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 1^st is proportional to the player's chip count.^[9]
• Otherwise, if player k finishes 1^st, then player i finishes 2^nd with probability
  $${\displaystyle \mathbb {P} \left[X_{i}=2\mid X_{k}=1\right]={\frac {x_{i}}{1-x_{k}}}}$$

   
• Likewise, if players m₁, ..., m_j-1 finish (respectively) 1^st, ..., (j-1)^st, then player i finishes j^th with probability
  $${\displaystyle \mathbb {P} \left[X_{i}=j\mid X_{m_{z}}=z\quad (1\leq z<j)\right]={\frac {x_{i}}{1-\sum _{z=1}^{j-1}{x_{m_{z}}}}}}$$

   
• 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 1^st-place payout is 70 units and the 2^nd-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]

 

The FEM mesh for 3 players and 4 chips.

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		1^st	2^nd	3^rd
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. 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.

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.