What Is The Independent Chip Model (ICM) In Poker?

The more I play poker, the more I realize that the poker strategy boils down to one thing: EV (expected value). Simply put, whenever I am considering a decision, be it to shove or to fold, I need to calculate the expected value of either action and to carry out the action that yields the highest EV for me possible.
However, while this calculation may be straightforward in a cash game, things are a bit different when it comes to tournaments. This is why players, including myself, use the ICM, which is short for the Independent Chip Model, when we are trying to calculate the expected value of our actions.

What Is the Independent Chip Model (ICM)?

Simply put, the Independent Chip Model (ICM) is a mathematical model that uses the list of the tournament payouts along with the current chip stack sizes of each player still playing in the tournament to calculate a dollar value for each player based on their probability of winning the tournament.

I know. I know. This definition seems obscure at best and downright nonsensical at worst. I remember the first time someone told me what the Independent Chip Model was; I had no idea what they were talking about. It wasn’t until they sat down and explained it to me and how it worked that things clicked. Their explanation went something like this:

Cash Games: Expected Value vs ICM

When you are playing a cash game, how do you calculate the EV? “Simple,” I responded. “You consider each chip/unit as a dollar. Afterward, whenever you find yourself facing a decision, you calculate the odds of the different outcomes as well as the payouts associated with these outcomes.

For example, let’s say I hold in my hand the King-Queen of hearts, and the flop came ten of hearts, jack of hearts, and three of spades. Subsequently, the turn came the seven of diamonds. Now, the pot is 1000 dollars, and my opponent puts me all in by betting an extra 500 dollars. There is no one else but me in the pot, and poker rules say I have one of two options: I can either call the raise or fold. The question is what should I do?

Seeing as I am on two different draws, a straight draw and a flush draw, there are 15 different outs that can make my hand. With 15 outs, the probability of me winning the hand is around 30 percent.

Bearing this in mind, I know that if I fold, my EV is equal to 0 because I forfeit the hand and get nothing. However, if I call, I will win 1500 dollars, the 1000 dollars already in the pot plus my opponent’s 500, three out of every ten times. On the other hands, I stand to lose my 500 dollars the seven other times. So, the total EV for calling can be calculated as follows:
1500 x 0.3 – 500 x 0.7 = 100.

In other words, should I be faced with this same decision over and over again, I can expect on average to make 100 dollars each time I call, and since 100 bucks is obviously larger than nothing, which is what I would have received had I folded, I ought to make the call every time.”

Poker Tournaments: ICM vs Expected Value

That is all good and well, but you can’t apply that same logic when playing in Texas Hold’em poker tournaments: Think of it this way, when you are calculating the expected value of your decision, you assumed that each poker chip represented a dollar, which it did because you were playing a cash game. However, when playing a tournament, each chip does not represent a dollar, so you can’t calculate the EV in the same way.

I was still unconvinced, so my very generous mentor went on to give me a simple example:
Let’s say you’re playing a tournament with three players, and the buy-in is around 100 dollars. Now, for your 100 dollars, you will receive 1000 chips to play the tournament. Let’s also assume that the payout structure goes like this:
First place gets 185 dollars.
Second place gets 115 dollars.
And, third place gets to go home empty-handed.

Now, a few minutes into the game, player 2 decides to go all in, and player 3 decides to call. The flop, turn, and river are played, and player 3 ends up with the win and an extra 1000 chips to add to their stack. So, this is the million dollar question: Is player 3’s 2000 chip stack worth 200 dollars?

Even though, in a cash game, the answer would have been an easy yes, in a tournament, the answer is a resounding no. As a matter of fact, if you remember that the payout structure dictated that first prize got 185 dollars, you should realize that no matter how many chips player 3 accumulated, their equity, i.e. how much their chips are worth, can never exceed 185 dollars, the top prize.

To make matters clearer, given that you are still playing, that you have 1000 chips in front of you, and that you are sure to at least win second place along with the prize money of 115 dollars, your equity can never go below 115 dollars or 38 percent, which is the prize money of second place, 115 dollars, divided by the total prize pool, the 300 dollars contributed by you and the two other players.

