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Harrington (close to bubble)
Problem 21 of Harrington’s Workbook
blinds are 400/800
50 left, 40 paid
An ABC player opens under the gun to $2500.
Hero has $4800 and AsQh in the cutoff. The button, sb, and bb are still to act.
– fold
– call $2500
– raise to $4800
Before reading the rest, try to answer the question above.
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DH goes through the math of how to make this decision.
assumptions:
— players behind can still squeeze with monsters with (~5% of hands).
— the opener could have a strong hand (90%) and a weak hand (10%).
estimations:
— vs the squeezer, hero’s AQ will win 25% against AA, KK, QQ
— vs the strong opener, hero’s AQ will win 47% against 88+, AK, AQ, AJ, KQ
— vs. the weak opener, hero’s AQ will win 60% against JT, T9, 98, 87
calculations:
1) When the hero jams, what pot odds are offered to the opening raiser?
— amount to call (4800 – 2500) = 2300
— size of pot = (400+800+2500+4800) = 8500
— direct pot odds = (8500 / 2300) = 3.7 to 1
— pot size after opener calls the hero = 8500 + 2300 = 10,800
2) What is your all-in equity against the opening raiser?
— how often do you expect to win w/ AQ when called?
— win 30% — equity = (30%)(10,800) = 3240 — net loss of 1560 from 4800
— win 40% — equity = (40%)(10,800) = 4320 — net loss of 480 from 4800 stack
— win 50% — equity = (50%)(10,800) = 5400 — net gain of 600 from 4800 stack
3) What is the pushing the correct play?
– fold
– call $2500
– raise to $4800
Use the estimations and assumptions above,
against the squeezer:
— 5% of the time, you’re up against a monster AA, KK, QQ
— 25% of the time, AQ beats AA, KK, QQ
— an extra (4800, 4400, or 4000) will inflate the pot from the (button, sb, or bb)
button squeezes
= (5%)(25%)(400+800+2500+4800+4800) = $92
sb squeezes
= (5%)(25%)(400+800+2500+4800+4400) = $91
bb squeezes
= (5%)(25%)(400+800+2500+4800+4000) = $88
no squeeze
against the strong opener:
— 90% of the time, you’re up against a 88+, AK, AQ, AJ, KQ — you win 47%
— 10% of the time, you’re up against JT, T9, 98, 87 — you win 60%
— the pot size will be 8500 + 2300 = 10,800 after the opener calls
strong opener = (90%)(47%)(10,800) = $4568
weak opener = (10%)(60%)(10,800) = $648
THUS, add up the value of each situation
strong opener + weak opener = TOTAL
$4568 + $648 = $5216
Doing the math for the (calling the bet) is little value because the assumptions about players calling behind and raising behind all change. This math was done assuming that the hero calls and it goes heads-up.
strong opener
= (90%)(47%)(400+800+2500+2500) = $2622
weak opener
= (10%)(60%)(400+800+2500+2500) = $372
total = 2622 + 372 = $2994
Folding earns you $0 on average — (4800 – 4800 = $0)
Calling loses $1800 on average — (2994 – 4800 = -$1806)
Raising gains $400 on average — (5216 – 4800 = +$416)
The math narrows down to a few keys:
— effective stack size
— assumptions about their range
— estimations on how often you win
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2 Replies
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I wrote a python script to get some quick results.
Here my pushing EV against the opener with two variables:
— size of the opener range (chance)
— hero’s equity vs that range (win)
Each calculation assumes a 2.5bb open and a 3-bet to 10bb.
For those of you stomping your feet about my 3-bet size, I’ll include data for a 7bb and 8bb….so take a chill pill. I was trying to keep things simple 🙂
In general, I found that the best EV spots for the hero occur:
— as %win increases (more EV when the hero expects to win more often)
— as %chance increases (more EV when the opener has a wider range)
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open 2.5bb, 3-bet to 10bb
vs. opener Win = 10 Chance = 20 EV = 43.0
vs. opener Win = 20 Chance = 20 EV = 86.0
vs. opener Win = 30 Chance = 20 EV = 129.0
vs. opener Win = 40 Chance = 20 EV = 172.0
vs. opener Win = 10 Chance = 40 EV = 86.0
vs. opener Win = 20 Chance = 40 EV = 172.0
vs. opener Win = 30 Chance = 40 EV = 258.0
vs. opener Win = 40 Chance = 40 EV = 344.0
vs. opener Win = 10 Chance = 60 EV = 129.0
vs. opener Win = 20 Chance = 60 EV = 258.0
vs. opener Win = 30 Chance = 60 EV = 387.0
vs. opener Win = 40 Chance = 60 EV = 516.0
vs. opener Win = 10 Chance = 80 EV = 172.0
vs. opener Win = 20 Chance = 80 EV = 344.0
vs. opener Win = 30 Chance = 80 EV = 516.0
vs. opener Win = 40 Chance = 80 EV = 688.0
How do you use this?
