By the way, above where I wrote "it is exactly obvious to me" I meant to say "it is NOT exactly obvious to me."

I can't rule out that I might write something like that and mean it, because as my wsop housemates will tell you I can sometimes be a jerk. But here I just missed a word.

Here's one. Re: correct vs incorrect plays.

Is a mistake (or incorrect play) considered any play that is EV- or not optimal, or is a mistake only a play that is going to have a negative EV?

OK, rant a second. What's the matter with plays that are EV- ? Maybe they are better than some other play that is even more negative. Maybe sometimes you get into spots where no matter what you do, you are losing money, and so you just try to lose as little as possible. OK, I feel better. Don't think about things as EV+ or EV-, that's needlessly calibrating to some imaginary zero. Just think about taking the MOST positive action.

Now your question...well, it's not really clear, because "mistake" is kind of ill-defined. I imagine that this is why you have this question. There are a lot of tautologies here.
"What is the goal of poker?"
"To make more money."
"OK, how are you going to make more money?"
"By making better decisions."
"Which are the better decisions?"
"The ones that make more money."

etc blah blah.

So maybe the thing to do is to define a method of play by which you can calibrate your decision making:

"My goal is to make more money in poker. I will do this by attempting to play optimally. If I could achieve this, I would ensure myself the game value and in practice make a bunch of money."
Now a mistake is a deviation from optimal, or more practically, playing some sequence in an exploitable way. Introspective processes are things like reading my own hand, investigating sequences, looking at my distributions and making sure I am balanced, etc.

or maybe...

"My goal is to make more money in poker. I will do this by estimating my opponents strategies, observing their deviations from optimal and exploiting them. If I could achieve this, I would be maximizing my money made against their strategy in a metagame sense."
Now a mistake is a deviation from maximally exploiting. Introspective processes are things like doing data mining on my regular opponents in order to explore and understand their strategic approach, using HUDs to provide realtime data on what they are doing, examining hand histories looking for more opportunities to exploit, or areas where I tried to exploit unsuccessfully and trying to figure out if they were adapting or if I was wrong about their strategies, etc.

or maybe...

"My goal is to make more money in poker. I will do this by establishing a strong base strategy which is close to optimal or alternatively, tends to exploit random opponents in my universe of common opponents. Then after I have played against players, I will deviate sometimes in order to exploit them, but if I sense counter-exploitation or my exploitation doesn't go well, I will snap back to my base strategy."
Now this is some kind of combination of the two above.

So now a "mistake" is basically one of two things -- when your actions are contrary to your self-stated approach to poker (obviously this is something you say to yourself; noone else cares and if you aren't honest with yourself you will certainly fail at poker)...and when you are unable to put your approach into practice (maybe you can't figure out how to be unexploitable in some sequence, or there are certain players who always seem to be one step ahead of your exploitation).

Anyway, that's a way of thinking about it that at least gives you something to think about in terms of what a mistake is.

Looks like im late to this surprise party thanks for doing this 1st off great book! but wish my dumbass could understand it.

My ? is if I am trying to be more GTO in my play so that i can spot and exploit my opponents better. So if i am trying to map out my ranges in NL 100 bb's on various boards, it seems pretty easy to know which hands to place in my value bet range. but i have no clue where to start with my bluffs.

for easy example sake lets say i have 18 value betting hands on the river and i am going to bet pot all in. I know i should have 6 combos of bluffs, but i have no clue which hands to include.

Really sorry would like to give a better example, but i don't really understand this topic much at all I know you don't want to solve anything in this thread but i would very appreciate where to start trying to learn how to do this correctly. thanks sooo much for your time

Let's take your example. You are going to bet pot allin on the river, and you have 18 value combinations. Now you need to bluff such that the ratio of bluffs to value bets is s/(1+s), where 1 is the pot size and s is your bet size. Since you are betting the pot, you need to have one bluff for every two value bets. So you will bluff nine combinations on the river. Apart from card removal concerns, you would just pick the nine worst card combinations you could have going into this situation.

Now those hands should have basically no showdown value. If you don't have nine such hands, that probably means that you made a mistake on a previous street by not bluffing or semibluffing. Often these hands are like small missed flush draws.

Sometimes it may be right to bluff (or even not value bet) certain hands that are a little better than your worst hands because they make it less likely that your opponent has a strong hand to call with. I'll give you an example. If you solve a river that is like raggedy small cards, like J8532, something interesting occurs. If the first player checks, the second player will usually check with hands like Kx. So when you solve the river and look at the first player's value betting range, he will like value bet AA and QQ but not KK, because checking with KK induces a lot of bets and bluffs (because it blocks the opponent from having Kx, a frequent checking hand).

Notwithstanding this card removal stuff, if you just pull your value bets from the top of your range and your bluffs from the bottom, you'll be taking a strong first step toward unexploitability.

Sorry for disrupting your ongoing discussion but
just want to make it clear so I have a reason to stop play them. Is there any way this structure is beatable?

Buyins 5-75$ 5man Hyperturbo 1500 startstack blinds increase every 3 min.
10/20, 20/40, 40/80, 80/160, 120/240, 200/400, 300/600.
10% rake, no rakeback whatsoever. The skill-level of the players are very, very low but still? The rake is ridic.

Payout structure?

