Translated with the help of AI. We apologize for any errors and would appreciate your help in correcting them.
Translated by order of the educational portal poker.university
Original source: GTO Wizard
As a tournament player, you may have heard of the concept of “ICM”, but what does it really mean? The Independent Chip Model (ICM) is a mathematical formula that converts your tournament stack into monetary value. This formula was first applied to poker by Mason Malmuth in 1987. The model uses only stack sizes to determine how often a player will finish in each position (1st, 2nd, etc.) and then allocates tournament capital based on the payouts for those positions. Tournament capital is your expected share of the prize pool, taking into account the payout structure, your position in the tournament and the size of the stack.
1. Why ICM is important
In a cash game, each chip has a monetary value, and the value of the chips scales linearly; doubling your stack means double its value as well. However, in tournaments, the value of chips does not change linearly; doubling your stack does not double its value. If the value of the chips does not scale linearly, we need a method to convert the value of the chips into monetary value. We need to find the real expected value from winning or losing chips to make strategic decisions. We need a utility function to convert chip EV to $EV. This is where the Independent Chip Model (ICM) comes into play.
2. How to calculate ICM
ICM assumes that all players are equally qualified and therefore the probability of winning is solely a function of the stack size. It calculates the probability that each player finishes 1st, 2nd, 3rd, etc., and then multiplies these probabilities by the payouts for each position. To calculate the probability of a particular player finishing first, divide their chips by the total number of chips in the game. The calculation of the 2nd and 3rd positions requires more complex mathematical calculations.
- Example: Tom, Amy and Bill play 3max SnG.
The stacks and payouts are as follows:
What is the tournament equity of players
Let's start by calculating their 1st place equity. This is the easiest step, since the probability of any player winning is simply his stack divided by the total number of chips in the game. Multiply the probability of winning 1st place by the 1st place prize to get the 1st place equity.
There are 1000 chips in the game:
Now we need to calculate their equity for the 2nd place. This step is more complicated, but it can still be done manually.
To calculate the probability of getting 2nd place, we take the following steps:
- Suppose that one of the other players has already taken 1st place. Completely exclude his chips from the game and divide the chips of the remaining two players by their amount.
- Repeat the 1st point of the remaining two players.
- Multiply each result obtained from the 2nd point by the probability of winning the first place of another player
Okay, that sounds complicated, but it's not that bad. Let's get started:
Now multiply the 2nd place equity by the probability of each scenario.
As you can imagine, this process becomes exponentially more complex as we add more players and payouts. Fortunately, there are plenty of quick and easy ICM calculators available online. It is almost impossible to calculate the ICM during the game. The process is too complicated. Tournament professionals hone their intuition of understanding ICM by studying thousands of spots in different depths of ICM and varying different variables.
Here is the same calculation made with the free online calculator ICM (Holdem Resources Calculator):
Just like in the previous example. Player 1, 2 and 3. They have 500, 300 and 200 chips.
Shares: 50%, 30% and 20% and ICM in $ and %
3. How to use this information
Let's ask ourselves how this information can affect the strategy. Amy dumps on BTN, Tom puts all-in with SB, Bill calls on BB. What will be their tournament capital if Tom or Bill wins?
You can calculate this by entering their stacks in any ICM calculator:
| Tom is winning | Tom is winning | ||
| Tom's equity | $58 (+$12.82) | Tom's equity | $30.85 (-$14.33) |
| Bill's equity | $0 (-$22.57) | Bill's equity | $38.29 (+$15.72) |
In other words, Bill risks $22.57 to win $15.72. Tom risks $14.33 to win $12.82. Thus, the ICM causes the pressure on the short stacks to increase unevenly and become stronger the shorter the stack. In this situation, Bill risks more tournament equity, so he needs a stronger range for protection. This gives a large stack an advantage edge.
4. Risk Allowance
Simply put, winning chips are worth less than losing ones. This uneven risk/reward ratio creates a “risk premium” effect. You risk more than direct settlement in a chip EV environment would suggest.
