Going All-In with a Marginal Hand!?
[dialogue]
Q-chan: "Aria, while I was studying , something came up that I wanted to ask you about."
Aria: "What kind of spot was it?"
Q-chan: "It's the situation I always study (6max, 100bb, NL50). In a 3bet pot, UTG vs HJ, the flop comes 654 and the pot is 14.5bb. UTG donk bets 4.8bb — 33% of the pot — and HJ responds with a raise to 18bb, or 55%."
Aria: "True, a donk is plausible on that board. So what caught your eye?"
Q-chan: "The thing is, the core of HJ's raising range was strong overpairs like QQ and better. Like this."
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[dialogue]
Q-chan: "So I checked UTG's reaction, and somehow QQ was shoving all-in...
QQ is definitely a strong hand, but since the opponent's value is AA and KK, shoving all-in with QQ feels kind of weird. Here, take a look."
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[dialogue]
Aria: "I see. That's a really interesting spot! The value core of HJ's 55% raising range is AA and KK, so from QQ's perspective, it looks like a pretty scary opponent — and in fact, QQ only has about 55% equity here. So it's natural to feel uneasy and think, 'Can you really shove all-in with QQ against such a strong range?'"
Q-chan: "55% equity is completely marginal, though."
Aria: "Right. But in GTO strategy, QQ does sometimes shove. And if you look at the equity relationship after the shove, it takes a really strange shape."
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[dialogue]
Q-chan: "Huh? Usually the side shoving all-in is the polarized one, but here UTG — the one who should be shoving — has its equity concentrated near 50%. What a weird graph."
Aria: "Right? So this time, I'll explain 'why an all-in with a middling hand is allowed' using a heavily simplified game."
[/dialogue]
Think Through a Simplified Game
[dialogue]
Q-chan: "What kind of game is it?"
Aria: "The setup is really simple. In this game, the hand and the board each have only one card. First, Q-chan has a K out of position. And I have 1 combo of A and x combos of Q in position. strength is A>K>Q."
Q-chan: "So I have a middling range, and Aria has a polarized range."
Aria: "That's a fair way to put it. Each of us has a stack of 1 point. The pot is also 1 point, and the only bet size is a 1-point all-in."
Q-chan: "Super simple. Any other conditions?"
Aria: "In this game, there are two actions: the turn and the river. On the river, a Q comes on the board with probability p, and my Q outdraws your K. Otherwise, a 2 comes and nothing happens."
Q-chan: "So there's an outdraw element. To sum up: OOP holds a marginal hand, and IP holds the nuts A or a Q that's behind now but can outdraw later?"
Aria: "Right. Here's what it looks like as a diagram."
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[dialogue]
Aria: "The question I want to think about is whether OOP should shove all-in now or check."
Q-chan: "Hmm, that's a bit tricky."
Aria: "Then let's set up a situation like this.
: 7328
OOP's range
TT (1 combo)
IP's range
AA (1 combo)
KQ (x combos)
Thinking about the possibility of a T on the river makes this complicated, so let's assume that a T never comes on the river. On the river, a K or a Q comes with some probability and outdraws TT. Otherwise, the hand strength order is AA>TT>KQ."
Q-chan: "So TT represents my K, AA represents your A, and KQ represents your Q."
Aria: "Right. Here, the probability p that KQ outdraws TT is 14.3%. Think of it as the 'two-overcard outdraw category'."
Q-chan: "I see. But OOP is afraid of IP's AA, so maybe just check for now?"
Aria: "That's what makes this problem interesting. When you actually do the math, the strategy changes depending on the number of KQ combos, x. Specifically, if x>3.18, TT shoves all-in; if x<3.18, it checks."
Q-chan: "Wait, what!? If I shove, I just get caught by AA and make KQ fold — wouldn't that be the worst bet?"
Aria: "I totally get that feeling. But when you weigh all the factors, you should reach the conclusion that all-in can be the right play."
Q-chan: "Hmm. I don't understand it at all."
Aria: "Then let me ask you, Q-chan. If you don't bet, it becomes my turn. What do you think I'd do?"
Q-chan: "Hmm, probably polarize and bet?"
Aria: "Exactly! IP uses the nuts AA and the drawing KQ to torment TT with a polarized all-in. So even if you check, TT is doomed to be tormented."
Q-chan: "Right — checking doesn't mean you get to go to showdown."
Aria: "Right. The only way to know which action is more profitable is to compare bet and check . So in the next section, let's actually do the comparison."
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Check EV Isn't Decided by "Being Ahead Right Now"
[dialogue]
Aria: "Here, let's compare bet EV and check EV.
First, let's look at bet EV — it's simple. If AA is 1 combo and KQ is x combos,
with probability x+11, AA calls and we lose 1
with probability x+1x, KQ folds and we win +1
so bet EV=x+1x−1."
Q-chan: "So the question is check EV?"
Aria: "Right. Just because TT is 'ahead right now' doesn't mean it can collect that win as is.
For TT to realize its equity (i.e., beat KQ at showdown and take the pot), two things are needed:
IP gives up and checks back on the turn
and then TT doesn't get outdrawn on the river
Both are required."
Q-chan: "Even if you're ahead now, it's meaningless if you can't hold on in the future."
Aria: "Exactly. And IP doesn't check back all of its KQ combos — it can use some of its KQ as a turn bluff-shove."
Q-chan: "That's the annoying part."
Aria: "Yeah. To strip TT of its equity, IP just needs to shove 0.636 combos of KQ for every combo of AA. That way, TT gets pushed to the edge between calling and folding, and its EV is driven to zero. That number comes from the required equity of 33%."
