I need a hand getting to grips with one of the ideas behind GTO strategy. It may be very simple, but I am currently confused about it and would like some help. I will give numbers to the questions I have so as to enable easier reference in answering posts.

I believe that I have a firm grasp of the reasons for which you would want to try to create an unexploitable betting strategy according to your bet size, such as betting pot on the river with a ratio of 2:1 value bets to bluffs.

I understand how to choose which hands to bet working backwards from this. So I've practised laying out a complete board, working out the number of value combinations I have on the river, then working out how many bluffs I get on each street (and how to pick them). [I play heads up and usually work with a 2/3 size bet as standard, so this means I get about a 2.5:1 ratio of value to bluffs on the river, twice as many bluffs on turn as on river, twice as many on flop as on turn.]

What I feel confused about is the rest of my hands. On any given river the number of combos you have that you can bet for value differs widely. So for some rivers you might have 30 value combos, and therefore 15 bluffs for a pot-sized bet, whilst on others you might have 150 value combos, and therefore need 75 bluffs. I understand the basic concepts behind how best to pick my bluffs (eg the debate between picking weakest hands vs picking best removal). If you are able to bet with such a balanced range, on the river the value of betting will be the size of the pot, for you (and 0 to your opponent if we treat this river situation in isolation from the rest of the hand).

What I feel I do not understand is the value of the rest of your hands. (1) Is there an ideal % of the total hands with which you reach the river that you would want to be able to bet? For example, would it be better to have a river range which would be composed, on average, of ONLY 33% bluffs and 67% value hands, if you are betting pot? So no hands with which you do not bet? In other words, you could work out what % of all starting hands, on average, you are able to bet for value on the river, and then add to that 33% bluffs and play those hands?

(2) And could you change your bet sizing for each situation to enable you to reach such an ideal/ to adapt to the % of value hands you have in any particular spot, and therefore to enable you eg to always bet?

(3) What is value of holding that part of a GTO range on the river which is hands with which you do not bet?

(4) Why is it mathematically necessary to have high betting/ barrelling frequencies from street to street?

(5) How does the EV of those bluffs with which you give up on turn and river factor in to determining correct play?

(6) How does the desire to hold a particular range on the river relate to the opening range you would want to construct (if it is ever useful to calculate it this way)?

I think for some of these questions I might just be tired and have tied myself in knots thinking about them when I really know the answers (I didn't have these questions last week!), but they seem to have slipped out of my grasp. I have read Ed Miller, Janda, Will Tipton, etc, and for whatever reason going back to them isn't helping me resolve these questions right now. What I'm looking for is a clear and pedagogic hand with my confusion, not any book/article recommendations. Thank you!

Like the minimum defense frequency, the indifference principle is useful in some areas and can be mathematically correct for toy games, but is grossly misused when applied to spots where you don't actually know imbalanced ranges are or are supposed to be.

I'm honestly not sure if its every river, but my guess that the vast majority of rivers (probably over 99%) involve having a checking back range in GTO play. The idea isn't so much that those hands are bad bluffs, its that if you're bluffing with the correct frequency your give-ups should have as much equity as possible. I really don't think there's some sort of shorthand way to estimate what percentage of your river range that would be, and even if there was, it would be dangerous to apply.

4) I don't think its mathematically necessary to have a high betting or barreling frequency. It depends a lot on what specific poker spot we're talking though, but I feel like in most spots proper preflop and flop play means ranges won't by widely imbalanced on later streets. The player with weaker range should get progressively stronger on later streets as the weaker holdings are folded out.

5 and 6 are just too complicated to answer. It seems correct to me that preflop and river are strategically tied to each other. You can't just look at allin equity to determine starting hand strategy, but its basically impossible to establish that connection without a super computer. You're just fooling yourself if you're trying.

Hi mate,

Thank you very much for this response. I agree with everything that you have said. I think it worth noting only that the indifference principle seems more properly applicable in heads up than in other forms of poker, where it may well be more often misused. But it tends to feature quite heavily in both HUNL books and player strategies (Doug Polk's Advanced Heads Up Mastery course for Upswing is a good example).

