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Independent Chip Modeling (ICM) determines your equity share of the prize pool in a tournament based on the stack sizes of the remaining opponents and the probability of your finish. With these probabilities, a dollar value can be associated with your stack size.
ICM is usually much more prevalent in SNGs, but can also apply when there are large payout jumps at the final table of a tournament. Most experienced SNG players are very familiar with ICM and it’s use, but may not understand the mechanics behind it, or how to calculate it by hand.
The need for ICM stems from a non-linear value of tournament chips. In all SNGs that are not winner-take-all, or in the heads-up stage, the chips that you risk will be of a lesser value than the chips you stand to gain.
This tournament equity calculator uses the 'Independent Chip Model' (ICM) to determine results. A discussion on how the results are obtained can be found at Two Plus Two forums here. Enter chip amounts and payouts between 0 and 1,000,000 using only numbers and decimal point. 147k members in the poker community. Shuffle up and deal! Official subreddit for all things poker. I created my own ICM calculator in a spreadsheet (link to. ICMIZER is a poker calculator designed to assist in making the right preflop decision in the latter stage of the tournament, when only two options remain: to fold or to go all-in.
For example, in a 6-man $20 SNG, with prizes of $84 for first place, and $36 for second place, you start with 1500 chips worth $20. If you win you’ll finish with 6 times as many chips as you started with (9000), which are worth 4.2 times as much money as you started with ($84). That may not seem fair, but all of the prize money is distributed at the end, so if you know how to use ICM as a weapon, you will end up with the lion’s share of the prize pool much more often than 1 in 6 times.
The Mechanics behind Calculating ICM
Let’s look at the mechanics behind calculating ICM using an example situation:
- $20 No Limit Hold’em SNG – 6-man.
- Prizes: $84 / $36
- Starting stacks: 1500 (total chips in play = 9000).
- Rick has 4500, Stu has 2700, and Mark has 1800.
- Blinds: 150/300.
The formula starts by calculating how much equity each player has in 1st place money, by looking at what percentage of the total chips a player has in play, and multiplying that by first prize ($84).
From there, the formula has to determine how much equity each player has in second place money ($36).This is a bit more complicated than determining first place equity, but it’s still doable by hand.
If Rick doesn’t win, there are a couple of ways he can take second place. His equity in second prize is the sum total of his equity for the times he doesn’t win, but he beats the remaining player. We will calculate the equity he has in second prize:
We should do the same for Stu:
and Mark:
…and if you add it up, this is what you get:
Armed with this information each player can use it to their advantage. Rick should know that he can shove on Stu’s big blind quite frequently. Stu will know he is risking $38.70 when he calls, and will get barely half of that back when he wins, so he will have to win more than 60% of the time to make a call if Rick pushes on him, and since only some hands win that often against Rick’s wide pushing range, he will fold a lot. Mark will know that if the others are in a lot of pots together, he may end up getting second by default, which drives his equity up from $27.09 to at least $36 (second prize) on a hand where he folded before the flop!
The Limitations of ICM
Independent Chip Modeling (ICM) is one of the most accurate ways to analyze the current value of your chip stack in a Sit & Go. But because it’s a purely math-based formula it misses some intangibles that the math doesn’t account for that should affect your decisions.
Lets look at some of these intangibles.
ICM Underestimates the Chip Leader’s Value
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This is by far the biggest limitation of ICM. If we’re playing a $20 SNG with a total of 13,500 chips in play (9-man, 1,500 starting stacks), and the remaining stacks are as follows; 6750, 2250, 2250, 2250, then ICM calculates the equity of the 6,750 stack by assuming that since he has half of the chips in play (6750/13500 = 50%), then he will win half the time. However, even if these players are equally skilled, the 6,750 stack will be able to use his leverage to dominate the table, and perhaps win as often as 60-65% of the time. This means that being the chip leader is an advantage that ICM does not account for.
How to adjust your game…
If you’re the chip leader, call a push less often than ICM would suggest that you do to protect your lead. If you’re a middle stack and calling would give you the chip lead, then call wider than ICM would suggest so that, should you win, you get to reap the benefits.
