game theory river play

It is a very nice article that I'll study in detail. In practice, if you can do this while playing, it is probably better than pure randomization versus all but the super-sophisticated opponents (or GTO robots).

However, and this should be made clear, utilizing card removal effects (CFR for Calldown Frequencies River) is an exploitative measure, it is not GTO. There is an (actually explicit in the article) assumption that the bluffer will bluff more or less according to the blockers you hold, and not according to the size of his bet relative to the pot size (though the bluffer is actually unaware of this, he might think he is playing GTO).

It is seemingly so that CFR is GTO since the bluff catching range has ha fixed size on a given board that is the correct GTO size. But this is an illusion, since there are hands outside the bluff-catching range stronger (versus every range in the world) than certain hands within it. What's really going on is that we aren't really playing GTO by assuming the opponent is bluffing too much/too little according to the hands we hold.

There is a way that the bluffer could counter an opponent playing CFR. He could use his wrist watch to determine when to bluff. Sooner or later the CFR player will make a bad call with A2 and a bad fold with A8 (from an example in the article where A2 is a call but A8o is a fold). It is probably possible (in theory) to go further. (The bluffer could consider his blockers to the callers blockers and adjust his bluffing frequency accordingly. This is just a silly mind game, but it illustrates that every deviation from GTO can be exploited.)

Having said that, the article and method described therein is excellent, and much more stimulating to the human mind than true GTO, and it probably works better versus humans (that don't use their wrist watch to randomize bluffs:)). CFR is close enough to GTO and probably deviates from it in the right way.

Originally posted by YohanN7
However, and this should be made clear, utilizing card removal effects (CFR for Calldown Frequencies River) is an exploitative measure, it is not GTO. There is an (actually explicit in the article) assumption that the bluffer will bluff more or less according to the blockers you hold, and not according to the size of his bet relative to the pot size (though the bluffer is actually unaware of this, he might think he is playing GTO).

No. Not at all. The bluffer is betting the exact same range the entire time. This range is affected by your hole cards, creating the effect of him bluffing too much, so that you can call, or too little, so that you have to fold, based on your blockers.

Originally posted by lnternet

Originally posted by YohanN7
However, and this should be made clear, utilizing card removal effects (CFR for Calldown Frequencies River) is an exploitative measure, it is not GTO. There is an (actually explicit in the article) assumption that the bluffer will bluff more or less according to the blockers you hold, and not according to the size of his bet relative to the pot size (though the bluffer is actually unaware of this, he might think he is playing GTO).

No. Not at all. The bluffer is betting the exact same range the entire time. This range is affected by your hole cards, creating the effect of him bluffing too much, so that you can call, or too little, so that you have to fold, based on your blockers.

If you read my post once again, carefully, you will see that this is precisely what I wrote;). The bluffer is not playing GTO, he is bluffing slightly too much or slightly too little. A players range and holecards has imperfections as a randomization device because the opponent knows what you don't have.

By using an external randomization device (i.e. not the hole cards) he can play true GTO. This would mean (referring to the example in the article) that A7s and A5s could be turned into bluffs. This makes his range truly balanced, and your bluff-catching range incorrect.

Further, if the bluffer knows what you are doing, he could chose to bluff less than game theory dictates when holding a hand like 8 :heart:7 (that has missed flush draws and gutshot), and effectively be exploiting you.

Still, I like the CFR strategy, because most players would be exploited by it and it is very close to GTO.

There's a pending friend request in your com-tool.

Ok, lets try it this way.

I am going all-in for a pot sized with all combos AA and x combos of bluffs with 43 on AK982. Villain's range is 1 combo of AK, 1 combo of KK and 1 combo of T9.

How many bluff combos should I have alongside my 3 value combos to be playing GTO?

You should have 1.5 bluff combos if you have 3 value combos (which is the answer to the original question in this thread). This makes it 2:1 against that you are bluffing, which is correct if you want to play GTO. Villain's range is then irrelevant if you play GTO. (The last sentence applies to GTO calling as well. This is the crux of the matter.)

