I'm having some trouble understanding a couple of key concepts and was hoping someone could clear up my confusion. I was looking at a Pio sim where UTG raised, and the bb defended. The flop was A54 with two spades. When checked to, pio wanted UTG to bet this spot pretty infrequently. Hands such as AJ and AQ were mixed strategies, betting about 10% of the time, and checking the other 90%. However, the EV of betting and checking was only a few cents different, which I realize is a common result in many of these solutions.

After that node locked UTG's strategy to force him to bet all his AJ+, as well as a bunch of back door flush draws that pio had him checking. This increased his overall flop betting frequency from roughly 20% to 40%. The BB responds with the maximally exploitative strategy, check raising slightly more frequently and attacking UTG's weak checking range on the turn. When this happens the EV of betting AJ on the flop goes down significantly, and many of the hands in the bb's range such as Ax go up significantly in value, since they can now comfortably value bet the turn after the flop checks through.

My first question is, if we know at equilibrium UTG should only bet AJ roughly 10% of the time and check the other 90, if the bb keeps his strategy unchanged and continues to play GTO, will the EV of betting AJ go down when UTG implements his new strategy of betting AJ+ and extra bluffs 100% of the time? The way I always understood this was since PIO told us in the first sim that betting and checking AJ had same EV, it wont matter how we chose to play this specific combo, unless our opponent knows what we are doing and decides to change his strategy and exploit the change in our betting frequency.

My second question is, if after we decide to bet all our AJ+ and extra bdfd bluffs on the flop, will many hands in the bb's range naturally increase in EV on the turn, when he again sticks to GTO and does not adjust to exploit UTG's weakened checking range? From the bb's perspective this makes sense to me, since now when he gets to the turn after check check, UTG is either faced with the choice of either calling down wider with weaker hands, or over folding the turn.

At this point I accept this to be true. Where I get lost is, its easy to visualize when we look at piosolver, or a grid of all our combos and assume they are all being played at once VS the same player. But when I apply this kind of range based thinking to my actual play, where I get dealt one hand at a time, and the way I play that particular hand should have no direct effect on the EV of other combos in my range, I get very confused. Again I am going on the assumption that my opponent sticks to a GTO strategy and does not adjust by betting the turn more frequently and thinner for value, and adding bluffs.

So if its true that when I sit down tomorrow, and every time I get dealt a hand that has a very similar EV as a check and bet, I always decide to bet, I will somehow have less money at the end of the year if my opponents play GTO and don't change their strategy, exactly where is this loss coming from? Am I losing future EV, invisible EV etc? The concept of losing money on a branch of the tree that we didn't arrive at (the turn) because I bet the flop too frequently makes my head hurt. How should I be thinking about it? In the past I always thought that doing something such as always betting here opened myself up to exploitation, but that an unchanged GTO strategy would not punish me naturally when this occurred. I was recently told my thinking is incorrect, but I cant make sense of why. Thanks for any help.

If a certain hand has the same EV as a check than as a bet, then it doesn't matter which action you choose, it's as simple as that.

Even though in reality, no one is stupid enough to just keep playing the same strat over and over vs you, without ever changing it. If you keep betting with all of those hands, regs will start exploiting you pretty fast, even at micros.

when you make an EV mistake (choosing an action that has less expected value) you will have less money at the end of the year. if you make a strategy mistake (same ev in a vacuum but unbalanced) you will probably also have less money at the end of the year because villain will most definetly exploit you. but if villain only has half a brain then strategy mistakes dont really matter.

If a hand has precisely the same theoretical EV as a bet or check, then it doesn't matter which option you choose. You'll just get your EV in a different way, depending on what you do. e.g. with a value hand you can bet and get called by worse, or you check back and pick off bluffs.

If you're more inclined to check back than bet, then villain will have more hands that see a turn card (because they didn't have a chance to fold on the flop), but his range won't increase in EV, because his turn probes (as bluffs) will get called by your stronger check backs.

As it turns out, most good players think that typical players make bigger mistakes vs c-bets than versus check backs. e.g. They fold too often to c-bets, and don't check-raise the flop, or lead the turn with air, anywhere near as often as the GTO strategy.
Thus, you can increase your win rate by exploitatively c-betting at higher than GTO frequencies. i.e. If the GTO strat is to mix between betting and checking, you'll probably do better vs most opponents in the real world by betting.

