In poker, the Independent Chip Model (ICM), also known as the Malmuth–Harville method,[1] is a mathematical model that approximates a player's overall equity in an incomplete tournament. David Harville first developed the model in a 1973 paper on horse racing;[2] in 1987, Mason Malmuth independently rediscovered it for poker.[3] In the ICM, all players have comparable skill, so that current stack sizes entirely determine the probability distribution for a player's final ranking. The model then approximates this probability distribution and computes expected prize money.[4][5]
Poker players often use the term ICM to mean a simulator that helps a player strategize a tournament. An ICM can be applied to answer specific questions, such as:[6][7]
- The range of hands that a player can move all in with, considering the play so far
- The range of hands that a player can call another player's all in with or move all in over the top; and which course of action is optimal, considering the remaining opponent stacks
- When discussing a deal, how much money each player should get
Such simulators rarely use an unmodified Malmuth-Harville model. In addition to the payout structure, a Malmuth-Harville ICM calculator would also require the chip counts of all players as input,[8] which may not always be available. The Malmuth-Harville model also gives poor estimates for unlikely events, and is computationally intractable with many players.
Model edit
The original ICM model operates as follows:
- Every player's chance of finishing 1st is proportional to the player's chip count.[9]
- Otherwise, if player k finishes 1st, then player i finishes 2nd with probability
- Likewise, if players m1, ..., mj-1 finish (respectively) 1st, ..., (j-1)st, then player i finishes jth with probability
- The joint distribution of the players' final rankings is then the product of these conditional probabilities.
- The expected payout is the payoff-weighted sum of these joint probabilities across all n! possible rankings of the n players.
![{\displaystyle \mathbb {P} \left[X_{i}=2\mid X_{k}=1\right]={\frac {x_{i}}{1-x_{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc56ddd1f641fa6f13704cfa63426ec535981cde)
![{\displaystyle \mathbb {P} \left[X_{i}=j\mid X_{m_{z}}=z\quad (1\leq z<j)\right]={\frac {x_{i}}{1-\sum _{z=1}^{j-1}{x_{m_{z}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/050d68f30f63e921a1c87cbba11cd9005483fdff)
For example, suppose players A, B, and C have (respectively) 50%, 30%, and 20% of the tournament chips. The 1st-place payout is 70 units and the 2nd-place payout 30 units. Then
![{\displaystyle \mathbb {P} [A=1,B=2,C=3]=0.5\cdot {\frac {0.3}{1-0.5}}=0.3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac20f13e88da1c7c3069d224a3ed1d191bff960)
![{\displaystyle \mathbb {P} [A=1,C=2,B=3]=0.5\cdot {\frac {0.2}{1-0.5}}=0.2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e118d6c327e63d5613ebc26780fac52fb53b22f)
![{\displaystyle \mathbb {P} [B=1,A=2,C=3]=0.3\cdot {\frac {0.5}{1-0.3}}\approx 0.21}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2070b6b52abde3ea7f3af5d3ba76b64cd22b7ec0)
![{\displaystyle \mathbb {P} [B=1,A=3,C=2]=0.3\cdot {\frac {0.2}{1-0.3}}\approx 0.09}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f75e83f1210b922d6f8d70a1587d221c4ebeb26)
![{\displaystyle \mathbb {P} [C=1,A=2,B=3]=0.2\cdot {\frac {0.5}{1-0.2}}\approx 0.13}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ee97614640b44f1fa42e5669a90724bf66012a)
![{\displaystyle \mathbb {P} [C=1,A=3,B=2]=0.2\cdot {\frac {0.3}{1-0.2}}\approx 0.08}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47e30a7302a1e447bae01498c13d18312e4a7c2b)
Comparison to gambler's ruin edit
Because the ICM ignores player skill, the classical gambler's ruin problem also models the omitted poker games, but more precisely. Harville-Malmuth's formulas only coincide with gambler's-ruin estimates in the 2-player case.[9] With 3 or more players, they give misleading probabilities, but adequately approximate the expected payout.[10]
For example, suppose very few players (e.g. 3 or 4). In this case, the finite-element method (FEM) suffices to solve the gambler's ruin exactly.[11][12] Extremal cases are as follows:
| Current stacks | Data type | P[A finishes ...] | Equity | ||||
|---|---|---|---|---|---|---|---|
| A | B | C | 1st | 2nd | 3rd | ||
| 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|/FEM | 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 | 0.0355 | 0.0355 | $0.35 | |||
| |ICM-FEM|/FEM | 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 25/87/88 game state gives the largest absolute difference between an ICM and FEM probability (0.0360) and the largest tournament equity difference ($0.36). However, the relative equity difference is small: only 1.42%. The largest relative difference is only slightly larger (1.43%), corresponding to a 21/89/90 game. The 198/1/1 game state gives the largest relative probability difference (4900%), but only for an extremely unlikely event.
Results in the 4-player case are analogous.
References edit
- ^ 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. 68 (342 (June 1973)): 312–316. doi:10.2307/2284068. JSTOR 2284068.
- ^ Malmuth, Mason (1987). Gambling Theory and Other Topics. Two Plus Two Publishing. pp. 233, Settling Up in Tournaments: Part III.
- ^ Fast, Erik (20 March 2012). "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.
- ^ Selbrede, Steve (27 August 2019). "Weighing Different Deal-Making Methods at a Final Table". PokerNews. Retrieved 12 September 2019.
- ^ Card Player News Team (28 December 2014). "Explain Poker Like I'm Five: Independent Chip Model (ICM)". cardplayer.com. Retrieved 12 September 2019.
- ^ Walker, Greg. "What Is The Independent Chip Model?". thepokerbank.com. Retrieved 12 September 2019.
- ^ a b 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". arXiv:2011.07610 [math.PR].
- ^ Either, Stewart (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer. pp. Chapter 7 Gambler's Ruin. ISBN 978-3-540-78782-2.
- ^ Gorstein, Evan (24 July 2016). "Solving and Computing the Discrete Dirichlet Problem" (PDF). Retrieved 9 June 2021.
Further reading edit
- Harrington, Dan; Robertie, Bill (2014). Harrington On Modern Tournament Poker. Two Plus Two Publishing LLC. ISBN 978-1-880685-56-3. 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.