many times they achieve said place. Once we have these figures, it's easy to calculate how much money each player will win on average by using the tournament payout structure. According to the ICM, this "average prize" is the monetary value of a given stack.
Taking the situation above as an example, with a quarter of all chips you will win the tournament 25% of the time. Your probability of finishing in any other position can't be calculated as easily though.
|The ICM evaluates chip stacks by estimating the probability of each player finishing in a particular place.|
Evaluating stacks according to the ICM
Let's take a look at how it's possible to assign a monetary value to chip stacks using the ICM.
As you have seen, calculating the probability of finishing in first place according to the ICM is relatively easy. However, calculations regarding further places are far more complicated.
Mathematically, there is a closed-form solution for obtaining the probabilities of placement using conditional probabilities, but it’s quite complicated and certainly cannot be used for calculations during the game.
That’s why it is crucial to have an intuitive understanding of the ICM. Of course, you will never be able to know the exact value of stacks during the in-game, but you will have the ability to intuitively make educated estimations in every situation, and that's all you need.
This ability can be obtained by calculating various stacks' values in special programs and practising with these calculations using examples from the game. You will learn now how it can be done in general.
As you already know, it's very complicated and time-consuming to calculate stacks' values manually. However, there are numerous programs that calculate the monetary value of stacks automatically.
One of the most practical can be found here. All you have to do in order to get the monetary value of your chips is to enter the appropriate data and click "calculate". The results look like this:
Examples to hone your ICM instincts
Take a look at the following example from a 9-man $10+$1 SNG with a 50/30/20 payout structure. With five players left, the stacks are as follows:
According to the ICM, your probabilities of obtaining places 1/2/3/4/5 are:
|Probability of finishing in place
In this scenario, the value of your chips according to the ICM is $20.70:
Monetary expected value of your stack according to the ICM ($EV) = probability of winning the 1st prize * value of the 1st prize + probability of winning the 2nd prize * value of the 2nd prize + probability of winning the 3rd prize * value of the 3rd prize
probability of winning the 1st prize: 22.22%
value of the 1st prize: $45
probability of winning the 2nd prize: 23.43%
value of the 2nd prize: $27
probability of winning the 3rd prize: 24.34%
value of the 3rd prize: $18
$EV = 0.2222*$45 + 0.2343*$27 + 0.2434*$18 = $20.70
To have a greater understanding of both how the ICM works, and the relationship between a stack's size and its value, take a look at the complete evaluation of all stacks from the above example (numbers have been rounded to two decimal places):
|Stacks evaluation according to the ICM
||Probability of finishing in place 1/2/3/4/5
||Stack's monetary value
|1||1000||7.42% / 9.33% / 12.94% / 22.05% / 48.4%||$8.17|
|2||3500||25.92% / 25.73% / 23.74% / 17.25% / 7.5%||$22.87|
|3||3000||22.22% / 23.43% / 24.34% / 20.05% / 10%||$20.70|
||1500||11.12% / 13.53% / 17.74% / 27.95% / 29.7%||$11.84|
|5||4500||33.32% / 28.03% / 21.44% / 12.85% / 4.4%||$26.42|
In the next lessons, you will learn when and how you should use the ability of approximating stacks' values in your game.
Assumptions and limitations
The ICM is based on certain assumptions; to simplify the process and make such equations possible, it assumes that:
- All players are equal in skill.
- Current position is irrelevant.
- Table image of players is irrelevant.
- Blinds are irrelevant.
These assumptions make the calculation process possible, but also show some limitations of the ICM. Since the factors it neglects are of vital importance for your decision-making, they will be the subject of separate lessons in which you will learn how to make correct adjustments. Although the ICM - due to the factors mentioned above - may be sometimes limited in terms of accuracy, it still provides valuable and useful results.
When it comes to the ICM, there are several versions of the stack-approximation algorithms, the most popular of which is the Malmuth-Harville model. This is the model used in the calculations made in this lesson. In more advanced lessons concerning the ICM, you will also learn about other algorithms and the differences as well as the relationships between them.
In this lesson, you have learned the fundamentals of the Independent Chip Model:
- The Independent Chip Model (ICM) is a model for giving monetary value to chip stacks in tournaments.
- The ICM evaluates chip stacks by estimating the probability of each player finishing in a particular place and calculating their equity in the prizepool.
- The ICM is based on specific assumptions, therefore it has certain limitations.
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