Independent Chip Model
The independent chip model or ICM is a mathematical model for play in the late phase of a tournament with relatively small stacks. It is based on the idea that every chip represents a chance to land in the money and thereby to make a profit. Each additional chip improves a player's position in the tournament and betters his resulting profit. The chips in a stack of 10000 are simply worth more than those in a stack of 100, which will be all-in by default if a blind of that size must be paid.
In a game for real money, every action has an EV that says what kind of gain or loss can be expected to result from the action. In a tournament, this translates to how many chips one can expect to gain or lose.
But in a tournament the chips are just a means to an end, i.e. they are just capital used to win money. For this reason, the EV of an action must be modified to consider how much actual money will be made as a result of the action, and this is called $EV.
An action with positive EV could have a negative $EV, especially in cases where an action would put a player all-in. Losing an all-in means elimination from the tournament and a loss of the buy-in. Hence, the ICM is useful in the push-or-fold phase of a tournament, where a player must decide whether to push or call all-in or to fold.
The $EV depends on many factors:
- the size of your own stack
- the type and strength of your opponents
- your own gaming skills
- your position
- the distribution of stacks at the table
- your table image
Using these as starting points, we can make some assumptions about the probability with which one will obtain an arbitrary placement in the tournament. These probabilities and the payoff structure allow us to compute the $EV.
$EV = P(1st place) * Payoff(1st place) + P(2nd place) * Payoff(2nd place) + P(3rd place) * Payoff(3rd place) + ...
The payoffs for 1st, 2nd, and 3rd places in an SNG are 50, 30, and 20 dollars, respectively. Suppose a player knows that his probabilities of obtaining 1st, 2nd, and 3rd places are 10, 15, and 20 percent respectively. Then:
$EV = 10% * 50$ + 15% * 30$ + 20% * 20$ = 13.5$
Using the ICM
In a given situation, then, one must calculate for each possible action the $EV and compare the results. If the possible actions are all-in or fold, then one must calculate the $EV for the cases where:
- you fold
- you go all-in, are called and lose
- you go all-in, are called and win
- you go all-in and are not called
The probability that you will win or lose against a particular opponent must be calculated by estimating the all-in calling range of that opponent. The sum of every possible outcome of an action weighted by its probability will tell you what the $EV of the action is, and this will tell you whether to fold or stay.
These calculations are too complex to make during play. They are meant to provide a theoretical basis for play, and in particular to comment on with which hand ranges one can go or call all-in. There are also programs that will perform ICM calculations for you.
Expected Value, Equity, Turnier, Range