Hey there,

I spent a lot of time lately, figuring out the theory behind GTO. There's one thing that bothers me, that I'm just not able to figure out. Maybe you guys have that knowledge.

GTO calling frequencies allows you to make your opponent indifferent between calling and folding.
The procedure looks as follows
1. Figure out all your hands, that beat bluffs.
2. Determine your call ratio, which is [1 + (1+s)], where s being the pot ratio (e.g. on a pot sized bet s would be one).
3. Call with the top percentage (calculated in 2) of hands that beat a bluff (figured out in 3).

The critical issue is point 1. WHAT IS A HAND THAT BEATS A BLUFF? This totally defines the amount of hands you're calling with.

An example:

Assumptions
- A bluff-hand never wins at SD (when checked)
- Suppose the pot is $2 and villain makes a pot sized bet, which means that we call 50% of our hands that beat a bluff. In theory this means:

Suppose Villain Checks his Bluffs
100% of the time he wins $0 = $0 (since when he checks his bluffs he'll never take down any pot)

Suppose Villain Bluffs
50% of the time you fold and he wins $2 (the pot)
50% of the time you call and he loses $2 (b/c I only choosed hands that beat a bluff in the first place)
0.5 x $2 - 0.5 x $2 = $0

==> Until this point everything is fine. It seems that villain is indifferent in bluffing or checking his bluff hands. Further he's playing at 0 EV.

HOWEVER, we do make a very strong assumption here (which I'm not sure if we can...), which is that we ALWAYS beat his bluffs.

What if he turns a made hand into a bluff? So Let's assume he has a bluffing range of 77,88,99 (18 combos, that he turns into a bluff on the river) and 45o (16combos). You hold 66. Clearly, this hand can beat bluffs, which means that we "add" it to the hands that can beat bluffs. Of these now 50% should call to make him indifferent.


Suppose Villain Checks his Bluffs
18/34 of the time he wins $2 = $ 1,06
(times we are beat with 66, when he has 77,88,99)
16/34 of the time he looses nothing (but also doesn't win anything) = $0 (times we beat him with 66 when he has 54o)
On avg. he wins $1,06

Suppose Villain Bluffs
50% of the time you fold and he wins $2 (the pot) = 1$
50% of the time you call
- 16/34 of that he looses $2 = $-0,94 * 0,5 = $ -0,47
- 18/34 of that he wins $4 = $2,12 * 0,5 = $ 1,06
On avg. he wins: $1 + $1,06 - $0,47 = $ 1,59

Although we applied the theory, villain is NOT indifferent (he should bluff), NOR is he playing 0EV.

I just can't figure out how I can make him indifferent in such a spot? Obviously 66 in this spot doesn't count to the hands that beat a bluff. So again, how can I make him play indifferent in this spot and how do I decide/calculate which hands belong to the hands that beat a bluff?

Looking forward!
- Cheers

Depends on the board, # of players in the hand, preflop action. All this stuff is going to affect ranges, so in some cases where ranges are narrow you have to turn "SDV" into a bluff. But, you need to know all the criteria/variables that lead to that point to make an accurate decision. Also, don't confuse making your opponent indifferent with choosing the highest EV line. Sometimes it is better to overfold than call enough to make our opponent indifferent because folding a ton is going to be higher EV.

Anyways, yeah interesting question and there is no concrete answer. For example, today I turned KdQx into a bluff on a AQd910dXd board OTR

Anyways, your example is bad because it doesn't give a board texture or an entire range. Basically you made up some random game that isn't poker. Lol, sorry.

Your mistake is assuming that in position bluffs have zero showdown value. Does the out of position player never check fold junk?

Here's a hand I played once upon a time:

4/8 limit holdem

bunch of limpers, I check in the big blind with 42o.

35Kr

checks to middle position, he bets, couple calls, I call.

To

checks to middle position, he bets, folds to me, I call.

To

I check intending to fold, he checks back.

Still I'm not sure how I should approach in order to find the hands that beat a bluff?
Is there any practical way to do it, or is there a common heuristic to approximate that?