Total required equity:
Let's use the example above to calculate Bill's risk premium in the case of Tom's push. The usual bank odds calculation shows us how much equity Bill needs to call in a cash game. We can call this amount “equity in chips”.
- Volume: 500 chips – Bill: 200 chips.
Bill needs to equalize another 150 chips, and if he wins, his stack will be 400. 150/400 = 37.5%.
In a cash game, Bill needs to call any hand that has an equity of at least 37.5% against Tom's range. But, as we know, the chips received are not equal to the lost chips. Bill needs to compare the cost of the fold with the risk of a call. There are 3 situations: Bill folds, calls and wins, or calls and loses.
Here is the tournament equity in each case:
Bill can fold his cards and keep the tournament capital of his chips equal to $17.93. Thus, Bill risks $17.93 by calling, and in case of winning, his stack will be $38.29. Let's perform a new calculation of the pot odds: $17.93 / $38.29. In other words, Bill actually needs about 47% equity of his equity hands for the call! Due to pressure from ICM, Bill needed an additional 12% equity. This additional 12% represents his risk premium. This extra allowance allows Tom on the WB to open up much wider, as Bill will always have to make an overfold on his pot odds. This is the edge of a large stack and the minus of a short one. The risk premium is a variable that is different for each pair of stacks at the table. In general, we can derive the following heuristic:
Your risk premium is higher relative to the stacks that cover you and lower relative to the smaller stacks that you cover.
5. General heuristics and effects
The ICM calculator cannot be started during the game. However, practicing the skill of understanding these effects outside the game can greatly help develop your instincts at the table.
- In tournaments, choose a narrower range than in cash games.
- Avoid marginal spots. A marginal plus spot in a chip EV environment is usually a minus spot in an ICM environment.
- Medium stacks should play more secretly near the bubble.
- Larger stacks can threaten smaller stacks as they take less risk when bouncing. This is especially true around bubble.
- The value of the won chips is less than the value of the lost chips of the same quantity.
- Pay attention to the payout structure. Larger jumps mean a higher risk premium.
- When the short stack is about to fly out, which will lead to a jump in payouts, all players, with the exception of players with the largest stacks, should usually significantly narrow down their game.
6. Limitations of the ICM model
ICM has some general limitations:
- ICM assumes that all players are equally qualified. In fact, we expect more experienced players to win in a larger percentage of cases relative to their stack.
- ICM calculations for large fields are computationally complex. Recent algorithms have made progress in this area, but it is still very difficult to correctly calculate the ICM for MTTs of large fields.
- ICM does not take into account the positions of players. A 3bb stack is much more valuable on the button than on the blinds.
- ICM ignores the increase in blinds. If you know that the blinds are about to grow, this can affect the optimal strategy, especially if there are short stacks.
- ICM underestimates the edge of chip leaders. Larger stacks can often intimidate smaller stacks due to ICM pressure, resulting in a higher win rate for larger stacks than ICM says.
This is a system based on mathematics that ignores many features of the real game.
7. Alternative models
There are some alternative models to address these limitations. One of the most popular is Future Game Simulation, which is essentially an ICM recursion. He calculates several rounds in advance to take into account the position, the increase in blinds and the future development of the game. This model is used in combination with the usual ICM calculations to more accurately determine tournament equity and is considered the main one in most tournament programs. There is also a “dependent chip model” that aims to solve the ICM problem that underestimates the edge of a large stack. Although this model tends to be more complex and overestimates the edge of the short stack.
8. Conclusion
ICM is a sophisticated tool that is used to convert stacks and payouts into real tournament capital.
While these in-game calculations are not possible, learning ICM out of the game will greatly increase your chances of success! The ICM reflects one of the most important aspects of the tournament game: the value of survival. The essence of tournaments is not to maximize BB/100, but to maximize your tournament equity. Studying these concepts, you will encounter situations in which you need to play much more tight than your instincts suggest, or on the other hand, where you should play much more loose than you think. Now you have learned how to use the pressure of survival to your advantage and avoid suicide in the ICM environment.
