first calculate your Z (optimal bluffing frequency )

Z= betsize / total pot
Z 1/5 = 20%

your betting range should have 80 value and 20 bluff

u know u have 20 value and this should be 80 ouf your betting range
then u can calculate u should bluff 5

your total betting range would be 25
where 20 is value (80%)
and 5 is bluff (20%)

in second point i imagine u try to calculate the villian fold frequency to make your bluff Ev=0

EV Fold = %fold * pot
EV Call = %call * ((pot * eq ) - cost)

• %call is the same than = 1-Fold%
• we are tying to calculte the point where Bluffs are indiferent then eq = 0

Ev fold = %F * 3

EV call = (1 - %F) * (( 0 ) - 1)
EV call = (-1 + %F)

Ev call + Ev fold = 0
then
%F *3 + ( -1 + %F ) = 0
4F = 1
F = 0.25
villian should fold 25% to make your bluffs indiferent

check this post:
https://www.runitonce.com/chatter/gto-simplified/

@OP: I admit, the formulas and the explanation for that spot are pretty atrocious, so I'll try to re-formulate it in a more understandable way.

First of all - let's imagine we have a baseline, the "clairvoyant" defense strategy of B. That means, B always calls, when he has the best hand (A bluffs) and always folds if he has the worse hand.

The MES' value now will be compared to exactly that value!! B's strategy will be predefined, always folding if A bluffs < 5%, always calling if A bluffs more than 5%.

Now, the following scenarios are possible:

=> A bluffs 0%. B never calls (the <5% rule applies). That makes an EV of 0. What's the EV if B had seen A's cards and always called if A had bluffed? Same, it's zero - as A never bluffs. This means, the difference between B's strategy and the "potential" EV he could've realized is zero. (First line in the chart).

=> A bluffs 2%. B never calls (the <5% rule still applies). That makes an EV of 0. What instead, if B had called everytime, A had bluffed (if for whatever reason B had known about the bluffs, like a tell)? Well, than he had won 3 bets in 2% (x), which makes a total EV of 3x (= 0.06). As he actually folds (instead of owning A), he loses 0.06 in value.

=> A bluffs 4.8%. B never calls (the <5% rule still applies). That makes an EV of 0. Had B owned A by calling everytime A had bluffed, he had won 4.8% times 3 (= 3x), so he actually forfeited 0.144 by folding all the time.

=> A bluffs 5%. B now start calling (according to his rules, even though we know it's irrelevant at this special point). That makes an EV of (5% x 4 + 20% x -1) = 0 for every call. Had B "correctly" only called in case A had bluffed, his EV would've been 5% x 3 = 0.15. This means, he's losing 0.15 in value (= x - 0.2).

=> A bluffs 10%. B still calls all the way. That makes an EV of (10% x 4 + 20% x -1) = 0.2 for every call. Had B "correctly" only called in case A had bluffed, his EV would've been 10% x 3 = 0.3. This means, he's losing 0.1 in value (= x - 0.2).

You see, the formulas do nothing else than showing the "calculated" difference between a "clairvoyant call-when-I-am-best" strategy and the real MES strategy. And B's loss (or win) is A's win (loss) and vice versa. So, A's best option is to find the point where B's optimal defense strategy (EV) shows the biggest difference to his actual hot-cold-equity (which he can't realize as cards are hidden).

In the first formula (always fold strat), this difference is defined by the times when B is not catching a bluff (and lets his opp. get away with it) which is defined by the bluff frequency of A.

In the second formula (always call strat), it's defined by the times, when B is "incorrectly" calling (and losing one bet), which is fixed at 20%.

The formulas simply were "born" from re-constructing the original EV-formulas, up to the point where nobody really understands anymore what happend (sth. the authors were really proud of obviously).

Got clear(er) now?

BigFiszh

Hi, yes thank you it's way more clear now. However, in the first formula, I would think the pot to be 4 bet, since it seems to be described like the pot is 3 bet. So I would think the EV should be 3bet + 1 bet (4 units). And in the second formula, you seem to consider the pot to be 4 bet (since you make it 4x - 0,2) .

And in the book the formula is not even correct since it does not even say to multiply x by pot size in the second formula.

Hey buddy, you're partly right, partly wrong. :) Right in having caught me, my explanation is a bit fuzzy and in parts simply wrong.

Wrong in claiming that the book is mistaken.

But as I am coaching that stuff, let me restore my honour and take another attempt. :)

So, grab a cup of coffee, free your mind and fasten seat-belts, we are starting the engines …

There's 3 bets in the pot. Player B has 20% equity (hot-cold), means, if it's just checked down, he will win 0.2 x 3 bets. That's the baseline. Let's call it "calculated EV".

Now, player A starts betting. He starts with value-only and then adds more and more bluffs.