I have been thinking about my question number (4) for the last couple of hours, and I suppose the answer can be gestured at/ sketched out by looking at the problem another way:

Why would it be necessary to bet at all, in a GTO solution to HUNL?

It is a function of the fact that for some % of hands it is more profitable to bet than to fold, and vice versa. And this is because when you bet you can *expect your opponent sometimes to be folding, and sometimes also to be calling with a worse hand* (a worse hand vs your actual holding, if they are playing GTO, but perhaps not vs your range). Both of which are profitable outcomes for you.

If you were always folding from the SB the value of the game would be -0.5bbs. If you were always betting…well, it would depend on the big blind’s response, but their GTO response would surely not include either all folding (-1bb) or always calling/ 3betting (then the SB could always bet only pretty strong hands and win money).

So, preflop from the SB, you play hands with a total EV of more than -0.5bbs (what you lose if you fold). Moreover, you will never win more than 1bb if the big blind is playing optimally (since the bb will never call in spots where his EV is worse than folding). So the value of the game of HUNL to the small blind is -0.5<x<1. Similarly the value of the game of HUNL to the bb is between -1 and +0.5 (though it is unlikely to be positive).

It can never be worse than always folding, and an optimal strategy clearly does not involve that from either position! So some degree of betting and calling is required.

So now for postflop. Let us consider betting opportunities in the SB (ignoring the possibility of the BB playing back at you at all). If your opponent is folding 100% of the time you should always bet and will be hugely profiting. If your opponent never folds you can profitably bet all hands in which you have over 50% equity against your opponent's range on any given board and check back the rest (this is clearly not an exactly optimal exploitation since it doesn’t take into account anything to do with bet sizing, but it would still be winning). From the SB this is likely to be just under 50% in principle, and more in practice in any real life situation without these imagined strictures, since any opponent will be less likely to be able to realise his full equity out of position.

The optimal BB strategy would, for this game in which he cannot be aggressive, be somewhere between never and always calling, and the optimal SB strategy will be somewhere between always betting and betting hands with over 50% eq. I would imagine that this holds true for the actual game of HUNL.

This is part of why a GTO solution involves some reasonable betting %.

The other part of why GTO would involve betting concerns our value hands. The river is the only point where value is fixed (excepting the immensely rare cases when you have the technical nuts that no river can change, as on AKQ all diamonds where you have the JT of diamonds in your hand. It seems worth nothing that these cases are rarer even than they might appear: if you hold KK on AAAA the river can still be a K to split the pot). If we always bet only value our opponent will (exploitatively) fold always and we will win the pot--a good result. But we can do better. If we mix in some bluffs we can bet more often, getting him to fold a bunch more (since in any spot he has some % folding if he is not always calling) and still making money when we are called, since we have a value heavy range (eg 2:1 for a pot sized river bet). So we are mixing in just enough bluffs to keep our opponent indifferent to calling/ folding when we bet, so we can bet more of the time (but betting in any higher frequency would entail a different value:bluff ratio, so make our opponent non-indifferent to calling).

This is the best we can do when it is assumed that we arrive at the river with a particular range. What it does not answer is: what is the value of the hands we are not betting? And how do we construct that part of our range optimally?

So we are betting our best and our worst hands, to give us the best chance to win when we check. So we have a middling checking range:

VALUE
<<cutoff hand (possibly mixed strategy hand)
[checking range]
<<cutoff at top of required bluffs
BLUFFS

Our value range will almost always beat our GTO opponent if he calls, our bluffs will almost never. There will be a cutoff value hand where checking has higher EV than betting (which might actually be a hand for which we should have a mixed strategy). If we (for the sake of simplicity) pick the absolute bottom of our range to bluff we can work out how many hands we are checking between these upper and lower bounds. So let’s suppose we are at the river with 20% value hands, always betting pot. We therefore need 10% bluffs, and 70% will be checking:

VALUE 20%
[checking range]:70%
BLUFFS 10%

I *think* that our checking range can possibly be a bit more losing than it is winning (given that we have folded out our opponent’s less good hands on earlier streets). If it were even then the value of the river to the small blind in this situation would be the pot, which is surely too much. What do others think here?

