Independent Chip Model
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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
- 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.
![{\displaystyle \mathbb {P} \left[X_{i}=2\mid X_{k}=1\right]={\frac {x_{i}}{1-x_{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc56ddd1f641fa6f13704cfa63426ec535981cde)
![{\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}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/050d68f30f63e921a1c87cbba11cd9005483fdff)
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.
![{\displaystyle \mathbb {P} [A=1,B=2,C=3]=0.5\cdot {\frac {0.3}{1-0.5}}=0.3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac20f13e88da1c7c3069d224a3ed1d191bff960)
![{\displaystyle \mathbb {P} [A=1,C=2,B=3]=0.5\cdot {\frac {0.2}{1-0.5}}=0.2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e118d6c327e63d5613ebc26780fac52fb53b22f)
![{\displaystyle \mathbb {P} [B=1,A=2,C=3]=0.3\cdot {\frac {0.5}{1-0.3}}\approx 0.21}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2070b6b52abde3ea7f3af5d3ba76b64cd22b7ec0)
![{\displaystyle \mathbb {P} [B=1,A=3,C=2]=0.3\cdot {\frac {0.2}{1-0.3}}\approx 0.09}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f75e83f1210b922d6f8d70a1587d221c4ebeb26)
![{\displaystyle \mathbb {P} [C=1,A=2,B=3]=0.2\cdot {\frac {0.5}{1-0.2}}\approx 0.13}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ee97614640b44f1fa42e5669a90724bf66012a)
![{\displaystyle \mathbb {P} [C=1,A=3,B=2]=0.2\cdot {\frac {0.3}{1-0.2}}\approx 0.08}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47e30a7302a1e447bae01498c13d18312e4a7c2b)