Player A wants to develop a strategy where - whatever player B does - the profit for A gets maximized (and minimized for B).

OK?

Let's call the EV that results from B's actual strategy the "realized EV". So, A's target is to maximizing the difference between calc.EV and real.EV.

Now, if B would fold anytime A bets, B would "lose" the pot anytime A had actually bluffed. In terms of current EV B's EV when folding is zero, but actually he gave up his share, which had been +3 (the pot), in case he had 100% equity (against a bluff). And here we are not talking about gaining even more, if B had called the bluff - we are simply talking about B not winning the naked pot which he originally was deserving.

That means, B's loss when he folds is 3 times the percentage, A bluffs. If A never bluffs, B's loss is zero (0% x 3 = 0). A valuebets 20% (B folds), A never bluffs - so B everytime takes down the pot, when he has the best hand. If A bluffs 2%, B will "lose" (= give up) 3 bets in 2% (as he always folds to a bet) for a total of 0.06. And so on. Still with me?

That leads to the first formula. As long as B sticks to the "always fold"-strategy, his EV-difference between "realized EV" and "calculated EV" would be (bear with me that I will substitute the "X" from the book by "B" - as bluff percentage - as it would otherwise collude with the multiplication-sign later on):

3B (-> according to 3x in the book)

We can "prove" that by subtracting "calc.EV" from "real.EV":

real. EV = (1-0.2-B) x 3 (-> The bet% of A is 20% + B, and as we are always folding, we only win the pot when A does not bet, which is 100% - 20% - B)

calc. EV = (1-0.2) x 3 (-> B's original share - namely hot-cold-equity = 80% times pot)

delta EV = (1-0.2-B) x 3 - (1-0.2) x 3 (-> simple subtraction)

Now we do some formula work to simplify, which eventually leads to:

delta EV = (3 - 3x0.2 - 3B) - (3 - 3x0.2)
= 2.4 - 3B - 3 + 0.6
= 3B

Nice, huh? It gets better! :D

OK, that was the warmup, let's go to the more complicated part. What happens if B always calls?

B wins the pot, when A checks, wins the pot + one bet when A bluffed and loses his bet when A valuebet.

Let's put up the entire EV formula:

EV (real.B) = ((1-bet%) x pot) + bet% x (EQ x (pot + 2 x bet) - bet)

Now let's define some of the variables before we continue:

bet% = V + B (-> bet% = value% + bluff%)
EQ = B / (V+B) (-> B's EQ is the ratio of bluffs to total range)
V = 0.2 (-> 20%, the value% of player A)
pot = 3
bet = 1

OK, here we go:

EV (real.B) = (1- 0.2 -B) x 3 + (0.2+B) x (B/(0.2+B) x 5 - 1)
= 3 - 3 x 0.2 - 3B + (0.2+B) x 5B / (0.2+B) - 1)
= 3 - 3 x 0.2 - 3B + (0.2+B) x (5B / (0.2+B)) - ((0.2+B) x 1)
= 3 - 0.6 - 3B + 5B - (0.2 + B)
= 3 - 0.6 - 3B + 5B - 0.2 - B
= 3 - 0.8 + B
= 2.2 + B

Now, what's the delta.EV?

Remember, delta.EV = real.EV - calc.EV:

delta.EV = (2.2 + B) - ((1-0.2) x 3)
= 2.2 + B - (3-0.6)
= B - 3 + 0.6 + 2.2
= B - 0.2

Now - substitute my "B" by "X" (as it is used in the book) and you'll recognize the formula from the book. Yieeehaaa!

So, what you have now, are (brutally) shortened formulas to calculate the EV-loss for B, that A's bluff-strategy results in, for each counterstrat of B, always calling or always folding.

Obviously, B will always take the strat (call or fold) that yields the highest EV in, depending of the bluff% of B. That leads to the strat-switch at 5% bluff frequency.

And against B optimally defending (switching from folding to calling just in the right moment), a bluff frequency of 5% leads to the biggest delta.EV (= EV-cut) that A can hope for.

And that was what the authors wanted to proof.

Everything "clear" now? :)

BigFiszh

PS: Everybody who reads this line - and did not skip the rest - earned my serious respect! :D

when "B" (or "X" in the book) is bigger than 20% results start to be positive while when B is lower than 20% results are negative

it is because?
Delta EV = Ev real - Ev calculated

Delta EV = (1-0.2-B) x 3 - (1-0.2) x 3

seems to be always negative result (except if B is a negative number wich cant be posible)

im missing something?

haha im stupid
when Bluff is bigger than 20% A is bluffing more than valuebeting then B is always calling
right?

It's completely okay that deltaEV is negative!! That's the "right" of the aggressor. B's only target is to keep the "negative number" still as big as possible. That's all he can do.