Independent Chip Model

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 m1, ..., m**j-1 finish (respectively) 1st, ..., (j-1)st, then player i finishes jth with probability \mathbb{P}\leftX_i=j\mid X_{m_z}=z\quad(1\leq 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. [[File:3PlayersIcmFem.png|thumb|The FEM mesh for 3 players and 4 chips.]]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.

References

References

1. Bill Chen and Jerrod Ankenman. (2006). "The Mathematics of Poker". ConJelCo LLC.
2. Harville, David. (1973). "Assigning Probabilities to the Outcomes of Multi-Entry Competitions". Journal of the American Statistical Association.
3. Malmuth, Mason. (1987). "Gambling Theory and Other Topics". Two Plus Two Publishing.
4. Fast, Erik. (20 March 2012). "Poker Strategy – Introduction To Independent Chip Model With Yevgeniy Timoshenko and David Sands". cardplayer.com.
5. (7 June 2016). "ICM Poker Introduction: What Is The Independent Chip Model?". Upswing Poker.
6. Selbrede, Steve. (27 August 2019). "Weighing Different Deal-Making Methods at a Final Table". PokerNews.
7. Card Player News Team. (28 December 2014). "Explain Poker Like I'm Five: Independent Chip Model (ICM)". cardplayer.com.
8. Walker, Greg. "What Is The Independent Chip Model?". thepokerbank.com.
9. Feller, William. (1968). "An Introduction to Probability Theory and Its Applications Volume I". John Wiley & Sons.
10. Persi Diaconis & Stewart N. Ethier. (2020–2021). "Gambler's Ruin and the ICM".
11. Either, Stewart. (2010). "The Doctrine of Chances: Probabilistic Aspects of Gambling". Springer.
12. Gorstein, Evan. (24 July 2016). "Solving and Computing the Discrete Dirichlet Problem".