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.

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 $${\displaystyle \mathbb {P} \left[X_{i}=2\mid X_{k}=1\right]={\frac {x_{i}}{1-x_{k}}}}$${\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) 1st, ..., (j-1)^st, then player i finishes j^th with probability $${\displaystyle \mathbb {P} \left[X_{i}=j\mid X_{m_{z}}=z\quad (1\leq z<j)\right]={\frac {x_{i}}{1-\sum _{z=1}^{j-1}{x_{m_{z}}}}}}$${\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 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,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} [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=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} [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=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 \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} (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} (B)=70(0.21\cdots +0.09\cdots )+30(0.3+0.08\cdots )\approx 32\approx 110\%}$${\displaystyle \mathrm {ICM} (C)=70(0.13\cdots +0.08\cdots )+30(0.2+0.09\cdots )\approx 22\approx 110\%}$${\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.

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:

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ドル.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.

• 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.

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