I'm understanding now that the above assumes that you are using non-dominated strategies and would therefore break even versus a theoretically GTO player.
But how correct is that assumption? And what exactly defines dominated/non-dominated strategies in poker? If we were able to compare 'winning LLSNL strategy' with a GTO strategy do you not think that many of the choices we are making are in fact dominated strategies?

I'm not assuming anything. I'm just saying that if you knew what a GTO strategy for poker was, and it was a mixed strategy, you would not need to play GTO to break even against GTO. You could play any non-dominated strategy.

Your question is making a few leaps in logic. Remember, we don't know what a GTO strategy for poker is. However, if we did know, we would also instantly be able to know which strategies are dominated and which aren't. (Any strategy that is included in a GTO strategy is not dominated, and any strategy that is not included in a GTO strategy is iteratively dominated.)

So to try to make this slightly more concrete, let's say that you could solve for a GTO strategy and determined that it was a random mix of 45 different strategies. You could pick *any one* of these strategies and it would be 0EV against the mixed GTO strategy; but any strategy that included one that was not one of the 45 would lose to GTO. (So for example, if you were in a 3-handed game where one player could be exploited by one of the 45 strategies, and the third player was playing GTO, you could play the exploitive strategy and you'd be the big winner in the game long-term. The GTO player might still beat the 3rd player while breaking even against you, but you would break even against the GTO player while beating the third player for more than the GTO player does. In fact it goes even deeper than that, but this parenthetical is already long enough.)

Now, on the other hand, if we could solve for a GTO strategy and determined that it was unique and static, *then* the assertion that non-GTO strategies always lose to "the" GTO strategy would be correct.

And of course our knowledge of GTO is so incomplete that we really don't have any idea whether any of the "winning strategies" used at LLSNL are dominated or not. It is possible that they are; it is possible that they are not. I don't think anyone knows. (My opinion is that they are not dominated, but they are iteratively dominated. But that's just my opinion, and it's not informed at all.)

I won't get into too much debate, but my point was that GTO begins to work when you know the starting conditions. For example, in the prisoner's dilemma, you can only create the GTO solution when you know the potential results of your decision. If the police in your example say, "I won't tell you what your sentence will be if you are convicted, how much time you'll get off if implicate your partner and what the sentence will be if he implicates you," you can't construct any analysis. Janda's point (if I got it right) is you can use bet sizing where you know the villain's range and the villain knows your range to make him indifferent to calling or folding. A simple example is that your range is a 2:1 favorite against the villain's range. If you make a PSB on the river, the villain is indifferent to folding or calling, because the result over the long term is identical.

Without misrepresenting him (I hope), he doesn't see playing GTO as a goal for an entire session, but rather a tool at times to put your opponents in a no win situation.

Good read.

But that has nothing to do with a Nash equilibrium or a GTO strategy, and calling it such is a gross misrepresentation of game theory.

Furthermore, think about it for a second. If you think that the Villain thinks he is a 2:1 favorite against your range, and he is correct to think this, why wouldn't you make a bet size that entices Villain to call or fold incorrectly? For example, if Villain thinks he is good 33% of the time, and you overbet, he will fold, right? So now your bluffs are profitable and your value bets will still win the pot (taking advantage of some previous misplays).

My point here is not to analyze this spot; that's a digression. My point is that range analysis is inherently exploitive. There is no getting around this. The instant you mention your opponent's range, you have firmly left the realm of GTO play.

I thought these were one and the same. From janda:

Am I missing something? I know he doesn't say "GTO" but I assumed "theoretically optimally" is the same.

One scenario people want to discuss is where they may have taken an exploitative line but, having gotten to the river, are unsure of what the exploitative play should be and are worried about being exploited, so they want to be able to determine the GTO play for those specific conditions, where you might have an idea of your opponent's range based on previous streets.

It's silly for them to talk about GTO preflop with multiple streets of betting because that is just too complex of a game. It might make sense for them to consider it when you can reduce the hand to a narrow game with one round of betting because it is the river with no more cards to come or because any bet will be all-in.

My basic takeaway from all this is: approximating GTO is a defensive strategy. When I encounter a player who is worse than me, I alter my strategy to exploit theirs. When I encounter a player who is equal to me and has position, or who is better than me, I try to play "closer to GTO."

