I'm not sure how to explain this any better than I did in the video. If we bet our entire range for <1/3 pot then villain can profitably call his whole range, yes. But if villain calls some non-zero amount vs a <1/3 pot bet then we can increase our EV by betting some of our value hands (but no bluffs!) for the smaller size and betting the remaining value + bluffs for a bigger size (while still not allowing him to profitably call the bigger bet). Since we can adjust to increase our EV, he is not playing GTO. Therefore, the GTO call frequency vs the smaller bet is 0.

What you are saying is true if STACKS are 1/3 pot or less. It is no longer true when stacks are deeper than 1/3 pot. I think I give an example in the video, I know I give a detailed one in the thread you linked showing that villain having a non-zero call frequency vs a small bet is exploitable.

for that bluff to value ratio, betting 1/4 pot is optimal and yields 1
P in EV AND Villain must call 80% for our bluffs to be indifferent to
checking of betting.

There are infinite GTO strategies here. Betting 1/4 pot with range is one of them, but not the only. Proof that calling 80% vs the 1/4 pot bet is not GTO below.

Assume villain should call 80% vs a 1/4 pot bet, and fold to a larger one. If we bet 1/4 pot with range, our EV is just P (we'll use P = 1 to make things simpler). Then we adjust and bet pot with 2 value and 1 bluff, and bet 1/4 pot with remaining 2 value. Then 3/5 we win the pot (we pot it and he folds), 2/5 we win the pot 20% of the time when he folds and 80% of the time we win the pot + 1/4 pot bet.
EV = 3/5 * 1 + 2/5 * (0.2 * 1 + 0.8 * 1.25)
EV = 1.08
We have increased our EV by adjusting, therefore villain's strat was not GTO.
Villain's GTO strat is to fold to any betsize.

Hey Steve, thanks for taking the time to answer.

You did indeed a good job in the video explaining your point.

Assume villain should call 80% vs a 1/4 pot bet, and fold to a larger one

That is not a valid assumption : if you bet a larger bet, let's say Pot and your bet is balanced like in your example, GTO opponent will call optimally (50%) to make your bluffs indifferent (to folding or calling). However, because you are balanced, his calling frequency doesn't change your expectation, which is always 1 (Pot).

Regarding your example : the problem in your reasoning is your adjusted strategy is only valid if the opponent doesn't counter adjust. But a GTO solution implies you cannot unilaterally increase your expectation by changing your strategy, knowing that your strategy is exposed or opponent is clairvoyant (Nemesis) and opponent WILL adjust.

So against your strategy, the Nemesis will adjust by never calling your 1/4 bets and by calling optimally your Pot sized bets. Your EV will now be decreased (< 1 Pot) instead of increased.
If you bet a 1:4 bluff to value ratio range, the GTO (maximally exploitative) bet size is 1/3 Pot; your EV is 1 P and you cannot increase that EV against a nemesis.

I you are right, you should be able to find a GTO solution for your example, playing against a clairvoyant opponent that will continually adjust to your known strategy.

So here, if we bet a 1:4 bluff to value ratio,
- V. should always fold IF we bet bigger than 1/4 Pot (EV call always < 0)
- V. should always call IF we bet less than 1/4 Pot (EV call always > 0)
- at exactly 1/4 Pot, Villain should call 80 % to make our bluffs indifferent to betting or checking (EV call always 0)

a mistake here in my first reply : the GTO bet sizing is 1/3 Pot for a 1:4 bluff to value ratio range.
Also, the Optimal Calling Frequency is 75%, not 80 %.

I should've refer to my own GTO simplified (OTR), lol.

edit: oops double post, removed the first one.
Robert, you are confusing GTO and nemesis strategies. The nemesis of GTO is GTO, but GTO is not the nemesis to all other strategies. GTO is a fixed strategy.

I will try to explain what Steve is saying. What does it mean that you dont have enough bluffs? It means that there is some betsize (usually large), that allows you to bet all your bluffs and appropriate number of value hands and you still have some value hands left. If this is the case, then you can bet those excess value hands for whatever sizing you want (even 1bb or you can even check them in theory, but its stupid in real game) and you do not need to balance those (smaller) bets with any bluffs - your EV is the pot with the "nuts only" range and this is the maximum you can get. And you do not miss any bluffing opportunities, because all of your bluffs are already in the large sizing range. Now the tricky part: if the situation is as above, then you can also transfer some of the bluffs to those smaller sizing ranges, making sure you use all of them (and of course in a balanced way). This forces your opponent to fold to any bet you make. To see why, imagine this: you use all of your bluffs in the range that bets the maximum possible sizing and bet some other sizing with the rest of the value hands. What is his best response to this strategy? He has to fold to the smaller sizings and against the max sizing, he can do anything as his EV is always the same. But only the strategy that folds everything even to this max betsize is GTO. Because if he ever calls, you can exploit him by transfering some bluffs from the max sizing range to those lower sizing ranges. Your EV on the lower sizings will not change and your EV on the max sizing will go up = you exploited your opponent (while remaing unexploitable).
So there are infinitely many Nash Eqs, all of the form: he folds everything to any bet, you bet one or more sizings so that each range corresponding to a sizing is either nuts only or a balanced mix (with respect to its sizing) of bluffs and value AND you bet all of your bluffs in the proces. He doesnt want to deviate from " fold everything " because he cannot gain by doing it and he can loose, you dont want to deviate because you win the pot with every hand.

