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# The Independent Chip Model

The Independent Chip Model (or ICM for short) is a recursive mathematical equation which, assuming equal skill (and equal luck) for every player, predicts the equity for a player in a tournament given their total number of chips, and the number of chips of every opponent of theirs. Unlike cash games, where every chip has a specific value (the red chip is worth $5, for instance), in a tournament the value of a chip will vary based on the specific stacks of all the players. This is because in a tournament, you can lose all of your chips, and still win money, or you could win all the chips, but you won't win the entire prize pool (unless, of course, you are playing in a winner take all tournament).

## What is meant by "recursive mathematical equation"?

A "recursive mathematical equation" is any equation that is run over and over again, until a "base case" is reached. An example of this is simple multiplication. If you want to evaluate x*y, you could look at it as (x+(x*(y-1))) - you would repeat this process until y=1. For example, 2*5 would be expressed as (2+(2*(5-1))), or (2+(2*4)) - which can be expressed as (2+(2+(2*3))) - you finally end up with (2+2+2+2+2), or 10. While the math involved in ICM is a bit more complicated (hence the need for a calculator), the same basic principle applies.

## So what is the actual math?

First you need to find the odds of any given player finishing in 1st place (determined by player chips/total chips in play), and then multiply this result by the first place prize money. Then, for each player's first place finish, you need to find the odds of every other player finishing in 2nd place. This is determined by removing the first place finisher's chips from the total chip count, and using the formula of chips/total chips. However, you have to multiply this by the odds of your first place finisher taking first. You continue with this process until all in-the-money finishes have been accounted for for each player (our base case is a finishing-position-payout of 0).

## Wait...I'm confused!

Let's start with a very basic example. You are in the end game of a 6-player SnG. There are 3 players left, and the top two get paid, $100/50. You have 3000 chips, villain 1 has 2500 chips, and villain 2 has 3500 chips. We first start with odds of finishing in first place - your odds are 1/3, villain 1's odds are 5/18, and villain 2's odds are 7/18. Multiplying these odds by the first place payout, our initial (first place only!) expected payouts are as follows:

Player | Odds, First Place | Expected Payout |

You | 1/3 | $33.33333 |

Villain 1 | 5/18 | $27.77778 |

Villain 2 | 7/18 | $38.88889 |

Now we need to determine the expected 2nd place payouts. Let's start with the assumption that you will take first place (which will happen 33% of the time). This leaves Villain 1 with 5/12 of the chips, and Villain 2 with 7/12 of the chips. Multiply these odds by 1/3 (we have to remember the odds of every place preceeding us), and we get odds of 5/36 and 7/36, respectively. Multiply these odds by the $50 2nd place payout, and add them to our running totals:

Player | Odds, 2nd Place (You take first) | Expected Payout |

You | N/A | $33.33333 |

Villain 1 | 5/36 | $34.72222 |

Villain 2 | 7/36 | $48.61111 |

Now we move on to the 2nd place odds, with Villain 1 taking first. This leaves you with 6/13 of the chips, and Villain 2 with 7/13. Multiply these by 5/18, and our odds are 30/234 and 35/234. Multiply by $50, and add to our running totals to get:

Player | Odds, 2nd Place (Villain 1 take first) | Expected Payout |

You | 30/234 | $39.74359 |

Villain 1 | N/A | $34.72222 |

Villain 2 | 35/234 | $56.08974 |

Finally, we move on to 2nd place odds, with Villain 2 taking first. This leaves you with 6/11 of the chips, and Villain 1 with 5/11 of the chips. Multiply these by 7/18, and our odds are 42/198 and 35/198. Multiply by $50 and add, and our final running totals are:

Player | Expected Payout |

You | $50.34965 |

Villain 1 | $43.56061 |

Villain 2 | $56.08974 |

## That's a lot of math - Is is really worth it?

Well, if you are using the ICM simply to find out what you should expect to make at a given point in a small tournament or SnG, it's not worth it. Used alone, it doesn't do anything more than give you an idea of where you stand (which you should know already). However, it is a good tool for reviewing hands you've played in a tournament, if you would like to know if it was really worth it to take a gamble or two. For instance, using the ICM will prove it is unprofitable to shove with AKs against 22 on the first hand of a tournament - our expected value actually goes down in a move like this.

It can also help us with bubble hand analysis. Let's take our example chip stacks, and assume blinds are 100/200. We have the button (Villain 1 is SB, Villain 2 is BB), and raise to 600 with AQs. Villain 1 folds, but Villain 2 goes over the top all in. Should we call? Let's start off by filling in the base case in our tool. We will set our base case to be the chip stacks **assuming we fold** - 2400/2400/4200. We will then set scenario A and B to the various chip stacks that could occur if we were to win or lose the hand. If we win the hand, stacks will be 6100/2400/500; if we lose it will be 0/2400/6600. Playing around with the various likelihood's of winning, we find that a call is a good move only when we figure to be a 52.61% or better to win the hand - so if we figure villain has so much as 22, we should fold our hand. However, if we believe we have villain dominated (or if he's just shoving with two live cards, like 89s), we should call.

Continuing with this example, it can teach us how NOT to over-commit ourselves to a hand with a preflop raise. In our previous example, let's say instead of raising to 600, we raised to 1000 instead preflop, before villain raised. Now, we are committed to calling if we believe a pocket pair under QQ is included in villain's range - we are put in a position where it's "profitable" to call off all of our chips as an underdog, which is never a position we want to be in.

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