Independent Chip Model
In poker, the Independent Chip Model (ICM), also known as the Malmuth–Harville method, 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; in 1987, Mason Malmuth independently rediscovered it for poker. In the ICM, all players have equal skill, so current stack sizes entirely determine the probability distribution for each player's final ranking. The model then approximates this probability distribution and computes expected prize money.
Poker players use ICM to estimate their tournament equity, negotiate final-table deals, and evaluate strategic decisions. Poker solvers use ICM as a utility function to optimize strategies in terms of expected prize money.
ICM calculations require the chip stacks of all remaining players and the payout structure. However, enumerating every possible finishing order becomes computationally intractable as the number of players increases. For large fields, ICM can instead be evaluated using methods such as Monte Carlo simulation or numerical integration.
Model
The original ICM model operates as follows:
- Every player's chance of finishing 1st is proportional to the player's chip count.
- 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.
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 where the percentages describe a player's expected payout relative to their current stack.
Comparison to gambler's ruin
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. With 3 or more players, they give misleading probabilities, but adequately approximate the expected payout.
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. 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
- ↑ 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. 7 June 2016. Retrieved 12 September 2019.
- ↑ Inariba, Wataru (5 October 2026), Near-Exact Computation of Independent Chip Model Placement Probabilities for Thousands of Players, arXiv, doi:10.48550/arXiv.2610.06028, arXiv:2610.06028, retrieved 29 October 2026
- 1 2 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
- 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.