How do we decide what to bluff and what to give up on the river?

The frequencies of bluffs vs strong hands is the most important thing, IMO. If you study game theory there is something called an indifference principle.

If you are betting and representing a very strong hand that your opponent cannot beat, the only thing that incentivizes them to call is the chance that you might be bluffing. There's an optimal ratio of strong hands to bluffs, which varies based on the bet size, that makes your opponent indifferent to calling or folding. At that point they will win or lose the same amount on average whether they call or fold.

If you bluff too much they can call every time and make additional money on average in the long run. If you don't bluff enough they can fold every time and you won't make as much with your strong hands on average in the long run. So an optimal ratio of value hands to bluffs maximizes profit against a rational opponent who will adjust correctly to your strategy.

So ultimately GTO is trying to mimic game theory optimal ratios. Now assuming you are going to bluff with an optimal number of combos, it makes sense to find any other factors that might increase your EV a small amount. Blockers are basically one such factor. You have to have bluffs, you might as well choose the best hands to bluff with.

That's not to imply blockers are unimportant, as in certain spots they can be very important (possibly even changing the ratios). For example if your opponent is making a big bet representing a flush and we have the A of whatever flush suit is out on the board. In that case in some scenarios something like half of their total flushes will include the A, because they play many suited aces in comparison to other suited combos. So having the A flush blocker can be very important, whether we are calling with a bluffcatcher or bluffing ourselves on a flush board.

Anyway if you disregard blockers, you would likely want to bluff with the highest equity hands that you would otherwise be folding (this is true for streets prior to the river). Like say we are betting AK-AT for "value." It would not make sense to then use 32o as a bluff, we would probably want to use the next strongest hand that we would otherwise fold like A9.

What the poster above is talking about is that sometimes our hands have equity when they check. They're not really a pure bluff, they also occasionally might be the best hand. So in that case it makes sense to showdown the hands with some value, even if they rarely win and are not strong enough to bet with for value, and use the hands with no chance of winning as bluffs. This is most true on the river when the winning hand has already been determined.

Anyway I put a lot of info in this post and made a lot of simplifications just trying to get some points across. Forgive me if anything is inaccurate.

I'm sure this came across as being confusing. If you want to learn more Janda's book, "Applications of NL Holdem, " does a good job of explaining this stuff in more detail if you're interested. Or for a more modern take in the post-solver era Modern Poker Theory is also good and explains the same concepts, but with charts and computer-derived results that might be more practical for your games.

The confusing part is there's actually two theories (that are really one theory):

1) the GTO, optimal strategy that maximises all bluffs in context per nash equilibrium of range vs range, blockers, frequencies etc as discussed ^^^
2) the best optimal strategy versus the idiot human you're playing who is usually making a pig's ear of point 1

EG: often you'll get to river as the BB in a SRP and the BU has bet flop, checked turn, and bet river. Does that seem like a strong range? No, usually not. Therefore it is highly vulnerable to attack. BU's bet/call hands are often unintuitive. So we can start extending our GTO bluff range to include hands that are never good 'in theory', but may nonetheless print solid EV against someone incapable of readjusting their own range on the fly to aggression.

It's all highly situational, so you need a decent understanding of the theory to anticipate your opponent's mistakes before flying too far off script yourself. But the reality is the game as played favours well justified exploits (a.k.a. 'fuck gto' - a kind of silly paradoxical saying given gto would make those exact plays if it knew how badly your opponent reacted), and all because humons are naturally terrible at balancing ranges.

Great responses. Recently it "clicked" for me that the way to poker mastery is through understanding theory and using it to exploit your opponents.

Ever since then, I have been obsessed with learning everything about theory that I can find. I've read Modern Poker Theory a year ago, before this "revelation", and now I think I need to re-read it. I'll read Applications of NL Holdem as well.

The next "click" that I want to achieve is how to put all these concepts together consistently and reliably, which will be the hardest piece of the puzzle.