# 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:
* 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, 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 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}}\approx 0.21$$\mathbb {P} [B=1,A=3,C=2]=0.3\cdot {\frac {0.2}{1-0.3}}\approx 0.09$$\mathbb {P} [C=1,A=2,B=3]=0.2\cdot {\frac {0.5}{1-0.2}}\approx 0.13$$\mathbb {P} [C=1,A=3,B=2]=0.2\cdot {\frac {0.3}{1-0.2}}\approx 0.08$$\mathrm {ICM} (A)=70(0.3+0.2)+30(0.21\cdots +0.13\cdots )\approx 45\approx 90\%$$\mathrm {ICM} (B)=70(0.21\cdots +0.09\cdots )+30(0.3+0.08\cdots )\approx 32\approx 110\%$$\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. 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:
3 players; 200 chips; $50/30/20 payout :
| Current stacks | Current stacks | Current stacks | Data type | P[A finishes ...] | P[A finishes ...] | P[A finishes ...] | Equity |
| A | B | C | Data type | 1st | 2nd | 3rd | Equity |
| 25 | 87 | 88 | ICM | 0.125 | 0.1944 | 0.6806 | $25.69 |
| 25 | 87 | 88 | FEM | 0.125 | 0.1584 | 0.7166 | $25.33 |
| 25 | 87 | 88 | |ICM-FEM| | 0 | 0.0360 | 0.0360 | $0.36 |
| 25 | 87 | 88 | ⁠|ICM-FEM|/FEM⁠ | 0% | 22.73% | 5.02% | 1.42% |
| 21 | 89 | 90 | ICM | 0.105 | 0.1701 | 0.7249 | $24.85 |
| 21 | 89 | 90 | FEM | 0.105 | 0.1346 | 0.7604 | $24.50 |
| 21 | 89 | 90 | |ICM-FEM| | 0 | 0.0355 | 0.0355 | $0.35 |
| 21 | 89 | 90 | ⁠|ICM-FEM|/FEM⁠ | 0% | 26.37% | 4.67% | 1.43% |
| 198 | 1 | 1 | ICM | 0.99 | 0.009950 | 0.000050 | $49.80 |
| 198 | 1 | 1 | FEM | 0.99 | 0.009999 | 0.000001 | $49.80 |
| 198 | 1 | 1 | |ICM-FEM| | 0 | 0.000049 | 0.000049 | $0 |
| 198 | 1 | 1 | ⁠|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.