# The Independent Chip Model – Examining a Bubble Hand

In the first article of this series entitled The Independent Chip Model – An Introduction, we investigated the roots of the Independent Chip Model (ICM), and we also explored an example of when ICM contradicts a decision justified by the conventional method of calculating pot odds. Now, let's expand on this idea further, and evaluate a situation with four players remaining. This stage of the game is referred to as bubble play, because in most Sit-n-Go ten player tournaments, there are three places paid out, and with four players remaining, none of them want to finish in the non-paying position.

Bubble play is also considered the most important part of the Sit-n-Go tournament for two reasons. For one reason, it's the determining factor in whether you will get rewarded for the time you've invested. However, it's also the time when the most money is awarded out. When there are three players remaining in a standard Sit-n-Go ten player tournament, each player left is guaranteed 20% of the prize pool, which means that 60% of the prize pool has already been awarded, and there's two more places to award still. Since Sit-n-Go tournaments pay out such a large portion of the prize pool at the beginning of the final three, it's even more important not to finish one out of the money.

With that being said, let's set up a theoretical bubble situation. Let's say you have 2400 in chips, the players to your left, in order, have 2200, 3600, and 1800, before paying the blinds. The blinds are 200/400, and you are the small blind. Action folds to you, and you have the monster of 98s, a suited connector, which you typically don't like to play heads up. What do you do?

First off, with 6BB, you're basically looking to either push or fold in this situation. If you make a minimum raise to 800, and the big blind comes over the top of you and pushes all-in, you are getting odds to call this. If you're getting odds to call it with the minimum raise, the question arises on why not just push all 2400 in to start, and make the big blind decide if they like their hand or not. So, we need to evaluate both decisions, and figure out which decision gives us a higher expected value, based on the ICM.

If we fold, we know what the chip stacks will be every time. You will have 2200, and to your left will be 2400, 3600, and 1800, respectively. When you punch these numbers into an ICM calculator, these chip stacks represent an ICM value of 0.2354.

Now, we evaluate all of the potential outcomes when we push. There are four outcomes that we need to investigate: the big blind folds to your push, the big blind calls and you win, the big blind calls and you lose, and the big blind calls and you tie. In most preflop situations, I do not calculate pot equity based on a tie, because it happens so rarely, that you're not losing much accuracy by not including it. Of course, this implies that we need to know how often that the big blind is going to fold to your push, and how often he's going to call. Generally, a tight player will call with around 15% of their hands and a loose player will call with around 30% of their hands. It's up to your read on the big blind what this percentage is, and it's merely an estimate, but in this situation, we'll assume they're a tight player, and will call with 15% of the hands they're dealt. I personally feel this is still a bit loose, and 10% could be much closer to correct, but 15% is a fair assumption.

So, we'll say 85% of the time, they're going to end up folding. When they folds, the chip stacks will end up as 2600, 2000, 3600, 1800. From these stacks, you have an ICM EV value of 0.2623. This is great, we've gained an extra couple of percent of the prize pool on average, just by this one push.

Now, suppose he's calls. It's definitely not what you're hoping for, but it will happen three out of twenty times. When you do get called, however, all is not lost. You'll win this hand about 36% of the time, on average. This is based on running a simulator to compare 97o versus a range of hands. Tables exist with this kind of data for quick reference, or you could use a poker odds calculator like the one at ITH to calculate a range of hands.

So, the probability of him calling and winning is 15% * 64% = 9.6%, and the probability of him calling and losing is 15% * 36% = 5.4% of the time. In the first case, the chip stacks will be 0, 4600, 3600, and 1800, and your ICM will be 0, since there's no chance for you to make the money. If you win, the chip stacks are 4400, 200, 3600, 1800, and you'll have an ICM value of 0.3639.

So, we have three ICM numbers, and the probability that each of those situations happens, so all we have to do now is average them. We then have the equation ICM EV = (0.85 * 0.2623) + (0.096 * 0) + (0.054 * 0.3639) = 0.2426.

So, if we fold, we have an ICM value of 0.2354 if we fold, and an ICM value of 0.2426 if we push. We're gaining on average about 0.7% of the prize pool if we push here, instead of folding, or in other words, about 72 cents on a \$10+1 tournament. If we're looking to make back two to three dollars on every tournament we play at this level, adding another 72 cents on average is a huge addition.

How can we apply this at the table? In contrast to the first article, in which we were getting away from a race once we were in the money, in this situation we're pushing into a situation where we're a race at the very best, and more than likely are behind by a good amount of 33% to 67% if not 20% to 80%. What's the difference in this situation? Well, for one, you're competing for an additional 20%-50% of the prize pool, as opposed to at 10%-30% share, but more importantly, you can win the hand in two ways: by either having the best hand at the end, or getting your opponent to fold. In contrast, when calling, you can only win the pot when you have the best hand. This is known as the gap concept, the idea that you must have a better hand to call with than to raise with.

Now, after seeing these two examples, you should be able to calculate any situation using the Independent Chip Model. However, there are exceptions to all models, and times where these models can fail. In the final article of the series, we will address these failures, in addition to other caveats of the Independent Chip Model.

Continue to the third article in the series, The Independent Chip Model - Caveats and Further Learning.

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0.7 % of \$10 is 7 cents, not 75 cents ...

The prize pool is \$100 in this scenario, not \$10. The actual gain is 72 cents, and not 75 cents, so I fixed that rounding error in the article.

Okay, at first you say we have 98s and have the BB covered by 200 chips, then when you do the calculations near the end, we have 97o and the BB has us covered by 200 chips... Errors?