singdean

but it does matter, Bigfiszh, on second thought.

In such toy game, saying that you have 90% equity on the flop is synonymous with saying that you are losing to 10% pure nuts and beating 90% dead hands. Likewise. saying you have 10% equity means you are losing to 90% pure nuts and beating 10% dead hands.

So when villain has exact optimal number of bluffing hands in his beginning range, say 19/27 or 70%, you will have 70% equity for all your bluff-catchers, and you can call optimally till the river and break-even at 0 EV. But when he has fewer than optimal, say 18/27 or 66%, you will have 66% equity, and defending with your bluff-catchers will be -EV. So you will have to fold on the flop immediately. Which means

having enough equity against his pure nuts&dead hands
= villain having enough bluffs in his beginning range
= 0 EV when playing optimal defense

The compounding effect of bluffing is quite astounding; you actually just need between 5 to 6 streets of geometric betting to have a beginning range of 10 value hands and 90 bluff hands

5 street betting: 1-(2/3)^5 = 87% (13% value hands, 87% bluff hands)
6 street betting: 1- (2/3)^6= 91% (9% value hands, 91% bluff hands)

Very deep and insightful example, Bigfiszh

Pepe, such an interesting point you highlighted about the compensation for our call this street coming from the air-hands that villain gives up on the next street.

It does seem like the flop call of -$100 is compensated by the 1/3 time villain goes check-turn&check-river (1/3X$300 = +$100), and the turn call of -$300 compensated by the 1/3 time villain goes check on the river (1/3X$900 = +$300), hence making every node 0 EV for the caller if both parties are playing GTO. A salient point not being mentioned at all in MOP , if I recall correctly.

Going back to Peibol's question, that means hero can defend with hands that have less than 33% equity on the flop if he believes that villain has more-than-optimal number of bluffing hands which he will give up on future streets. However, wouldn't that be indirectly saying villain's flop betting range is unbalanced with too much bluffing hands in it?

Pepe thanks your calculation really clarified bigfiszh's example

I wonder if taking peibol's scenario to an extreme will help clarify the situation:

What if even the very best hand in your defending range has less than 33% equity? Will you still try to call with 50% of your range just to be unexploitable? No. I believe you will fold all hands and just walk away from the game.

What if you have 100 hands in your defending range, and only the very best hand has >33% equity, while the rest have have <33% equity (hence calling with them makes it -EV)? Again, I believe that defending with 50% can't be a Nash equilibrium, since you can always unilaterally improve your situation by folding, which is 0 EV.

Since calling with a hand that has less than 33% equity is -EV, and that you can always improve your situation unilaterally by folding such a hand, that should mean that your worse bluff-catcher in your defending range should have at least 33% equity. Hence those hands with less than 33% equity should not be considered bluff-catchers, and should not be in your defending range.

Yeah, except when Villain has so many nuts vs nut-blocker hands that villain can just simply claim the pot.

But back to the question, just wanted to point out that hero will never be in that difficult situation which Phishyfishy described where hero will only be calling with only nuts and folding all 2nd nuts when raised by villain, as long as villain only bluffs with the netblocker.

cos when hero has the ace, hero actually blocks the nut-blocker, so villain will never be raising him.

"Assuming he has equal combos of A and K high flushes, then am i right in thinking he only ever needs to call with A high flushes?""However, if hero is 100% certain that villains raising range will always contain the nut blocker (either as a bluff, or the nut flush)"

When Hero has an A-high flush, he will never get raised by villain. Since the ace card is in Hero's hand, villain will never have the nut-blocker (you mentioned that villain raising range always contained the nut blocker). Hence there will be no scenario where Hero will be pondering a call with a calling range consisting of 50% nut flush and 50% 2nd nut flush.

When villain raised, hero will always have 100% K-high flush in his calling range. Hence if Hero's goal is to be unexploitable (i.e. make villain indifferent to bluffing), the calling range should just simply be the GTO ratio 1 minus alpha - 50%.

The villain's range should not come into consideration at all when pondering a call with a 100% K-high calling range.

Hi GT/Ben - I see your point. Very interesting. So you guys will bet the universe just to pick up a "guaranteed" dime, as long as you are convinced you are playing GTO.  Von Neumman, what have you started!?

Hi Ben, I'm not very clear what your thought experiment is
trying to elucidate. I assume what you are trying to say is that in the long
run even having very small edges is sufficient to decimate your opponent, if
the two of you play long enough? If that’s your point, I totally agree with it.



But what I am instead asking is an epistemological question
- how sure are you that you do in fact have a small edge? How do you know that
you know? 



Game Theory came to
the conclusion, laden with assumptions, that the edge is $200 and the standard
deviation +-$40,000 in this spot (being a gambler he doesn't worry about the
standard deviation). However, making a slight change in his assumptions may
lead him to conclude that the edge is -$500 with another standard deviation or
+$800 with another standard deviation. He can’t be sure of his edge.

I think polar raised a valid question about how small your edge is in the situation you had described. Your calculation showed that you could win 0.5% over the long run by risking 100% now (win $200 but risk$40,000), but you used so many assumptions about your opponent to reach that conclusion. my question is, what is the standard deviation of your forecast, or how confident are you that you really have a small positive edge? if changing your assumptions slightly will lead to vastly different conclusion, that means the standard deviation of your forecasted edge is large. an example I can think of, used in Nassim Taleb's book "fooled by randomness", is if someone tells you the Mars is on average 26 degree, but one std dev of his forecast is -200 to +300 degree, will you make the trip there? I think there's a saying in the financial market that goes " it's better to be roughly correct than precisely wrong".

I came to this site cos I love Phil Galfond's videos,always insightful, well-articulated yet never veering from common sense.