Independent Chip Model

In poker, the Independent Chip Model (ICM), also known as the Malmuth–Harville method, 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; in 1987, Mason Malmuth independently rediscovered it for poker. 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.

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:

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, 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 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

Model
The original ICM model operates as follows:


 * Every player's chance of finishing 1st is proportional to the player's chip count.
 * Otherwise, if player $k$ finishes 1st, then player $i$ finishes 2nd with probability $$\mathbb{P}\left[X_i=2\mid X_k=1\right]=\frac{x_i}{1-x_k}$$
 * Likewise, if players $m_{1}$, ..., $m_{j-1}$ finish (respectively) 1st, ..., $(j-1)$st, then player $i$ finishes jth with probability $$\mathbb{P}\left[X_i=j\mid X_{m_z}=z\quad(1\leq z<j)\right]=\frac{x_i}{1-\sum_{z=1}^{j-1}{x_{m_z}}}$$
 * 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 $$\mathbb{P}[A=1,B=2,C=3]=0.5\cdot\frac{0.3}{1-0.5}=0.3$$$$\mathbb{P}[A=1,C=2,B=3]=0.5\cdot\frac{0.2}{1-0.5}=0.2$$$$\mathbb{P}[B=1,A=2,C=3]=0.3\cdot\frac{0.5}{1-0.3}\approx0.21$$$$\mathbb{P}[B=1,A=3,C=2]=0.3\cdot\frac{0.2}{1-0.3}\approx0.09$$$$\mathbb{P}[C=1,A=2,B=3]=0.2\cdot\frac{0.5}{1-0.2}\approx0.13$$$$\mathbb{P}[C=1,A=3,B=2]=0.2\cdot\frac{0.3}{1-0.2}\approx0.08$$$$\mathrm{ICM}(A)=70(0.3+0.2)+30(0.21\cdots+0.13\cdots)\approx45\approx90\%$$$$\mathrm{ICM}(B)=70(0.21\cdots+0.09\cdots)+30(0.3+0.08\cdots)\approx32\approx110\%$$$$\mathrm{ICM}(C)=70(0.13\cdots+0.08\cdots)+30(0.2+0.09\cdots)\approx22\approx110%$$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. 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. 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 ($0.36). 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.