While a tight game is the cornerstone of early-game success, don’t over do it by making drastic changes to your ranges by playing much tighter than you would in a cash game or MTT. Simply play slightly tighter in a SNG than these other formats. By accumulating chips in the early going, you are setting yourself up to have the chip lead going into the bubble phase of the game, and you’ll enjoy this benefit.
ICM and the Position of the Blinds
ICM doesn’t account for the position of the blinds. This is important to remember especially when the blinds get high, or when two players have very short stacks. Here is an example. Stacks 7100, 4900, 600, 400, with payouts of $45 ,$27, $18. Blinds are 300-600. Here is the ICM calculation:
In this situation, with the blinds at 300-600, ICM shows that Mark’s stack is worth $12.99 and Doyle’s is worth $8.73. The truth of the matter is that the biggest consideration is which player will be in the big blind next. Should Mark have to play from the big blind before Doyle, then Doyle’s equity is higher than Mark’s because Doyle is facing elimination first.
How to adjust your game…
If you are going to eat a big blind that is more than one-third of your stack soon, before anyone else will, then shove wider to protect your stack. You don’t want to be forced to call with a bad hand in the big blind.
If you’re short stacked but someone else will face elimination before you within the next few hands, play a very tight range.
ICM Assumes Fixed Blinds
ICM assumes fixed blinds and does not account for blind increases, or fold equity. One of your largest weapons in poker, other than your chips and your cards, is your fold equity, which is your ability to win an uncontested pot when you push your chip stack. Your fold equity diminishes as your chip stack diminishes, but should you drop below 5 big blinds, your fold equity diminishes drastically, as your opponents will often be committed to call any push you make.
If you have 5-9BB or 8-12BB with the blinds about to increase, you are in a position where your stack will likely drop. In this situation you are in immediate danger of falling below this 5BB threshold. ICM calculates the number of chips you have divided by the number of chips in play to determine your equity, so it has no knowledge of this.
How to adjust your game…
If you are using an ICM calculator to determine your pushing range and you are going to lose your fold equity if you do not push, then push wider than recommended.
If a solid player is in danger of losing their fold equity, and they push, realize they may be shoving a wider range, and call slightly wider than ICM recommends.
If the other effective stacks are dropping below 5BB as well, then do not push wider, because even if you gain chips, your fold equity is gone anyway, as your opponents with 4BB and less will call you with a similar range whether you have 4BB, 6BB, or 14BB.
Conclusion
Understanding the concept of ICM fundamental to success in SNGs. You don’t need to be a math geek to understand ICM – just being aware of the concept is the most valuable knowledge a successful SNG player can possess. If you can find a hand history of your own, then take an example of two, and work through it yourself. Play around with ICM calculators and study situations away from the table. It’s a great way to learn.
As was said at the beginning of this lesson – due to the non-linear value of tournament chips, the chips that you risk will be of a lesser value than those you stand to gain. Therefore the most important thing to remember is that your last chip is always going to be your most valuable chip. If you’ve fully grasped this concept then you’ll know that your chips and the leverage they provide is your biggest weapon.
Related Lessons
By Jennifear
Jennifear is an online professional MTT and SNG player. She is a well-regarded, highly experienced and hugely popular poker coach who has taught hundreds of successful players.
Contents
- Evaluation
Introduction
We have developed a new algorithm for multi-table tournament (MTT) ICM calculations that provides highly accurate approximations of Malmuth-Harville ICM for field sizes up to thousands of players. The deviation of the calculated values to exact Malmuth-Harville ICM is orders of magnitude smaller than in the previous HRC MTT model. A web calculator featuring this new method is available here and as of February 2020 the latest HRC version includes an updated MTT mode that uses the new model.
Background
The underlying Malmuth-Harville ICM model is extremely inefficient for large player fields. Naive implementations of ICM can handle about 15 players, and even optimized versions can't calculate exact Malmuth-Harville values beyond 25-30 players. So the standard ICM chips-to-equity mapping that is used for single-table calculations can't be directly applied to calculations with larger fields.
