If theres a spot you are value betting with very few bluffs in your
range, it would make sense to bet very small given the odds villain is
getting on a call, or else he wont have the right pot odds to call vs
your range and will never pay off.

I think this is a commonly held belief but not really correct. Say we get to the river in nuts/air vs bluffcatcher with 4 nut combos and 1 bluff combo (ie we have very few bluff combos). No matter what betsize we choose, our opponent should always fold. I think it's worth taking a minute to think about why this is.

Say we bet pot with everything. Well obviously our opponent just folds since he's getting 2:1 but only good 20% of the time. Our EV is just the pot.

Say we bet 1/3 pot with everything. Well now our opponent is getting 4:1 and is good 1/5 so he's indifferent to calling. His strategy is irrelevant, our EV is just the pot still.

Say we now bet 1/4 pot. So villain is getting 5:1 and has a profitable call if we bet everything. He doesn't want our bluffs to profit, so he calls enough to make them indifferent - 80% of the time.

Well now we can improve our EV vs this strategy. We just pot our bluffs and 1/4 pot our value hands. Maybe you're worried about being exploited so let's pot 2 value and 1 bluff and 1/4 pot the other 2 value, same idea. So now 3/5 of the time we win the pot (when we pot it). The other 2/5 of the time, 20% he folds and we win the pot, 80% he calls and we win 1.25pot. So our EV is now:
EV = (3/5)(1) + (2/5)((0.2)(1) + (0.8)(1.25)) (for pot size of 1)
EV = 1.08
So no matter how we mix our betsizing, if we don't have enough bluffs our opponent's GTO response is to fold everything.

Lastly, exploitatively speaking you're making a big mistake betting smaller. Take the same spot and look at betting pot vs betting 1/3 pot. When we bet 1/3 pot, no matter what his strategy is our EV is the whole pot. If we bet pot then our EV is the whole pot if he always fold, but if he EVER calls then our EV is greater than pot. That is, betting pot is >= betting 1/3 pot.

Remember the above is all nuts/air vs bluffcatcher so it's not perfectly transferable to real poker, but the idea still applies in river situations.

Steve, you wrote this

"I think this is a commonly held belief but not really correct. Say we get to the river in nuts/air vs bluffcatcher with 4 nut combos and 1 bluff combo (ie we have very few bluff combos). No matter what betsize we choose, our opponent should always fold"

if we bet 1/3, he is getting 4:1, and by having 1 bluff for every 4 value hands, he is indifferent from calling and folding, so he should not necessarily fold.

So if our range is made by x value hands and y bluffs, we can size to (x/y) - 1 : 1 to give him x/y : 1 odds on the river and to have him indifferent to fold or call, and so to be unexploitable (even if the strategy sucks, because if we get to the river with too few bluffs we are not maximizing our EV because we have to bet small to be unexploitable)

Also, if we are sizing even less than that (even worse strategy), the opponent should always call.

If we size more than that, he should always fold

so the opponent decision can change depending on our sizing

You didn't read the rest of my post...if we bet really small with 4 value, 1 bluff he has a profitable call, but his call is exploitable as my post above yours explains.
(the short version is if he calls 1/10th pot all the time, then we can pot 2 value/1 bluff, bet 2 value for 1/10th pot and do better than if he folded all the time regardless of sizing. His unexploitable strategy is to fold to any bet, regardless of size. If we play an exploitable strategy then it may be that he wins more by exploiting it, but he will become exploitable himself)

I guess I do not understand why you said that his unexploitable strategy is to fold to any bet, regardless of size. that does not look like an unexploitable strategy to me, because it exists a counter strategy (we bet super small) that exploits that (and yes, we are exploitable ourselves, but who cares, our counter strategy to exploit him doesn t have to be gto)

But we're not exploiting him by betting super small. If we bet big, he always folds so his EV is 0. If we bet small and he always folds his EV is still 0. But if he ever calls vs a smaller size then he is open to exploitation.