And how, then, does this extrapolate from the river to earlier streets? I suppose that whenever you barrel your opponent can’t always be folding (you profit by always betting) or always calling (you profit only betting value). If you bet pot they can’t fold more than 50%, or you can profit with ATC. But they can’t call any more than this or you can bet only value and always profit. They must therefore be doing *some* folding (about 50% to a pot-sized bet). This is clearly profitable, so you want to be making them fold the maximum amount you can without opening yourself up to exploitation by betting too frequently (then they can start to call less with more value).

So on the river you want to be betting as much as possible to win the pot as much as possible, without betting so much as to become exploitable, and the same is just as true on earlier streets. The notion that you will profit because your opponent will call with -EV hands is, *on average*, not primarily true; of course in practice the SB has advantages that will give him an edge and force the BB to lose more, but a BB playing GTO will be calling with the correct hands. You are really making your money because you expect the BB to be folding some hands when you bet, which wins you money.

So you want your frequencies to enable you to do as much betting as possible, without it being toooooo much.

Can somebody explain to me why, for a pot sized bet, you would need twice as many bluffs on turn as on river? It’s temporarily gone out of my head. Not a troll—I wish I was joking. I’m just quite new to all this. Also corrections and input please!

You will have a higher ratio of bluffs to value on the turn because you treat your river value hands+ river bluffs on the turn as the same, you treat them like river betting range, so on the turn you will bet you turn bluffs+value hands+river bluffs, this is why the ratio of bluffs is higher on the turn, but this theory is more useful for toy games, in real game things are different

+1. I think Janda used the twice as many hands in an online thread a long time ago to demonstrate the misapplication of the results from the AKQ game in Mathematics of Poker.

He was trying to illustrate the ridiculous number of hands you would need to bet on the flop to carry through the 3 streets on the river.

Thanks guys. So the point is that since your opponent, when you bet the river, will be indifferent to calling that range, we can work on the turn as if his calls corresponding to that part of your range = -1turn pot-sized bet EV, so we need more bluffs to make him indifferent?

IE on the turn when you bet:

2/3 of the time you will bet again on the river, when he will be indifferent to calling (and is therefore -his call of a pot sized bet in EV)

1/3 of the time you won't bet again, and he will (possibly) win the pot and your bet.

So calling your balanced range on the turn, 2/3 of the time he is -1 (his bet); 1/3 of the time he is +2 (the pot and your bet), which sums to 0 (ie indifference).

Is that correct?

i think "possibly" is quite important. just because you check back river does not mean Villain wins the pot. same as you dont always win when you bet river with a value hand.

you can not "plan" on the turn, how many combos you will bet on the river because you dont know the river card yet. on some river you will be able to bet a lot , sometimes even your entire range and sometimes you will have to check back a lot...

I don't think you can generalize it that way. On the turn villain has equity vs some or all of your range meaning villain may win at showdown regardless of the strength of your hand on the turn.

In otherwords at the decision point on the turn there is no way to guarantee indifference to a river bet because the river has yet to come. Your equity on the turn captures the indeteminant nature of that choice in the form of probabilities, which may or may not become actual reality.

That's why in order to solve your preflop range you need a super computer as another poster suggested. You literally have to test/play each hand to end of the hand (opponent folding or showdown) to determine whether it's overall positive EV to play preflop.

If these hands mentioned above make it to the turn through making the play with the highest expectation, these hands may fold the turn at 0ev if faced with a bet.

This is false. Checking always has a positive expectation relative to folding.

I think you’ve misunderstood the point...and why would I ever open fold when checked to from the small blind on the river? Bizarre comment!

