I recently purchased Polks HU series. I was shocked to learn after all of these years of playing I really can't figure out the fast math required to figure out optimal bluffing frequencies and spots based on mine and villians ranges. Don't get me wrong I know a good turn card to barrel but I was amazed that Polk had taught himself to know the math required on the fly. Does anyone have any effective study techniques to learn how often a bluff needs to be called to be profitable on the fly??

I don't know how exact Doug wants to be in the HU series.

Some bluff ratios are easy to remember with bet sizing.

If Doug is using the actual combos he assigns to the value range vs the bluff range, and then computing the exact bet size, then that would be the exact bet size required to profit the size of the pot before the bet was made, assuming Doug's ranges are correct, or any range errors are conservative and errors favor the bettor (such as exploiting nitregs).

What level of accuracy is Doug teaching in the material?

Getting more exact than these would be really hard to do on the fly.

Bluff/(pot+bluff+call) is the formula for break even percentages.

So the above is just break even, and is just with bluffs. To earn money with the bluffs, such as to profit the size of the pot, you would want your pot size bluffs on the river to be called only 25 percent of the time. At that point you will profit the size of the pot just by bluffing, over the course of your bluffs.

and if sizing being a little bigger increases the likelyhood of them folding TPWK/Good Under pair/middle pair etc by a % that greatly increases the likelyhood of you taking the pot down, the "Fold Equity" is a hard to calculate but a defo important factor in bluffing, if you have 0 fold equity, you should be bluffing 0% of the time.

But if you think he will fold 25% of the time with a 1/2 pot bluff, but folds 50% or more with a bigger size, than the equity gained from taking it down more than ur share adds up. and I don't mean much bigger, a slightly not exact half pot bet can work better than some exact half pots.

but as everything in poker, Villian dependent. (but MATH is MATH)

thoughts?

I’m not sure this is directly answering OP but it is related. As I read the OP, there are two different questions.

1. How does hero determine optimal bluffing frequency?
2. How often does villain need to call for hero to not profit with a bluff?

Assume hero has only 100% value hands and 0% bluff hands and villain has only bluff-catchers. If P is the pot after hero bets B on the river with villain either calling or folding, we use the indifference principle to arrive at the following:

1. To make villain indifferent to calling or folding, hero bluff frequency = B/(P+B) =1/PotOdds.

If hero’s value hands have less than 100% equity he needs a greater frequency of value bets. If his bluffs have some equity, he can bluff more.

2. For villain to prevent hero from profiting with a zero- equity bluff, he needs to call at frequency >= P/(P+B). This is called the minimum defense frequency (MDF). Conversely, villain needs to fold with frequency of B/(P+B) or more for hero to profit with a pure bluff. This has been termed Alpha.

I think Janda's Applications book has the formula for correct bet sizing if villain calls with frequency = MDF (no raises), not something to do on the fly however.

The above is well known. With the given assumptions, the simple equations shown allow one to do the math on the fly. With less than 100% value equity and/or more than 0% bluff equity the situation is more complex and the equations are more complicated. You can develop tables for various equity combinations that can hopefully guide you towards reasonably correct math-based GTO-like decisions. I suppose one can use a GTO solver to see how well the straight-forward EV math results compare.

Let's assume your bluff has no equity. Then the EV of your bluff is:

EVbluff = F*P - (1-F)*B

Where:

F = villain's folding frequency
P = pot size before your bet
B = beside

Now we want our bluffs to be profitable. Mathematically that means we want

EVbluff >= 0

To hold true. We can now solve for F in terms of our pot and bet size:

F*P - (1-F)*B >= 0
F*P - B + F*B >= 0
F*(P+B) - B >= 0
F*(P+B) >= B
F >= B/(P+B)

So the amount villain folds needs to be greater than the ratio of your bet to the pot plus your bet. You can make this substitution to get a number based on common betsizes expressed as fractions of the pot:

B = X*P where X > 0

Examples:

Half pot bet:

F >= .5P/(P+.5*P) = .5/1.5 = 33%

75% pot bet:

F >= .75*P/(P+.75*P) = .75/1.75 = 42.85%

Pot sized bet

F >= P/(P+P) = 1/2 = 50%

If you want to know the calling frequency instead of the folding frequency you can just subtract the folding frequency from 1 or 100 if you're using percents:

Examples:

Half pot bet:
1-.33 = .67 or 67%

75% pot bet:

100% - 42.85% = 57.15% or .5715

Pot bet:

1 - .5 = .5 or 50%

Now keep in mind we ignored any equity your hand might have when it is called, or EV on future streets if called, or rake considerations. All of those can effect how successful your bluff needs to be on this street to be profitable. There is no mathematical formula that will get you that information which is why Bob has suggested this model is not well suited for streets of play prior to the river.


Edit: technically B is the amount you risk so if villain has a shorted stack than you B would be the remaining stack size for villain regardless of what you actually bet into the pot.

I'm not sure if OP just wants the most basic equation, but... ...is as simple as this:
Folding frequency required to break even with air = bet / (bet + pot)

Calculating whether villain will actually fold as often as you want/need him to obviously requires you to understand what his range looks like. I presume the course is designed to teach you how to range your opponents more accurately, such that you don't underbluff or overbluff.

I’m puzzled why you claim this.

For the case you specifically considered, a hero bluff, assume the bluff has equity EQbl. Then

EVhero-bluff = F*P+(1-F)*(EQbl*(P+B)-(1-EQbl)*B),

and you can solve for F in the same way you did when EQbl=0.

F= (EQbl*(P+B)-(1-EQbl)*B)/( EQbl*(P+B)-(1-EQbl)*B – P),


which reduces to B/(P+B) when EQbl=0.


Of course, the equation also provides much more information. For example, a bluff has to be successful at least 1/3 of the time if you make a pot size bet that is called.

That is still a model that doesn't encapsulate the whole scenario with more streets to play.

The new model is a better model than the one where we assume no equity, but what I was trying to elaborate on is that there is no purely mathematical formula that can solve this problem for you with multiple streets of play.

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I agree. As I did in my earlier post, I should have specified a river bet with villain calling.

Ah yes then your model is much improved and quite excellent for that scenario!

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Yeah I should pick up Jandas book. In Polks coarse he shows a giant action tree that he had someone create on Asianflushie. Ill be honest that part of the course was totally over my head. The equations u guys talk about above I have tinkered with but in all honesty I never put too much work into it because I figured "there is no way I can apply this in real time as I just don't have the ability to apply the equation in real time. It seems Polk has that ability. I guess that's why i will forever be a midstakes player. What's a good starting point for applying this in real time??

This is actually very helpful. Thanks!!

Where would be a good place to get started on studying these formulas?

I begun reading mathematics of poker and was insta too complicated for me, I want to try applications next I hope I can make through it fine

Most of the important stuff can be derived from a basic EV equation:

EV (some event) = P (some event)*Payout(some event occurred)

From there the only other thing you really need to know is that you can add EVS together to get the payout for more complex decisions and an EV equation can become the payout portion of other EV equations.

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