To give an example of a silly hand. Suppose you have AJ and the flop comes A72r. If the GTO strat was to sometimes check back, it would partly be because the opponent should - in theory - sometimes overbet bluff on the turn (to attack your "weak" check back range).
But if your real life opponent never overbet bluffs on the turn, there's no point in checking back the flop. You should just c-bet as normal.

Thanks a lot for the responses. What you guys have collectively said so far is how I've always understood it. I recently had a coaching session where I was told otherwise. His two main points were, AJ specifically decreases in EV vs a gto response when UTG decides to bet it at 100% frequency. This seems clearly wrong.

His second point was that once UTG no longer has any AJ+ in his checking range, many hands in the bb's range will naturally increase in value without any adjustment from him. As I mentioned in my original post his reasoning was that UTG will get to the turn with a much weaker range, and he either needs to overfold or call down wider. That is where the last part of my original question came from. If this is true, and checking AJ makes us lose money on a different branch of the tree (on the turn after the flop checks through), how do I conceptualize and relate this to real poker when I'm just playing one hand at a time. He tried to illustrate this by creating a third simulation where UTG's flop strategy was node locked to bet AJ+ at 100% frequency. Then he node locked bb's response to this all well, and instead of letting the program maximally exploit this, he copied and pasted the gto strategy and forced the bb to continue with the same range as he did in the gto sim. Then he we looked at one specific turn, the 8d. The bb was again node locked to play an unchanged gto strategy vs UTG's now weaker turn betting range. So he's not leading thinner for value with Ax hands after the flop checks through, etc. The result was that many of the bb's hands went way up in value on the turn, specifically Ax. One obvious problem was that he did not go through and force the bb to continue to play gto on all the possible rivers. So on that street, pio naturally switches back to the maximally exploitative strategy and gains massive EV. If I'm understanding you guys correctly, the only reason the the bb's overall EV was increasing, is because he did not force the bb to play gto on the river. So just because he force UTG to get to the turn with a weak range, in order to gain EV, the bb will have to deviate from GTO in some way to gain.

So it is fair to say that in some situations, a GTO strategy does not automatically gain when an opponent deviates away from gto, but sometimes it does? An easy example would be on river, when player A bets a balanced range of the nuts and bluffs, and player B chooses to always call. Player A doesn't gain unless he adjusts his strategy to bet bigger with the nuts, or stop bluffing all together. But in other situations, say one player starts bets far too large for value on the flop, and narrows his opponents calling range too significantly, when his opponent defends at the right frequency he automatically gains from this mistake. Or if player A chooses to bluff the river with bad blockers, and player B responds in a GTO fashion, he gains. So if all this is true, when UTG chooses to always bet AJ+ on this flop, and the consequence of that is a much weaker turn range when the flop checks through, the bb is not naturally gaining EV when he sticks to the gto strategy?

Arty, I had some questions about this comment. This is assuming a SRP, right? And if so, how does one decide their sizing for overbet bluffing the turn, considering you can't get all in by river

Yeah, I was assuming a SRP. If the player in position checks back the flop, it's often the case that you can't get all in by the river. Indeed, that's kind of the point of "taking a street off" by checking back with mid-strength hands, so that you never get stacked with a marginal hand. If your opponent wants to play for stacks, he'll have to make two HUGE overbets, and you'll rarely pay him off. Checking back to keep the pot small (and to prevent yourself from getting stacked) is one of the great advantages of being in position.
FWIW, when I'm OOP and the flop checks through, I'll usually go with 2x pot if I choose to overbet. It's the size that I learned with Snowie, but there are various viable sizing regimes you can utilize, or solve with software.

If I were putting it in the solver to solve for bet sizing, does something like 110,150,200 seem reasonable? Would you ever bet a 1/3 or 2/3 size in the same situation?

Yeah, I sometimes bet smaller. Some hands just want to get some thin value or want to deny equity cheaply. The over-large sizing usually only makes sense for a small part of your range. i.e. You don't always want to get all in, because you could just be value-owning yourself.
So if you're solving it, you should give the software the choice of two different sizes to see how it decides to split its range.