As I mulled over this example, I asked, “So, what you’re saying is that my 1000 chips are worth 115 dollars because I am currently in second place whereas my opponent’s 2000 chips are worth 185 dollars or around 62 percent of the total prize pool because they are in first place?”
“Not so fast,” was the response.

Despite the fact that your opponent has more chips than you, they still don’t have a lock on this thing; after all, they aren’t certain to win the tournament. The only thing their larger stack buys them is a higher chance of winning the tournament. In other words, each player’s chip stack size is used to calculate the probability of them winning the tournament, and by using this probability and multiplying it by the different payouts, we can finally arrive at a more accurate representation of their equity.

Going back to our example, your opponent’s equity must be less than 185 dollars, whereas your equity must be more than 115. As a matter of fact, if you use an Independent Chip Model (ICM) calculator, you will find that their equity is around 162 dollars and your equity is approximately 138 dollars.”

I began to understand a bit at this point.
“So, going back to your example, because I have one-third of the chips, I have 33 percent to win first place and 66 percent to win second?”
“Exactly.”
“And, according to these probabilities, my equity can be calculated as follows:
.33 (probability of winning first place) x 185(first place money) + .66 (probability of winning second place) x 115 (second prize money) = 137 dollars”

“And, as you can see the Independent Chip Model only cares about the chip stack size and the payout structure. What’s also interesting to note is that the value of a chip is always changing according to the Independent Chip Model (ICM) in poker tournaments: Chips are only valuable in Texas Hold’em poker tournaments in so far as they increase your chances of winning said tournaments.”

“But how does any of that help me when I’m playing in an actual tournament?”

Understanding the Independent Chip Model (ICM)

Understanding the Independent Chip Model (ICM) can be integral to your poker strategy, especially if you plan to play Texas Hold’em poker tournaments. Just as your calculations of EV are integral to your decisions in a cash game, your calculations of EV during a tournament are instrumental to becoming a better player. The difference is that in order to calculate the EV of any decision during a tournament, you need to rely on a model like the ICM, especially if you are planning to cash.

For instance, let’s go back to the example you gave earlier during the cash game, and let’s imagine that that exact hand has played out during our imaginary tournament, the one where you had 1000 chips, your opponent had 2000 chips, and the payout was 185 for first and 115 for second.

So, by putting the two examples together the final result is as follows:
You hold the King-Queen of hearts, and the cards on the felt are the ten of hearts, the jack of hearts, the three of spades, and the seven of diamonds. Also, there are 1000 chips in the pot, and your opponent just raised you 500 chips, putting you all in. What do you do?

If you remember, because you were on two draws, you had a thirty percent chance to win the hand. If you win the hand, you’ll have 2000 chips, whereas your opponent will only have a thousand left. Accordingly, your equity will be 162 dollars by ICM poker calculations. On the other hand, if you call and lose, your equity will be 115 dollars, which is the payout you will receive for coming in second place.

Having calculated the equities resulting from both possible outcomes from calling, your EV for calling can be calculated by multiplying the equity of each scenario by the probability of that scenario happening, and adding the results together:
162 (your equity according to ICM poker if you win the hand) x 0.3 (the probability of winning the hand if you call) + 115 (your equity if you lose the hand) x 0.7 (the probability of losing the hand) = 129.1 dollars.

The alternative is folding, in which case, you will be left with 500 chips and a total equity of (500/3000) x 185+ (2500/3000) x 115 = 126.6 dollars.
Seeing as the EV of calling, 129.1 dollars, is higher than that of folding, 126.6 dollars, you should call that shove.

Now, even though conventional calculations and ICM poker calculations yielded the same result, i.e. a decision to call the shove, this won’t always be the case, and, in many scenarios, ICM poker will dictate that you should play tighter than you normally would during a cash game.

See you at the WSOP!
♠ pokerjournal.org

Independent Chip Model – FAQ