I think you are worrying about something that can't happen:

In the simplest case, on the river, where you have already checked and villain bets, and you are trying to figure out your calling range...

1. A hand which beats >= 50% of your calling range is a value bet.
2. A hand which beats some but < 50% of your calling range is always better off checking than "bluffing".

Such a "bluff" gains nothing when we fold since that hand beats our entire folding range anyway. And when we call that hand is not happy either since it loses more than 50% of the time to our calling range.

Let's say villain bets pot.

1. Start with a 50% calling range. That is the widest possible range you will call.
2. Find villain's value-range which is any hand that wins at least 50% of the time when called.
3. Find what % of his entire range is his value range. For example it might be 30% of his range.
4. Find his bluffing range which will be 1/2 the size of his value range, in this example it will be the bottom 15% of his range.
5. Find the best hand in his bluffing range and look at the EV(check) vs EV(bluff) with that hand. If he is better off checking than bluffing, then you are calling too much so you have to lower the calling% somewhat (it's a guess) and return to step 2. If he is better off bluffing than checking, then you need to increase your calling% somewhat and return to step 2. If EV(bluff) = EV(check) then you are done.

There is also a possibility that his range is strong enough that you must fold your entire range. Also, this method doesn't take into account blockers.

Well isn't the question that should concern me, if I'm playing GTO, not villain? So on the river villain may certainly decide to either turn hands into a bluff or bluff hands that don't have SD value at all.

Isn't the key behind the calling frequencies to add only those hands that beat a bluff? Because only those are candidates for calls (if I don't beat his betting hand, then I don't want to call obv.)

So we have 3 situations
1.) A hand is ALWAYS ahead of his betting range => then all of the hand's combinations are included into the "bluff catcher/hands that beat a bluff"-range
2.) A hand is ALWAYS behind vs. his betting range ==> I don't add it to the "hands that beat a bluff"-range
So if my hand is ALWAYS better than villains it's worse than I include this hand into my "bluff catcher/hands that beat a bluff" range, since I would make a mistake when folding the better hand.
3. Imho it get's tricky when a hands beats some, but also looses some of villain's betting hands. In this case I would want to include that hand partially into my bluff catcher range (hand combination wise)

So I think the right approach is to look at how often the hand is beaten and how often it beats villain's betting range.
Let's assume the following: our range comprises of JJ, 23o and villain bets only AA,KK,TT on a 233 board. Then I should only include 1/3 of the 6 JJ-combinations (2 combos, cause 2/3 of the time I don't even want to consider calling since I'm behind...) into my "bluff-catcher" range, and the full 6 FH combinations (since it would be wrong folding it everytime).
This leaves me at 8 hand combinations that actually "beat a bluff". So, in order to make him indifferent in betting/checking with his bluffs when he makes a pot sized bet, I need to call 50% of those 8 combinations, which means 2/3 of my 23o hands.
Obviously that's not the optimal play, however this should make him indifferent in checking/betting at least.

I don't understand what you are trying to do. Your GTO calling range in this example should be your six 23o combos which is 50% of your range. His corrent value/bluffing ranges are empty. If he want's to bet AA,KK,JJ into your FH's let him.

GTO is not dependent on what villains does. You don't calculate GTO based on what a non-GTO villain might be doing.

All I want to do is to find a way to have at least a rough feeling for my balancing. And I think that you can NOT call 50% of your entire range, since GTO calling frequencies concern bluffs and not bets. That said you want to call with 50% of our "hands that beat a bluff" (not of your range). The critical question is, WHAT IS A BLUFF and that's what I tried to outline above.
1) Hands that always beat is betting range are fully counted to "hands that beat a bluff" (we always make mistake folding them to their betting range)
2) Hands that always are beat by his betting range are not counted to "hands that beat a bluff" at all (we always are right to not continue at all)
3) Hands that partially beat his betting range and partially are beat by his betting range should be partially counted as a "bluff catcher", thus partially counted to "hands that beat a bluff" (we s.t. make a mistake folding them & s.t. folding is the right play) ==> Only some combos are counted as "hands that beat a bluff"


Do you know what I mean? Pretty difficult to explain, but I try to find a range that I want to call with when he bluffs (in order to be balanced), but these hands are only consisting of hand combos that are truly behind when he bets (which can be partial for some hands - category 3 hands).