I balance my bluffs and my value bets, check some strong hands, etc. So if I encounter an ABC fit-or-fold player, I'll cbet ~100% of the time and show a profit. Against a smarter player I'll cbet less often as a defensive measure. Of course, I don't know what exactly a GTO strategy looks like (no one does), and it's a rich, complex topic, but that doesn't mean we can't have a general idea of what it looks like. The really interesting and complex part of this topic is bet sizing imo. I guess the bigger we bet, the less often we should be bluffing, when speaking of GTO. Obviously this is not how we should be playing against many exploitable players.

Actually, the bigger we bet the more we can be bluffing.
If we bet $100 into $100, our opponent becomes indifferent to calling and folding when our value:bluff ratio is 2:1
If we bet $300 into $100, the ratio becomes 4:3, so we get to have 1 extra bluff for every 4 value combos.

Game theory is not only useful for examining how you are playing, but also finding ways to exploit your opponent. If we bet pot, our opponent has to call at least 50% of the time on the river or we can bet 100% of our range. If our opponent bets pot with a range of more than 2 value hands for every bluff, we can fold all of our bluffcatchers. etc etc, just know what the mathematically optimal frequencies are and look for spots where your opponents frequencies are out of whack.

Interesting post. Let me take a stab any why game theory is so difficult to apply to poker. Let's assume that everyone has one betting unit and nobody folds (in other words, the only bet is pre-flop). Everyone's EV in that game (assuming no rake) is zero. Over the long run everyone gets the same card distribution. This is like the RPS game where nobody deviates from the 1/3 R, 1/3 P, 1/3 S strategy. So the only way to make money is to deviate from the strategy of never folding. Of course this assumes that none of the other players deviate pick up on what you're doing. How does your strategy change if one player starts folding? Two players? Now how does your strategy change when we start adding multiple betting units? Now we take all of this and do it again after the flop, turn and river, each time taking into account all of the previous actions by all of the players. Now factor in that each player may deviate in different ways from a particular strategy. Now add in the fact that the same player may deviate in different ways from a particular strategy for reasons that have nothing to do with the game.

The variables may not become infinite, but they quickly approach that level. And without taking all of these variables into account, it's nearly impossible to "solve" the game. Maybe you get close to solving it for heads-up, limit hold'em (or at least coming up with a very solid, difficult to exploit strategy), but once you introduce multiple players and the option to bet various amounts on each street, you create an environment too dynamic to "solve."

I don't play tournaments but have always thought that early in a tournament, because the blinds are so small in relation to one's stack size, that one should play tighter than one's hand value warrants. This is because villains will be making so many and such large errors that are exploitable by playing tighter. However, later in a tournament, when the blinds are bigger in relation to one's stack size, one must play in a way that is much more closely based on one's hand value(I guess because the size of villains' errors will have shrunk). It seems that what I thought of as, playing based on one's hand value, is GTO, or something like it.

OK, I think I have time to write a reply to all of this now:

So, I didn't say anything about poker itself in my OP because I wanted to limit it to the bare-bones mathematical facts of game theory, but now I want to sort of give my opinion on game theory and poker.

Both of you seem to echo a sentiment that I've heard before, which is that the reason we shouldn't care about GTO play in poker is that poker is too hard to solve. The implication is that if we knew how to play GTO, we would, but since no one can, we don't need to worry about it.

Both of you are missing the point.

I am not claiming that the reason we don't want to try playing GTO poker is because it's too hard. I'm claiming that even if we knew how to play GTO poker, we probably still shouldn't want to.

To show where I'm coming from, I'm going to give an example of a third game that illustrates a concept that I didn't get to show in my OP--iteratively dominated strategies. Let's say you are playing a game where, similar to RPS, you both make your "move" at the same time and compare results. Only this time, let's change the payout matrix. Let's also change the names of the strategies since this isn't RPS anymore. Imagine each player has 3 strategies: let's call them C, E, and G.

