ICM in one worked example
Tourney Manager's finish-distribution report kept showing people a tall 4th-place bar. The explanation was usually the same: they were playing the bubble as if chips had a fixed cash value. The Independent Chip Model (ICM) is the standard correction, and it is easier to understand from one worked example than from a formula.
The situation
Four players left in a 9-max $10 SNG. Prize pool $90: $45 / $27 / $18. Stacks:
| Player | Chips | Share of chips | "Chip-EV" value (share × $90) |
|---|---|---|---|
| A | 5,000 | 50.0% | $45.00 |
| B | 3,000 | 30.0% | $27.00 |
| C | 1,500 | 15.0% | $13.50 |
| D | 500 | 5.0% | $4.50 |
The chip-EV column is wrong, and obviously so: A cannot win $45 with certainty, and D with 500 chips has better than a 5% chance of sneaking into third when B and C collide.
The model
ICM assumes the probability of finishing 1st equals your share of chips. Given who finished 1st, the probability of finishing 2nd is your share of the remaining chips, and so on. Your equity is the sum over every finishing order of (probability of that order × prize for your place).
For D to finish 3rd, for example, one path is A 1st, B 2nd, D 3rd:
P = 5000/10000 × 3000/5000 × 500/2000 = 0.5 × 0.6 × 0.25 = 0.075
There are six such paths to 3rd for D, three to 2nd, one to 1st. Summing them all (the script has a ten-line recursive version):
| Player | Chips | Chip-EV | ICM equity | Difference |
|---|---|---|---|---|
| A | 5,000 | $45.00 | $34.00 | –$11.00 |
| B | 3,000 | $27.00 | $28.16 | +$1.16 |
| C | 1,500 | $13.50 | $20.30 | +$6.80 |
| D | 500 | $4.50 | $7.54 | +$3.04 |
The chip leader's chips are worth 24% less than their face value; the short stack's are worth 68% more. That asymmetry is the whole content of ICM.
What it does to a decision
D moves all in for 500. A, on the button with 5,000, considers calling. Ignore the blinds and suppose it is a pure flip (50/50) for simplicity.
| Outcome | A's stack after | A's ICM equity |
|---|---|---|
| Fold | 5,000 | $34.00 |
| Call and win (D busts) | 5,500 | $35.84 |
| Call and lose (D doubles) | 4,500 | $32.08 |
Calling: 0.5 × $35.84 + 0.5 × $32.08 = $33.96. Folding: $34.00. A coin flip that would be neutral in chips is a small loser in money — and it is a loser because busting D is worth only +$1.84 to A while doubling D up costs A $1.92. The gain from the knockout goes mostly to B and C, who did nothing: their equity rises to $29.88 and $24.28 when D busts.
Make the same call a 55/45 favourite and it becomes marginally profitable ($34.15). Make it 60/40 and it is clear. That is the practical rule ICM gives you: on the bubble, the chip leader needs a real edge, not a flip, to call an all-in — and the short stack, whose equity gain on a double is large ($7.54 → $12.79), should be shoving wide precisely because the big stacks are correct to fold.
Limits
- ICM knows nothing about blinds, position or who is about to be forced all in. Tools like ICMIZER and HoldemResources Calculator add a blind structure and solve push/fold ranges under it; use those for actual ranges.
- The "share of chips = probability of 1st" assumption overrates big stacks slightly in real play and underrates skill edges entirely.
- It is a bubble and final-table tool. Early in a tournament chip-EV is close enough.
Further reading
- Malmuth–Harville method: the original chip-value model, described in Mason Malmuth's Gambling Theory and Other Topics; the racing analogue is D. A. Harville, "Assigning probabilities to the outcomes of multi-entry competitions", Journal of the American Statistical Association 68 (1973).
- Wikipedia, Independent Chip Model.