Imagine a situation where we get dealt AA right on the bubble of a big tournament. We are shallow stacked and have been hoping to last just long enough to make the money, before trying to build our stack by ramping up the aggression.
In most cases, we are nearly always pleased to get dealt Aces, but perhaps we don’t feel so excited in this case. Sure, there is a great chance that we get to double our chip-stack. But, in order to do that, we have to put all of our chips on the line, right on the bubble.
We know that if we continue to fold, we have a very high chance of making the money. Could folding AA ever actually be correct? Surely, it has to be +EV to get the money in when we have the preflop nuts?
Chip-EV vs $-EV
Perhaps, we are familiar with the previous scenario and wonder what the correct decision would have been. Naturally, it depends on a number of factors, such as the exact number of chips we have, our position in the tournament, and the prize structure.
Let's say that the tournament pays the top 20 finishers equally. There are 21 players remaining; we are in second place. The chip leader is shoving all-in, forcing us to put our tournament life on the line. It is correct to fold Aces here. If we purely look at the chips, however, it’s hard to see how folding Aces can be the correct option. After all, we clearly have the best hand.
We need to make a differentiation between the two following methods of calculating EV in a tournament setting:
(EV = expected value)
Tournament players should purely be interested in the $EV of their decisions and not the chip-EV. There are some scenarios where the chip-EV might be positive, but the $EV is negative. We can see this illustrated in the above example.
Chip EV just tells us whether we will increase the size of our chip stack on average. This fact is always the case when we get all-in with pocket Aces. The $EV tells us how frequently we ensure ourselves a bigger tournament payout on average. To figure this out correctly, we would need to consider the tournament payout structure, not just the current stack sizes.
ICM – Independent Chip Model
In these types of situations, we can use the Independent Chip Model (ICM). ICM attempts to assign a monetary value to our remaining tournament chips, allowing us a more accurate way to make +EV decisions.
The calculation for ICM is complex and is outside the scope of this article, but there are a number of ICM calculators available for free online. All we need to do is plug in the numbers.
Let’s consider a basic example and see what an ICM calculator tells us about the situation.
Prize Pool of $1000.
Places 1-3 pay out 50%, 30%, and 20% respectively of the entire prize pool.
According to ICM, what is the $EV or monetary value of each stack size?
We need a little bit of extra information first to work this out, as follows à
There are 10,000 chips in play, and the 5 remaining players have the following stack sizes:
Player 1 - 4,000
Player 2 - 2,500
Player 3 - 2,000
Player 4 - 1,000
Player 5 - 500
To get the monetary value of each of the remaining stacks, we will need to plug the numbers into an ICM calculator. The following results would be produced:
Player 1 - $328.238
Player 2 - $256.797
Player 3 - $222.929
Player 4 - $126.029
Player 5 - $66.007
One of the things we can observe when playing around with an ICM calculator is that the more top-heavy the prize pool, the more valuable being the big stack becomes. Let’s run the same numbers but, instead, say that the prize structure is now winner takes all. (First prize takes all the money, everyone else goes home with nothing.)
Player 1 - $400
Player 2 - $250
Player 3 - $200
Player 4 - $100
Player 5 - $50
Notice that the distribution here is proportional to the number of chips we have. In other words, the closer a tournament becomes to a winner-takes-all structure, the closer our $EV calculations become to chip-EV.
Let’s imagine now that the top 4 places paid exactly 25% of the prize pool and see what our ICM calculator tells us.
Player 1 - $245.084
Player 2 - $235.938
Player 3 - $228.357
Player 4 - $185.109
Player 5 - $105.512
Notice how each of the $EV values is converging on each other now – getting much closer together. Being the chip leader is less of an advantage in this scenario. Now, if we decided to say that the payouts were 20% for the top 5 players, then every player would have a $EV of $200 (given that there are only 5 players). In that instance, being the chip leader would be completely irrelevant.
ICM in Practice
But, how does any of this help us at the tables? Understanding the value of our stack in terms of real $ allows us to make stronger EV calculations.
Let’s do a little experiment using our ICM calculator, following on from our original question.
The prize structure assigns 25% of the prize pool to the top 4 finishers – the 5th place finisher gets nothing. This might not be an overly realistic scenario, although some tournaments do follow this structure. (Mostly in situations where a fixed prize is being given out, such as tournament tickets to a larger event.) We are choosing this specific example because, as we have discovered, our $EV will be significantly different from our chip-EV.
