the fact that something is exploitable means it is not GTO. so if GTO requires different betsizing, it will require that you bet full pot only SOME of the time with your top 25%, and you would bet half pot SOME of the time with your top 25%, enough to make it breakeven at best for your opponent to push back against your weaker value range.

also, just because a particular move with a particular hand is +EV, does not mean it is part of a GTO strategy. you pass up +EV moves sometime for the sake of balance. in other words, precisely because you would be turning your hand face up, you pass up a move that seems to be +EV in a vacuum when it would enable your opponent to take that EV away by knowing what you have.

+1

So many posts itt take the form of "what I don't get is if you just had a big enough rocket I'm pretty sure you could exceed the speed of light ldo".

I don't really know what a "gto calling strategy" is, although I suspect it has something to do with what the opponent does with hands that are worse than your worst value hand and better than your best bluff.

Where did anyone get any idea that game theoretic strategies "don't make assumptions" about ranges? I debunked this idea earlier in the thread.

Anyway, you said that the best strategy with the nuts is to bet big, but the best strategy with a hand of medium strength is to bet small. But then your opponent can exploit this by value-raising you very hands with hands that beat your medium strength hand. Suppose he did this. What would be your response exploitively? To bet small with the nuts in order to smash his thin value raising. But maybe you should do that anyway? I don't know where you got the idea that betting big with the nuts is the best play anyway; it's true in some contrived situations where your opponent can never raise you.

So now what do we have? Maybe three hands for each bet size. A strong hand, a medium-strength hand, and a bluff, mixed in the right ratio. The strong hand gets value when the other guy raises, the medium-strength hand gets value when the other guy calls, and the bluff gets value when the other guy folds. If we mix these appropriately, we can profit from the opponents' deviations from optimal.

This statement can be taken in two ways; one correct and one incorrect.
INCORRECT:
If you mix appropriately so that you can't be exploited, you profit from your opponents' deviations.

This is only rarely true. In your example, if your opponent calls too much, your expectation for your bluffs is less but it is more for your value bet. If your opponent folds too much, vice versa. If your opponent raises too much, your expection for your medium hands is less but it is more for your strong hands. Your net expectation is the same because it is this equilibrium that defines the correct mix. You do not profit.

CORRECT:
You can mix these appropriately to profit from your opponent's deviations.

If you know that your opponent is deviating you can suppress or augment that part of the mix to exploit the deviation. If he calls too much, you don't bluff. If he folds too much, you bluff more. If he raises too much, you suppress the thin value bets.

---------------------------------------------------
GTO strategies can't be exploited; however, they generally don't exploit imbalances in your opponent's strategy. You must unbalance yourself to take advantage of your opponent's imbalance; if he figures it out, he can exploit you! This is what 'feel players' do instinctively.

All of this has happened before, and all of it will happen again.

Oh, I see now. Thanks for clearing all that up for me.

What I meant that to say was that by mixing properly any deviation from optimal will be more profitable for one of the hand types, thus the opponent can't exploit this type of strategy (in an obvious way). I got distracted and wrote something silly instead.

Do you think the solved GTO/equilibrium strategy uses call multipliers?((not my term/concept btw) - as in extra% hero needs to continue vs a bet when only calling to avoid villain cbetting his entire range and freerolling with his marginal equity hands)

I try and incorporate call multipliers(1.2x OTF and 1.1x OTT) when doing simple GTO math for my ranges(as in calling 60% or w/e range each street), but sometimes i struggle to reach the full amount of combos needed(simply don't have enough continuable holdings). I wonder if i might be trying to overdefend wrongly..

No, "call multipliers" are a human construct that may or may not help you play closer to equilibrium. Also, it seems likely to me that there will be situations where you just can't stop your opponent from "cbetting his entire range and freerolling with his marginal equity hands". IIRC MOP covers an analogous situation where you're folding the river > X/(P+X) where X=opponent's bet and P=Pot. Sometimes because of position or previous action you're just at an inherent disadvantage even if you play the equilibrium strategy.

u cylon bro?

Thanks Jeff. So i guess your answer is nobody knows?
What got me thinking about it was a situation where i call blind vs blind 35% range in 6max NLHE 100bb stacks, and my GTO strategy with call multiplier 1.2x requires me to call A8o or 77 on Th2sJs flop for SDV, but my friend thinks those hands are really weak and are both folds. I thought maybe i shouldn't be adding a call multiplier after all. Infact since the pfr's equilibrium strategy is probably to c/f 30% air on each street(to let my weakest flop and turn calls see showdown), the equity A8 has(stove says 30% but it's probably just the A pairing that counts, so maybe 10% or so) vs his betting range, is like a freeroll also. So both partys have equity.. i wonder if they both cancel each other out?

