There have been some questions just like this nowadays. I think you can still find those threads.
We don't need to defend minusEV hands for the sake of not folding more than X%, our goal should be to play every hand the best way possible. Also, equity is not the primary thing that should be considered with borderline hands. There are hands with 25% equity which will make more money than hands with 40% in some situations. So what matters is how much of our EQ we can realize with a hand.

The nice thing about OTF folding is, villain doesnt know what we fold. So in this case he doesnt learn too much about our play and activly exploit us.

Thank you for the answer,i understand what you are talking about realization of equity but imagine the same question in the river, where equitys are polarized, what can we do if we have to defend hands with less than 33% equity in order to not being exploitable?

Thank you Will, so u are saying that if our opponent have a balance range our worst bluffcatcher is going to have at least a 33% equity?

its my understanding (and basically what Will said) that when we don't know what our opponent is doing we can try to defend 1-A to make him indifferent with his bluffs.

if we suspect he is underbluffing then we exploit him by overfolding.
if he's overbluffing then we exploit him by bluffcatching a ton.

in the situation where he is underbluffing, we only need to defend hands that we suspect can realize x% eq, where x is our pot odds.

There is no way to translate EQ into EV - pre river. You can easily create a scenario where you have a hand with 60% equity on the flop and still can't call a psb. ;)

Above that - listen to Will, he nailed it.

i don't understand what you're getting at w/, "no way to translate EQ into EV - pre river."

consider a situation where we defend from the bb, check the flop and face a bet.

i think the better our backdoor draws are (gutters/straight draws/flush draws to the nuts v non-nut draws) the more eq we can expect to realize. hands in this category may have less eq on the flop than some hands that do a shitty job of realizing eq because there are so few turn/river combos on which they can profitably call another bet.

also, if we have some sort of bluff catcher, the better our hand is at blocking his value region and not blocking his bluffs the more eq we can expect to realize. hands in this category just have more eq in general than other candidates to defend and make for reasonable calling hands on future streets because of their blockers/unblockers.

these seem fairly obvious and useful and I'm often considering R (equity realization) issues when playing. it looks to me like you're making an argument that some of these considerations are useless. seems more likely that i just don't understand what you're trying to say. help a brotha out.

Equity correlates strongly but doesn't cause EV. So when you're confused, it's reasonable to call top X% sorted by equity. However, in the large number of cases where EV and equity diverge, this heuristic will fail you.

You should think about what being exploitable means, and what he can do to you. Haven't read any of the comments so ppl might have said this

Reverse engineering from river and assuming Villain is barreling us with GTO frequency on every street (or attempting to) we need to demonstrate one of two things;

-accounting for indifference, villain's air should have EV = 0. So if his value has flop EV > 300 then we are better off always folding
-villain needs > 8 combos of air to make us indifferent with a pot size on the flop, which immediately makes our optimal strategy to just always fold.

I reverse engineered from river to turn (by which I mean did some very easy algebra with some difficulty) and found that villain is barreling 2 combos of T9 on river (fairly obvious) and 5 on turn (less obvious) in order to make our bluffcatcher indifferent and calling him 1/2 the time. At that point I got a bit lost and couldn't manage to go from turn to flop. Any tips? Is it very similar to the first step?

Not posting my work because it's fairly (needlesly) lengthy and messy but I got following EVs on the turn:

Assuming villain gets there with 8xT9 and 4xA2, we should have:

EV(air) = 0 according to indifference
EV(nuts) = 3001/2 (they fold turn) + 6001/4 (they call turn and fold river)
+ 1500*1/4 (they call turn and call river)
= 675

Overall Bettor's EV is thus:

EV = 2/3(0) + 1/3 x 675 = 225

Being a zero sum game, Caller's EV is 75.

Any tips on going from turn to flop? I feel a bit out of my depth figuring out the Bettor GTO value/bluff range on flop knowing the optimal turn range.

After some trouble, I get to 2/6 river bluffs, 5/9 turn bluffs and 19/27 flop bluffs. Considering villain gets to flop with 2/3 aka 18/27 bluffs, we should fold since his value range is >1/3 of his possible betting range. Therefore villain just bets pot with range and expects to always win 100. He does better if we don't fold. Do those numbers seem correct?

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.

Pepe thanks your calculation really clarified bigfiszh's example

Thanks, though really I didn't post how I got to those numbers.

@ bigfizsh: is the shortcut you're referring to, assuming 'b' the bluffing region of our flop betting range and 'v' our value betting portion on river and same size on every street, the following?

b = 1-v^3

In our example, that would give us:

b = 1-(2/3)^3
b = 1-8/27
b = 19/27

Had some trouble getting to this and probably the generalization only works if we bet the same on every street. For different sizings I guess we have to do reverse induction from river to flop to figure out the optimal contination frequency from one street to the next?