There are 5 players remaining with the following stacks, and 20,000 chips remain in play.
Player 1 – 7,000
Player 2 - 6,000 ß Hero
Player 3 – 4,000
Player 4 – 2,000
Player 5 – 1,000
As we can see, we only need player 5 to bust out, and we are guaranteed 25% of the money. Let’s assume that that prize pool is the same as before ($1,000) and calculate the $EV of each chip-stack.
Player 1 - $243.047
Player 2 - $240.177
Player 3 - $227.935
Player 4 - $184.352
Player 5 - $104.490
Let’s imagine a scenario where the SB big stack decides to shove all-in for 7,000. We have pocket Aces and want to establish whether calling for our remaining $6000 is correct. For simplicity’s sake, let’s ignore any blinds.
We are not interested in how many chips we make on average, but how much calling effects our $EV on average. Firstly, let’s imagine that the Villain is shoving about 7% of hands and see what our equity looks like. (Note that we are not suggesting this is a reasonable shoving frequency; this is just what we think this specific villain might be shoving with).
|88+, Ats+ KQs, Ajo||15.38%|
We know that our $EV chip stack is worth about $240. This means that 15.38% of the time we are going to lose about $240 in $EV and 84.62% of the time, we will win. But, how much? Again, we need to think in terms of $EV, and the only way we can really do this is to redo the calculation with the ICM calculator.
In this particular case, it will be very easy. We know that the value of everyone’s stack will be $250 in $EV. This is because once that 5th player busts out, all remaining competitors will receive 25% of the $1000 prize pool, worth $250 in $EV.
(N.B. Assuming the tournament doesn’t end there, we should do a separate calculation with our ICM calculator, factoring in how the stack sizes will look after we call and win. This will tell us how much we stand to make in $EV. It was also pretty straightforward in this example to see that we lose $240 in $EV when we lose our all-in, since we will be out of the tournament. Assuming calling does not bust us, we should run another ICM calculation for the scenario where we lose, see what our projected $EV would be, and hence calculate how much $EV we lose when we call the all-in and lose.)
We have enough information now to see what the $EV of calling will be. There are 4 key components that make up an average simple EV calculation.
Probability of Winning – 84.62%
Amount We Can Win – About $10 (Difference between current $EV stack and $250)
Probability of Losing – 15.38
Amount we can lose – About $240 (our entire $EV stack)
Already we can see that this is not looking great in terms of calling. Let’s plug the numbers into the EV formula
(Probability of winning * Amount-won) – (Probability of losing * Amount-lost)
(0.85 * $10) – ( 0.15 * $240)
$8.50 - $36 = -$27.5
Oh dear. If we call with our pocketaces, we are giving away $27.5 on average! It may seem counter-intuitive at first that folding Aces would ever be correct, but we can see that under the right circumstances calling is horrible. We should fold and simply wait for player 5 (or someone else) to bust out.
Naturally, the example was somewhat contrived, and we are not suggesting for a minute that it is usually a good idea to go around folding pocket aces. The example is here to help us understand the relevance of ICM in tournament decision making.
In closer spots, we would also like to see the $EV of folding so we can compare it to the $EV of calling. (Sometimes calling might still lose money but lose less than folding on average). To find the $EV of folding, we re-run our ICM calculations with the adjusted chip stacks, taking into account that we folded away some money, while our opponent gained a little money. The difference between our original $EV and our $EV after folding will be the overall $EV of folding.
ICM is Not Everything
ICM is a theoretical way of analysing various tournament situations. Sometimes heavy ICM users will give the impression that ICM is the “perfect” solution to every tournament situation, and that any deviation from it will result in $EV losses.
But the truth is that ICM calculations are imprecise, and there are also many different types of tournament players out there. If our opponent is very tight, we should potentially fold more often. However, if they are very loose, we should be calling more often. Also, most ICM calculators will tell us that our $EV due to our stack size is the same regardless of whether we are on the BTN or UTG. This just can’t be the case in practice. When we are UTG, we are right on the verge of paying for a fresh round of blinds. ICM doesn't account for this.
Some of us are also probably wondering how on earth we can use ICM calculations while at the table. Surely, it takes too long. There is a lot of truth to this. Trying to use ICM during an actual hand is not practical. ICM calculations are typically used as a way to review hands after a tournament is over. Even though it won’t help us in the tournament we have already finished, it should help to refine our decisions in future tournaments.