I've done the research and know the equation. The problem is, I can't get it to make sense in my head.

To solve for alpha it's S/(P+S). So if we're betting $50 into a $100 pot we want to have 1 bluff for every two 1/2 pot sized value bets, or 1 out of 3 bets, which is .33. Where S = the % of the pot you're betting and P = the size of the pot. In this case, $50/($100+$50) = .33.

I realize this is all about making an opponent indifferent. Bluffing at S/(P+S) makes sense to me. 2/3 of our bluffs fail for a loss of $100 for every one that succeeds for a gain of $100. Break-even. But I've also heard that our opponent needs to call the inverse of this and that's what isn't making sense to me....

Once we bet, our opponent is now receiving $150 to $50. If his equation is P/(P+S), which is I believe is the correct equation for calling, then that's .75. He needs to call with 75% of his best bluff catchers, while folding the worst 25%. This is not the inverse of our bluffing frequency, which would be .66.

When solving for calling frequency, don't we need to consider the last bet that gets put into the pot? Or do we ignore it? If we ignore it, it makes sense because then P still = $100, so P/(P+S) would be $100/($100+50) or 66%, which is the inverse of our bluffing frequency. But that's not the reality. We are receiving $150 to $50 odds, which to me, means we could lose 3 out of 4 times to break even and should therefore call with 75% of our bluff catchers.

I apologize because I'm sure I'm making some dumb error. Do we not take pot odds when considering GTO or optimal calling frequency? That's the only answer or way this makes any sense to me.

Thanks

If X is how often we are called, then our profit when bluffing is how often we are not called (1 - X) multiplied by the size of the pot 1 minus how often we are called (X) multiplied by the size of our bet.

In your example betting .5PSB an opponent needs to call X to force our indifference where

1(1-X) - .5(X) = 0

1 - X - .5X = 0

1 = 1.5X

X = 1/1.5 = 2/3

In general if we bet a B fractional pot sized bet then

X = 1/(B+1)

Or if we want to use actual bet sizes B and pot sizes P it would be

X = P/(B+P)

So if we bet 50 into 100 our opponent needs to call

X = 100/(100+50) = 2/3

I think your mistake is assuming we win the pot P plus the bet B when bluffing when really we just win P.

Hope this helps.

I really appreciate the time you took in explaining this. I'm sure you're right. I just have a really fuzzy mind when it comes to math and not even familiar with basic algebra. Can you please stick with me on this?

So:

This makes sense. Again, I'm not even familiar with algebraic terms, but if we let X = 2, then I think this means something like: 1 multiplied by (1-2) = -1 minus .5 multiplied by 2 = 1. So -1 minus 1 = 0. Correct?

Here I'm confused. Why does 1 = 1.5? Is 1 the size of the pot after first player (let's call him Player A) bets? Then 2/3 makes sense.

I play mainly limit poker so it's a bit easier in that I can just look at static bet sizes. So if there are 2 big bets in the pot Player A can only bet 1 unit. So 1/(P+1) where P = the pot size. 1/3 = .33

I think my mistake is trying to look at both players perspective simultaneously. Player A will optimally have 1 bluff for every two value bets (or 1/3) to keep Player B indifferent. But once Player A bets, then Player B is receiving 3:1 and can be wrong 3 times for every time he's right (75%). So this is not inverse to Player A's optimal betting frequency.

If Player B uses 2/3 of his bluffing range to pick off Player A's bluffs, he loses 2 bets when Player A is value betting and wins 3 bets the 1/3 of the time the bet is a bluff. That's a net gain of 1 betting unit. This is what I don't understand. Also, once player A bets there are now 3 bets in the pot and Player B can call with 75% of his bluff catchers because he only has to win once (25%) to break even.

You're probably wondering how I even play poker being this bad at math and I don't blame you. But I really want to understand this. I get why Player A has to bluff 1/(P+1) to keep player A indifferent, but I don't get why Player B shouldn't call 75% of the time after the pot reaches 3 bets and it only costs him 1.

When you bet half pot, you give villain calling odds of 3:1, so you should only be bluffing a quarter of the time (1 in 4). (The relative price of calling is cheap, so your bets will be called more often, so you should be more weighted towards value).
A pot-sized bet lays odds of 2:1, and then you need a third of your range (1 in 3) to be bluffs.

