Expected Value (EV) issue

B

Brewhaus

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I am attempting to create a basic shove chart but I'm getting way different results then normal shove charts and I don't know why. I don't know if the formula I am using is missing something... With a 10% ante and 9 people at the table my formula is saying that the small blind should be shoving every hand as long as they have 17BB or less. Clearly this is insane. Could anyone please tell me what I'm doing wrong? Here is the formula and values I'm using and an example with 72o.

Formulas:

Fp = 1 - Cp
Wa = 1.5 + (Ap * Pdc)
Ev = Fp * Wa + Cp * (Eq * (Wa + Bb) - (1 - Eq) * Bb)

Cp = Call percent - The percentage of hands the big blind will call your shove
Fp = Fold Percent - The percentage of hands the big blind will fold
Ap = Ante Percent - Then percent of the big blind the ante is (usually 0%, 10% or 12.5%)
Pdc = Players dealt cards - The total number of player that were dealt cards (and therefor that anted)
Wa = Win amount - The amount in the pot when you shove (Small blind (.5) + Big blind (1) + antes)
Eq = Equity - The chance you win a showdown against the big blinds expected hands
Bb = Big blinds - The number of big blinds you have left to shove
Ev = Expected value - The number of big blinds you can expect to win on average per shove


Example:

My hand: 72o
My stack: 15 big blinds after ante and posting of small blind
Big blind call range: 22+, A2s+, K8s+, Q9s+, J9s+, T9s, 98s, 87s, A8o+, K9o+, QTo+, JTo (this seems really loose to me for a 15 bb shove but the tighter the range the higher the EV due to the fold percentage)

Cp = 25.19% (The above listed hands make up 25.19% of all possible hands)

Fp = 1 - Cp
Fp ​= 1 - 25.19%
Fp ​= 74.81%

Ap ​= 10%

Pdc ​= 9

Wa = 1.5 + (Ap * Pdc)
Wa ​= 1.5 + (10% * 9)
Wa ​= 1.5 + .9
Wa ​= 2.4

Eq ​= 28.18% (Run in Equilab)

Bb ​= 15 (Specified above)

Ev = Fp * Wa + Cp * (Eq * (Wa + Bb) - (1 - Eq) * Bb)
Ev = 74.81% * 2.4 + 25.19% * (28.18% * (2.4 + 15) - (1 - 28.18%) * 15)
Ev = 1.79544 + 25.19% * (28.18% * 17.4 - (71.82%) * 15)
Ev = 1.79544 + 25.19% * (4.90332 - 10.773)
Ev = 1.79544 + 25.19% * -5.86968
Ev = 1.79544 - 1.478572392
Ev =.316867608



So shoving 72o from the small blind with 15bb has a positive EV? This can't be right. What am I doing wrong?
 
Phoenix Wright

Phoenix Wright

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I didn't check the math because I'm not a math person by any means lol

However, I suspect this math is correct, or at least somewhat close to it. I suspect the "weird" thing is having an ante for this calculation and the fact that the SB is already "invested" into the pot (hence giving better odds to call).

According to the CardsChat 30 Day Course, the guideline presented was to shove any "magic range hand" or better (course "magic range" was any pocket pair and any suited Ace, but naturally we also would like to shove with hands like AKo) if at 10 or less bb and 15 bb or less with an ante in play.

Obviously, this is a guideline simplification. For instance, it doesn't take into account how large the ante is etc.

Upon a superficial glance, but not looking at the math itself, I don't think this is too far off with SB at 17 bbs or less, but 72o sounds way too loose to me.

Perhaps this similar scenario I posted would be worth your quick glance. Maybe your 17 bb shove is too loose because you assumed Villain ranges too loose? After all, our opponents do not shove anything 100% of the time :D

https://www.cardschat.com/forum/learning-poker-57/should-we-be-calling-stations-479223/
 
Collin Moshman

Collin Moshman

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Nice work trying to do these calculations by hand, you'll get a lot out of this process!

To answer your Q: You're giving the big blind a calling range that's too tight.

For a 15bb effective stack with an ante, when you shove he's supposed to call with hands as weak as K5o, Q6s, and A2o.
 
D

DegenerateQuant

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I agree with Collin. The BB calling range you've used here is too tight. BB should be defending closer to 37% of their range here.

Your EV equations are correct for the most part. Though you are ignoring card removal effects. 72o will have a different CP/FP compared to AKo, as the AK blocks more of BBs calling range.

Regardless, when BB is overfolding this much it becomes profitable to shove any two cards.
 
B

Brewhaus

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Thanks guys, I will adjust the expected hands and see if that fixes my problem!
 
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