Example: My range is 6 combos of nuts and 2 bluff combos and we are 2 pots deep. I bet 1 combo of nuts for 1bb, 2 combos of nuts for pot, and 3 combos of nuts + 2 combos of bluffs for 2 pots. What will happen if you try to bluffcatch against the 2 pot range? I can change my strategy to bet 1bb with 1 nut combo, bet pot with 2 combos of nuts and 1 bluff combo and bet 2x pot with 3 nut combos and 1 bluff combo. This is balanced and exploits you. Anytime you bluffcatch against any part of my range , I have the tendency to deviate, meaning we are not in a Nash Eq. I do not deviate only when you fold against any betsize.

"You're saying that a 1/2 pot bluff has to work 67% of the time to show profit?"
If he wasn't clear I'm sure he was just saying the villain has to call more than 67% of the time to make the bet unprofitable. We just need to to work more than 1/3 of the time to show an immediate profit in that spot.
.5*3 = 1.5, we bet 1/2 pot 3 times, we get 1 fold and 2 calls we break even. win pot 1 time, lose 1/2 pot bet twice

I dont get it also shouldn't the formula be b/(b+1) instead as when P1 bets with sizing b it will show a profit for P1 if it works more than b/(b+1)

Good catch, just had it backwards in the slide. The bluff needs to work b/(1+b) and so bluffcatcher calls 1/(1+b). I fix my mistake in the P1's river EV slide 2 slides later, where I have P2 calling 1/(1+b) implying that P1's bluff works only b/(1+b) of the time.

Thanks!

well, in this game all of your bluffcatchers are equivalent and there are no card removal effects. Villain bets enough bluffs at equilibrium so that your EV of calling is 0. So if when he bluffs 39 combos your EV of calling is 0, then when he only bluffs 38 combos your EV of calling is negative and the exploitative adjustment is to never call.

In practice, your bluffcatchers are not all equivalent. Say you get to the river bb vs bu on K94r33 where your opponent has bet flop, bet turn and now bets river. Your range is going to be very bluffcatcher heavy and your bluffcatchers are mostly Kx and 9x. Hands like K5 and A9 are equivalent in that he's never vbetting worse than K5 but better than A9, but K5 is a much better bluffcatcher thanks to the K blocker. So here if villain bluffs 39 combos (random number) to make A9 indifferent to calling, K5 is still going to be a very profitable call if villain bluffs 38 combos.

Thanks for your answer Steve I now get it. I didn't fully understand that this toy game situation is either call or fold and what the calling threshold tells us is how much we profit from calling rather than how much we should call to make his bluffs indifferent. I was probably already thinking how to apply it in real game because I still have no idea how to approach the bluffcatch situation and I think that if I solved this problem my winrate at NL50-NL100 would skyrocket. Since you know a lot about the subject do you recommend any easier practical read or video that covers the subject in real game situations?

here's the link: http://www.runitonce.com/nlhe/theory-behind-bet-sizing/

No, the mdf is ok, but you have to chose the right spots, its not universal. One of the first theorems in any gto textbook is this: IF you mix between several actions in a gto strategy, then those actions have the same EV. In our case: if you sometimes bluff and sometimes not, then ev(bluff)=ev(giveup)=0. From that you derive the mdf.
The important part is the IF. When we do not have "enough" bluffs, we bet all of them, no mixing occurs and you dont have any equation to derive the mdf from.

Luke,

Thanks for your comment. So if the board overwhelmingly favor villain, and he simply smashes it so hard that he won't have enough bluffs, the MDF goes out the window and we fold everything. (Eg 4 bet pot in PLO on A94r)

If he has some bluffs that he will give up, we have to use MDF to prevent him from profiting with ATC.

Am I on the right track?

Bernard: yes, thats the general idea. If he has few bluffs and many many value hands, you really cant stop him from bluffing profitably...if he has a lot more bluffs than value hands, then you need to prevent him from bluffing them (all) profitably.

Hey, it's been a couple weeks since I made the video, can you be more specific about which formula you're interested in?

That's the tough part unfortunately.

equity is your probability of winning the pot if there's no more betting (like if you're all in)

betting 10 value hands means 10 combinations of hands. So AcKs is one combination, AcKs and AsKc is 2 combinations, etc.

Just finished and submitted it. Not sure when it will be released. Thanks for the reminder!

Not that I'm aware of. Excel is pretty useful if you have any experience with it.

Careful with order of operations. (1+b)/(1+b) is 1 for all values of b. 1 + b/(1+b) is only equal to 1 for b = 0.

So to give an example, take b = 2. What you are doing is:
[(1+2)/(1+2)]/2 = [3/3]/2 = 1/2 = 0.5
What you should be doing is:
[1 + 3/(1+3)]/2 = [1 + 0.75]/2 = 1.75/2 = 0.875

Steve

oops yes not sure why I changed b to 3 in the second part there...

I see one minor one talked about in the comments, are there others people should be aware of?

Very old video, so but one I put a fair bit of work into (along with part 2) so would appreciate it if you have fixes to point out.

yeah that's the mistake pointed out in the thread, just a typo that is fixed later in the video.

You are correct that we use 1 because we call the pot size 1 and then the size of the bet is expressed in % of the pot (so if you bet half pot b=0.5). You could change it so that p is the money in the pot but then b would have to be put in dollars as well (so if betting $5 into $10 then change 1 to $10 and b = $5).

https://www.runitonce.com/poker-training/videos/betsizing-part-2-2/

If you call less/more than indifference P1 can increase his EV by bluffing always/never. Player 2 reduces player 1 overall EV by calling to make bluffs indifferent. Of course if somehow player 2 knows how often P1 is bluffing then go ahead and deviate.

(I'm assuming you're talking about the 10nuts/10air example, not the 10nuts/5air, since in that one indifference doesn't matter)