Sure I will try. First let me comment that ArtyMcfly is correct in that if you are trying to determine your bluffing frequency to make villain indifferent to calling, with a .5PSB it would actually be 3:1 value to bluffs since you are offering villain 3:1 pot odds. In general if we offer villain pot odds of X:Y the optimal bluffing frequency is Y/(X+Y). For instance we bet 2PSB we offer odds of 3:2 and our optimal bluffing frequency is 2/5 or 40%.

I missed this in my initial response as I was focusing on answering the question of determining bluff catching frequency.

This is technically correct but we aren't plugging in values for X, rather we are solving for X. But both sides of the equation are 0 for indifference, yes.

Think about it like this. If we bluff a certain amount vs. a non optimal opponent, then our profits are greater than our losses and we can be exploited by villain always calling. And if our profits are less than our losses then villain can exploit us by always folding.

For indifference we need Profits - Losses = 0

And if we bluff and are called at frequency X then our Profits are (1 - X) multiplied by the size of the pot, which fractionally is just 1(1 - X) or (1 - X).

Similarly our Losses when called at frequency X are X multiplied by the fractional bet size B. For example a 70% bet size puts our Losses at .7X

So Profits - Losses = 0 becomes

(1 - X) - BX = 0

We can then solve for villain's calling frequency X in terms of our fractional bet size B.

1 - X - BX = 0

1 = X + BX (moving terms to RHS of equation)

1 = X(1 + B) (factoring the RHS)

1/(1 + B) = X (dividing both sides by (1 + B) )

X = 1/(1 + B)

So if we are looking at this from the perspective of the bluff catcher, if our opponent bets B=3/4PSB then X = 1/(1+3/4) = 1/(7/4) = 4/7. So we would bluff catch 4 times out of 7 or roughly 57%

1 = 1.5X (not 1 = 1.5 !)

1 is the size of the pot in terms of pot sized bets which is how I'm framing bet sizes (as fractional bets of the pot). Therefore the pot size is always 1.

I'm not that familiar with limit. I think this is wrong though. If there are 2 big bets in the pot and we are looking at absolute pot size then the pot size is 2 big bets or 2 units (not sure if a unit is a big bet?) So the bluff catch frequency would be 2/(2+1) = 2/3

If player A can only bet half the big bet then it would be X = 2/(2+.5) = .8

Again not really sure what the standard bet sizing is but I looked it up and I think player A bets the small bet which is half a big bet so the bluff catch frequency would be 80%

Player A's bluffing frequency is determined solely by the pot odds a given bet size offers. A .5PSB offers 3:1 and therefore A has 3:1 value to bluffs.

Player B's bluff catching frequency is determined solely by the fractional bet size.

Your calculation of Player A's bluffing frequency is off. He is bluffing 1/4 of the time and therefore value betting 3/4 of the time (assuming he has bet half the pot).

So if there's a .5PSB then Player B loses .5 PSB when player A is value betting, which occurs at frequency 3/4 not 2/3. And he wins the pot plus the bet when A is bluffing so wins 1.5PSB at frequency 1/4. And we can check

1.5(1/4) - .5(3/4) = (3/2)(1/4) - (1/2)(3/4) = 3/8 - 3/8 = 0 as expected.

Player A bluffs determined by the pot odds he gives player B. A half pot bet gives 3 to 1 odds so the bluff frequency would be 1/(3+1) or 1/4.

Most poker players don't understand math well so you're certainly not unique in that regard but at least you're attempting to clarify what confuses you.

And I hope this post cleared it up but it's possible there's a miscommunication based on using limit as an example where I've never played limit.

Personally I find these equations easiest to understand the more abstract they are but if you can clearly explain to me the limit scenario and exactly how much Player A is betting and the pot size I'm sure I can reframe it in that way.

This is a tremendous help and I really appreciate it! Yes, I "used" to correctly think that the optimal value bet to bluff ratio was 3:1. But I was watching a Matt Hawrilenko video: https://www.youtube.com/watch?v=VHcrsMPQtgo and at 21:18 he shows this exact equation:

* Call = 1/(1+S) of hands that can beat a bluff

* alpha = S/(1+S) = ratio of bluffs to value bets

He explains that 1 = the size of the pot and that S = the percentage of the pot being bet.

Limit poker is fixed bet sizes. So in a 20-40 game you have two blinds of $10 and $20 with the bets and raises in $20 increments preflop and on the flop. On the turn and river the bets double and are in $40 increments. There is usually a maximum of 3 raises per street unless it's heads up, in which case the raises are unlimited (but must be in the correct increments). So a preflop limp or a flop bet would be $20. These are considered small bets. On the turn and river bets are $40 and these are considered big bets.

