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Independent Chip Model (ICM)

The Independent Chip Model (ICM) is a concept every serious Poker Player has to be familiar with, in order to play Tournaments (especially Sit & Go Tournaments) profitably. Players who are not aware of this concept will take non profitable (or less profitable) decisions during all phases of a tournament. It is especially vital for Bubble Play (the phase just before reaching the prize money), where one often sees terrible calls due to non awareness of the Independent Chip Model.
The ICM is a model, which allows you to calculate the probability of each remaining player in the tournament to finish first, second, third etc. What can this be used for?
1. Your two remaining opponents of a big tournament offer you a deal and you want to calculate whether you are being offered a fair deal or not.
2. You want to decide whether to call an All-In of your opponent on the bubble of a Sit & Go Tournament.

The mathematical principles of the ICM: The model assumes that the mathematical chances of winning a tournament is as high as the proportional amount of a players chips to the total amount of chips in the Tournament.
Example:
Four players remain in a one table Sit & Go Tournament, in which each player started with 1'000 Chips. Player A has 5'000 chips, Player B has 3'000 chips, Player C has 1'500 Chips and Player D has 500 Chips. According to the ICM the probability of each player winning the tournament are as follows:
Player A: 50 % (5'000 Chips divided by the total of 10'000 Chips)
Player B: 30 % (3'000 Chips divided by 10'000 Chips)
Player C: 15 % (1'500 / 10'000 )
Player D: 5 % (500 / 10'000)

The ICM also enables you to calculate the probability of each player finishing the tournament in second, third, fourth position. If you now multiply each probability with the payout of that specific position, you have calculated the expected value (short: EV, also often referred to as equity) of that specific tournament situation. To calculate the expected value (or equity) all kind of tools exist. For Sit & Go Tournaments we recommend the following simple ICM Calculator: Short Description of an ICM Calculator. Very important to know is that the ICM calculates the expected value only by the current chip distribution. The skills of each player are neglected as are also the current position of the blinds. When the blinds are very high compared to the players chip stacks, the position of the blinds can have a huge impact on the expected value.

Example: calculation of the expected value (using the same tournament situation of the previous example)
Assumption: It is a 10 + 1 $ Sit & Go Tournament. The winner wins 50$, second place pays 30$ and the third finisher receives 20$.

To be honest, as Player D has 500 chips and Player A has 5'000 chips, most novices would assume that player Ds expected value must be ten times lower than player As expected value? That is wrong! In the picture alongside you can see the result of the current situation that proves that each additionally won chip has a smaller value than previously won chips. A short explanation of the columns:
Prob 1st = Probability of finishing in first place
Prob 2nd = Probability of finishing in second place
Prob 3rd = Probability of finishing in third place
Equity = Expected Value of the current situation (probability of finishing in first place multiplied with the prize money for first place, Probability of finishing in second place multiplied with the prize money for second place etc.)
Remark: Only if one single player gets a prize (Winner takes it all), the chip value stays linear during the whole tournament.

Now what do these numbers tell us?
Example 1
Let us assume that you are Player B with 3'000 Chips. The blinds are 100/200. You just posted 200 Chips because you are in the big blind. On the Small Blind is Player A. Player C and D fold, Player A (with 5'000 Chips) pushes All-In. You have JJ. Player A has been playing a solid game and never pushed All-In with complete 'Junk Hands'. Now we can calculate the following with the ICM: before the hand your expected value is 31.29$. There are three different possible outcomes of this hand (we shall neglect the probability of a split pot when you decide to call):
1. possible outcome: You fold. After this hand you still have 2'800 Chips and Player A has 5'200 Chips. Your expected value sinks to 30.52$. Your expected value has diminished by 0.77$.
2. possible outcome: You call and win. You now have 6'000 Chips. Player A has 2'000 Chips. Your expected value now is 40.59$. You have increased your expected value by 9.30$.
3. possible outcome: You call and lose. You are eliminated from the tournament. Your expected (and certain!) win is 0! You lose 31.29$ on expected value.
The result is pretty sobering! You can lose very much, but can hardly win anything! Which hand do you have to put your opponent on to make this a profitable call?

