I'm certain many people have read through the online forums on occasion, and have seen a glimpse of something referred to as ICM. Upon reading further into a jumble of decimal numbers, people typically end up doing one of two things: scroll to the bottom to find the actual decision decided through ICM in plain English, or give up altogether, and assume that since there was so much heavy math involved, it wasn't of too much use at the table.
ICM is short for Independent Chip Modeling, and while on the surface looks at best complicated, in actual practice is quite simple as long as you understand pot odds, and understanding the math behind it will improve your Sit-n-Go Game.
The fundamental problem with using your chips to calculate pot odds in a tournament is that tournament chips have no fixed value. In addition, different chips have different values to them. That is to say, if you were to compare the value of a person's last remaining chip, or the 10,000th chip in a stack of 10,000, the last chip has much more value than the 10,000th chip, because once you lose that last chip, you are out of the tournament, while when you lose the 10,000th chip, you still have 9,999 chips remaining. This concept is similar to the idea of diminishing returns to capital, for those economists out there. Because of this, calculating pot odds based on chips does not fully capture the amount of money won or lost on average based on a decision, which is really what we're doing when we're calculating pot odds.
The ICM has foundations from several different sources, from usenet groups to Mason Malmuth and others. The theory behind the ICM can be described as complex at best. The main assumption is that your probability of winning is based on the number of chips you have compared to the number of chips in the tournament. Then, you calculate your probability of finishing in second, which is based on how many chips you have compared to the number of chips in the tournament after the winner's chips have been removed from the tournament. This iterates down for the number of places that pay out. You then calculate how much you win by multiplying your probability of finishing in a certain place, and you'll arrive at a decimal, which represents your expected share of the prize pool, commonly referred to as your expected value.
As expected, the math above can end up being quite complex, and the above description is still simplified. For that reason, several calculators have been written to do the work instead of having to calculate it on your own, including the ICM Calculator hosted at this very site. If you want to actually calculate it out, there are pages out there that have the mathematical formula for ICM as well, but the number of computations that are made by hand just aren't worth the hassle, in my opinion.
Now that you have the calculator, let's try a situation. Suppose you have three players left, the button has 2000, and you and the small blind have 4000 chips. Payouts are a standard Sit-n-Go ten player structure of 50% for the winner, 30% for 2nd place, and 20% for 3rd place. The blinds are 150/300, and you're on the BB. You look down, and you get QQ, and fireworks go off, you have a monster of a hand. The small blind pushes all-in, and we'll make the assumption that you also know from playing with him that he'll only make that kind of bet with AK. What do you do?
So, you have to calculate your ICM expected value (EV) for three situations: if you were to fold, if you were to call and win, and if you were to call and lose. We will ignore splitting the pot in this situation since it will occur less than one percent of the time. I'll address splitting the pot in the next article.
We will start with the first situation. If you were to fold, the button would still have 2000 chips, you would have 3700 chips, and the small blind would have 4300 chips. When you enter these numbers into the calculator, it will output the ICM value of 0.3482. If you were playing a $10+1 buy-in tournament, you'd have an expected value of $100*0.3482, or $34.82, with that chip stack against those other two chip stacks, specifically. In other words, in the long run, you'd expect to win $34.82 on average if you were put in this exact situation every time. Not bad, eh?
Now, let's calculate the second and third situations. If you call and lose, the math is very simple; you would end up with 0 chips and 3rd place, or an ICM expected value of 0.2000. When you multiply this number by the prize pool, you end up with $100*0.2, or $20 in the long run, which is the payout for 3rd place. If you call and win, you will have 8000 chips, and the button will have 2000 chips. When you punch this value into the ICM calculator, the calculator outputs a value of 0.4600.
What's the expected value of calling? QQ will win 56% of the time versus AK. We can figure these percentages out using the help of an odds calculator, such as ITH's Poker Odds Calculator. So, 56% of the time, we'll have an ICM of 0.4600, and 44% of the time, we'll have an ICM of 0.2000. So if we average this out, we have (0.56*0.46)+(0.44*0.2) = 0.3457.
This gives us a EV of calling at 0.3457, and an EV of folding of 0.3482. So, the more optimal play is to fold your QQ. If we wanted to quantify this, at a $10+1 tournament, the difference between the two decisions on average is $100*(0.3482 – 0.3457), which is equal to a quarter. A quarter doesn't sound like much, but that is an increase of your ROI by 2.3% by folding instead of calling in this situation every time.
This may sound strange, but there are reasons for this. First off, this is assuming that you know your opponent is playing AK exactly. Of course he could make this push with a wider variety of hands, and that could (and most likely would) make it a callable all-in. However, the main idea is that you do not want to be taking risks and calling off your chips (note the emphasis on calling) with three players remaining. You don't want to risk calling into races, mostly because of the small increase from third to second place. This is contradictory to the style of most players, who are thrilled that they just made the money, and are willing to take risks, loosely calling all-ins after playing tight on the bubble, in hopes of getting lucky and possibly winning first place.
This is the end of article one in a series of three articles. In the next article, we'll focus on the other side of the decision - whether to push all-in or fold. Then, in the third article, we'll evaluate the pitfalls of the ICM model, and the realistic applications it has at the table.
Continue to the second article in the series, The Independent Chip Model - Examining a Bubble Hand.