^ his answer was pretty unambiguously no, it's actually the first word of his post

As best I understand it, the concept probably has some relevance to dynamic hand value games, but really only applies directly to static hand value games.

The concept is for dynamic hand value games. The idea is you take the principle from static hand value games that you have to defend with pot/(pot+bet) frequency and multiply that number by some arbitrary coefficient to get a frequency you should defend in a dynamic hand value game.

Oh. I think I'm about 1/6 in this thread, heh. Well if that concept did have any validity, it would only need to be true averaged across all boards, right? So thinking about it at the table while faced with a specific board would be pretty useless anyway. Maybe. Always glad to be told why I'm wrong in any case :/

One thing: Any time you play poker, whether live or online, you will have extra information about your opponent's play. This could include stats, timing, situational information, as well as physical tells and even "hunches."

It looks to me like some of the gto poker extrapolations or whatever you want to call them are based on totally information neutral scenarios, for instance two insentient computers playing against each other.

In a real game, there is always a vast and unquantifiable amount of extra information floating around.

Another thing: I might be wrong about this, or completely missing the point, but there seems to be a peculiar kind of recursion that could happen in a poker game because of the random element of the cards.

Let's say I realize that my opponent is playing a "game theory optimal" brand of poker. Since I know that he is, let's say, balancing all his actions, then I could get a pretty good idea of what he is doing, say if I reverse engineer like "what would a gto guy do?" and if I understand the correct gto strategy and know that he's playing it, I could predict. THEN if I play not-balanced, say I do such and such with my aces, such and such with 5-7 suited, such and such with AK, then, and I mean I obviously haven't worked this out, but it seems like I would have an advantage.

Like if an unbalanced player 4 bets me preflop, he got it and I easily fold. But if the "balanced" gto guy 4 bets me, then maybe my all-in fold equity or racing equity is that much greater, because I know he is balancing so he must have air some percentage of the time, and that air goes straight into my equity column.

Obviously I see the value of balancing, however it seems like if deception is lacking, than the opponent could snatch back the advantage.

Your line of logic starts from a fallacious standpoint that there is any way to be profitable against a GTO strategy. There is none.

Yes, 5bet bluffing vs a GTO bot might be more profitable than 5bet bluffing a certain unbalanced player, but it won't have a net positive winrate for you. If you adapt perfectly to the GTO bot, then you will have a winrate of 0bb/100.

What you say about information is true, insofar as that a GTO strategy could never really be used in live poker, due to the inherently exploitable human facets of the game.

This entire thread is pointless. Obviously GTO play in poker has already been solved.

https://www.youtube.com/watch?v=IZL7uiLGsO0

One of the main strong points of a GTO strategy is that you can tell your opponent exactly what your strategy will be in every situation, and he still won't be able to exploit it, because there are no unbalanced plays to take advantage of.

Also, since we really have no idea what an actual GTO strategy in NLHE would look like, it's very likely that you wouldn't recognize it even if you saw it.

Two of you have already responded to anilyzer's "reverse engineering" comment, but I want to try another way of saying it. It's been said in this thread before, but bears repeating.

He's actually totally right that one could "reverse-engineer" the exact perfect strategy to play against the GTO guy. But this is the power of a GTO strategy pair. Player X says "I will play strategy x. And if you play *perfectly* against me (which is actually easy to do, once I tell you my strategy, see "calculating a nemesis" earlier in this thread) I will make an EV of exactly z.

So far, this is not very interesting. I mean, if I'm in the small blind in a 1/2 NL heads up game, I can say "I will fold every hand, and make at least -1 every hand, no matter what you do."

The amazing thing about a Nash Equilibrium is that it says the best strategy for Y will have a value that is *exactly equal* to the negative of z for X's best strategy. So to repeat:

X says "If I play x, and you do *the best you can* I will make z."
Y says "If I play y, and you do *the best you can* I will make -z."

