ICM, worked by hand
What Is ICM in Poker? A Three-Player Example You Can Check by Hand
ICM, the Independent Chip Model, also called Malmuth-Harville, turns tournament chips into prize money. It works out each player's chance of finishing in every paid place from the chip counts, multiplies each chance by that place's prize, and adds them up. The total is the stack's ICM equity.
Three players are left and you have half the chips. First pays $5,000, second $3,000, third $2,000. Half of that $10,000 pool is $5,000, but ICM says your stack is worth $3,839. With three players the whole calculation fits on a page.
The ICM formula (Malmuth-Harville)
- Your chance of finishing first is your chips divided by all the chips.
- Once some players have been placed, your chance of taking the next place is your chips divided by the chips of everyone not yet placed.
- The chance of one whole finishing order is those fractions multiplied together. For A, then B, then C: A's chips over all the chips, times B's chips over everything but A's.
- Your equity is your chance of each place times that place's prize, added up. A place that pays nothing counts as $0.
The whole model rests on the first line: hold half the chips and you win half the time. It knows nothing about cards, skill, position or the blinds. The best player at the table and the worst get the same price for the same stack.
How to calculate ICM by hand: three players
A has 5,000 chips, B 3,000 and C 2,000, so 10,000 in all. The prize pool is $10,000: $5,000 for first, $3,000 for second and $2,000 for third.
Step 1: the chance of winning
A wins half the time, 5,000/10,000 = 50.0%. B wins 3,000/10,000 = 30.0% and C 2,000/10,000 = 20.0%.
Step 2: every finishing order
Three players can finish in six orders. Each order's chance is the winner's share of all the chips, times the runner-up's share of what's left after the winner is gone. The last player is whoever remains. The working is in thousands of chips, so 5/10 is 5,000 of 10,000.
| Order | Working | Chance |
|---|---|---|
| A B C | 5/10 × 3/5 | 30.00% |
| A C B | 5/10 × 2/5 | 20.00% |
| B A C | 3/10 × 5/7 | 21.43% |
| B C A | 3/10 × 2/7 | 8.57% |
| C A B | 2/10 × 5/8 | 12.50% |
| C B A | 2/10 × 3/8 | 7.50% |
| Total | 100.00% |
Take B A C, the third row. B wins with probability 3,000/10,000. With B gone, A and C hold 7,000 chips, and A's share of those is 5,000/7,000. Multiply the two and you get 21.43%. Check that the six add up to 100%. If they don't, one of your fractions is wrong.
Step 3: each player's chance of each place
Add up the orders that put each player in each place. A finishes second in B A C and C A B, so A's chance of second is 21.43% + 12.50% = 33.93%. Every row and every column of this table adds up to 100%.
| Player | 1st | 2nd | 3rd |
|---|---|---|---|
| A (5,000) | 50.00% | 33.93% | 16.07% |
| B (3,000) | 30.00% | 37.50% | 32.50% |
| C (2,000) | 20.00% | 28.57% | 51.43% |
Step 4: multiply by the prizes
A's money is each prize times A's chance of finishing there: 50.00% × $5,000 + 33.93% × $3,000 + 16.07% × $2,000 = $3,839.29. Do B and C the same way and the three stacks add up to $10,000.00, the whole pool.
| Player | Chip share | ICM equity | Money share |
|---|---|---|---|
| A (5,000) | 50.0% | $3,839.29 | 38.4% |
| B (3,000) | 30.0% | $3,275.00 | 32.8% |
| C (2,000) | 20.0% | $2,885.71 | 28.9% |
| Total | 100% | $10,000.00 | 100% |
To check these on the site, open this spot with the stacks and prizes filled in.
Why the big stack is worth less than its chips
A has 50.0% of the chips and 38.4% of the money. C has 20.0% of the chips and 28.9% of the money. A can never win more than $5,000, however many chips A piles up. C can never get less than $2,000, because third place is paid.