Just_grindin and Zuko : thank you very much for your responses. Of course what you say is true; you can’t plan with certainty for the river on the turn, nor with certainty for any ensuing streets on earlier ones. But you could say a similar thing of the river, in fact: unless you have the technical nuts (by which I mean the one best hand for any given board) you cannot be certain about your equity vs villain’s holding (it is just that it can only be 100%, 0% or 50%). As, in fact, you also observe:

Of course I agree with this as well.

But I do not see that these observations, which presumably are only to say “we can’t be sure”, affect how we should attempt to create the best plan given our limited knowledge. Just because we don’t know everything doesn’t mean we shouldn’t do anything! What I am wondering is simply whether what I have sketched is the idea as to how one should envisage play given all the possibilities of future alteration and limited knowledge. Do I have it right?

I also imagine that the facts you mention, that (1) we sometimes win when we check, and (2) they sometimes win when we think we are value betting, should do some work to balancing each other out, and therefore go further to suggest that the suggested method of planning is the best possible.

I agree we should plan but I don't think the method of planning for indiffernce by the river is the best or even a good way to plan your hand. You just have to adapt your strategy to the holding you have as information on the board is revealed.

You have to do work with software off the tables to see what insights you can gain from equity calculators or more importantly poker solution analysis tools such as pio, gto+, monker, poker snowie, etc.

Since folding is free, it serves as the constant we use to describe profitability.

The point I was getting at is that checking ranges should not lose money in the long run if we’re playing correctly.

Bob148: thanks for the clarification, but I am still not sure you have got what I was saying. I was suggesting that checking back can permissibly be a strategy in which we lose the hand more than 50% of the time, given that we are profiting when we bet. I’m not talking in terms of EV, and since it would be insane to open-fold the river it’s meaningless to say that checking has a higher EV. Of course it does!

Just_grindin: thanks for the response. I haven’t used any of those solvers tbh; I have so far taken an approach closer to Doug Polk’s in Advanced HU Mastery/ Janda’s Applications: Miller’s 1%, which all seem to advocate planning as I suggest. Why do you think it’s not such a good approach?

I am only vaguely familiar with Janda's work through MoP and posters sharing/paraphrasing of Applications of No Limit hold 'em. I have heard Newall is good but haven't read him and I am inly familiar with Doug Polk's thoughts via analysis videos on YouTube on Doug's channel.

I guess what I am trying to say is I am not familiar enough with all of their work to say if what you are describing here:

Is a correct application or interpretation of their work.

My comments were specifically about this thought process. I don't think you can extrapolate river play to the flop in this fashion. Not without gto solution software any way. I just don't think this rough estimation of deriving a preflop range from your river range works.

Ok, that's a far cry from the bit I copied from post #3 in this thread:

because it's not "losing" in any way.* Sure, when you check, your opponent's share in the pot increases naturally, but this is a necessary evil of not betting too often.

*actually your checking range is gaining if compared with a strategy that bets with hands that should check for higher ev.

Thanks just grindin. Would like to have Zuko’s input also.

Bob, just give it a rest man—I don’t feel like any of your responses are trying helpfully to answer my question.

Ok, I think you should reconsider this stance as it's the crux of the matter, but since you wish me to stop, I will stop for now.

That's because your questions are rooted in misconception.

Here is the way in which I believe you have misunderstood what I'm saying:

-It is important to talk about EV, obviously. However, since we are looking at the profitability, on average, of different segments of an overall strategy, stating that the EV of folding the river is 0 is completely irrelevant, and also untrue. It is undoubtedly the case that EVfold=0 if we consider the river separately from the rest of the hand, but that is exactly not what we are doing here. Further, as Will Tipton points out in his excellent books, many people have rote-learnt that the EV of folding=0, and misapply it (eg by treating SB open-folding as EV=0 when it is more useful (I think) to treat it as -0.5). To avoid this confusion I prefer to treat EV as stack size or the difference in stack size, as Tipton does, from the beginning of the hand. That helps to avoid mistakes like calling EVfold=0 when considering a particular part of the tree as an outcome in an overall strategy.

