Independent Chip Model

In poker, the Independent Chip Model (ICM), also known as the Malmuth–Harville method,[1] is a mathematical model that approximates a player's overall equity in an incomplete tournament.

David Harville first developed the model in a 1973 paper on horse racing;[2] in 1987, Mason Malmuth independently rediscovered it for poker.

The model then approximates this probability distribution and computes expected prize money.

An ICM can be applied to answer specific questions, such as:[6][7] Such simulators rarely use an unmodified Malmuth-Harville model.

In addition to the payout structure, a Malmuth-Harville ICM calculator would also require the chip counts of all players as input,[8] which may not always be available.

The Malmuth-Harville model also gives poor estimates for unlikely events, and is computationally intractable with many players.

The original ICM model operates as follows: For example, suppose players A, B, and C have (respectively) 50%, 30%, and 20% of the tournament chips.

where the percentages describe a player's expected payout relative to their current stack.

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.

[9] With 3 or more players, they give misleading probabilities, but adequately approximate the expected payout.

In this case, the finite-element method (FEM) suffices to solve the gambler's ruin exactly.

[11][12] 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).

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.