60/40

I have never studied much game theory but I find parts of this thread that I can understand interesting.
One point that was brought up earlier that I wanted to reiterate on was where you talked about expanding your lag style by calling wider in spots to semi bluff turns or rivers or to simply call to represent a nutted hand and act with aggression. Does game theory support using this strategy more in cash games or tournaments? Or is this all dependent on the player, stack sizes, situations, cards etc?
It seems like some very successful players have utilized the limp reraise in tournaments with some stack sizes. I've seen good players limp with big stacks>100 bbs to trap ss players on a table, or keep pots smaller at a 3bet happy table. That kind of makes sense 2 me, but what doesn't make sense is when the limp reraise is used in the 40-70 bb range. First off is there an optimal bb range to do this with under 80-100bbs? And secondly is this move used to reduce variance? What I mean by that is I've seen some very successful players limp reraise with blocker type hands, so instead of raising and getting 3bet and getting stuck in a raise/4bet jam/commit mode they limp reraise with a nutted/blocker type range every so often to force the opponent to make the higher variance play. Does game theory support doing this at an optimal bb range under 100bbs?

Lastly, I consistently beat midstakes mtts online but struggled to beat high stakes mtts. This was due to a lack of experience and creativity or precision mainly. One concept I struggled with was merging hands against tough players at this level. Is there optimal board textures to merge your hand against once you've hit a piece of the flop against aggressive opponents?
The only success I've had with merging hands is when I raise in ep-mp with q-10,k-10 or hands with weaker kickers where I'll get called by a villian with a wider range and I have 25-40 bbs. I usually cc flops here when I hit top/mid pair and merge by c raising a lot of turns, especially where there's wet flop textures. I do this to avoid get called down more often when im out kicked and to avoid being exploited when villains raise the flop or turn vs me and put me in tough spots. Is this an example of merging a hand/range, and if so does game theory support these types of plays? Know it's a lot, thanks for any response you can give.

This should be quoted for awesomeness. Very good post about having a solid meta game mind set! Did you author a book about game theory and poker? If so when was it written?

The Mathematics of Poker by Bill Chen and Jerrod Ankenman. 2006.

Widely considered the bible of Poker Game Theory.

Thanks for making this thread. Some interesting stuff so far.

Is GTO solved for betsizing in 3-street games where SB minraises and BB calls pre with 100bb?
Or is there some kind of solved game for 3 street games that will give a hint as to how your betting/checking distribution should be on each street?
I´m ok with ignoring check-raises and just looking at ranges for betting once, twice and three times for now if that makes it easier to say something meaningfull about..

This is exactly the kind of stuff that is hard for me to answer; knowing what "game theory supports" in a NL strategy in some sense requires calculating such a strategy, which is impossible at the moment. In fact, any strategies that anyone is calculating are based on very significant simplifications of the real game. Now sometimes in practice these simplifications might work, but there are no guarantees. In fact, there was a paper by UAlberta people which proved that under very normal circumstances, a "more accurate" simplified game (ie one which more closely approximated the real game) can yield a strategy that is significantly worse when mapped to the real game. So not only are there no guarantees, sometimes you can shoot yourself in the foot by doing a "better" simplification.

I do have the following things to say about some of your questions: first, game theory is very hard to formally apply to multiplayer poker games, and the results aren't that useful. So when we talk about multiplayer game theory, generally we are talking about "game theory inspired" strategies where we try to play unexploitably, and then at some point it gets heads up and we try to play optimally in the subgame that results from the preflop multiplay...or somethng like that.

Tournaments aren't really any different from cash games from this perspective, except that instead of betting "chips," it's more reasonable to think of betting "equity," and the so the value of bets aren't necessarily symmetrical (due to chip nonlinearity). Additionally, in tournaments the stacks are normally just shallower than in cash games, which is an artifact of the tournament process.

When you think about how to be unexploitable, you start to plan ahead a lot more, and think about your entire distribution. A good exercise is to pick a sequence preflop and just assign distributions to the two players however you want. Now sketch out methods of play for both of them on this particular flop (maybe the way you would ordinarily play). There's no need to give them hands, just write down "check with XYZ hands, bet halfpot with ABC hands, etc etc". Then consider each guy in turn and now assume he knows the other guy's strategy and can exploit it maximally. Do the best you can to do this. Now think of how the first guy could improve his strategy so that it can't be exploited so much. Go back and forth a few times. You'll find yourself moving toward optimal play, even if you are still far away. You'll realize just how balanced you have to be; if you leave yourself not enough strong hands in some situation, it's bad. If you don't have enough bluffs, it's bad.

Now the result of this is just one sequence. But when you start thinking about how a villain who can read your soul (strategywise) would exploit you, you can't help but play closer to optimally.

About limpreraising...I don't know. I don't do it, except in the small blind. But that doesn't mean anything.

Only for the clairvoyant game, where one guy is polarized. Then it's geometric growth like we detailed in the book. As for real games, it's unclear which model really holds: is the real game more like the [0,1] game, where you bet different amounts with different hands, carefully balanced among various strengths of hands in order to not be exploited? Or is information hiding worth enough that you merge large groups of hands into very few bet sizes? Or maybe it's the former on the river but the latter on the earlier streets? I don't know. I think that the work that has been done on bet sizing in the literature is far from satisfactory on this topic, and just reflects that the problem is hard.

Then if the field plays bad it's for sure beatable