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Profitability of a Shove —- Math
I love math…if you don’t, stop reading.
Here is the scenario, you are playing a tournament at the Final Table. The big stack has attempted to steal your BB the last 3x, all of them you folded. On his/her 4th attempt, you plan to use your image to fight back.
Key Questions:
Start by making some estimates about your opponents range:
1) How often is my opponent trying to steal? (% opening range)
2) How often will they call my re-raise with that opening range? (% calling range)
3) How often will I win against that calling range? (% win when called)
Next, evaluate the stack sizes:
4) How much will I win when I raise and my opponent folds? (pot size + opening bet)
5) What is my net loss when called and I lose? (effective stack)
6) What is my net gain when called and I win? (effective stack + blinds + antes)
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Example #1:
6 Players Left.
You have 159,000 chips in the BB and the villain has you covered with 200,000 and the Button. The stack sizes give you leverage because you cripple the villain if they call and lose.
The blinds are 4000 / 8000 and 1000 ante. The pot size before any action takes place is (4000 + 8000 + 1000 * 6 = 18,000 chips).
The villain opens to 3x the bb or 24000 chips.
Assume that the villain is opening 50% on the button to steal the blinds. You estimate that he will call 20% of those hands when you shove all-in. You estimate that you’re semi-strong shoving hand will win 33% against that calling range.
You’re in the BB so you already have 9000 chips in the pot with 150,000 left.
- If you shove 150,000 and they fold, you will win a pot of (18000 + 24000) or 42000 chips.
- If you shove 150,000, they call, and you lose. You bust out losing 150k in chips.
- If you shove 150,000, they call, and you win. You double up to win (150,000 + 18,000) or 168,000 chips gained. Your chip stack would become 318,000.
Calculation #1
How often should they fold to the shove?
They open 50% and only call 20% of those hands and fold 30% of those hands. That means that they will fold to my shove = (50% – 20%) / (50%) = 60%. This means that they will call the raise 40% of the time.
Calculation #2
What is your expected value when they fold?
= (%fold) * (amount win when they fold)
= (0.60) * (42,000) = 25,200 chips
Calculation #3
What is your expected value when they call and you lose the hand?
= (% lose) * (amount lose when they call and beat you)
= (0.67) * (150,000)
= 100,500 chips
Calculation #4
What is your expected value when they call and you win the hand?
= (% win) * (amount win when they call and you win)
= (0.33) * (168,000)
= 55,440 chips
Calculation #5
Incorporate how often they call the shove with calculation #3 and #4.
= (% call) ( call/win – call/lose)
= (0.40) * (55,440 – 100,500)
= (0.40) * (-45,060)
= -18,024 chips
Calculation #6
What is your Total Expected Value?
= Fold EV + Call EV
= 25,200 – 18,024
= 7,176 chips
Conclusions:
With the assumptions of our opponents range and the relative stack sizes, the math suggests that shoving is +EV by almost 1 full bb.
- If our opponent calls our raise more often, our EV will likely drop.
- If our opponent folds more often, our EV should rise.
- If our hand equity is weaker and we win less often, our EV will drop.
- If our hand equity is stronger and we win more often, our EV will rise.
- If the effective stack size is larger, we are risking too many chips for a smaller net gain which means our EV tends to drop.
- If the effective stack size is smaller, we are risking less chips to win more which means our EV tends to rise.
Try it yourself:
Example #2
— Since @SteveFredlund is likely a habitual blind stealer, we will pretend that he is our villain. He opens on the button as chip leader and you have simply had enough of his ****. Time to punish him with a re-raise. Let’s assume that Steve is opening 40% of hands but will only call your shove only 15% of the time. Let’s assume that you make this move with a small pocket pair and expect Steve to fold AK but call with all over-pairs. Maybe a stretch but let’s go with it. This means you will win about 20% of the time when you get called. The blinds are 150/300 + 50 ante and it’s 8 handed. Steve started the hand with 15,000 chips and he opens to 900 chips. You have 7,000 chips and are in the BB. Is this a profitable shove?
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4 Replies
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Dude, I love this but I’m too tired tonight to give it enough though. I like that you think I could fold AK in that spot 🙂 — super fun bud.
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To me, I think it’s definitely possible for you to fold AK in some spots but maybe not that spot. It depends on your assumptions about my range and your equity against that range. If you think my range very tight (AK and any pair), you are at best flipping for your tournament life. As seen on TV, some pros like Humberto Brenes will even fold JJ in a similar spot when pressure is applied.
I built some simple ranges below:
169 starting hands
13 are pocket pairs (AA – 22) ~ 7.7% of total —- 6 combos each
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78 are suited hands (AKs – 32s) ~ 46.2% of total —- 4 combos each
78 are unsuited hands (AKo – 32o) ~ 46.2% of total —- 12 combos each
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Let’s say Steve is opening X% of hands…
10% — (22+, AKs – ATs)
20% — (22+, AKs – A2s, AKo – A7o)
30% — (22+, AKs – A2s, AKo – A2o, KQs – K9s, KQo – K9o)
40% — (22+, AKs – A2s, AKo – A2o, KQ – K2s, KQo – K2o)
50% — (22+, AKs – A2s, AKo – A2o, KQ – K2s, KQo – K2o, QJs – Q8s, QJo – Q8o)
These ranges are only guesses. To build them, I was just choosing an additional ~16.9 hands to add to the range for each extra 10%. Obv, it makes more sense for a player like Steve to be selective PF, thus you can replace weaker holdings like A7o with something like QJs if you wish.
In my example, when Steve opens 40% and then only calls a raise 15% of the time. I estimate that Steve might be only calling my raise with (22+, AKs-A7s, KQs-JTs). However, this calling range might shrink in a tough spot from some well-placed leverage. As Steve’s opponent, the idea is that you are trying to pull a quick one with the weak end of your range. To pull it off, make sure that:
- Steve is likely opening wide
- Steve is likely calling thin
- Your table image is assumed to be tight, well-respected, and never out-of-line.
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I love this – @ARW always knows how to make the point clear with math. RecPoker Nation, solve his example for Steve’s button and you will see how the actual cards in the foe’s hands are less important the the range they are playing with – and how the better you can make your assumptions about their range, the better your poker decisions will become.
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He’s a wizard I tell you, a wizard! This is why @ARW is up for our forum poster of the year award…. will he win the RECie ?
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