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!