The Malmuth-Harville formula behind ICM

Finishing probabilities from chip stacks, then prizes: the whole model in four numbers per player.

Updated 2026-10-23 · exact ICM math (Malmuth-Harville)

Every ICM calculator, including ours, rests on a single idea. It is short enough to do by hand for three or four players, and it explains why ICM behaves the way it does.

The rule

The chance that a player finishes first is his share of all chips. Remove that winner and repeat the same rule for the players who are left: the chance of finishing second is the chance that someone else won first, times your share of the chips that remain. And so on for third place. This is the Harville model for ranking probabilities (it comes from horse racing) and it was brought into poker tournament analysis by Malmuth, hence the name.

For a player with s chips among T in total, the chance of first place is s / T. The chance of second place is the sum over every other player j of (s_j / T) · (s / (T − s_j)).

Worked example: the chance of second place

Stacks 4000, 3000, 2000 and 1000, so T = 10000. What is the chance that the 3000 stack finishes second? Someone else must win, and then he must beat the rest:

• The 4000 stack wins (40.0%), then 3000 takes the next place among the remaining 6000 chips: 40.0% × 50.0% = 20.0%
• The 2000 stack wins (20.0%), then 3000 takes the next place among the remaining 8000 chips: 20.0% × 37.5% = 7.5%
• The 1000 stack wins (10.0%), then 3000 takes the next place among the remaining 9000 chips: 10.0% × 33.3% = 3.3%
• Sum: 30.8%

The full table

Doing this for every player and every place gives the finishing probabilities:

From places to prizes

Multiply each probability by its prize and add. With payouts of 50/30/20/0 percent the 3000 stack gets 0.5 × 30.0% + 0.3 × 30.8% + 0.2 × 26.2% = 29.5% of the prize pool. The whole result:

The equity values add up to 100% of the prize pool. The largest stack has less than its share of chips (ratio 0.84) and the smallest has much more (1.33). That is the whole reason why risk matters: the curve from chips to equity bends.

Properties worth remembering

• More chips always means more equity, but each extra chip adds less than the one before.
• Winner takes all is the one case where equity is exactly proportional to chips.
• The model has no memory of how a stack was built, only of its size, and no idea of skill, future blinds or pay jumps ahead.
• The number of orderings grows as n! (362,880 for nine players), which is why exact ICM is instant for a final table but needs approximations for a hundred players.

Put your own stacks into the ICM calculator to see all of this for any payout structure, and read when ICM misleads for what the model leaves out. To practise decisions with these numbers, use the ICM trainer.

Known limitation of ICM. ICM models predict tournament results more accurately than a simple baseline model, but in one large study of tournament results they underestimate the results of players with larger stacks and overestimate those who are short-stacked (Kim 2025, arXiv 2506.00180).

Keep exploring

• Check the bubble factor of any all-in
• How much extra equity ICM asks for: the risk premium
• Browse the push/fold charts for 3 to 15 BB
• What the solution book knows and where it still differs

ICM by payout structure

• ICM at a nine-handed final table
• ICM with a flat payout structure
• ICM with a top-heavy payout structure
• ICM on the bubble: four players, three paid
• ICM three-handed with 50/30/20 payouts
• ICM heads-up with a 70/30 split
• ICM in winner-takes-all
• ICM in a satellite: seven players, three seats

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