Is it really an exploit if you don't increase your own payoff? I guess it's a way to decide if the rake or your opponent gets more EV lol

Do solvers find a true Nash equilibrium though, or just an approximation? If it's just an approximation, then it is exploitable surely.

Well if you solve down to perfect accuracy then they find (a) true equilibrium for whatever subgame you're representing.

But it's still just a subgame. You're approximating the true gamespace with limited bet sizes and you've probably ignored card removal from the bunching effect. So no, it won't be perfect. In that sense, every solution is exploitable.

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Reading back my posts were kind of long-winded so I'll summarize here:

1. If you know your hand and your opponent’s exact (fixed, unchanging) strategy, then you can always find the highest EV action(s) for that hand. You don’t need information about your own range to find the best move. You’re essentially playing your hand in a vacuum vs their strategy.
   
2. If there's rake involved, then your mixing errors can transfer EV between the opposing player(s) and the house, without changing your own EV.
   
3. GTO only gains against pure mistakes (actions that should never be taken AND lose EV). It does not punish mixing mistakes. Mixing mistakes can be exploitable if your opponent will adjust.
   
4. If a hand mixes actions, those actions should theoretically have the exact same EV. If they don't, that's due to solver noise and can be fixed by solving to higher accuracy.
   
5. If villain is allowed to adjust their strategy, then you need to start thinking about your overall range construction. If you underbluff they can stop calling. If you cap yourself they can punish you. If you fold too much they can bluff more. All of a sudden you start playing range vs range. But this only happens against a dynamic strategy.

Thanks for the reply, pretty good post actually, summarising a lot of the functioning of solvers wrt Nash equilibria.

How can you calculate the EV of future streets if the strategies remain static? If the strategy on the flop changes the composition of our range, that would then change our turn strategies and river strategies, how can you then calculate the EV of future turns and rivers, and therefore our flop hands, without "recalibrating" the strategies in the first place? How can you prove that the EV of the individual hands on the flop exist independent of the context of our range on future streets if those future streets can't be recalculated without adjusting the opponents strategy to reach a new equilibrium?

I'm just missing a piece of the puzzle that hasn't made any of this logically click for me I guess

If villain plays Nash Equilibrium for the entire NLHE game, and not for the toygame that you set up with ranges that enter a given street, then our hand has a fixed EV regardless of our range.
If both players recalibrate strategies when entering future street because of ranges, then it's not Nash Equilibrium for NLHE, but for that particular toygame.

It's very simple really, if you try to ignore all of the solver baggage for a second. If my opponent folds exactly MDF on turn, and we know exactly how much he folds on every single river runout (fixed strategy), then we can calculate the EV of bluffing turn versus this player regardless of our range construction on turn or river (as long as we know what we're going to do with this particular holding).
We would only have to care about our range construction if villain is able to recalibrate on the river based on the input ranges to that street, which is an exploit on his part in of itself.
It's up to you to decide if your opponent is able to deliberately recalibrate like this depending on what part of your strategy is imbalanced, but by no means your imbalances would get automatically exploited by a static player. (Unless you're taking suboptimal EV lines)

Those future streets don't need to be recalculated. You already know their strategy and your EV with every individual hand/action against that strategy.

It's easy to get lost in the abstract woo woo of theory. The truth is that the EV for any hand/action is simply a function of the opponent's strategy. Imagine we know exactly how often they will raise/call/fold in any line on any runout. We know how often they'll fold in certain spots given our blockers. We know what they call with, and our equity against them at every showdown point in the game tree. Every hand is trying to independently maximize its EV against the opponent's strategy in a vacuum. If their strategy won't change, then our EV won't change.

How do you think solvers calculate the EV for actions they don't take? Let's say 55 bets the flop when it's supposed to always check. How does it know the EV of betting 55, if 55 doesn't exist in the betting line? By your logic, this would lead to infinite complexity. What if our hand takes the bad action 1% of the time? Or 10%? Or 11.5431%? Or 20.3%? Each of those would need a brand new solution. There are literally infinite frequencies that a hand can mix. The answer is that you don't need to recalculate the solution to find the EV. You just fix the opponent's strategy and calculate the EV of betting 55 against that strategy. The solver takes turns fixing one player's strategy and letting the other player exploit it, which wouldn't be possible unless you can find the EV of each hand in a vacuum against a fixed strategy.

What about CREV? This program is perfectly capable of finding Max EV lines against a fixed strategy. It does this without any GTO magic. It simply finds the maximum EV action with every hand vs whatever fixed strategy you've plugged in. Unsurprisingly, it finds the same EV and same actions against a fixed strategy even if you change hero's range.

This seems like a good summary, and is consistent with what I have read.

I think the first point causes a lot of confusion though, because in the real world against human opponents we are always playing against dynamic strategies, and never against fixed ones.

There's a reason solvers include so many mixed strategies. It's because our "true" highest EV option in the real world is dependent on our opponent's frequencies.

This is where a lot of players seem to miss the bigger picture. For example they will select a hand that is "solver-approved" as a bluff some percentage of the time, and bluff with it almost every time. Then they make similar plays with other "solver-approved" equilibrium bluffs over and over.

Against an equilibrium strategy, those bluffs are all neutral EV, even if the player's bluffing frequency is way higher than it should be.

However against a human who recognizes someone is bluffing too much, and starts calling down light, those originally"neutral EV" mixed-strategy bluffs suddenly become massively - EV.

I think this goes back to game theory. The EV for our bluffs is generated by our value hands. Our bluffs get through because our opponent has to worry about paying off our value hands.

So many players get caught up in, "How should I play X hand," which is important, especially for newer players.

Eventually though it comes down to how often someone is bluffing vs how often they "have it." That's where the great players exceed their competition. With all the different flops and all the different run-outs, it's a monumental challenge for a human to construct reasonable ranges across all the different run-outs. The great players are better than average, and are also better than average at recognizing and exploiting their opponents' imbalances.

Anyway, I wasn't trying to call you out. What you wrote all seems correct to me. It just really confused me when I was first getting into game theory.

If you or anyone else has anything to add to what I wrote, or if anyone sees anything wrong with my perspective, then I would appreciate the input.

There are cases where human opponents have static strategies, like in anonymous pools, or people who don't care about HUD use and don't pay attention to opponent leaks, but rather just try to play their own game.
Against such opponents, the only thing that matters is taking the best EV line with each hand, and if two lines are indifferent, then it doesn't matter.

Good point. I shouldn't have said, "never." Especially with the proliferation of bots and RTA, there are certainly some spots where our opponents play a fixed strategy. Even in live poker there are occasional players who just never seem to adjust.

I suppose my main point was that many players misuse solvers. I see many players use solvers to justify bluffs or calls that in many cases are just bad.

I made the mistake myself of opening suited connectors from early position too often, which was just losing money. Eventually I realized these hands do not perform well. In some of these spots, the solver recommends opening 10ish percent of the time (for board coverage, I think). Many players seem to see that a solver makes a play some low percentage of the time and assume that means that it was "fine" for them to play their hand the same way. However when a low frequency play is made too often, it can become very bad, and many people seem to underestimate how often they are pulling the trigger when a low frequency call, bluff or whatever is "solver-approved."