Independent Chip Model
In poker, the Independent Chip Model (ICM) is a mathematical model used to approximately calculate a player's overall equity in a tournament. The model uses stack sizes alone to determine how often a player will finish in each position (1st, 2nd, etc.). A player's probability of finishing in each position is then multiplied by the prize amount for that position and those numbers are added together to determine the player's overall equity.[1][2]
The ICM is also known as the Malmuth-Harville method.[3] In 1973 David Harville published a method to calculate the probability for a horse to finish 1st, 2nd, etc. in a horse race.[4] In 1987 Mason Malmuth adapted Harville's method to calculate the probability for a tournament player to finish 1st, 2nd, etc.[5].
The term ICM is often misunderstood to mean a simulator that helps a player make decisions in a tournament. Such simulators often make use of the Independent Chip Model but are not strictly speaking ICM calculators. A true ICM calculator will have the chip counts of all players, as well as the payout structure of the tournament, as input and each player's equity as output.[6]
The calculation of ICM can be elaborated as below:
- Assume every player's chance of finishing 1st is proportion to chip count
- If player i did not finish first, given player k finish first, player is' chance of finishing second is P(Xi,2|Xk,1) = xi/(1-xk）
- Following this logic, given m1 finish 1st, m2 finished 2nd, mj-1 finish j-1th, the chance of Player i finish jth place is P(Xi,j|Xm1,1, Xm2,x.....Xmj-1,j-1) = xi/(1-xm1-xm2-....-xmj-1)
- Sum of the value in each permutation (in proportion to n!)
The ICM can be applied to answer specific questions, such as:[7][8]
- The range of hands that a player can move all in with, considering the action so far and the stack sizes of the other players still in the hand
- The range of hands that a player can call another player's all in with, and recommends either calling or moving all in over the top, considering all the stacks still in the hand
- When discussing a deal, how much money each player should get
ICM precision
2-players case
For any one of the 2 players the probability to finish 1st is exactly equal to its share of the tournament chips.[9] The ICM gives perfect results.
3-players case
The Finite Element Method (FEM) can be used to compute for any player its exact probabilities to finish 1st, 2nd etc. and its exact tournament equity.[10] Those exact values allows for an evaluation of the precision of the ICM.
The FEM is used to compute the exact values for all the repartitions of 200 big blinds between 3 players. The table hereafter summarizes the comparison of the approximate ICM values versus the exact FEM values.
| Big blind
repartition |
Data type | 1st player
finishes 1st |
1st player
finishes 2nd |
1st player
finishes 3rd |
$50/$30/$20
equity |
|---|---|---|---|---|---|
| 25-87-88 | Icm | 0.125 | 0.1944 | 0.6806 | $25.69 |
| Fem | 0.125 | 0.1584 | 0.7166 | $25.33 | |
| Icm-Fem | 0 | 0.0360 | -0.0360 | $0.36 | |
| (Icm-Fem)/Icm | 0% | 22.73% | -5.02% | 1.42% | |
| 21-89-90 | Icm | 0.105 | 0.1701 | 0.7249 | $24.85 |
| Fem | 0.105 | 0.1346 | 0.7604 | $24.50 | |
| Icm-Fem | 0 | 0.0350 | -0.0350 | $0.35 | |
| (Icm-Fem)/Icm | 0% | 26.37% | -4.67% | 1.43% | |
| 198-1-1 | Icm | 0.99 | 0.009950 | 0.000050 | $49.80 |
| Fem | 0.99 | 0.009999 | 0.000001 | $49.80 | |
| Icm-Fem | 0 | -0.000049 | 0.000049 | $0 | |
| (Icm-Fem)/Fem | 0% | -0.49% | 4900% | 0% |
The big blind repartition 25-87-88 gives the largest difference between an ICM and a FEM probability (0.0360) and the largest tournament equity difference ($0.36 for tournament payouts $50/$30/$20). The relative difference between an ICM and a FEM tournament equity [(ICM- FEM)/FEM)] is 1.42%
The big blind repartition 25-87-88 gives the largest relative difference between an ICM and a FEM tournament equity (1.43%).
The big bling repartition 198-1-1 gives the largest relative difference between an ICM and a FEM probability (4900%). However that large relative difference has no impact on the tournament equity.
Although the ICM does a poor job when computing the exact probability of a player to finish 1st, 2nd , etc., it gives fairly good tournament equities in the 3-players case.