Change the payouts to winner-take-all and the gap disappears. With all $10,000 going to first, A's stack is worth $5,000, exactly its share of the chips. The chip leader article gives the gap for every chip share from 9 players left to heads-up.
Four players
A fourth player turns 6 orders into 24. Say the stacks are 5,000, 2,500, 1,500, 1,000 and the prizes are $4,000, $3,000, $2,000, $1,000. The first order, A B C D, has a chance of 5,000/10,000 × 2,500/5,000 × 1,500/2,500 = 15.00%. Work out the other 23 the same way and A comes out at 50.00%, 31.05%, 14.83%, 4.12% for first to fourth, so A's equity is 50.00% × $4,000 + 31.05% × $3,000 + 14.83% × $2,000 + 4.12% × $1,000 = $3,269.23. You can also open these stacks.
| Player | Chip share | ICM equity | Money share |
|---|---|---|---|
| A (5,000) | 50.0% | $3,269.23 | 32.7% |
| B (2,500) | 25.0% | $2,672.62 | 26.7% |
| C (1,500) | 15.0% | $2,205.77 | 22.1% |
| D (1,000) | 10.0% | $1,852.38 | 18.5% |
| Total | 100% | $10,000.00 | 100% |
When a place pays nothing
Pay only the top three, $5,000, $3,000, $2,000, and fourth counts as $0, so you only need each player's chances down to third. D's 1,000 chips are still worth $1,453.78, because D finishes in the money whenever someone else busts first. This is the bubble: four players, three paid. Open it and compare the stacks.
Why exact ICM gets slow past a final table
The number of finishing orders is the number of players multiplied down: 3 × 2 × 1 = 6 for three players, 362,880 for a nine-handed final table, and 479,001,600 for twelve. With twelve players there are only 4,095 groups of players who could still be left, and the ICM calculator on this site works through those instead of the orders. The answer is the same to the cent.
Where ICM comes from: Harville, Malmuth and Plackett-Luce
The rule is R. Duncan Luce's choice model from probability theory, which David Harville used in 1973 to price every finishing order in a horse race from the win odds. Statisticians call it the Plackett-Luce model. Mason Malmuth brought it to poker in a 1986 article in Poker Player Newspaper, collected in his 1987 book Gambling Theory and Other Topics, which is why poker players call it Malmuth-Harville.
Malmuth's book also has a second method, credited to Mark Weitzman, that works up from the bottom instead: each player's chance of busting next is in inverse proportion to their stack. Some study tools still offer it. Malmuth-Harville is the one everyone means by ICM.
What ICM gets wrong
It prices the stacks as if the next hand didn't matter. In the four-player example, D's 1,000 chips are worth $1,852.38 whether D is on the button or about to post a 500-chip big blind that takes half the stack. Study tools patch part of that with future game simulation, which plays the next hand out before applying ICM.
A 2025 study built from more than ten thousand WSOP.com and PokerNews events found that ICM underrates big stacks and overrates short ones, so real chip leaders do better than the model says. Persi Diaconis and Stewart Ethier modelled three players trading chips on fair coin flips and found ICM roughly right for likely finishing orders and badly off for unlikely ones. Players and casinos still typically price final-table deals with it, because nobody at the table has to agree on who the better player is.
What to do with it
Near a pay jump, the chips you could lose are worth more than the chips you could win, so calling an all-in takes more equity than the pot odds alone say. At the 2019 WSOP Main Event final table, six-handed, Alex Livingston folded pocket queens to a three-bet and said ICM was why. Garry Gates, who made the three-bet, had pocket tens, so Livingston folded the best hand. His 50 million chips were worth about $3.97 million by ICM, with $1.85 million already locked up for sixth, so getting it in and losing would have cost him about $2.12 million.
The bubble factor calculator measures that pressure, and the bubble factor chart turns it into the equity a call needs. Part ten of the lessons plays through these spots hand by hand. The MTT strategy guide runs ICM on the 2025 Main Event final table.