Possible holdings of your opponent and the effects on your expected value:

1. possible holding, your opponent has two higher cards, AK, AQ or KQ.
According to the Poker Odds Calculator you are roughly a 57% to 43% favorite to win this hand. Remark: If this where a cash game and your opponent showed his cards face up this would be a clear call! In a cash game there is no payout structure you have to think about because if you lose the hand you can think about how brutal poker is on the way to the ATM but also assure yourself of having made the correct move. Let us calculate the Sit & Go Situation: In 43% of the cases you lose and your expected value drops to 0. 57% of the time your expected value will rise to 40.59 $. That means, your expected value when making a call is 40.59 $ * 57/100 (57 %), which is 23.13 $. In this case a call is not profitable, because your EV is 30.52$ if you fold. Statistically you lose 7.39 $ each time you make this call, although you have the better hand!!! Calling in this situation is a terrible play. Think about it: the Buy-In was 11$, and you are losing 7.39$ with one single call! It should slowly be dawning on you how important it is to bully your opponents when you are the big stack on the bubble... they need excellent hands to call you profitably!

2. possible holding, your opponent has a high pair, i.e. QQ, KK or AA
We do not have to make huge calculations. You are a 1 to 4 underdog. Let's do the maths anyway: Let's assume your opponent hat KK. (QQ would be worse, because your straight chances sink, AA would be slightly better. Let's assume you are an 81 to 19 underdog. In 81% of the cases your EV is 0. In 19% of the cases your EV will be 40.59 $. A call will average an EV of 7.71 $. A fold would leave you with an EV of 30.52$, thus on average a call would reduce your EV by 22.81$.

3. possible holding, your opponent has a lower pair. This time you are an 81 to 19% favorite. In 19% of the cases your EV sinks to 0. 81% of the time it increases to 40.59$. If you call, your EV will increase to 32.88$ (40.59*81/100). If you fold, your EV will be 30.52$. In this case a call on average would increase your EV by 2.36$. Wow, you make an average profit of 2.36$ although you are dominating your opponent 4 to 1!

Conclusion of example 1
If you are on the bubble of a 'normal' Sit & Go tournament and have a neat chip lead on the third and fourth placed player, you need an excellent holding to call the chip leaders All-In. On the other hand you have to put a lot of pressure on your opponents when you are the chip leader! In the example above the correct move is to fold! Let's assume that in the following hand the chip leader eliminates player D. Now the Chip distribution looks like this: Player A: 5'800 Chips. Player B (you): 2'700 Chips, Player C: 1'500 Chips. According to the ICM your EV is now 33.20$. Although you have 300 Chips less then you had two hands ago, your EV has risen by 2.68 $!

Example 2
In the mean time a couple of hands have been played. Player A has 5'500 Chips, Player B (you) has 3'000 Chips and Player C has 1'500 Chips. The blinds are still 100/200. Once again you are the big blind. Player A on the SB goes All-In. The last hands he has often pushed All-In, he probably is pushing All-In with any two cards. Your hand is as before JJ! Hey, this is the same situation we experienced in example 1! Not at all! In case you are eliminated your EV drops to 20$ and not to 0$. We calculate again. Hey, we never claimed poker does not have a mathematical side!
If you fold your EV is 32.61 $. This is a minus of 0.59 $. If you call and lose, your EV is exactly 20.00 $, namely the prize money for third place. If you win, you have 6'000 Chips and Player A has 2'500 Chips. In this case your EV is 41.06 $.
Possible Holdings and effects on your EV:

1. possible holding, your opponent has two higher cards, AK, AQ or KQ.
As mentioned previously, you are a 57% to 43% favorite. Initial Situation: 43% of the time you lose and your EV is 20$. In 57 % of the cases you win and your EV is 41.06 $. Let's calculate:
41.06$ * 0.57 + 20.00 $ * 0.43 = 32.00 $
In this case a call would make your EV drop to 32$. If you fold your EV is 32.61$. You always have to calculate the EV to your EV in case of you folding! So, if you call your EV is reduced by a mere 0.61 $.

2. possible holding, your opponent has a higher pair like QQ, KK or AA
From example 1 we know that you are a 19 to 81 underdog. Like in holding number 1 we do the maths:
41.06$ * 0.19 + 20.00 $ *0.81 = 24.00 $.
Your EV would sink by 8.61 $.

3. possible holding, your opponent has a lower pair. This time you are the 81 to 19 favorite. We calculate:
41.06$ * 0.81 + 20.00 $ * 0.19 = 37.06 $.
Your EV has risen by 4.45 $.

4. possible holding, Player A pushed All-In with something like A8 or A9. The Odds Calculator tells us that we are a 72 % favorite to win the hand. Time to do the maths again:
41.06 * 0.72 + 20.00 * 0.28 = 35.16$.
On average your EV would increase by 2.55 $.
5. possible holding, player A has low cards (let's give him credit for not pushing with complete trash but maybe have 87s (Suited Connectors). Again the Odds Calculator has to help us and calculates that this time we are a 78 % favorite to win the hand. Again we calculate:
41.06 * 0.78 + 20.00 * 0.22 = 36.43$.
On average your EV will rise by 3.82 $ in this situation.