BTW, I found an interesting take away from these calculations is proof of a game theory consequence. Assuming we have a strict bluff catcher vs strict drawing dead air / unbeatable nuts situation, then the bluff catchers we continue with are 0EV on every decision point of the game tree. When reverse inducing though, it's clear that considering that getting to a future 0EV calling/folding spot costs a bet on the present street, then we have to be compensated for the investment. That compensation comes exclusively from villain checking the optimal part of his air to us, making that bet we called neutral. So essentially, as pointed out earlier ITT, we are exclusively dependant on villain having enough bluffs in his range that he has to give up with. If he doesn't then we are paying a premium on one street for the opportunity to flip for neutral EV on a future street, which is obviously always -EV.

I just thought it was interesting to stumble upon mathematical proof of this by hand. I'm sure it's obvious to anyone who's been less lazy than I in their study of game theory :) Just one little toy game and we get to special identities of flop frequencies WRT sizings, proof of the compounding bluffs effect and also a very clear illustration of why early street equities are far from the whole picture. I learned/confirmed a lot from the homework assignment so thanks to bigfizsh as well as OP for starting the thread :)

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?

Yeah, you got it!!

That's exactly the point I wanted to show. Once Villain bets every street with GTO frequency (balanced with 100%/0% hands) and x/f the appropriate airpart of his range on any street, a hand with 90% equity on the flop is as good as a hand with 10% (from our perspective) - as we ONLY win $$ once Villain gives up his air on later streets.

Well done, Pepe!

Very deep and insightful example, Bigfiszh

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)

Solid Thread!!

Hi all and thanks for posting this.

With my words i would say that it will always depends of : how many direct pot odds we have addin with or without implied odds on a present decision, and how implied odds on future street could influence/compense our EVcall on the present street.

Exemple : hero has 78o vs AK (effectifs stacks 150 each) and card exposed for hero on the turn
board : A562 (pot 100) equity (8/44) = 18,18%

AK gonna always put all money on the river, 78 could fold if no draws comes up.AK PSB on the turn which gives (neutral EVcall turn would have been 100)
- EVcall-fold w/o implied odds = 300 * 18,18% + 0% x 50 = 54,54
- EVcall-call w/ implied odds = 18,18 % x 300 + 100% x 50 =104,54

So without implied odds we cant justify a call, with implied odds we are dependant how often he does gonna put the remaining stacks. In my case is 100%, too much optimist...
If vilain check/fold more than
18,18%x300 + X % x 50 = Ev neutral turn = 100
So Vilain needs to bet more than X > 90,92% with his all remaining of stacks (50) on next street to have EVcall turn w/implied odds > 0

It's different that the example that u gave. In the case of big fish we are made hands vs made hands in a perfect polarise (nuts/airs) vs bluffcatch game. In my case, we are made hands vs draws. slightly different coz made hands vs made hands have both always showdown value on river...

for this formula :
b = 1-(2/3)^n between n = 1 and n =3 It's only right with ur assumptions with GTO frequencies on each street (foldin 33%). But if ur opponent does fold less than that then u give him odds to make profitable call(float). Obviously, if Vilain call 100% on each street is better to get n = 1 (maybe 0) than n = 3 coz as u said u give him much more compensation than u should by check/fold. I think it's what we called reverse implied odds... So n must be a variable of frequencies on MDF per street.

Last point, I'm readin slowly and hardly MOP, on chapter 7 figure 7.1, it talks about it.
Bet's EV overall function of the pot size, depends of 2 functions linear,
1) K1X + C1 (avec K1 > 0) direct pot odds
2) K2X + C2 (K2 < 0) implied odds on future street

It shows how the sizebet could increase or decrease ur EVoverall :
So u can start calculate :
- the critical amount of the pot sizebet where ur EV start to decrease (when 1) = 2) X=...)
- The region of amounts where u will have 2 pot size for the same EV's
- Maximum pot size to keep EVoverall > 0

PS: with some algebra u could find K and C, of course im'not using this yet at my micro level but it helps me to understand the effect of the sizebet and implied odds

don't get trapped looking for equity only and not equity+playability

for instance on HU on Ah7c2s you have a lot more equity with 55 than 8h9h but you will choose 8h9h to bet flop and a lot of turns rather than 55

from the defensor point of view, you will fold 55 on AT8r but call J9 even if 55 has more equity here vs a wide range