What is each stack worth?
Enter the chips each player holds and what each place pays.
Only stacks and payouts count: every player is assumed to be equally skilled, and each place is drawn in proportion to chips. How this is computed
How ICM works
The model
ICM, the Independent Chip Model (Malmuth-Harville), values a stack by the prize money it should win. A player holding s of the S chips in play finishes first with chance s / S. Given who finished above, the next place is drawn the same way among the players left.
equity(i) = sum over places p of payout(p) x chance(i finishes p)
For stacks of 5,000, 3,000 and 2,000 chips and payouts of $50, $30 and $20, the chances of each finishing place are:
| Player | 1st | 2nd | 3rd |
|---|---|---|---|
| Player 1 | 50.00% | 33.93% | 16.07% |
| Player 2 | 30.00% | 37.50% | 32.50% |
| Player 3 | 20.00% | 28.57% | 51.43% |
Worked example
The prize pool is $100.00. Each player’s equity is the payout of each place times the chance of that place, added up.
| Player | Chip share | ICM equity | Gap |
|---|---|---|---|
| Player 1 (5,000) | $50.00 | $38.39 | $11.61 below |
| Player 2 (3,000) | $30.00 | $32.75 | $2.75 above |
| Player 3 (2,000) | $20.00 | $28.86 | $8.86 above |
Prize equity differs from chip share because the pool pays in steps. The big stack’s chips are worth less each, and the short stack’s chips are worth more.
The bubble factor
What it measures
BF = (E(h) - E(h - a)) / (E(h + a) - E(h))
E(x) is the hero’s prize equity at stack x, with the villain’s stack moved by the same chips, and a is the all-in size. A call risking R to win G needs R / (R + G) equity in chips, and BF x R / (BF x R + G) once prize money counts.
Player 2 (3,000 chips) calls player 1 (5,000 chips) all in for 3,000. Equity is $32.75 now, $20.00 after a loss and $41.00 after a win. The bubble factor is 1.55, so the call needs 60.71% equity instead of 50.00%.
Common mistakes
- Counting chips as money: a chip you win is worth less than a chip you lose.
- Ignoring the pay jumps: the next place up or down decides what a chip is worth.
- Using ICM for a single hand without the bubble factor: the factor turns the chip price of a call into a prize-money price.
- Entering payouts in the wrong order: first place goes first, and payouts never rise.
How this is computed
Method
The calculation is exact. Every finishing order is counted through a bitmask over the set of players already placed, which is at most 2^N states for N players (N is at most 10), with no simulation. The equities always add up to the prize pool.
Only stacks and payouts count. Every player is assumed to be equally skilled, and position, blinds and the order of action are ignored.
Structures compared
| Structure | Biggest | Smallest |
|---|---|---|
| 3 players, even stacks | 33.33% to 33.33% | 33.33% to 33.33% |
| 3 players, one chip leader | 66.67% to 42.74% | 11.11% to 25.87% |
| 3 players, one short stack | 44.44% to 37.44% | 11.11% to 25.11% |
| 4 players, stacks 4,000 down to 1,000 | 40.00% to 33.60% | 10.00% to 13.32% |
Computed exactly by the GeorgeLiPoker math library.
Book a session to work through a spot of your own.
Questions
What does ICM mean in poker?
ICM stands for the Independent Chip Model. It turns every stack into a share of the prize pool by working out each player’s chance of every finishing place from chip counts alone. With stacks of 5,000, 3,000 and 2,000 chips and payouts of $50, $30 and $20, it values the stacks at $38.39, $32.75 and $28.86.
Why is the chip leader’s equity below its chip share?
Prize money pays in steps, so each extra chip adds less than the one before it. The 5,000 stack holds 50.00% of the chips but only 38.39% of the pool, while the 2,000 stack holds 20.00% of the chips and is worth 28.86%. A chip you lose costs more than a chip you win earns.
What is a bubble factor?
The bubble factor is the prize equity you lose when you lose an all-in divided by the prize equity you gain when you win it. For the 3,000 stack calling the 5,000 stack all in it is 1.55, so the call needs 60.71% equity instead of the 50.00% that chips alone suggest.
Does ICM account for skill or position?
No. ICM uses only the stacks and the payouts. It assumes every player is equally skilled and gives each place out in proportion to chips, so it ignores position, blinds, the order of action and anyone’s edge. Use its numbers as a baseline for tournament decisions, then adjust for the opponents you face.
How many players and places can I enter?
Enter 2 to 10 players, each with a whole-number stack, and payouts for up to as many places as there are players. Payouts must stay the same or fall from first place down. A place you leave out pays nothing, so a nine-player table that pays three places needs three payouts.
Related pages
Play the bubble by the numbers
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