Take your opponents range and structure it as strong and weak. If you look at combos, you can make better estimates of %chance but here, I’ll say 80% of their range is strong and 20% is weak.
After you build these two ranges, calculate your equity.
Imagine having pocket 55 — @FiveByFive
You are deciding whether to push or fold over the top of an open.
— 80% of the time, they have an over-pair and you have a 20% chance to win
— 20% of the time, they have two overs and you have a 40% chance to win
vs. opener Win = 20 Chance = 80 EV = 344.0
vs. opener Win = 40 Chance = 20 EV = 172.0
The EV of this spot is 344 + 172 = 516
I hope you’re looking at my answer and wondering “what the heck”, that should be losing proposition. You would be correct!!!
This answer needs to be compared to the original stack size (before acting) to make sense out of it. In these calculations, I assumed that the 1 bb=100, open size was 2.5 bb = 250, and the 3-bet size was 10 bb = 1000.
This means that pushing pocket 5’s in this scenario is a negative EV play because you’re investing 1000 and returning only 516. This is a net loss of 484.
As promised
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open 2.5bb, 3-bet to 7bb
vs. opener Win = 10 Chance = 20 EV = 31.0
vs. opener Win = 20 Chance = 20 EV = 62.0
vs. opener Win = 30 Chance = 20 EV = 93.0
vs. opener Win = 40 Chance = 20 EV = 124.0
vs. opener Win = 10 Chance = 40 EV = 62.0
vs. opener Win = 20 Chance = 40 EV = 124.0
vs. opener Win = 30 Chance = 40 EV = 186.0
vs. opener Win = 40 Chance = 40 EV = 248.0
vs. opener Win = 10 Chance = 60 EV = 93.0
vs. opener Win = 20 Chance = 60 EV = 186.0
vs. opener Win = 30 Chance = 60 EV = 279.0
vs. opener Win = 40 Chance = 60 EV = 372.0
vs. opener Win = 10 Chance = 80 EV = 124.0
vs. opener Win = 20 Chance = 80 EV = 248.0
vs. opener Win = 30 Chance = 80 EV = 372.0
vs. opener Win = 40 Chance = 80 EV = 496.0
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open 2.5bb, 3-bet to 7bb
vs. opener Win = 10 Chance = 20 EV = 35.0
vs. opener Win = 20 Chance = 20 EV = 70.0
vs. opener Win = 30 Chance = 20 EV = 105.0
vs. opener Win = 40 Chance = 20 EV = 140.0
vs. opener Win = 10 Chance = 40 EV = 70.0
vs. opener Win = 20 Chance = 40 EV = 140.0
vs. opener Win = 30 Chance = 40 EV = 210.0
vs. opener Win = 40 Chance = 40 EV = 280.0
vs. opener Win = 10 Chance = 60 EV = 105.0
vs. opener Win = 20 Chance = 60 EV = 210.0
vs. opener Win = 30 Chance = 60 EV = 315.0
vs. opener Win = 40 Chance = 60 EV = 420.0
vs. opener Win = 10 Chance = 80 EV = 140.0
vs. opener Win = 20 Chance = 80 EV = 280.0
vs. opener Win = 30 Chance = 80 EV = 420.0
vs. opener Win = 40 Chance = 80 EV = 560.0
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Hero has only 6bb in his stack. Against an EP open of >3bb Hero has zero fold equity if he jams. The three players left to act are getting a good price to call off if Hero jams. Also, ICM pressure could be coming into play here. With 6bb Hero may be on the outside looking in. Fold equity is so, so important when short stacked. We need a powerhouse to jam here.
It’s unwise for Hero to call off for over half his stack with almost any holding. I may do it with KK+ if I don’t think V or the players yet to act are good enough to smell a rat here. But that’s a really strong assumption.
To be honest, I don’t think I’d defend with AQo here, even against an open limp.
This is a fold.
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