EDIT: You need to have a rough picture of villain's range (not his frequencies though), just to get an approximation of what his "bluff hands" are...

Once you find GTO strategies you can verify them by the definition of GTO: Neither player can unilaterally improve their strategy.

Your solution was "I need to call 50% of those 8 combinations, which means 2/3 of my 23o hands." So you are folding the nuts? That's not GTO by definition. You can do better by calling all your 23o. Clearly something is wrong with your methodology. What is wrong is this: You started with his betting range as AA,KK,JJ but that's not a GTO betting range. You can't deduce a GTO calling range from a totally non-GTO betting range.

My solution was
You: call all 23o full houses and fold all JJ.
Villain: check all hands.

Neither player can improve. So that is the GTO solution.

All bluff-catchers fall into this category. They beat the bluff parts of villain's range and lose to the value part.

Bluff,value bet, bluff-catcher are terms used to describe hand vs range match ups for both villain and hero and not range vs range.

The idea behind GTO is that given an exact picture of villain's range you can calculate an equilibrium strategy for yourself even if villain's frequencies are unknown. In the calculation process, there's no step where you sort bluff hands from value-bets because all possibilities are considered. Books and articles use very simplified toy games because otherwise all the math wouldn't be able to fit on the page. With wide (fixed) ranges, you'll be dealing bluffing off split situations and blockers and probably other issues that make these sorts of problems extremely difficult to do by hand, though it would be possible via computer.

So, I do have to consider my entire range, instead of the "hands that beat a bluff"?
This irritates me, cause Hawrilenko (hoss_TBF) says exactly the latter (45:50): http://ocw.mit.edu/courses/sloan-sch...cision-making/



Or is it all hands that beat a bluff, no matter how often they actually are beat vs villains value betting range?

You can consider hands that beat a bluff. But initially you know nothing about villain's correct bluffing range so you need to guess and adjust.

You can approach it like this. Let's use pot-sized bets for this example:

1. Initially guess that your entire range beats a bluff.
2. Call with 50% of that.
3. Calculate villain's value and bluff range base on your calling range.
4. Now that you have an approximation of his bluffing range, you can find which part of your whole range beats a bluff.
5. Change your calling range to 50% of your hands that beat his biggest bluff.
6. Go to step 3 until converges.

For example suppose both ranges are (0,1,2,3,4,5,6,7,8,9) with high hand wins.

Tentatively assume your whole range beats his bluffs.
Your calling range is (5,6,7,8,9) i.e. 50% of your range.
His value range is (7,8,9) and bluffing range is (0:50%, 1).

Your range that beats a bluff has changed to (2,3,4,5,6,7,8,9).
Your new calling range is (6,7,8,9) i.e. 50% of your range that beats a bluff.
His new value range is (8,9) and bluffing range is (0).

Your range that beats a bluff is now (1,2,3,4,5,6,7,8,9).
Your new calling range is (5:50%, 6,7,8,9).
His new value range is (8,9) and bluffing range is (0).

Nothing changed so your done.

Final answer is:
You call with (5:50%, 6,7,8,9)
He value bets with (8,9) and bluffs with (0).

You should never call more than the naive indifference, but sometimes have to fold 100% of your range. It just depends a lot.

If there are multiple betsizes possible otr, it's also possible that villain can turn some hands with showdown value into bluffs.

That's when you are oop- when you are ip you can sometimes (actually quite often) call slightly wider than naive indifference because villain can face a bet after checking as well.

