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Originally Posted by daviddem
OK I will solve that AKQ game first, then I will try to show why I think the situation is similar in your scenario for this hand. Then you can prove me wrong.
Hero is IP with a K, villain is OOP with an A or a Q. Pot size is 1, bet size ( all-in) is b. Villain can check or open shove b. Hero always checks behind when checked to and can either call b or fold when shoved into. It makes sense for Hero to always check behind when checked to, because Villain always calls with an A and always folds a Q, so there is no sense in betting. Now this game may seem unfair because Hero always has a K and Villain knows it. In real poker that would correspond to a Villain who is good at putting Hero on a range and the action on previous streets tells Villain that Hero has neither a nutty type of hand, nor complete air. Hero is not bad himself and he knows that Villain knows that, and he also figures that the range of Villain when he bets is made up of nut hands and bluffs.
Villain's strategy: always bet an A and bet a Q a fraction x of the time
Hero's strategy: fold a fraction y of the time
Preliminary remark: since villain bets all his A and a fraction x of his Q, it means that, whenever villain bets, he holds an A a fraction 1/(1+x) of the time and he holds a Q a fraction x/(1+x) of the time.
Good so far. This is a good way to model facing a polarized range.
Originally Posted by daviddem
If both players play unexploitably ("at balance"), Villain's EV of betting a Q is zero (betting or checking a Q has the same expectation for him), and Hero's EV of calling is zero (calling or folding has the same expectation for him).
Villain's EV of betting a Q:
EV = y*1 - (1-y)*b
EV = y - b + yb
EV = y*(1+b) - b
Hero's EV of calling:
EV = (x/(1+x))*(1+b) - (1/(1+x))*b
EV = (1/(1+x)) * ( x*(1+b) - b )
When we zero both EV's, we find:
x = b / (1+b)
y = b / (1+b)
Which is the familiar bet / (pot+bet) ratio.
You were doing well until you got to this part. Nobody knows what x and y are at this point. I'm going to assume that y is how often Hero folds? If so, you have Villain's EV of a bluff being:
<Villain, Bluff> = (Hero folds)(win pot) + (Hero calls)(lose bet)
And that's right. You defined x as the percentage of the time that Villain bluffs with a Q, so you have Hero's EV of calling:
<Hero, Call> = (Villain's bluffing)(win bet+pot) + (Villain's value betting)(lose bet)
And this is right. You set both EVs to zero, and you get.
0 = y*(1+b) - b
0 = (1/(1+x)) * ( x*(1+b) - b )
Note that he's "pulled out" 1/(1+x) in the second equation if you're trying to follow along. If you solve these equations, you find that x = y = b/(1+b) like he's said. This is correct. So far, so good.
Shortcut: Villain's balanced bluffing range is always an alpha:1 ratio of bluffs to value bets in river all-in spots (or y:1 in the above). Hero's balanced folding range is to always fold the bottom alpha % of hands (x in the above). This would have saved you a lot of time and chances for mistakes.
Originally Posted by daviddem
So now with your scenario for the hand above, the pot OTR after your $6.75 turn bet is $16.84 and the bet size OTR is $12.66. That gives us a bet/pot ratio b of 0.7518.
To be unexploitable, villain should bluff x=42.9% of the time with the bottom part of his range and check the rest of that bottom part.
The bet size is $12.66, and the pot size is $16.84. The alpha value is 12.66/(12.66+16.84) = 0.429 or 42.9%. Villain will bluff with a ratio of 0.429:1 to value bets when balanced here. Along similar lines, Hero will fold 42.9% when balanced.
Originally Posted by daviddem
It also means that when Villain plays unexploitably and he bets, he has almost exactly 30% bluffs and 70% value hands in his range (a ratio of 0.429:1, just like in Spoon's signature's post).
Shortcut: You can do bet/(2*bet + pot) to get what percentage of Villain's balanced betting range will be bluffs. Here it's 30.0%.
Originally Posted by daviddem
To be unexploitable, Hero should fold y=42.9% of the time, and so he should call a seemingly massive 57.1% of the time (that's where I completely messed up when I said above that Hero should only call 12.9%).
Yep. That was your big mistake, and it makes an absolutely huge difference. Note that now Hero can still call like 15-20% of the time and be exploiting Villain to a large degree without over-exposing himself to be exploited in return. In effect, you're giving up a small amount of EV currently to protect the larger part of your EV in the long run.
Originally Posted by daviddem
As an aside, it intuitively seems like an awful lot of calling when we are crushed 70% of the time, but this is exactly made up for by the 30% times we win his bet AND all the dead money in the pot. Some reasons it may feel weird to call so much is 1) we are used to try and make fat +EV plays rather than 0 or near 0 EV plays (especially at microstakes), which is another way of saying that we tend to try to play closer to an optimal exploitative strategy than close to balance and 2) we are not used to play against Villains who have very balanced ranges themselves.
It's more about not understanding how a small degree of balance will help your EV more than it hurts it. Poker players who start studying and trying to get better get geared to think in terms of making all of the plays that they can that are +EV in a vacuum and avoiding all of the plays they they believe are -EV in a vacuum. It's hard to see beyond that since it's so easy to question it and stay in the comfort zone of trying to play optimal exploitative poker.
More to the point: I'm not advocating a balanced strategy in this hand on the river. I'm advocating a not-quite-optimal exploitative strategy.
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