# Independent Chip Model

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

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 ${\displaystyle \mathbb {P} \left[X_{i}=2\mid X_{k}=1\right]={\frac {x_{i}}{1-x_{k}}}}$
• Likewise, if players m1, ..., mj-1 finish (respectively) 1st, ..., (j-1)st, then player i finishes jth with probability ${\displaystyle \mathbb {P} \left[X_{i}=j\mid X_{m_{z}}=z\quad (1\leq z
• 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 ${\displaystyle \mathbb {P} [A=1,B=2,C=3]=0.5\cdot {\frac {0.3}{1-0.5}}=0.3}$${\displaystyle \mathbb {P} [A=1,C=2,B=3]=0.5\cdot {\frac {0.2}{1-0.5}}=0.2}$${\displaystyle \mathbb {P} [B=1,A=2,C=3]=0.3\cdot {\frac {0.5}{1-0.3}}\approx 0.21}$${\displaystyle \mathbb {P} [B=1,A=3,C=2]=0.3\cdot {\frac {0.2}{1-0.3}}\approx 0.09}$${\displaystyle \mathbb {P} [C=1,A=2,B=3]=0.2\cdot {\frac {0.5}{1-0.2}}\approx 0.13}$${\displaystyle \mathbb {P} [C=1,A=3,B=2]=0.2\cdot {\frac {0.3}{1-0.2}}\approx 0.08}$${\displaystyle \mathrm {ICM} (A)=70(0.3+0.2)+30(0.21\cdots +0.13\cdots )\approx 45\approx 90\%}$${\displaystyle \mathrm {ICM} (B)=70(0.21\cdots +0.09\cdots )+30(0.3+0.08\cdots )\approx 32\approx 110\%}$${\displaystyle \mathrm {ICM} (C)=70(0.13\cdots +0.08\cdots )+30(0.2+0.09\cdots )\approx 22\approx 110\%}$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.[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:

3 players; 200 chips; $50/30/20 payout 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

1. ^ Bill Chen and Jerrod Ankenman (2006). The Mathematics of Poker. ConJelCo LLC. pp. 333, chapter 27, A Survey of Equity Formulas.
2. ^ 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.
3. ^ Malmuth, Mason (1987). Gambling Theory and Other Topics. Two Plus Two Publishing. pp. 233, Settling Up in Tournaments: Part III.
4. ^ Fast, Erik (20 March 2012). "Poker Strategy – Introduction To Independent Chip Model With Yevgeniy Timoshenko and David Sands". cardplayer.com. Retrieved 12 September 2019.
5. ^ "ICM Poker Introduction: What Is The Independent Chip Model?". Upswing Poker. Retrieved 12 September 2019.
6. ^ Selbrede, Steve (27 August 2019). "Weighing Different Deal-Making Methods at a Final Table". PokerNews. Retrieved 12 September 2019.
7. ^ Card Player News Team (28 December 2014). "Explain Poker Like I'm Five: Independent Chip Model (ICM)". cardplayer.com. Retrieved 12 September 2019.
8. ^ Walker, Greg. "What Is The Independent Chip Model?". thepokerbank.com. Retrieved 12 September 2019.
9. ^ a b Feller, William (1968). An Introduction to Probability Theory and Its Applications Volume I. John Wiley & Sons. pp. 344–347.
10. ^ Persi Diaconis & Stewart N. Ethier (2020–2021). "Gambler's Ruin and the ICM". arXiv:2011.07610 [math.PR].
11. ^ Either, Stewart (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer. pp. Chapter 7 Gambler's Ruin. ISBN 978-3-540-78782-2.
12. ^ Gorstein, Evan (24 July 2016). "Solving and Computing the Discrete Dirichlet Problem" (PDF). Retrieved 9 June 2021.