I think anailyzer's mistake is in not realizing that when X says "I make at least z" that means *already taking into account the sort of manipulations he's describing." It's *easy* to play perfectly if the other guy tells you his strategy. Yet we find (making up numbers here) The small blind can say "I will play this way and *no matter what you do* I will average making .2/hand. If you play badly, I'll make more." And the big blind can say "If I play this way, I'll lose no more than .2/hand. No matter what you do, I won't lose more than that. If you play badly, I won't even lose that much."

So if you play a GTO strategy, you will get *at least* the value of the game. And if your opponent does so also, you will make exactly that value.

As has been pointed out many times, that is not necessarily a good thing. You won't exploit a bad opponent. And in fact, there are GTO strategies that make stupid plays (but only in response to a stupid opponent.) It's not all bad, though" in simple games, like rock-paper-scissors, if you play GTO your opponent can't make a mistake - however, poker isn't like this, there are plenty of opportunities to make plays that look ok but lose money to a GTO strategy.

Not trying to bag on anilyzer here, he put plenty of qualifiers in to say he wasn't sure. Just trying to help those still struggling with the math. [Hey, you're not in bad company, apparently durrrrr doesn't quite get it either ]

It just seems clear to me that gto/Nash equilibriums would have to be like functions, that would take something like the other players ranges as an argument. I glanced around the net and there are a ton of hardcoded nash equilibrium tables for poker that only take into account stack size, I don't see how these can possibly be correct.

People can vary their calling/raising/all-in ranges, they can play tighter or looser regardless of stack size. This should automatically alter the expectation of the other player, and if he isn't flexible to adjust, we can't just gloss over this by saying "ok, I will win X, but he will win -X, so we're all even and he's unexploitable."

Here. This is from Wikipedia:

["the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium."]

http://en.wikipedia.org/wiki/Nash_equilibrium

I mean, the "this works as long as the other player doesn't..." highlighted sections of this definition seem to perfectly describe exactly what poker players are constantly doing.

Hopefully I'm not being unbearably obtuse about this. To be honest my only exposure to this was in the "Raiser's Edge" (excellent book) section on it, but I just skimmed that part because I found the hand examples and strategy/meta-strategy discussions and hand examples far more interesting.

It just seems to me like there would be some theoretical questions about this to solve before we just jump in and start doing the math to come up with ranges. I mean with advanced stats, you could maybe plug that data into a program that would come up with some kind of adjusted equilibrium gto range for a specific situation...

but say there some specific game theory optimal range for pushing 20 BBs utg+1 with 6 players left at final table. Whatever that "optimal" range is, it seems like I could make it better by playing tighter than that in some situations, right? Or let's say I loosen it up in some situations.

Like if I am pushing 30% and everybody is folding unless they get AA/KK/QQ/AK-AQ, then all those times they fold, what difference does it make WHAT my hand was? If it's just a matter of frequency or percentages. When they call I have 25% or 15% equity usually, and when they fold I make a big blind. If they widen their range significantly than maybe my equity is one or two points higher because I have 22 and A2 in my range, but don't have J6 or T5 or whatever.

so like if I shove 20bbs and everybody folds, and then I'm like "yeah bitches I was gto" it's like, meh, ok, I guess. But it almost seems like I could shove the exact opposite low end of the range +AA and KK the same percentage of the time and get a very comparable result.

Ok, I'm now officially rambling. Later

Nash equilibriums do take player ranges into account, in the sense that they can't be -EV on the whole versus any range. The Nash tables that are on the net are only for shortstacked push/fold preflop poker, which is a gross simplification of actual poker, and vastly more easy to calculate.

Playing differently will indeed alter the expectation of the GTO player, it just won't make it negative. It's possible to skew your range so that you beat the GTO player in specific spots, but you will still be -EV on the whole. For example, you might decide to reraise preflop with only AA and nothing else: in this case, a GTO bot would certainly put in too much money versus your 3-betting range. Are you exploiting the GTO bot? No, because the value you gain from your aces is offset by the value you lose by never reraising your other big hands. GTO strategy, by definition, doesn't have to adjust; it beats, or ties with, every other strategy.

If I understand what you're saying here, I think you're misinterpreting what Wikipedia said. It's basically just defining a Nash equilibrium situation between two players: a situation where neither player can gain by adjusting their strategy. This, in fact, shows that one player can't gain an edge over GTO by playing looser, tighter, more or less aggressive, etc.