What is it that you don't agree with?

I agree with everything you have said so far (as long as you do not claim that either the bluffer or the caller are playing true GTO when using the holecards as randomizers).

Ok, so I bet 3 combos of AA and 1.5 combos of 43.

Then for villain we have:
AK is a +EV call, because when he has AK, you are value betting 1 combos AA and 1.5 combos 43s.
KK and T9 can do whatever, same EV for both actions, but calling half a combo of KK is advisable.

Since AK is blocking AA, it is a call, whereas a hand like 33 would bea clear fold, because if blocks 43. And KK/T9 are indifferent, with those you can flip a coin.

I understand precisely what you mean. But, read on ...

You must realize that what you are talking about is not GTO (which, after all, is the subject of this thread), and moreover, it is not a way of exploiting an opponent playing GTO since, by definition, GTO is unexploitable, and yet more, GTO equilibria exist as the theorem by Nash shows. You run into impossibilities here.

In your example neither player is playing GTO, because the value range consisting of AA is, according to GTO, obviously wrong so your example yields no counterexample to the impossibility of beating GTO. The value range for a GTO player is the exact range that beats his own range in the opposing seat (in the worst case scenario when the opponent plays GTO as well, any other range, and the GTO player profits, or at least, can adapt so that he profits, either mostly with bluffs or mostly with value).

The CFR strategy is nice versus opponents whose bluff range you can guess, provided they use their holecards as randomizers. That is, it is nice in practice, but has no effect versus a GTO player, but rather renders you exploitable.

Originally posted by YohanN7
You should have 1.5 bluff combos if you have 3 value combos (which is the answer to the original question in this thread). This makes it 2:1 against that you are bluffing, which is correct if you want to play GTO. Villain's range is then irrelevant if you play GTO. (The last sentence applies to GTO calling as well. This is the crux of the matter.)

This is true but badly formulated. Blockers versus the villains value range does matter when calling down a bluff. But since you see those in your own hand, you still don't care about the villains range even if it narrows the villains value range tweaking your calling range to become wider.

Another thing worth mentioning is that the 1:N bluffs rule is an approximation only, since it is based on the simplifying assumption that it's air + nuts versus bluff catchers. Especially when ranges are narrow, some adjustments probably need to be made to get closer to exact GTO.

Originally posted by YohanN7But since you see those in your own hand, you still don't care about the villains range even if it narrows the villains value range tweaking your calling range to become wider.

What does that mean? Do I have to call down with more care to play optimally? In other words: is it -EV to not care before I make the call?

Alright, now more seriously: What do you mean by "you don't care about villains range"? When do I care and when do I not care? What does it imply?

A full GTO solution is just a HUGE Nash type of table where it says something like Call 34.44%, Raise 63.66% (using a randomizer) in this exact game state. The game state is the previous action (this hand), the hole cards, and the board. Nothing relates to the opponents range. If that was the case, it would be an exploitative strategy, not GTO.

If you have blockers, say an A on a Awxyz board, then both common sense and that huge Nash table tells the same story. The opponent has (on average) fewer combinations containing an A, which he would conceivably would bet for value. He will have more busted draws and a larger percentage of his bets will be bluffs. Therefore you should call slightly more than the pot odds dictate. (The bluffer doesn't know that you hold the blocker and should bluff "as usual")

The CFR (discussed in this thread, covered in the gold article Game Plan - Calldown Frequencies River) strategy goes further and does make assumptions about which hands the opponent bluffs with. So, for example, if the hand that would have the most outs on turn were 8♥7 :heart:, and a part of that hand happens to be in your hole cards, then the chance that the opponent will be firing a bluff decreases a bit and accordingly you should call less frequently.

Taken to the extreme it might mean (on the same board with an A on it) that you call with A :club:2 and fold A :spade:8 :heart:. With the first hand you have a blocker to a value hand and a blank card. You call wide and call with the A :club:2 :club:. With the second hand you have a blocker to a value hand but also a blocker to a typical bluff hand. You call less wide and fold the A :spade:8 :heart:.