Say the sb raises and the bb calls. There are 4 small bets in the pot. The sb bets the flop and gets called by the bb. There are now 6 small bets in the pot. On the turn, I just divide the small bets by two, so would consider the pot now as containing 3 big bets. Say the turn gets checked through. There would now be 3 big bets in the pot. So how often does sb have to bluff to keep the bb indifferent?

I now (think?) I understand that sb needs 4 value bets for every bluff. This means that out of 5 bets... 1 of them is a bluff. That's 20% of sb's total bets. So does this mean the equation is 1/(P+2)? Dollar-wise there is $120 in the pot and the bet size is fixed at $40. If 1/(P+2) is correct that would be: $40/($120+$80) for a 20% bluffing frequency.

Now the corollary for BB finally makes sense! With P/(P+1) or $160/($160+$40), that equals 80% and jives. 20% optimal bluffing frequency opposed with an 80% optimal calling frequency reaches equilibrium.

I'm really sorry. All the Ss and Xs and 1s and Bs confuse me. Math is intimidating when you don't formally understand it. You've been extremely patient with me. If you could just confirm that my above example is correct, I can feel secure that I grasp it well enough for now. Thanks!

As someone pointed out in the YouTube comments 5 years ago, he made a mistake.

The ratio of bluffs to value bets should be bet : (pot+bet) (or S : (1+S) if you prefer those symbols).

So if the bet is $50 into a $100 pot (i.e. a half pot bet), the ratio is 50:150 = 1:3 => a quarter of the combos are bluffs.

First, the correct term is complement (1 – alpha), not inverse.

Here is my explanation.

Hero Bets: Alpha determines the minimum fold equity hero needs for a pure bluff (eq=0) to have positive EV. Alpha = Bet / (Pot+Bet). Therefore, if villain folds less than Alpha then hero will not profit. So 1- Alpha is the minimum call probability for hero to not profit, known as the minimum defense frequency MDF. It is equal to 1 – Bet / (Pot+Bet) = Pot / (Pot+Bet), the original Pot and Bet values, not those changed by hero’s bet.

If villain is the bluff bettor, then MDF will still equal Pot / (Pot + Bet), which can be obtained directly from hero’s EV equation, which is set to equal Pot and that makes villain’s EV = 0.

I haven't seen the video but worked this out independently (I have a math degree so for me it's not too hard). The YTer must jave made a mistake in his derivation of alpha. The equation for calling appears correct and is equivalent to where I derived X = 1/(P+1), we are just using different symbols. X is always a good symbol for an unknown and P seemed natural for pot size but use S if you like.

Thanks for clarifying the limit terms.

So in this river pot there are 3 big bets in the pot (120) and the SB can bet 40? In other words the fractional bet is 1/3 the pot.

So the SB offers his opponent odds of (4/3)1/3) and his optimal bluff frequency is (1/3)/(5/3) = 3/15 = 1/5 or 80% value 20% bluffs.

You could also solve this with the actual pot size: 160:40 or 4:1, which gives a value to bluff ratio of 4:1 as expected.

Right.

No I'm not sure where you came up with P+2. I mean your math is otherwise correct but you somehow have P=120 (yes) and 2=80 (you left something out because 2 is not 80).

I think you meant 1/(P+2B) where B is the bet size. And yes that appears equivalent to the pot odds derivation I used.

Well the call frequency is actually 1/(B+1) = 1/(1/3+1) = 1/(1/3+3/3) = 1/(4/3) = 3/4 or 75%.

The bluffing and calling frequencies do not sum to 1, which may seem confusing. They might sum to 1 for some specific bet size but generally they are independent. In this case SB bluffs 20% and we call with 75% of our bluff catchers.

We can confirm results for each player by seeing if it really makes the other player indifferent to bluffing or bluff catching.

From SB's perspective, BB wins 4/3 PSB 1/5 of the time (when bluff catching successfully) and loses 1/3 PSB 4/5 of the time (when calling a value bet)

(4/3)(1/5) - (1/3)(4/5) = 4/15 - 4/15 = 0 as expected. So BB is indifferent to calling when SB bluffs with frequency 1/5.

From BB's perspective, SB wins 1 PSB when BB folds 1/4 of the time and loses 1/3 PSB when BB calls 3/4 of the time

1/4 - 3/4(1/3) = 1/4 - 3/12 = 1/4 - 1/4 = 0 as expected. So SB is indifferent to bluffing when we bluff catch with frequency 3/4.

