In
poker
Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game w ...
, the Independent Chip Model (ICM), also known as the Malmuth–Harville method, is a
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
that approximates a player's overall
equity in an incomplete
tournament
A tournament is a competition involving at least three competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses:
# One or more competitions held at a single venue and concentr ...
. David Harville first developed the model in a 1973 paper on
horse racing; in 1987,
Mason Malmuth
Mason Malmuth is an American poker player, and author of books on both poker and gambling. He is the owner of Two Plus Two Publishing, which publishes books and runs an online gambling discussion forum.
Malmuth was born in 1951 and grew up in C ...
independently rediscovered it for poker. In the ICM, all players have comparable
skill
A skill is the learned ability to act with determined results with good execution often within a given amount of time, energy, or both. Skills can often be divided into domain-general and domain-specific skills. For example, in the domain of w ...
, so that current
stack sizes entirely determine the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
for a player's final ranking. The model then approximates this probability distribution and computes
expected prize money.
Poker players often use the term ICM to mean a
simulator
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the ...
that helps a player strategize a tournament. An ICM can be applied to answer specific questions, such as:
* The range of hands that a player can move
all in with, considering the play so far
* The range of hands that a player can call another player's
all in with or move all in over the top; and which course of action is optimal, considering the remaining opponent stacks
* When discussing a deal, how much money each player should get
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,
which may not always be available. The Malmuth-Harville model also gives poor estimates for unlikely events, and is
computationally intractable
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...
with many players.
Model
The original ICM model operates as follows:
* Every player's chance of finishing 1
st is proportional to the player's chip count.
* Otherwise, if player finishes 1
st, then player finishes 2
nd with probability
* Likewise, if players , ..., finish (respectively) 1
st, ...,
st, then player finishes j
th with probability
* The
joint distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
of the players' final rankings is then the product of these
conditional probabilities
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...
.
* The expected payout is the payoff-weighted sum of these joint probabilities across all possible rankings of the players.
For example, suppose players A, B, and C have (respectively) 50%, 30%, and 20% of the tournament chips. The 1
st-place payout is 70 units and the 2
nd-place payout 30 units. Then
where the percentages describe a player's expected payout relative to their current stack.
Comparison to gambler's ruin
Because the ICM ignores player skill, the classical
gambler's ruin
The gambler's ruin is a concept in statistics. It is most commonly expressed as follows: A gambler playing a game with negative expected value will eventually go broke, regardless of their betting system.
The concept was initially stated: A pers ...
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.
With 3 or more players, they give misleading probabilities, but adequately approximate the expected payout.

For example, suppose very few players (e.g. 3 or 4). In this case, the
finite-element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...
(FEM) suffices to solve the gambler's ruin exactly.
Extremal cases are as follows:
The 25/87/88 game state gives the largest
absolute difference
The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for ...
between an ICM and FEM probability (0.0360) and the largest tournament equity difference ($0.36). However, the
relative equity difference is small: only 1.42%. 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.
Results in the 4-player case are analogous.
References
Further reading
* Harrington discusses the ICM on pages 108-122.
*
*
{{Poker tools
Poker gameplay and terminology