You can't make such assumptions when playing GTO. That is to say, they don't lead anywhere except to possibly a slight loss versus an opponent playing GTO: Sometimes the bluffs will turn out to be value bets (the bluffer bluffs with A :diamond:3 and gets called by A :club:2 :club:) and sometimes bluffs succeed unexpectedly (the bluffer bluffs with A :diamond:3 and his opponent folds A :spade:8 :heart:).

Nobody knows exactly what a GTO solution looks like in all detail. It's possible that it shows the same qualitative tendencies as the CFR strategy, but surely not to the extreme. Say that 6 is a typical bluff blocker. Then perhaps A :club:5 would be a call, while A :spade:6 wouldn't.

In short, GTO doesn't care about ranges since it's supposed to work against every range, but you should care because it might bring you pretty close to GTO.

I am not exploiting a GTO strategy when I consider blockers. I am taking one strategy and finding it's best response strategy. If I then found a stable strategy pair, I have a solution.

We can never claim a full solution, as it's multi street with many different sizings, which can't even be written down. However, when I look at a simple enough river situation in a vacuum, we can find a solution for this single street. Like the {AA (3),43 (16)} vs {AK (1), KK (1), T9 (1)} you can find a solution.

Then if you blow that solution process up, you arrive at the CF:R/BF:R guides for building strong ranges. In CF:R you fix some parts of opponents strategy, namely sizing and vbet region size, and find a strategy pair for that. In BF:R we assume only the sizing be fixed, and some simplification to raises, to find a strategy pair.

Regarding what a solution to poker is like, you are confused about the frequencies. You need not play a mixed strategy on a single combo. When you call 34.44%, this will work out to an exact combo split, like these 43 combos of 125 call. Not select random 43 combos of 125 combos. Like you never raise 72o UTG. (You will split on single combos sometimes, but I think a strategy that never splits on a single combos has a very low exploitability.)

Originally posted by lnternet
Regarding what a solution to poker is like, you are confused about the frequencies. You need not play a mixed strategy on a single combo. When you call 34.44%, this will work out to an exact combo split, like these 43 combos of 125 call. Not select random 43 combos of 125 combos. Like you never raise 72o UTG. (You will split on single combos sometimes, but I think a strategy that never splits on a single combos has a very low exploitability.)

No, as explained in my post, I'm talking about an entry in a GTO solution, i.e. one single game state. For example, you have J 7♣ on a T :heart:8 :heart:4 board in position in a hand that went raise 2.5bb - call. This is a game state. Now you might have 66.44% bet and 33.66% call (or whatever).

Originally posted by YohanN7
No, as explained in my post, I'm talking about an entry in a GTO solution, i.e. one single game state. For example, you have J 7♣ on a T :heart:8 :heart:4 board in position in a hand that went raise 2.5bb - call. This is a game state. Now you might have 66.44% bet and 33.66% call (or whatever).

Yes, I think this doesn't occur very often. As J7cc = J7dd = J7dc ... you don't need to split on individual combos much. And if you would split on a specific combo, both options would by definition have the same EV against the other strategy.

I think a strategy that never splits on a single combos has a very low exploitability.

Originally posted by lnternet

Originally posted by YohanN7
No, as explained in my post, I'm talking about an entry in a GTO solution, i.e. one single game state. For example, you have J 7♣ on a T :heart:8 :heart:4 board in position in a hand that went raise 2.5bb - call. This is a game state. Now you might have 66.44% bet and 33.66% call (or whatever).

Yes, I think this doesn't occur very often. As J7cc = J7dd = J7dc ... you don't need to split on individual combos much. And if you would split on a specific combo, both options would by definition have the same EV against the other strategy.

I think a strategy that never splits on a single combos has a very low exploitability.

