Two GTO questions
Posted by papahook
Posted by
papahook
posted in
Gen. Poker
Two GTO questions
Hey guys,
Two questions about GTO approach.
There is a simple formula how much does we need to defend vs various villains bet sizes so he is not making automatic profit 1-(bet/(bet+pot). Let’s say he bets full pot, so we should defend (1-(1/(1+1))=50% of our range. So say that we defend only 40%. Out of 100 hands he makes 60 times +1, 40 times -1 (even if he loses every single time we call him), so he is +0.2/hand. Does that mean that he makes automatic profit? On the other hand, we win 40 times 2 (pot size 1 + 1 his bet) if we win every time, so we make +0.8/hand. My point is that his +0.2 is not automatic profit, it’s just small part of realized equity from the pot, which he build in the first place (if we call 40% of our range with 100% equity vs his betting range, we split the pot 20% for him and 80% for us). What am I missing everywhere is callers vs bettors equity in the formulas.
There is another formula which is ratio value/bluffs based on our betsizing. If we bet full pot on the river, we should have 2/3 value, 1/3 bluff. This works fine in polarized scenarios. Because our value range wins always. However this is usually not the case and again, we need to add equity of our value range vs villain calling range to equation. Let’s say, that our value range has 90% equity. If we get called 100 times we get:
33 * -1 + 67 * 0,90 * 2 = 88, way less than 100 which we get if our value range has 100% equity. So if our value range has less than 100% equity, we need to add less bluffs than the ratio we get based on our bet sizing or we are (way) overbluffing. Is there a formula which takes this into account?
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Hey Papa, the questions you put most likely reflect the confusion most players fight with when they start to think about GTO-stuff.
1) The goal of the defender is to keep Villain's worst hand indifferent (!) between his various options. Take a short break - and try to think about a) what that means - and b) what follows from that. How is that related to "auto profit"? Take some minutes before reading on ...
OK. Say, Villain has to options, he can check (and say he always folds after he checks) or he can bet pot. Say, his range is totally polarized, for nuts / air. As mentioned, our goal is to keep him indifferent with his worst hands - which is air in this case. 0% Equity. What is his EV when he checks? Right, it's 0, as we defined that he always gives up. Now, to keep him indifferent ,we have to call his bluffs often enough that the EV of the bet is 0 as well. And the solution for this equation (I won't bother you with the math here) - is the mentioned formula (1 - bet/(bet+pot)). When we call exactly with that frequency, his bets are 0EV.
Why is that important? Simple: when his bets make more than 0EV, he will never check (and fold) - because bet > x/f and he should always take the maximum EV decision. Same by the way if bet < fold (if we call too often), then he should never bluff, as bet < x/f.
Got it?
Now, your formula, where you calculated Villain's EV for betting and Hero's EV for calling is not complete. It does not count what Hero wins WHEN he calls. It's important what Hero's overall strategy's EV is. And as poker is a zero-sum-game, a profit of X bb/hand for ANY TWO in Villain's range means, Hero is losing that X bb/hand. Period. You can't just take the times Hero calls - and say - "oooh - he is winning!". :) What if Hero called only the nuts? Then he still wins 100% when he calls - but he loses the pot in 99%.
To finally get it clear - as you are obviously familiar with EV-calculations. Say, Villain has 10% nuts and 90% air. When he bets pot, he should bet 10% + 5%, and we should call 50%. OK?
Here are the EV-formulars for the overall-strategies:
EV (Villain) = 0.15(0.52p + 0.5-p) = +0.15p
EV (Hero) = 0.150 + 0.85p = +0.85*p
So, in this scenario Hero's EV is 85% of the pot. Which is anytime, Villain gives up. Once Villain bets, Hero has a break-even call (so he could fold as well, his EV is zero - and Villain wins the pot). Agree?
Now, what happens, if Hero defends only 40%? => Villain adjusts and bets any two now!!
EV (Villain) = 0.1(0.42p + 0.6p) + 0.9(0.4-p + 0.6p) = 0.11.4p + 0.90.2p = +0.32p
EV (Hero) = 0.4(0.1-p + 0.92p) = 0.41.7 = +0.68p
DUCY? Villain more than doubles his entire EV - when Villain only "slightly" deviates from his optimal calling frequency. At the same time, Hero loses 20% of his EV.
Everything understandable?
2) As to your second question, yeah formulas relate to polarized scenarios. "Real-game" scenarios that involve merged ranges and/or even refer to earlier streets are more complex by several dimensions. This would be by far too complex to shortly explain it in this post.
actually i think #2 is quite easy to answer
Big NO to your statement. Whoever told you this is simply wrong.
We want to make villain indifferent to calling with a pure bluffcatcher (aka a hand that beats 0% of our value range). You are with your logic somehow trying to make him indifferent to calling a hand that beats our value bets, which doesnt make sense ofc. (since he will just fold everything worse than that given percentile hand and all of our weaker value bets will just be burning money since they are never getting called by worse).
Long story short: you only want to make villain indifferent to call his weakest bluffcatcher, if he has a hand that beats parts of your value range then he is supposed to make a profit with that (which makes you have less than the EV of 100 that you expected us to have)
OMG, you are right - I reread what I wrote and actually I was somehow bound to bets on previous streets. Once we reached the river - where any hand has either 0% / 100% equity against Villain's actual holding we indeed strive to make him indifferent to call with his weakest bluffcatchers.
As an extension of this, when we're looking at river bet sizings:
If our our value range is getting called by a better hand say 20% of the time (So when called, 20% we lose 80% we win), and getting raised 5% of the time, do those two factors "push" the optimal bet sizing down with our betting range? For instance compared to a time where we'd get called by a better hand 5% of the time and never raised?
There seems to be a pushing and pulling factor to bet sizing. The push down effect is that some of our value bets lose some of the time and the possibility of facing a raise is very damaging to some of our non-nut value bets. The pulling up effect seems to be sizing such that we can add as much air as possible and win the entire pot with the highest frequency regardless of OPP's decision with his bluff catchers.
Is there a way to solve using these factors (aside from doing some giant M.O.P calculus sim)?
Thanks guys for your thoughtful responses.
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