Let's say we pot 2 value, 1 bluff so that he has 0EV vs our pot sized bet. Then we have 2 value left that we split among all other bet sizes. If he folds to all sizes then his EV is 0. If he puts money in vs any size then his EV drops (he never wins since we have nothing but value bets in all other sizes). We are not exploitable, we win at least the pot no matter his strategy. His optimal response is to fold everything.

what I was trying to say is that if we are playing gto, we bet 1/3 pot with 4 nuts and 1 bluff; if we do that, he is indifferent to call or fold, his EV = 0.
In this case, we are not exploiting him, we are playing gto.
His best response is to let us indifferent to betting or checking with our bluffs, that is, he should call 3/4 of the time, because the pot is laying us 3:1 on our bluffs.

if we deviate from our strategy of betting 1/3 pot, we are exploitable, because he is not indifferent between calling and folding anymore.
if we bet smaller than 1/3 he should call 100% time, because for him the EV of the pot on the 1 time we bluff more than compensates for the value bets losses
if we bet bigger than 1/3 he should fold 100% time for the opposite reason. (folding 100% has EV zero, calling has EV < 0)
In the last 2 scenarios nobody is playing gto, so we are both exploitable.

betting super small does exploit him if he folds 100% time, the same way it is for any bet size because he folds 100%. Folding 100% can not be optimal for every size, because we have few bluffs that will always succeed. Folding 100% is optimal only if we never have bluffs, or if we do have bluffs and bet more than optimal, which is what happens in your example.
in your example you say that 2 pot value and 1 pot bluff has 0EV, and then we have 2 value left with different sizes. but that is like saying we have 4 value and 1 bluff and we are betting more than 1/3 pot; that case folding 100% is the best option: but it is not for all the sizing

What you are saying is often true, but in this case is not. Your incorrect assumption is that the GTO strategy is to bet 1/3 pot because I have 4 value, 1 bluff. That is one GTO strategy, but it is not the only one. In fact I can bet my whole range for any size >=1/3 pot and that is a GTO strategy. Betting 2 value, 1 bluff for pot and 2 value for 1/3 pot? Also a GTO strategy, you cannot exploit it.

Now let's talk about the caller who we'll call C. Call the bettor B. If C's strategy is GTO then it is true that:
1. There is nothing B can do that will increase his EV
2. B's EV has to be less than or equal to the size of the pot (otherwise C would just always fold making B's EV the pot)

Hopefully we can agree on this. Now you're saying that GTO is for C to call a 1/3 pot bet 3/4 of the time. But this is VERY exploitable. B now bets 2 value, 1 bluff for pot (or larger) and C is indifferent (or worse) to calling that bet. B's EV for those combos is the whole pot. He bets his remaining 2 value combos for 1/3 pot. 1/4 of the time he gets a fold, 3/4 he gets a call.
EV = 1/4 * 1 + 3/4 * ( 1 + 1/3)
EV = 1.25

B's EV is greater than the size of the pot and so C's strategy is exploitable and thus not GTO. The ONLY strategy C can take that cannot be exploited by B is to fold everything to any and all sizes.

Hopefully this explains it. If you still disagree, feel free to propose a strategy for C that has an EV>0 and I will show you why that strategy is not GTO.

I agree that for B betting the whole range for any size >=1/3 pot is a GTO strategy, and C response is folding; it seems that every strategy from B that forces C to fold, is a GTO strategy.

my example about GTO strategy for C calling 3/4 was just if B is betting 1/3 pot, because the bet lays 3:1 on a bluff, but only if B bets that much with 4:1 value to bluff ratio.
If B is betting pot with 2 values and 1 bluff, then C should call 2/3, because the bet lays 2:1 on a bluff (so B is indifferent between checking or betting his bluffs), and so on.
That s what I was trying to say, and I am not sure that the C strategy I described is GTO, (it is when B bets that amount, but not all the time, so I am not sure if it qualifies as GTO)

About this:
" The ONLY strategy C can take that cannot be exploited by B is to fold everything to any and all sizes."

what I was trying to say is that if for any reason B bets < 1/3, B is not playing GTO anymore: best response for C is to call every time, because its EV > 0; C strategy is not GTO (and so it is exploitable), because he is exploiting a suboptimal B strategy.
so C should call every time only if B bets < 1/3; if that does not happen, this is not the case anymore.

I guess you were talking about a strategy for C that is unexploitable all the time, so I understand now your point that if C folds 100% is not exploitable; but should it be like that? C knows that B has 4 value and 1 bluff, so C can react to a small bet by using an exploitable strategy rather than a gto one

what I was trying to say is that if for any reason B bets < 1/3, B is not playing GTO anymore

You're assuming B can only use one betsize with his range. B can bet <1/3 pot with some combos and still be playing a GTO strategy (eg. bet 2 value, 1 bluff for pot and 2 value for 1/4 pot). If C ever calls any size bet, his strategy is exploitable.