Agreed. Never splitting a combo would take a years or more to significantly exploit. But the splitting explains why unexpected hands sometimes are used for bluffs providing the GTO defense versus CFR (or any other strategy).

Sorry to interfere, but where would a fish start learning GTO from scratch ?

... ♠_biggrin:... soooo, after reading this thread a couple of times, I have a million questions... ♠_confused:

But here's the most glaring one:

Ok, so I bet 3 combos of AA and 1.5 combos of 43.Then for villain we have:AK is a +EV call, because when he has AK, you are value betting 1 combos AA and 1.5 combos 43s.KK and T9 can do whatever, same EV for both actions, but calling half a combo of KK is advisable.

Is it even possible for the bettor to play GTO?! When he bets 3 combos of AA and 1.5 combos of 43, he THINKS he is playing GTO...but in reality when his opponent has AK he is not playing GTO since he has only 1 combo of AA due to card removal. So how can he play GTO without knowing what blockers his opponent has?! ♠_o:

Originally posted by Itsnevereasy
... ♠_biggrin:... soooo, after reading this thread a couple of times, I have a million questions... ♠_confused:

But here's the most glaring one:

Ok, so I bet 3 combos of AA and 1.5 combos of 43.Then for villain we have:AK is a +EV call, because when he has AK, you are value betting 1 combos AA and 1.5 combos 43s.KK and T9 can do whatever, same EV for both actions, but calling half a combo of KK is advisable.

Is it even possible for the bettor to play GTO?! When he bets 3 combos of AA and 1.5 combos of 43, he THINKS he is playing GTO...but in reality when his opponent has AK he is not playing GTO since he has only 1 combo of AA due to card removal. So how can he play GTO without knowing what blockers his opponent has?! ♠_o:

You are correct, but both are playing GTO (assuming (which we do) Internets calculations are correct). The opponent doesn't have blockers every time. This shows up in "real" poker too, especially if the ranges are narrow. Then you might to modify the N to 1 - 1 in N mnemonic. the bluffer should bluff a tad less without blockers (a lot less in Internets extreme example) and a tad more with blockers. I'm pretty sure it's a straightforward, but tedious calculation.

Originally posted by Itsnevereasy
Is it even possible for the bettor to play GTO?! When he bets 3 combos of AA and 1.5 combos of 43, he THINKS he is playing GTO...but in reality when his opponent has AK he is not playing GTO since he has only 1 combo of AA due to card removal. So how can he play GTO without knowing what blockers his opponent has?! ♠_o:

Yes, this is how poker works.

Every GTO bluffcatch call is a +EV call, otherwise it would not be a call. The Caller calls often enough, to not allow too many profitable bluffs, but all those calls are +EV. Every bluff is also a +EV bluff, and those +EV bluffs are exactly the right bluffing frequency, to not allow the caller less or more profitably than optimal.

If you think about how all hands block all ranges, you will see that you never have to "call 30% of your range" at random, but you only have to call the best blocking 30% of hands, which are +EV calls, and fold the worst blocking 70% hands, which are -EV calls. All while all opponents good blocking bluffs are +EV bluffs, and his bad blocking bluffs are -EV bluffs.

(+EV means "highest EV of all options", -EV means "not highest EV of all options".)
(Hands can be exactly indifferent between to options as well, so that two plays are +EV. That doesn't happen as often in GTO as the lay person may believe, there is almost always exactly one correct play.)

@unshpe
Unfortunatly, I don't know of a good beginners guide to GTO. David Sklansky's Theory of poker has a chapter on one-street bluffing and calling. It is a damned good but quite boring book. Every poker player should read it.

I'll be a bit nitpicky here. It's worth it to make an important point that makes it a little bit easier to understand GTO.

Every GTO bluffcatch call is a zero or +EV call. Even if GTO wins a bit of money against most (all not playing GTO) opponents, its biggest virtue is that it is it is unexploitable. The bluffer can do nothing, bluffing more or less wouldn't help. The situation is also symmetric; The caller can do nothing, calling more or less wouldn't help.