The way in which that applies here, of course, is that what I was suggesting is that our river checking back range will on average capture less than 50% of the pot. This is the way in which it is losing; if we find ourselves in this position (if that suggestion is correct) then we have finished the hand with less money than we started it. Our overall strategy is winning, but if, as part of it, we find ourselves in this spot, I posit that the hand is likely to be one on which we lose slightly on average.


No, it's because your answers aren't engaging in trying to explain or teach, but are just pretty irritating quips.

I'm not trying to quip at you. I prefer to keep things as simple as possible. You asked this question:

If you make the plays on previous streets with the highest expectation, then the current street will always offer ev greater than or equal to zero and there will be no loss of value if the current street is played correctly.

Just because you were the aggressor on the previous street, this doesn't mean that you get >50% of the pot on every possible river card with every hand in your range. You're correct that the checking range will capture <50% of the pot on average, but this does not mean that you made a mistake previously in the hand.

Sometimes you end up on the river and your stack will not be as big as it was at the beginning of the hand if you play the river correctly. This doesn't mean that you made a mistake.

This is fine, but don't be a dick when others use (fold = 0ev). Both ways work just fine.

Isn't this the same mistake as assuming EV fold = 0? You're looking at one part of the game tree and saying I don't win 50% in this node so that my overall startegy over the rest of the nodes is slightly negative.

I assume by overall strategy you mean up until the river?

If that is the case how can having +EV ply until the river and then having a positive EV from winning some fraction of the pot on the river result in a negative overall line?

Or do you mean our strategy across all hands is positive EV but we happen to have some slightly -EV hands mixed in so could do better by moving them into another part of our strategy?

Bob--I agree with everything you've written in your response; glad things are resolved. And I wasn't being a pedant/ dick about EV; it seemed like you had willfully misunderstood/ been unhelpful.

just_grindin: "Isn't this the same mistake as assuming EV fold = 0? You're looking at one part of the game tree and saying I don't win 50% in this node so that my overall startegy over the rest of the nodes is slightly negative." Not saying that the overall strategy over the rest of the nodes is slightly negative. Even in a positive strategy there can be end-nodes (leaves, Tipton calls them) where our EV is not positive. EG imagine (for the sake of simplicity) that villain folds to all turn bets except when he has, say, quad aces. Betting the turn will be +EV, but the 'checked river' node will be -EV for us, as seen from the beginning of the hand.

"If that is the case how can having +EV ply until the river and then having a positive EV from winning some fraction of the pot on the river result in a negative overall line?

Or do you mean our strategy across all hands is positive EV but we happen to have some slightly -EV hands mixed in so could do better by moving them into another part of our strategy?"

Neither of these. [I also want to stress that what I wrote was a hypothesis not an assertion (I'm not sure our checking range will automatically be less than 50% likely to win the pot)] I mean the strategy across all hands is +EV, but that, as a result of that overall positive strategy, when we nevertheless find ourselves checking the river, that *particular spot* might be negative EV. Do you see what I mean? Clearly you can have a positive EV overall strat but have to encounter individual spots which are - or 0EV (like whenever we fold!)

I agree you can have a node in a strategy that is negative but then the sum of all the EV's of paths through that node is positive.

But I am confused how you use this at the river to justify checking can have -EV?

If your strategy is overall positive until the negative checking node on the river, then how is it the node on the river is negative?

Your path up until the node is positive and you check and see showdown with some equity so you always end up with a positive payout on that node.

That's where I'm confused. Maybe I am just mixing up what you and Bob were talking about.

When I make a low equity bet on the flop in the hopes that my opponent folds, my bet is intended to be slightly profitable, as it receives profit from these ev sources:

Pair or better showdown value

Fold equity

Unimproved showdown value

Notice that if we’re called on the flop, our ev is almost entirely made up of pair value and backdoor draws (which benefits a little from turn semi bluffing value).

So because of that, when we miss on the turn and we check back, it’s a guarantee that we will not be recouping the previous investment in this spot.

The stronger your draw, the less this is true.