Until a few years ago, the standard procedure for calculations before the final table was to either use chip EV calculations and ignore any ICM considerations entirely or to setup single-table calculations with artificial prize structures like 80/20 to introduce small bubble factors to the calculation.
State of MTT ICM calculations
In the last few years, ICM tools started to automate the setup of approximative structures with dedicated modes for multi-table tournaments. In the background, the previous HRC MTT model would generate a set of about 10 hidden stacks and an adjusted artificial prize structure, aiming to achieve bubble factors similar to those in the original MTT situation, but in a compressed field of no more than 20 players where exact ICM values can still be efficiently calculated.
The accuracy of that approach is quite reasonable, estimates of the previous MTT model were typically off by around 3% compared to exact ICM values. So a chip stack with with actual Malmuth-Harville equity of 100$ would typically be predicted to have 100$ ± 3$ equity by the MTT model. That's much better than using chip EV and likely better than any manual approximations that can be carried out using single-table calculations. But, while the MTT model was a clear improvement over the then status quo, there's certainly room for improvement. The accuracy of our previous mode is inherently limited by the approach used. When fields with hundreds or thousands of players are compressed down to below 20 stacks, some details necessarily get lost.
New HRC MTT model
The new HRC MTT model pushes MTT accuracy to an entirely new level. Unlike the previous MTT mode, it no longer operates on a compressed game structure behind the scenes. The ICM estimates are calculated directly on the original stacks and prizes. The typical relative error is orders of magnitude lower than our previous model, the average is well below 0.01% for the test scenarios presented below. For realistic game settings we expect the new estimates to be, for all practical purposes, just as good as exact ICM values.
We use two slightly different variations of the new model: The full variant is fast enough for fields of up to 500 players in the HRC desktop version. For even larger fields, an additional approximative step is used in the calculation to speed things up further. But even this faster variant is still remarkably accurate for medium and large fields, while being fast enough to comfortably support calculations with thousands of remaining players.
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The online MTT calculator here uses full accuracy for up to 64 players and switches to the faster variant for larger fields. The HRC desktop version currently uses full accuracy for fields of up to 500 players before switching to the faster version.
Evaluation
The following section provides some additional details about the evaluation procedure used and lists the results for our main evaluation set. The selection of test scenarios presented here is quite limited, the model was actually evaluated on a variety of more extreme stack and prize functions, as well as a selection of actual tournament structures.
Methodology
If it is impracticable to calculate exact ICM values for large fields, how else can we evaluate the accuracy of the new model?
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Back in 2011 Tysen Streib (co-author of 'Kill Everyone') introduced a method to calculate ICM values by Monte Carlo sampling. The original post can still be found in the 2+2 Poker Theory forum: New algorithm to calculate ICM for large tournaments.
This method allows the approximation of ICM values to arbitrary accuracy by random sampling, but it didn't get widely adopted by any ICM software tools because the sample sizes required to achieve good accuracy are quite large. The method is too slow to use in a full fledged ICM calculator, where thousands of ICM estimates are needed to calculate a single hand. However, it's perfectly suited for evaluation purposes where we can spend several hours or days to calculate a few equities.
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Using Tysen's method, we simulated the finishing distributions for varying field sizes of 32 to 1024 players for 1010 tournaments each. The various tables below were then created by applying different prize structures to the same simulated finishing distributions, so the different tables are all based on the same set of samples.
