We’ve all been in this situation before: We’re on the bubble, our stack is down to 10 big blinds, it is folded around to us in the small blind, and we look down at J8o. Is it good enough to push or should we wait for a better spot? In this series we’re going to look at the factors that govern how you should play spots like this, and other situations as well.
There are several things to be aware of when making the decision to commit your stack against the big blind. Understanding the math behind the push is critical. Many players are simply too tight in the small blind, and pass up spots where pushing all-in has a positive expected value (+EV).
Relative stack sizes can play a role in your decision as well. For example, if the big blind barely has you covered, and calling would cripple him, he may not be so quick to call your push with his K2s this time. The presence of antes is a another big factor. When antes are in play, there are more chips in the pot to be won, and you should be more aggressive in trying to win them. Other things we’ll touch on are your image, or how the table perceives your recent play, and your reads on how the big blind has been playing. With that, lets get started – first up, the math.
Calculating EV for a SB vs BB Push
So how do we go about determining if a push has positive expected value? Note that this exercise is best done in post-game analysis. The math is much too involved to be able to calculate on the fly in a real time blind vs blind encounter. However, once you start analyzing scenarios, you will find that you will develop a feel for what is a +EV push and what is not, and can start applying what you’ve learned to in-game situations.
The first thing we need to know is how often the big blind is going to call our push. Unfortunately, we can never know this for sure, so this calculation is never going to be an exact science. However, we can make some fair estimations, and we can also run some different scenarios and see what the best play is in each of those situations. Lets say we assume that the big blind is calling us only with the top 10% of hands. That means that when we push, he is going to fold 90% of the time, and we will win the blinds (and antes, if there are any).
The other 10% of the time, the big blind calls and we’re going to see a showdown. Assuming that the big blind has us covered, we’re going to more than double up if he calls and we win, and we’re losing our entire stack if he calls and we lose. We now need to know our chances of winning (our equity) against his range of hands. There are several equity calculators available online. PokerStove is one of the more popular ones. Let’s assume that of the 10% of the time that the big blind calls us, we win 50% of the time and lose 50% of the time. This means that every time we push, we win the blinds 90% of the time, we get called and win 5% of the time, and we get called and lose 5% of the time.
To calculate our EV then, we need to assign chip values to each of these three possible outcomes, multiply by the chance that they will happen, and add the results. If the end number is positive, then we have a +EV situation, and the push is profitable over the long run. If we come up with a -EV situation, then we can wait for a better spot. Let’s run through an example with some specific numbers.
Blinds are 200/400 with antes of 50 at a full 9-player table. After posting the ante and the small blind, we have 4000 chips left, and the big blind has us well covered. The action folds around to us, and we look down at 44. We know the big blind has been playing tight, and assume he will call us with a range of ATo+, A8s+, KJs+, and 66+ (10% of hands).
There is currently 1,050 in the pot with the blinds and antes. So, if we push, 90% of the time we pick up this pot:
0.90 * 1050 = 945
Of the 10% of the time that the big blind calls, we take our equity calculator and run our 44 vs his range, and see that our equity is 38.9%. When we push and are called, we shove our remaining 4,000 and the big blind calls 3,800. This means that if we are called and win, we win 4,850 chips (3,800 + 1,050). So, taking in account that we are only called 10% of the time, and of those times, we win 38.9% of the time, we get:
0.10 * 0.389 * 4850 = 188.67
Of the 10% of the time that the big blind calls, we lose our entire stack 61.1% of the time (100% minus our equity of 38.9%). Running these numbers like we did above yields us the following:
0.10 * 0.611 * -4800 = -244.40
Adding these three chip results together yields us our EV for the push:
945 + 188.67 – 244.40 = 899.27
As you can see, the result is a positive number, meaning that if we make this play every time, we expect to gain about 900 chips on average, over time. This is a +EV push.
Note that in this example, the biggest number of the three possibilities comes from the times we raise, the big blind folds, and we pick up the blinds. In this case, this is because he’s calling us with such a tight range, and we get the blinds 90% of the time. If you open up the big blind’s calling range to include more hands, then this number goes down. However, since he’s calling us with a wider range, our equity in the hand increases, and the “call and win” number will go up while the “call and lose” number will go down. Whether a wider calling range will result in a +EV situation is a combination of his range and our hand’s equity against that range.
Another interesting bit of information you can derive from these calculations is your risk of busting out of the tournament. This can act as a minor factor in marginal EV situations and can help you determine if you want to make that close decision. The formula for your bust out risk is fairly straight forward. You just take the percentage of time you are called, and multiply by your negative equity vs the big blind’s range. In the example above, you get 10% times 61.1%, or 0.10 * 0.611 = 0.0611. This means that when you make the push in this scenario, you will lose your entire stack and bust out 6.11% of the time. Since this is a +EV push, and your risk is relatively low, you can feel comfortable making this push every time.
In the next segment, we’ll be looking at other scenarios where you might have a higher risk, for less potential gain, and learn how to deal with those situations. We’ll also be highlighting short stack pushes, the advantage of doubling up, and what to do if you have a good stack but the big blind is short.
Be sure to check out the other parts of this article, as it is a multi-part series!
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