Q-chan: "So checking gives KQ a chance to bluff."
Aria: "Right. As a result, check EV is the probability that IP checks back multiplied by the probability that a card other than K or Q comes on the river. You can find IP's check-back probability by thinking about how many KQ combos IP uses as bluffs, but it's a bit tricky — if you're confident in your math, feel free to try it."
Aria: "I'll skip the calculation and just give you the conclusion:
IP's check-back probability = x+1x−2−3p1
Probability that a card other than K or Q comes on the river =1−p
Multiplying these gives check EV=x+1x−2−3p1(1−p)."
Q-chan: "That's a seriously complicated formula..."
Aria: "You don't need to fully understand the formula. Remember, the point is comparing bet EV and check EV. I'll interpret the formula later."
Q-chan: "Okay. Let's compare bet EV and check EV for now."
Aria: "Sure. This time, under the assumption p=14.3%, let's see which is bigger, bet EV or check EV. Calculating this gives the following facts:
When x>3.18 bet EV > check EV
When x=3.18 bet EV = check EV
When x<3.18 bet EV < check EV
In other words, the strategy I mentioned earlier — TT shoves all-in when x>3.18 and checks when x<3.18 — is correct."
Q-chan: "Hmm, this is completely different from my intuition."
Aria: "I know how you feel. Let's organize what we've covered so far:
At first glance, TT's bet looks like it just pays off AA (a hand that beats us) and folds out KQ (a hand we beat). That's why it feels like 'aren't I just folding out the hands I beat?'"
But in reality, if you check, KQ bluffs on the turn, and even if it gives up the bluff, it outdraws with some probability on the river. Because of that, TT's rightful share of the win gets chipped away considerably."
Q-chan: "Ah, so it's not that betting is strong — checking is weaker than I thought."
Aria: "Exactly. You can't see this just by looking at current equity. You have to think about how much of that equity can be realized."
Q-chan: "When x=3.18, that means there are more than 3 times as many KQ combos as AA combos... so if you calculate it, TT's equity at that point is actually 65%."
Aria: "Right. But the key point this time is that having equity and being able to realize it are two different things."
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What the Formula Really Means
[dialogue]
Q-chan: "The formula from earlier was too hard, so I only got a vague sense of it. How should I interpret all those formulas?"
Aria: "There are two main points.
First, the larger x is — in other words, the larger the share of the opponent's weak hands — the more OOP tends to shove all-in.
That one's easy to see. The opponent is less likely to have strong hands, and it's easier to enjoy the benefit of folding out KQ.
Second, the larger p is — in other words, the higher the outdraw potential of the opponent's weak hands — the more OOP tends to shove all-in. In other words, the stronger the opponent's 'hands that are behind now but become a real nuisance later', the more easily the opponent can bluff, and"
Q-chan: "That kind of matches my intuition. It's like: 'The opponent has only a few strong hands, but getting outdrawn or tormented by bluffs is worse, so let's just shove!'"
Aria: "Nice interpretation. And here's the interesting consequence: there are cases where this all-in is correct even when equity is below 50%."
Q-chan: "Wait, what!? You can shove even when you're behind right now?"
Aria: "For example, let's look at the case p = 0.27. Here's the setup:
Board: 7328
OOP's range
TT (1 combo)
IP's range
AA (1 combo)
K9, K5 (1.5 combos total, p=27%)
In this case, plugging into the formula gives p=27% and x=1.5. If you calculate and compare bet EV and check EV, bet EV comes out higher, so shoving all-in is the correct play."
Q-chan: "It's already pretty strange at this point. 40% of the opponent's range is the strongest hand — shoving is basically self-destruction..."
Aria: "In actual GTO strategy, TT does shove. And what's even stranger is that TT's equity is only 44%."
Q-chan: "44%!? That means you're already behind!"
Aria: "And yet all-in is still correct. I think this is the most counterintuitive part of the whole article."
Q-chan: "Why does that happen?"
Aria: " reason. It's not that bet EV is high — it's that check EV is extremely low. If we check, the opponent bluffs on the turn and also outdraws on the river. In other words, our 'current 44% equity' won't realize as a full 44% just by checking."
Q-chan: "So it's not as simple as 'I'm behind, so I can't shove'."
Aria: "Right. What matters isn't current equity, but how much of it you can realize."
[/dialogue]
Summary
[dialogue]
Q-chan: "So what's the conclusion this time?"
Aria: "In one sentence:
Just because IP has a polar range doesn't mean checking is always correct for OOP
That's the takeaway. When IP has a polar range that includes nuts-level hands as well as plenty of hands that are behind now but carry decent equity, OOP sometimes shoves all-in, accepting that 'getting caught by the nuts is just one of those things' in order to make the opponent's draws give up their outdraw chances and bluff opportunities."
Q-chan: "ing out the hands you beat feels wasteful, but if you leave them alone, they'll take your equity, so you fold them out — got it."
Aria: "Right. With that feel for the game, GTO's seemingly strange all-ins become a lot easier to understand."
Q-chan: "Where would this be useful in real play?"
Aria: "The typical spots are low- situations where the opponent has a polar range. The clearest example is probably the strategy after calling a 4bet pot."
Q-chan: "Ah, like after 3betting TT or JJ from the SB, calling a 4bet, and the flop coming low?"
Aria: "Right. In spots like that, it's really hard to handle hands that are moderately strong but painful to play defensively. With this way of thinking, instead of just checking and passively taking the punishment, you also have the option of shoving all-in back — that widens your play."
Q-chan: "Got it, I'll try it in practice! Thanks for teaching me!"
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