4-players case
The FEM is used to compute the exact values for all the repartitions of 100 big blinds between 4 players. The table hereafter summarizes the comparison of the approximate ICM values versus the exact FEM values.
| Big blind
repartition |
Data type | 1st player
finishes 1st |
1st player
finishes 2nd |
1st player
finishes 3rd |
1st player
finishes 4th |
$40/$30/$20/$10
equity |
|---|---|---|---|---|---|---|
| 9-30-30-31 | Icm | 0.09 | 0.1176 | 0.1814 | 0.6110 | $16.87 |
| Fem | 0.09 | 0.1003 | 0.1387 | 0.6710 | $16.09 | |
| Icm-Fem | 0 | 0.0173 | 0.0427 | 0.0599 | $0.77 | |
| (Icm-Fem)/Icm | 0% | 17.20% | 30.78% | 8.93% | 4.80% | |
| 7-31-31-31 | Icm | 0.07 | 0.0943 | 0.1539 | 0.6817 | $15.53 |
| Fem | 0.07 | 0.0785 | 0.1108 | 0.7407 | $14.78 | |
| Icm-Fem | 0 | 0.0158 | 0.0432 | -0.0590 | $0.75 | |
| (Icm-Fem)/Icm | 0% | 20.17% | 38.94% | -7.96% | 5.06% | |
| 8-30-31-31 | Icm | 0.08 | 0.1062 | 0.1684 | 0.6454 | $16.21 |
| Fem | 0.08 | 0.0895 | 0.1251 | 0.7054 | $15.44 | |
| Icm-Fem | 0 | 0.0167 | 0.0434 | -0.0601 | $0.77 | |
| (Icm-Fem)/Fem | 0% | 18.66% | 34.66% | -8.51% | 4.97% | |
| 91-1-1-1 | Icm | 0.91 | 0.0294 | 0.000600 | 0.000006 | $39.69 |
| Fem | 0.91 | 0.0300 | 0.000036 | 0.000000003 | $39.70 | |
| Icm-Fem | 0 | -0.0006 | 0.000564 | 0.000006 | $0.01 | |
| (Icm-Fem)/Fem | 0% | -1.90% | 15.45% | 200000% | 0.01% |
The ICM gives fairly good tournament equities in the 3-players case.
References
- ^ Fast, Erik (2012-03-20). "Poker Strategy -- Introduction To Independent Chip Model With Yevgeniy Timoshenko and David Sands". cardplayer.com. Retrieved 12 September 2019.
- ^ "ICM Poker Introduction: What Is The Independent Chip Model?". Upswing Poker. Retrieved 12 September 2019.
- ^ Bill Chen and Jerrod Ankenman (2006). The Mathematics of Poker. ConJelCo LLC. pp. 333, chapter 27, A Survey of Equity Formulas.
- ^ Harville, David (1973). "Assigning Probabilities to the Outcomes of Multi-Entry Competitions". Journal of the American Statistical Association. Vol. 68, No. 342 (June 1973): 312–316 – via JSTOR.
{{cite journal}}:|volume=has extra text (help) - ^ Malmuth, Mason (1987). Theory and Other Topics. Two Plus Two Publishing. pp. 233, Settling Up in Tournaments: Part III.
- ^ Walker, Greg. "What Is The Independent Chip Model?". thepokerbank.com. Retrieved 12 September 2019.
- ^ Selbrede, Steve (2019-08-27). "Weighing Different Deal-Making Methods at a Final Table". PokerNews. Retrieved 12 September 2019.
- ^ Card Player News Team (2014-12-28). "Explain Poker Like I'm Five: Independent Chip Model (ICM)". cardplayer.com. Retrieved 12 September 2019.
- ^ Feller, William (1968). An Introduction to Probability Theory and Its Applications Volume I. John Wiley & Sons. pp. 344–347.
- ^ Persi Diaconis & Stewart N. Ethier (2020–2021). "Gambler's Ruin and the ICM". Retrieved 2021-05-22.
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Further reading
- Harrington, Dan; Robertie, Bill (2014). Harrington On Modern Tournament Poker. Two Plus Two Publishing LLC. ISBN 1-880685-56-6. Harrington discusses the ICM on pages 108-122.
- Collin Moshman (July 2007). Sit 'n Go Strategy: Expert Advice for Beating One-Table Poker Tournaments. Two Plus Two Publishing LLC. pp. 122–. ISBN 978-1-880685-39-6.
- Jonathan Grotenstein; Storms Reback (15 January 2013). Ship It Holla Ballas!: How a Bunch of 19-Year-Old College Dropouts Used the Internet to Become Poker's Loudest, Craziest, and Richest Crew. St. Martin's Press. pp. 17–. ISBN 978-1-250-00665-3.