Hey bobf,
thanks for the answer. I'm not sure why you do this iterating steps. Example

Board: KJJT,3
My hand: 22, 44-99, QQ, AA (9 hands)
Villain's range: Tx, 44-99, 22

So, the procedure would be:

50% call of my entire range: AA, QQ, 99, 88, 50% 77

Villain vbets: Tx (since more than 50% calls)
Villain Bluff range: 22,44-99

My hands that beats a bluff: AA, QQ ⇒ Calling 50% = AA

His vbets against AA = nothing
Thus his bluffs are: 22,44-99, Tx

My hands beats a bluff: AA, QQ ⇒ Calling 50% = AA
==> Nothing changes, so AA is our calling range

But doesn't that feel a bit tight?

After thinking and discussing this a lot, I more feel like that (same setup as above):
I have 9 hands, so his bluffs are defined as those hands that don't value bet right. In this case this would be 22, 44, 55, 66.
77 is on the edge, since he is beat by 88, 99, QQ, AA but beats 22,44,55,66. Anything better than 88 is a value bet, since he has more hands beat, than he's beaten.
So my hands that beats his best bluff is 77, 88, 99, QQ, AA. So 50% vs. a pot sized bet is AA, QQ, 50% of 99, which seems more realistic in my eyes.
What do you think?

Basically what I'm saying is that I don't need to know his bluffing range. If he makes a pot size bet, then his GTO play would be to bluff 50%, right? So, ASSUMING he plays GTO, I need to call 50% of his bluffs. And his bluffs are the complementary to my range I hold in this spot.
So, if i found out that on a PS-bet his bluffing frequency is not 50%, but rather 10%, then off course I won't call 50% of all my hands that beat a bluff (still defined as above), but only 10%.?

I do the iterating steps because I don't know any other way to do it. Value range is calculated as hands that beat at least 50% of calling range. Bluffing range is calculated as 50% the size of value range. Calling range is calculated to make best bluffing hand indifferent.

So its a circle Calling-Range --> Value-Range --> Bluffing-Range --> Calling-Range. Each range depends on the other. Since we don't have a known starting it seems the only way is to guess and iterate.

Can you tell me what you mean exactly by Tx? Do you mean
- 10 and any card smaller than a 10?
- 10 and any card smaller than a 10 that does not make 2 pair (i.e. exclude T3)
- 10 any any card e.g. JT?
....

Once I know that I can check the rest of your calculations and discuss the rest of what you wrote.

Oh sure, sorry. Tx means a single T, so basically a one pair, with a T The x indicates that the second card is irrelevant.

It appears, my algorithm doesn't converge on this example. Also this example involves blockers making it more difficult.

I ran a program I wrote on this setup. Here is the solution it gives:

Value Range: AT, QT (24 combos)
Bluffing Range: 22, 44 (12 combos)
Calling Range: AA, QQ, 99, 88 calls 52.3% of the time.

To verify if this is correct (no bug in my program) we would have to check each hand and make sure an alternate action is not higher ev.

GTO would be to have a bluffing range that is 50% the size of his value range.

Not necessarily. In the solution I gave above, 22,44 are bluffs. If instead he checks with 44 that will be +EV (about 13% equity). If you call 50% of the time his bluffs will be 0 EV so he will never bluff. So calling 50% is calling too much, not GTO.

Here is an improved (but tedious) algorithm.

The idea is to do a binary search on the size of the calling range. We guess at the calling range size, compute value range, compute bluffing range. Then we check and see if the calling range we guessed at was too wide or too small and adjust accordingly.

1. Let cmin = 0.0, cmax = 0.5 (lower and upper bounds on calling %)
2. Let c = (cmin + cmax) / 2 (current calling %)
3. Compute calling range from c.
4. Compute value range (>= 50% equity vs calling range)
5. Compute bluffing range (50% size of calling range)
6. If best bluffing hand does better checking then we are calling too wide so set cmax = c. Go to step 2.
7. If worst checking hand does better bluffing then we are calling too little so set cmin = c. Go to step 2.
8. done

Here it is applied to your setup. Pretty easy to make a mistake, but here is what I got:

Iteration 1
cmin = 0.0, cmax = 0.5, c = 0.25
calling range: AA, QQ, 25% 99
value range: empty
bluffing range: empty
worst checking hand 22 is better off bluffing so call more often: cmin = 0.25

