I'm currently working again through the book Applications in No Limit Hold'em. At the end of the chapter "defending by raising: the value bluff ratio on the flop" I came up with a confusing question/thing

I won't explain the parameters since I would take a lot of space and I guess most of you know the book anyways.

They gave an example of the equation of how many bluff combos you need to add if you raise with nut flush draws with 47% equity.

X*0,47 + 0.2*(1-X) = 0,389

Endresult is that we have a ratio of 2.3 valueraises for every 1 bluffraise.

I did the same equation with an OESD, where on a dry board against (for simplification reasons) 50% value hands and 50% bluffcatchers in villains range I get 44% equity.

Now here's the interesting part, when I solve this equation:
X*0,44 + 0,2 * (1-X) = 0,389
I get a ratio of 3.75:1 - that's quite astonishing for an equity change of - 4%!

I got a understanding questions now:
What does this equation actually tell me? Haha
Since ~39% of my flop bets have to be value bets, this equation tells me how many times I semi-bluff in these 39% when I have, in this example 44% equity?
Or does it simply tell me if I am raising here it needs to be 3.75 (or 79%) times for value with better hands and one time(21%) with a bluff? Here I don't know why the 39% are even in the equation.
I hope you understand my question and can help me out, I'm a bit lost of what I'm actually doing here

I'd just ignore that whole section in the book. You're not playing a toy game with perfectly polarized ranges. You're playing multiple-street no limit poker.
Read Janda's second book and/or get a solver. Those value:bluff ratios for the early streets don't have much real world utility, partly because the terms "value-bet" and "bluff" don't make much sense until the river.
Even if you moved towards more useful categories for hands in your range, such as "high equity hands" and "low equity hands", the ratios still vary from flop to flop, due partly to nut/range advantages or other asymmetries, and this means each flop/situation has its own solution for sizes and frequencies of betting. The five-year-old book that contained "best guesses" of what optimal poker would look like turned out to be quite wrong in many respects.

X(Equity of "value hands") + (1-X)(Equity of "bluffs") = Y
where:

X = percentage of flop bets that need to be "value bets" on the flop, given the equity assumptions above.

Y = (Turn Betting Frequency)(River Betting Frequency)(Percent of
River Bets Which Are Value Bets)

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What this equation tells you, is that if you plan to bet 57%-pot on the turn and river, with a polarized range consisting of "value" hands that have exactly 44% equity, and bluffs that have exactly 20% equity, then you'd need 70% of your bets to be "value bets" on the flop.

This simple model assumes ALL of your value bets have 44% equity, and ALL of your bluffs have 20% equity. If you increase the equity of your "value bets" (44% equity is pretty low for a value bet), then you could bluff quite a bit more.

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The 39% is given by Y (see above). If you don't know why this is here then you've skipped the math on every chapter so far. It's based on the bet size of future streets. See pg 112 for details on how to derive this number.

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p.s.

You're gonna get a lot of 2+2 regs telling you that this math is useless and outdated. While it's true that this model is VERY limited, you can use it to come up with reasonable frequencies in most situations. The trick is to focus on your entire range rather than specific hole cards.

Actually i figured this while reading it the first time - and thinking about the equations given in this book i'm not sure how to use them bc as you said there are so many more variables. Nevertheless i'll work through it, i guess it gives me a lot of very theoretical approaches to get a better understanding of Poker. I'm playing, with breaks, now over 10 years and i have a Solver, but i finally wanted to get deep into the theoratical aspect and understand what a Solver is acutally doing. Not sure this book or this section will bring me there but atm i'm quite enjoying it.

Thanks a lot for the explanation and clearing up my confusion - ok so its all based on the 57% Turn and River bets.

Nono, i do and did know where this number came from, just if my second assumption was acutally true, it didnt make sense.

I guess it could be, yeah. If i split my Range correctly in Value Hands and Bluffs and determine the equity it could help me to have a balanced strategy to the river. I already did it for some situations and im eager to try it. But as said above by ArtyMcFly, does every value hand at the flop, really stay a value hand on the river? Ofc not and thats quite worrying to me, there i see the limitation in it.
The second thing im really worried about is, that this model doesn't take in anyway the villain with his range etc along. I know for the purpose of it, making the villain indiffrent, its not really necessary, nevertheless, while actually playing poker i'm heavily making my desicions based on the opponent. But i guess i can come up with some really good frequencies on not too complex boards.

This book represents what GTO looked like before the advent of solvers. It's built on very abstract models, like most science.


Right, in actual poker equities can shift drastically from street to street. Some turn card might give someone a massive range advantage, but in general luck balances out.

If you really want to practice this I'd recommend testing your strategy on a site like RangeVsRange. This is poker played with ranges instead of hole cards, so you could test very precise strategies.

It does account for villain's range. Villain's range effects your equity, which directly affects X.

This model isn't designed to exploit villain. It's designed to give you a rough idea of your overall betting frequencies.

Stupid question: on the river, if you could bet an infinite amount, this would give your opponent 1:1 odds --> you should have 1 bluff for every 1 value. So is it correct to conclude then that in any river situation ever, we should literally never have more combos in our bluffing range than we do combos in our value range (since we can never lay worse than 1:1 for our opponent)?

yes, that's correct