It is indeed possible to make higher EV plays versus other players than a GTO player would, if those players are playing exploitably. For example, if you find an opponent who folds literally everything but AA preflop, and jams when he does have AA, you can make a lot more money than a GTO player by minraising preflop with 100% of your hands. GTO does not mean making the most money possible versus every strategy, GTO means not losing to any strategy, whether it's tight, loose, passive, aggressive, or just plain weird.

To give you an example of the value of GTO: suppose you are playing online poker against an unknown opponent. Behind your opponent's computer, however, is a room full of an unknown number of different players. Some are fish, some are maniacs, some are pros. You never know, in any given hand, which you are playing against. It could be a passive station one hand, a maniac the next; you could end up playing the entire match against durrr, you have no way of knowing from hand to hand. Since you have no way of adjusting your play based on opponent tendencies, what should you do? The answer is try to find a balanced strategy that does well against any opponent: reasonable preflop ranges, and solid postflop lines with a good balance of value betting and bluffs. If you were to play this way in the absolute most perfect way possible, then you would have found a GTO strategy.

Thanks for the very detailed response. I'm starting to get a clearer idea about this.

One thing:

It's hard for me to visualize how math could be used to find that perfect way of playing.

I mean, if we consider the possible number of two hand + flop combinations, [x][x] + [x][x] + [x][x][x] that right there is a large number. And that's just one hand heads up.

Now consider that we are playing hundreds, or even thousands of iterations or hands. And maybe that there are seven other players with various cards as well.

And then consider that I may be altering my strategy in different ways, based on the way I am responding to the opponents actions and strategy, and what I perceive that he thinks, or thinks that I am thinking, etc.

That's a lot of variables to consider, if we're going to use math to plan out a fixed strategy approach in advance and declare it to be at least breakeven no matter what happens, no matter what the opponent's actions are.

I mean, sure, on Wall St. the AIs can figure out that buying and selling the same stock 1,000 times a day in response to what their enemy AIs are doing will net a profit or prevent the enemy from netting a profit. It's really a place that humans can't visualize, but apparently its possible. Or maybe the AIs might've figured out that by colluding together to crash the market in 2008 that within 2 years they would be able to turn an even greater profit and consolidate far more of the dollars in play than they had before. It's possible that AIs can run so many deep permutations that they come up with conclusions that are completely alien to human thinking, but still work. I think it was in Wired magazine, there was an article about wall street AIs, how they will traverse this vast series of seemingly unnecessary actions and transactions just to arrive at a point of making one dollar, or a seemingly insignificant amount of money. But they will do these calculations and repetitive transactions a zillion times a day, that's just how they roll.

Like as I understand it, chess computers if they play each other will either totally smash their armies into each other and finish the game really quickly, based on a slight mathematical edge, or else will play extremely counterintuitive defensive games that could go for tens of thousands of moves and end in a draw everytime.

I like the idea of gto poker, but I have a doubt as to whether it is truly gto in
a reasonably limited sample size, for instance a 10 hour live session of ~appx 450 hands. Or in a single sng tournament.

I mean I can push 20bbs from middle position, get called and have 28% equity, lose, and then tell myself "I was gto right there and totally unexploitable. I may have lost but I was gto." But at some point what does that really mean?

I don't have time for a long response but I'll try to get a quick one off.

It means that, assuming your shove there was really GTO, there's no strategy your opponent can play which exploits your range there. Maybe they only call you with the nuts and sure, your equity is bad when you get called, but you're winning the pot uncontested most of the time; maybe they call you frequently, but then your equity is better when you get it in.

And as for your general point, yes, calculating GTO play is complicated. In fact, for NLHE, it's almost unfathomably complicated, and certainly beyond the capability of any of our computers for the foreseeable future. Computers have managed to calculate near-GTO strategy for HU limit poker specifically, because the number of possible situations is relatively small, and even then it requires a colossal amount of computations. If you add in more players, or switch to no-limit, the game tree quickly grows far beyond our abilities to compute. You won't be seeing truly GTO bots any time soon, or even probably within your life; but the point is that Nash proved that a GTO strategy does exist, even if it is far beyond our ability to calculate.

Not really sure if your posts are just an attempt at a stupid level or not. But if not, just go read the rest of this thread. I am sure your questions will be addressed.

I agree, it's not a very useful concept. Understanding that you need to defend a flop pot sized cbet more than half the time to make semibluffing ineffective is important. Deciding that your flop call multiplier is 1.2 and calling 60% of the time is just picking an arbitrary number and pretending that you've solved it.