So this confirms SB bluffs optimally at 1/5 or 20% frequency and BB bluff catches optimally at 3/4 or 75% frequency.

No apologies necessary. Math is confusing for everyone at some level. It took even me longer to explain this and confirm the results were correct than I'd care to admit, but it was a good exercise nonetheless.

Let me know if still confused about something.

I'm sure I'm close to the point where you throw up your hands and say this guy's just too dumb, but I think I'm almost there!

Really appreciate you pointing this out. I never thought to look at the comments and it caused me hours of anguish.

This makes sense to me. 1:3 means there are 4 bets total and one of them is a bluff, so 25%. Got it. I kind of view it backwards because I play limit and am more concerned with making sure I have enough value bets (while it is possible to successfully bluff rivers in limit, fold are much fewer and farther between due to the much larger pot size to bet ratios than in NL). So if there are 6 big bets in the pot, I want to make sure I have at least 7 value bets for every bluff.

But I'm still a bit confused because doesn't this go against what Stanmanhal just stated?

This would be $50/($100+$50) which leads us back to 1/3 or .33

OR...

Do we adjust the pot size to include our bet as in: $50/($150+$50), which would equal 25%?

Have mercy. It's tough going through life being this bad at math. I'm trying to learn.

Thank you! This seems simple enough. Then I had it right before my first post that alpha = 1/(P+1) where P is the pot size and 1 is the bet amount?

This really shouldn't be this hard even for me. If there are 3 betting units in the pot and I'm trying to determine alpha then 1/(P+1) = 1/(3+1) = .25 is that correct?

And calling frequency is P/(P+1). So if someone bets 1 unit into a pot containing 3 units, my calling frequency should be 3/(3+1) = .75, since .25 + .75 = 1 and appears that both players have reached equilibrium.

This is the way I've been doing it all along. All the letters combined with the misinformation in the video I was watching just confused the hell out of me. I really appreciate everyone's time. Thanks!

Thank you. As ArtyMcFly pointed out there was a mistake in the video! This really confused the heck out of me as it never dawned on me that one of the best limit players on the planet would make a mistake like that regarding such a simple, yet important poker equation.

I think I understand everything now (if so, it's the way I always thought it was). Maybe check out my last two posts to make sure? I'm not used to dealing in fractional pot sized bets. Even though bets in limit are also fractional, they are fixed units so we can just think of them as 1, as in alpha = 1/(P+1) and conversely, calling frequency = P/(P+1). At least I hope that's right!

Thanks!

No none of this isn't right. I think you're getting confused by the different terminology people are using. Alpha is the bluff frequency not defense frequency.

Minimum defense frequency MDF is 1/(B+1) or P/(B+P). This is how often to bluff catch to force the bettor to be indifferent to bluffing. In 1/(B+1) B is fractional bet size. In P/(B+P) P is pot size and B is bet size. You can use either method to find the MDF.

In your previous post you use the example of someone betting 1 unit into a pot containing 3 units. The fractional bet is B=1/3 since it's a 1/3 pot sized bet. So

MDF = 1/(B+1)

MDF = 1/(1/3+1)

MDF = 1/(1/3+3/3)

MDF = 1/(4/3)

MDF = 3/4, as I previously derived.

Equivalently, you can use B=1 unit, P=3 units, to get MDF = P/(B+P) = 3/(1+3) = 3/4

Same thing.

Now onto "alpha" or the bluffing frequency for the bettor to make the caller indifferent to bluff catching.

If one bets B into P this creates pot odds for the caller of (P+B):B, which we need to convert to a fraction for "alpha" so

alpha = B/((P+B) + B)

alpha = B/(P+B+B)

alpha = B/(P+2B)

So back to your example of betting 1 unit into a pot of 3 units, we have

alpha = 1/(3+2*1)

alpha = 1/5

These are exactly the results I proved in my previous post.

Summary of resulting equations

MDF = 1/(B+1) where B is fractional bet size

MDF = P/(B+P) where B is bet size and P is pot size BEFORE the bet.

alpha = B/(P+2B) where B is bet size and P is pot size BEFORE the bet

Okay I think I see where some of the confusion is coming from. You originally use "alpha" to mean the optimal bluffing frequency. However, statmanhal uses "alpha" to mean the minimum fold equity hero needs for a bluff to be +EV. In this definition yes alpha = B/(B+P)

But this is NOT the optimal bluff frequency this just tells us if B/(B+P) > 0 then a bluff is +EV.

