Independent Chip Model Explained

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 finishes 1^st, then player finishes 2^nd with probability $$\backslash mathbb\backslash left[X\_i=2\backslash mid\; X\_k=1\backslash right]=\backslash frac$$
• Likewise, if players, ..., finish (respectively) 1^st, ..., ^st, then player finishes j^th 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 possible rankings of the 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 $$\backslash mathbb[A=1,B=2,C=3]=0.5\backslash cdot\backslash frac=0.3$$$$\backslash mathbb[A=1,C=2,B=3]=0.5\backslash cdot\backslash frac=0.2$$$$\backslash mathbb[B=1,A=2,C=3]=0.3\backslash cdot\backslash frac\backslash approx0.21$$$$\backslash mathbb[B=1,A=3,C=2]=0.3\backslash cdot\backslash frac\backslash approx0.09$$$$\backslash mathbb[C=1,A=2,B=3]=0.2\backslash cdot\backslash frac\backslash approx0.13$$$$\backslash mathbb[C=1,A=3,B=2]=0.2\backslash cdot\backslash frac\backslash approx0.08$$$$\backslash mathrm(A)=70(0.3+0.2)+30(0.21\backslash cdots+0.13\backslash cdots)\backslash approx45\backslash approx90\backslash \%$$$$\backslash mathrm(B)=70(0.21\backslash cdots+0.09\backslash cdots)+30(0.3+0.08\backslash cdots)\backslash approx32\backslash approx110\backslash \%$$$$\backslash mathrm(C)=70(0.13\backslash cdots+0.08\backslash cdots)+30(0.2+0.09\backslash cdots)\backslash approx22\backslash approx110\%$$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.^[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 !! rowspan=2

Data type !				Equity
A	B	C	1^st		2^nd	3^rd
25	87	88		0.125	0.1944	0.6806	$25.69
				0.125	0.1584	0.7166	$25.33
				0	0.0360	0.0360	$0.36
				0%	22.73%	5.02%	1.42%
21	89	90		0.105	0.1701	0.7249	$24.85
				0.105	0.1346	0.7604	$24.50
				0	0.0355	0.0355	$0.35
				0%	26.37%	4.67%	1.43%
198	1	1		0.99	0.009950	0.000050	$49.80
				0.99	0.009999	0.000001	$49.80
				0	0.000049	0.000049	$0
				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.

Further reading

• Book: Harrington . Dan . Robertie . Bill . Harrington On Modern Tournament Poker . 2014 . Two Plus Two Publishing LLC. 978-1-880685-56-3 . Harrington discusses the ICM on pages 108-122.
• Book: Collin Moshman. Sit 'n Go Strategy: Expert Advice for Beating One-Table Poker Tournaments. July 2007. Two Plus Two Publishing LLC. 978-1-880685-39-6. 122–.
• Book: Jonathan Grotenstein. Storms Reback. 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. 15 January 2013. St. Martin's Press. 978-1-250-00665-3. 17–.

Notes and References

1. Book: Bill Chen and Jerrod Ankenman . The Mathematics of Poker . ConJelCo LLC . 2006 . 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 . 10.2307/2284068 . 2284068.
3. Book: Malmuth, Mason . Gambling Theory and Other Topics . Two Plus Two Publishing . 1987 . 233, Settling Up in Tournaments: Part III.
4. Web site: Poker Strategy – Introduction To Independent Chip Model With Yevgeniy Timoshenko and David Sands. Fast, Erik. cardplayer.com. 20 March 2012. 12 September 2019.
5. Web site: ICM Poker Introduction: What Is The Independent Chip Model?. Upswing Poker. 12 September 2019.
6. Web site: Weighing Different Deal-Making Methods at a Final Table. Selbrede, Steve. PokerNews. 27 August 2019. 12 September 2019.
7. Web site: Explain Poker Like I'm Five: Independent Chip Model (ICM). Card Player News Team. cardplayer.com. 28 December 2014. 12 September 2019.
8. Web site: Walker, Greg . What Is The Independent Chip Model? . 12 September 2019 . thepokerbank.com.
9. Book: Feller, William . An Introduction to Probability Theory and Its Applications Volume I . John Wiley & Sons . 1968 . 344–347.
10. 2011.07610 . math.PR . Persi Diaconis & Stewart N. Ethier . Gambler's Ruin and the ICM . 2020–2021.
11. Book: Either, Stewart . The Doctrine of Chances: Probabilistic Aspects of Gambling . Springer . 2010 . 978-3-540-78782-2 . Chapter 7 Gambler's Ruin.
12. Web site: Gorstein . Evan . 24 July 2016 . Solving and Computing the Discrete Dirichlet Problem . 2021-06-09.