Looking at the whole situation in most cases your EV will increase.

Conclusion of Example 2
Also when you have reached the prize money, it is worth keeping the ICM in your mind and acting accordingly! And exactly because it is not possible to go through all possible holdings and calculating the odds within 20 seconds (when playing online poker, you won't get much more time...), you have to prepare yourself. Think about specific situations or analyze your game you just played by looking at your hand history. It is of immense importance that you try to make positive EV decisions if you want to be a successful Sit & Go player. Many players consequently use the M-Concept also for bubble play in Sit & Go tournaments. Obviously the M-Concept also has its importance in Sit & Go tournaments, but it is a mistake to put too much importance on the M-Concept! Your Return on Investment (ROI) is going to rapidly increase if you master the ICM and also act accordingly! The M-Concept is absolutely vital in the middle phase of multi table tournaments, but not that important in Sit & Go Bubble Play. The ICM is important especially in Sit & Go Tournaments and in multi table tournaments with a steep prize structure. (Sit & Go tournaments usually have the steeper prize structure than normal multi table tournaments!) Also when you have reached the prize money, it is worth keeping the ICM in your mind and acting accordingly! And exactly because it is not possible to go through all possible holdings and calculating the odds within 20 seconds (when playing online poker, you won't get much more time...), you have to prepare yourself. Think about specific situations or analyze your game you just played by looking at your hand history. It is of immense importance that you try to make positive EV decisions if you want to be a succesful Sit & Go player. Many players consequently use the M-Concept also for bubble play in Sit & Go Tournaments. Obviously the M-Concept also has its importance in Sit & Go tournaments, but it is a mistake to put too much importance on the M-Concept! Your Return on Investment (ROI) is going to rapidly increase if you master the ICM and also act accordingly! The M-Concept is absolutely vital in the middle phase of multi table tournaments, but not that important in Sit & Go Bubble Play. The ICM is important especially in Sit & Go tournaments and in multi table tournaments with a steep prize structure. (Sit & Go tournaments usually have the steeper prize structure than normal multi table tournaments!)

Calculating Deals

Right at the beginning of this article we mentioned that the ICM enables one to calculate whether one is being offered a fair deal or not. Let us assume that you are playing at the Swiss Poker Open in the Casino of Berne. The winner takes down 80'000 CHF (about 80'000 USD), second prize gets 50'000 CHF and the third finisher wins a respectable 25'000 CHF. You are at the final table and only three players are left, congratulations! You are the chip leader with 400'000 Chips. The second place player, a ranked professional (let us conveniently call him Gus), has 250'000 Chips stacked in front of him and the shortest stack, also a famous professional (hmm... let's call him Scotty), has 160'000 Chips in front of him. The blinds are 2'000 / 4'000, so they are pretty low for this stage of the tournament. No one is forced to do some hair-raising All-In moves yet. (The M of the shortest stack is over 25!). The fourth player just was eliminated, the audience is still applauding, fotos are being taken of the fourth finisher and he still has to give a short interview for the local TV station. Scotty and Gus start talking to each other and then make the following proposal. No matter of the outcome of the tournament you receive 57'000 CHF, Gus gets 53'500 CHF and Scotty 44'500 CHF. The tournament of course will be played to the end, but only the trophy and the title is on stake. Do you accept the deal or do you decline it with thanks? You might think 'Come again? I almost have the same amount of chips as the other two guys have together, and they want to fob me off with a ridiculous sum of 57'000 CHF? You must be kidding! No way I am going to accept that!!!' Let's enter the numbers in the ICM Calculator, and we see the following calculated expected values:
You: 60'710.89 CHF
Gus: 51'402.50 CHF
Scotty: 42'886.61 CHF
As you can see, they are giving you about 3'710 CHF less (about 6% of the sum), as you actually deserve. Gus gets an extra 2'100 CHF and Scotty an extra 1'600 CHF. Do you accept the deal? Well, the Blinds are quite low still and good players have enough air to play their A game. Both of them are top professionals, so maybe the deal isn't even that bad? Depending on how you estimate your own skills, this might even be a good deal. The Independent Chip Model does not consider the skill of each player; it is just a snap shot of the current situation!

In our Sit & Go Strategy article we point out to an article where we analyze a Coin Flip situation in early Sit & Go Tournament stage with the ICM.

A short comment at the end: In a couple of poker forums we read statements saying that the ICM only works when playing 20+2 $ Sit & Gos o higher. That is absolute codswallop! The ICM is a mathematical base that works completely independently of any level of Buy-In. Especially in low Buy-In tournaments one often sees terrible plays because people do not understand the ICM. This makes low Buy-In Sit & Gos pretty easy to beat!

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