Iteration 2
cmin = 0.25, cmax = 0.5, c = 0.375
calling range: AA, QQ, 99, 37.5% 88
value range: empty
bluffing range: empty
worst checking hand 22 is better off bluffing so call more often: cmin = 0.375

Iteration 3
cmin = 0.375, cmax = 0.5, c = 0.438
calling range: AA, QQ, 99, 93.8% 88
value range: AT, QT
bluffing range: 22, 44
worst bluffing hand 44 is better off checking so call less often: cmax = .438

Iteration 4
cmin = 0.375, cmax = 0.438, c = 0.406
calling range: AA, QQ, 99, 65.6% 88
value range: AT, QT
bluffing range: 22, 44
worst bluffing hand 44 is better off checking so call less often: cmax = .406

Iteration 5
cmin = 0.375, cmax = 0.406, c = 0.391
calling range: AA, QQ, 99, 51.6% 88
value range: AT, QT
bluffing range: 22, 44
44 is better off buffing than checking. good!
55 is better off checking than bluffing. good!
so we are done

--------------------------------------

This algorithm can fail too, due to blockers. You can't necessarily compute a calling range from c because sometimes lessor hands are better candidates for calling than better hands if they block more of the value range or less of the bluffing range.

WOW bobf, it's now your third algorithm in this thread. Do they all say the same, or is it you playing around? And how sure are you about them? I'm curious, because I don't see the logic behind. Could you elaborate on that? Basically the "why" you are doing those 8 steps, or the other steps in the other algorithms that you suggested?

Algorithm 1 and 3 are really the same, but 3 explains specifically how to adjust the calling range. Algorithm 2 was an attempt to work from the idea that you call with x% of hands that beat a bluff based on the bet to pot ratio.

I'm pretty confident that Algorithm 3 is correct in a basic sense with the caveat that it does not deal with blockers or all the divergent cases. For example, one step says to make your bluffing range 1/2 the size of your value range. Well, what if my value range turns out to be 90% of my original range? Then I don't even have enough bluffs left to do that.

The basic logic is that:
(1) If I magically could know your GTO calling range, then I can determine my value range. I will bet any hand that beats >= 50% of your calling range. This ensures the EV(bet) >= EV(check) for my value hands.

(2) If I know my value range then I can determine my bluffing range as 1/2 the size of my value range for pot-size bets. This ensures that your bluff-catchers will be indifferent to calling/folding. (This is one step I'm not 100% sure of either. I'm not sure we always have to make your bluff-catchers 0 EV. Your bluff-catchers certainly must have EV >= 0 or they would fold.)

(3) If I know my value and bluffing range then I can verify whether or not your calling range (which we guessed at) was too wide, too narrow, or just right. Your calling range is too wide if any hand in my bluffing range has higher ev checking. Your calling range is too narrow if any hand in my checking range has higher ev bluffing. Your calling range is just right if neither of these is true.

In short
calling % --> calling range --> value range --> bluffing range
verify calling %. adjust if too wide or too narrow. stop if just right.

At any rate this algorithm is more for learning than being 100% useful because blockers are very important and this algorithm doesn't deal with them.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Here is an extreme example to illustrate how blockers can mess with things:

Board: Kc Ts 9c 3s
Pot: $1
Stacks: $20,000
Ranges: any hand
Player 1 can go all in or check.
Player 2 can call or fold or check.

Value bet range: black QJ (100%)
Bluffing range: QsQc (22%)
Calling range: black QJ (100%) other QJ (less than 1%) QsQc (21%)

Notice that QsQc is in the calling range!!! Seems crazy but it's correct -- we call a huge bet with middle pair while folding our straights. This is purely because it blocks the entire value range. It can't lose. In fact it can't even happen because when player 2 holds this hand player 1 never bets. Nevertheless it is a necessary part of player 2's strategy because without it player 1 can successfully bluff his JsJc hands.

I can't say I totally understand what you're looking for.

But this book is probably the best book on poker. It talks about visualizing the EV of one range against another range, and it has a tool to help with the process.

http://www.dandbpoker.com/product/ex...oldem-volume-1