The stack and prize setups were chosen to be easily reproducible. For the tables below, the following stack and prize setups were used:
- With n players, stacks are: 1, 2,...,n - 1, n
- With p spots paid, payouts are: 1st = p, 2nd = p - 1,..., pth=1
The model quality is evaluated using the absolute percentage deviation of the model values against the simulation results:
- mean: Mean APD, 100% * mean( Si - Mi / Si)
- max: Maximum APD, 100% * max( Si - Mi / Si)
- with Si being the simulated values and Mi being the model estimates
Result Tables
| New MTT Model | Previous | ||||||
|---|---|---|---|---|---|---|---|
| Full | Fast | MTT Model | |||||
| n | p | mean | max | mean | max | mean | max |
| 32 | 8 | 0.0018% | 0.0127% | 0.0192% | 0.0269% | 1.2% | 2.8% |
| 64 | 16 | 0.0028% | 0.0180% | 0.0053% | 0.0228% | 1.5% | 3.6% |
| 128 | 32 | 0.0023% | 0.0197% | 0.0020% | 0.0186% | 1.7% | 4.5% |
| 256 | 64 | 0.0024% | 0.0204% | 0.0025% | 0.0203% | 1.8% | 4.8% |
| 512 | 128 | 0.0023% | 0.0237% | 0.0025% | 0.0232% | 1.8% | 5.2% |
| 1024 | 256 | 0.0022% | 0.0424% | 0.0026% | 0.0425% | 1.8% | 5.0% |
| Linear stacks and prizes, top 25% paid, sample size 1010 | |||||||
| New MTT Model | Previous | ||||||
|---|---|---|---|---|---|---|---|
| Full | Fast | MTT Model | |||||
| n | p | mean | max | mean | max | mean | max |
| 32 | 16 | 0.0014% | 0.0121% | 0.0224% | 0.0303% | 2.5% | 4.4% |
| 64 | 32 | 0.0016% | 0.0151% | 0.0048% | 0.0159% | 3.0% | 6.8% |
| 128 | 64 | 0.0014% | 0.0108% | 0.0015% | 0.0131% | 3.6% | 10.1% |
| 256 | 128 | 0.0016% | 0.0216% | 0.0017% | 0.0211% | 3.8% | 10.8% |
| 512 | 256 | 0.0014% | 0.0311% | 0.0016% | 0.0313% | 3.7% | 11.5% |
| 1024 | 512 | 0.0014% | 0.0397% | 0.0017% | 0.0399% | 3.8% | 12.3% |
| Linear stacks and prizes, top 50% paid, sample size 1010 | |||||||
| New MTT Model | Previous | ||||||
|---|---|---|---|---|---|---|---|
| Full | Fast | MTT Model | |||||
| n | p | mean | max | mean | max | mean | max |
| 32 | 24 | 0.0006% | 0.0029% | 0.0222% | 0.0500% | 3.9% | 15.8% |
| 64 | 48 | 0.0010% | 0.0065% | 0.0047% | 0.0209% | 4.0% | 11.6% |
| 128 | 96 | 0.0009% | 0.0084% | 0.0012% | 0.0100% | 5.8% | 21.7% |
| 256 | 192 | 0.0010% | 0.0117% | 0.0011% | 0.0163% | 5.7% | 32.1% |
| 512 | 384 | 0.0009% | 0.0140% | 0.0011% | 0.0142% | 6.3% | 21.5% |
| 1024 | 768 | 0.0009% | 0.0225% | 0.0012% | 0.0223% | 6.2% | 28.1% |
| Linear stacks and prizes, top 75% paid, sample size 1010 | |||||||
| New MTT Model | Previous | ||||||
|---|---|---|---|---|---|---|---|
| Full | Fast | MTT Model | |||||
| n | p | mean | max | mean | max | mean | max |
| 32 | 32 | 0.0003% | 0.0015% | 0.0252% | 0.3977% | 5.3% | 19.7% |
| 64 | 64 | 0.0005% | 0.0023% | 0.0094% | 0.3491% | 9.6% | 27.6% |
| 128 | 128 | 0.0005% | 0.0038% | 0.0042% | 0.3212% | 9.6% | 40.5% |
| 256 | 256 | 0.0007% | 0.0428% | 0.0019% | 0.2380% | 10.7% | 49.8% |
| 512 | 512 | 0.0009% | 0.0953% | 0.0016% | 0.3441% | 11.4% | 55.8% |
| 1024 | 1024 | 0.0010% | 0.0616% | 0.0014% | 0.2742% | 11.5% | 62.4% |
| Linear stacks and prizes, top 100% paid, sample size 1010 | |||||||
Note: Keep in mind that the model quality is evaluated against a noisy reference. Even with samples of 1010 tournaments, the sampling error is still quite significant in comparison to the model error, so the tables above only provide an upper bound of the error levels in the tested scenarios. Although actual model accuracy might be better than indicated, even the listed accuracy is already more than sufficient for all practical purposes.