I don't know if "alpha" has some special meaning in poker theory, in math it's just another common symbol used like X or Y, so I apologize if I used "alpha" incorrectly to indicate optimal bluffing frequency (which I assumed was what it meant from context of your OP).

I want to be as clear as possible.

Optimal calling frequency MDF = 1/(B+1) where B is fractional bet size

Equivalently MDF = P/(B+P) where B is bet size and P is pot size BEFORE the bet.

Optimal bluffing frequency = B/(P+2B) where B is bet size and P is pot size BEFORE the bet

I've not actually read the famous book, but apparently it's used on page 113 of "The Mathematics of Poker" to mean the "minimum fold equity required to make a break-even bluff bet".

This Run It Once article might add to OP's confusion, but I found it was a useful summary of basic "GTO math" a few years ago: http://www.runitonce.com/chatter/gto-simplified/

You're right that the terminology and different letter variables people use just add to my confusion.

You defined variables B and P and assuming the 1 here is the pot size this makes sense. 120/(40+120) = .75

All this makes sense as well. Here, 1 can just remain 1, and I at least taught myself how to add, subtract, multiply, and divide fractions. So 1 divided by 4/3 is 1/1 * 3/4 = 3/4. Got this!

This is at the heart of what's been confusing me! If you bet 1 unit into a pot containing 3 units, I understand that you've bet 1/3 of the pot. However, I'd be receiving 4:1. This means I can call and be wrong 4 times (for -4 bets), and be right the 5th time (for +4 bets) to break even.

This is how I've always evaluated river decisions when facing a bet and deciding whether or not to call (let's pretend raising isn't an option). I include villain's bet when calculating P/(P+1) where P is the *new* pot size I'm facing and 1 is the betting unit that it costs me to call. This is 4/(4+1), which gives a .8 calling frequency. I can call 5 times and be wrong 4 (or 80% of the time) and beat a bluff or worse value hand once (20% of the time).

I know you guys are right, but what's wrong with my thinking?

So I've got it exactly backward! What I'm coming up with as a calling frequency is actually villain's alpha ratio of bluffs to value bets. Goddamn I'm dumb! But I think I got it now (still don't quite understand why I can't call with the upper 80% of the best bluff catchers in my distribution getting 4:1), but your perseverance is greatly appreciated!

fwiw - I do believe I have some form of dyslexia when it comes to math. I'll say things like 2-7 when I mean 7-2. I flip things around and can't keep anything straight in my head. I also have no more than mid 5th grade official education in math. I got the chickenpox when I was a kid and missed two weeks of school. Because math builds on itself I was completely lost when I came back. It became embarrassing to keep raising my hand and having the teacher stop and explain whenever I didn't understand something that the rest of the class already knew, so I stopped and somehow faked my way through math until I dropped out of high school at 16. I wound up taking a GED when I was 19 and to this day I don't know how I ever passed the math portion lol.

Thanks again!

There's some good stuff in this thread but the theory applications aren't that helpful as our overall range on certain textures plays a bigger role in determining our frequencies than simply "we should have x% of bluffs cause we bet x% of pot" and there's tonnes of spots where we aren't bluffing at the exact ratio due to our range advantage or range disadvantage. The same goes for our defending frequencies.

Is there even any practical use for these types of equations in a GTO model?

There is no reason to derive algebraic expressions in order to feel like you will always get the right answer. If you are the bettor with a hand that is either certain to be best or certain not to, the GTO ratio of value bets to bluffs to is the same as the ratio of the caller's pot odds. If you bet twenty dollars into an 80 dollar pot he is getting 5 to 1 so there should be five value bets for every bluff. (83.3%)

With one exception you use similar logic when deciding whether to call. Now you look at the odds the bettor is getting when he happens to be bluffing and you use the same ratio when calling. So if he bets twenty dollars into an 80 dollar pot he is getting 4-1 on a successful bluff and you need to thwart that by calling four times as often as you fold ie 80%.

Notice that the GTO calling frequency normally does not depend on the probability that the bettor, BEFORE he bets, has the best hand. Except for the aforementioned exception. That occurs when that probability is high enough that when you add in the GTO calculation for bluff frequency it adds up to over 100%. In that case the GTO calling frequency becomes zero per cent. For example if you think going to the river that there is a 70% chance you are beaten and he is aware of that, he should bluff the size of the pot another 35%. That means he bets every time, yet you still can't call.

You now know everything without using one capital letter.

Thanks! That certainly simplifies things.

Cool post. Thanks