You are correct that B can play a strategy such that C can make some exploitatively profitable calls.

well I think the simplest case is when you arrive at a spot where your opponent is capped and you have a pretty polarized range but not enough bluffs. There are only 2 ways you can screw this situation up:
1. Don't bluff. Every bluff you miss vs an opponent who has to fold to a bet costs you the pot. If your opponent is just not folding enough then you can choose to not bluff sometimes, but recognize that if he plays correctly by not bluffing you have cost yourself the pot. Most bluffs you make your optimal opponent will make them 0 EV, so vs said optimal player it doesn't really matter if you bluff or not, except with your very best blocker hands. This falls apart on runouts that are particularly strong for your range.
2. Bet too small. It's tempting to bet smaller so that your value hands get paid off, but if you start betting too small with your whole range you might accidentally cause your opponent to have a profitable call vs your actual range. Of course his theoretically correct play is still to fold to the small bet (see other post), so the small bet for value may be good exploitatively.

Situations where this actually comes into play are probably quite rare fwiw.

You can also find optimal strategies that perform the same against optimal oponents, but one of them is always better against nonoptimal oponents.

What I understood from [Lefort's video] was when our range is bluff heavy, its good to go large as we can bluff more often and be balanced.

I think this is another common misconception. Why is being able to bluff more often better if our opponent calls such that our bluffs are 0EV? The answer is it's not really. Our EV goes up when we bet bigger and we bluff more often but to say that bluffing more increases our EV is confusing correlation with causation.

We bluff more often when we bet bigger because of the price we're giving our opponents - if we bet bigger but don't add bluffs in a balanced fashion then his correct response is to always fold. We bet bigger because he is forced to put in more money on average vs a larger bet than a smaller bet.

An example using nuts/air vs bluffcatcher:
We get to the river with 10 nut combos and 10 bluff combos. When we bet (no matter how big!) our opponent has to call enough to make our bluffs indifferent, otherwise we can increase our EV by jamming everything.

Case 1: We bet pot. Villain calls 50% of the time to make our bluffs indifferent (ie our bluffs have EV = 0). Then with our nut combos we win the pot half the time and the pot + 1psb half the time. With our bluffs we win 0. Note that we're giving him 2:1 so we should bluff with 5 of our bluffs to remain balanced, but the EV of checking a bluff equals the EV of betting a bluff equals 0.
EV = 10/20(.5 * 1 + .5 * 2) + 10/20(0)
EV = 0.75

Case 2: We bet 2x pot. Villain calls 33% of the time to make our bluffs have EV = 0. Our nut combos win pot 2/3 of the time, and 3x pot 1/3 of the time. Here we're giving him 3:2 so we should bluff 6.6 of our bluffs, but EV(bluff) = EV(check) = 0
EV = 10/20(.66 * 1 + .33 * 3) + 10/20(0)
EV = 0.83

Note that we bluffed more in case #2 but our bluffs made 0 in both cases. Being able to bluff more did NOT increase our EV. But our opponent had to put in more money on average to prevent us from having profitable bluffs so the EV of our nut combos increased by betting bigger.

If we don't have enough bluffs to balance for a sizing - say we only have 4 bluff combos instead of 10 on the river - then it doesn't matter what we bet, our opponent should always fold and we win the whole pot (see my first post ITT for explanation)

And so, when we have a nut heavy range, we should go smaller.

If we have a nut heavy range and we bet big, our opponent folds and the EV of every hand in our range is just the pot. Betting smaller may increase the EV of our value hands (because he calls sometimes now) but it decreases the EV of our bluffs. In nuts/air vs bluffcatcher the EV of the nuts/air player is never greater than pot (if it was, then bluffcatcher could improve his EV by just always folding!). So betting smaller to get calls cannot improve our EV if the alternative is to bet bigger and always win the whole pot. Exploitatively it may be a useful strategy though.

Also don't get the second part, surely the smaller a get he faces the
more he has to widen his bluff catching range.

If the ranges and stack depth remaining are such that you can pick a sizing that he is forced to always fold to (ie you have few bluffs and enough money behind that you can bluff all of them in a balanced way) then facing a smaller bet should not affect his bluff catching range - see my example in my first post for an explanation.