The payout matrix, which I'll write as ordered pairs of strategies and payouts for the first player in a zero-sum game (because this format makes it really hard to align matrices), looks like this:

(C,C) = 0_____(C,E) = -15_____(C,G) = -6

(E,C) = 15_____(E,E) = 0______(E,G) = -3

(G,C) = 6______(G,E) = 3______(G,G) = 0

Given this payout matrix, how would we solve for the Nash equilibrium?

The first step, as always, is to look for dominated strategies. When we do, this is what we find:

-G is clearly not dominated. There is no other row where the third entry is at least 0.

-E is clearly not dominated. There is no other row where the first entry is at least 15.

-C clearly *is* dominated. No matter what column you are in, the payout for C is less than the payout for G, so C is dominated by G. (It's also dominated by E, but we only needed one "witness".)

That means a Nash equilibrium cannot involve C. So once we know that both players will only play E and G, we can "reduce" the game by making a 2x2 matrix where the only strategies are E and G. It obviously will look like this:

(E,E) = 0______(E,G) = -3

(G,E) = 3______(G,G) = 0

Now here's the next step: once we have this reduced matrix, it is now clear that a Nash equilibrium cannot include E! The reason is that once we eliminate C, E now appears to be dominated by G. We've thrown away the only strategy that stopped E from being dominated, so now we can treat it as if it is.

This means that the GTO strategy for this game is unique and static--throw G 100% of the time.

Let's back up and consider this for a second. Here we have a game where, if you play GTO and your opponent plays GTO, you will both have 0EV. But if your opponent deviates from GTO and you don't, you will have a positive outcome. This is exactly the "ideal" situation that many people erroneously believe GTO to be all the time (hopefully by this point in the thread you know why this is erroneous). In this case, G functions as a "perfect" strategy. And yet--and yet!--even in this "ideal" situation, it would be wrong to jump to the conclusion that you should always be playing G.

Why is that? Well, solving for the Nash equilibrium means making a leap in logic--one that clearly does not apply to a game like poker. The only reason we were able to conclude that we should not throw E was because we assumed our opponent would not throw C.

Clearly, in this game, if our opponent is deviating from GTO, but only doing so by mixing up E and G, we maximize our expectation by sticking to G. But if our opponent will not always avoid C, all bets are off and now we might do even better than GTO by putting E back on the table.

Here is an example, which I'll leave as questions and let you guys answer and interpret however you want. Suppose you are playing an opponent who is randomizing his play so that he will play C one-third of the time, and G the other two-thirds of the time. If you know your opponent is doing this:

1) What should you do to minimize your probability of a negative outcome?
2) What should you do to maximize your EV?

Great post, I'm enjoying working through these.

For 1), it seems obvious that you should always play G, because there are no negative results. Adding E or C would result in negative EV some of the time. Always playing G gives an EV of 2:

EV = 1/3 * 6 + 2/3 * 0 = 2

For 2), I wrote out this equation, where p_G is the probability of us choosing G, and (1 - p_G) is the probability of choosing E. We should never choose C because E and G are strictly better choices.

EV = 2/3 * (0p_G - 3(1 - p_G)) + 1/3 * (6p_G + 15(1 - p_G))
EV = -2(1 - p_G) + 2p_G + 5(1 - p_G)
EV = 3(1 - p_G) + 2p_G
EV = 3 - p_G

So this result says we should always play E, giving us an EV of 3. This seemed counter-intuitive to me at first but after thinking about it, makes sense: the reward for exploiting our opponent's mistake of playing C by playing E exceeds the value of playing the theoretically correct strategy, even though we'll lose 3 66% of the time.

+1

@thetruewheel Disclaimer: I didn't check your math, but am just trusting it because the different answer to the two questions is the point here.

And this answers your question. "Theoretically optimal" does NOT mean highest EV. It means "non dominatable." They are only equal when your opponent is ALSO playing GTO. When your opponent (or any of your opponents) are playing dominatable strategies, then your EV is higher when you are also playing a dominatable strategey if it dominates their strategy more than GTO dominates their strategy.

Hell, sometimes GTO won't dominate their strategy, it will just be incapable of being dominated. In your example GTO gave us an EV of 2, given V's dominated strategy, but it could give an EV of zero, given different payout matrices. It just can't have an EV of less than zero (given a fair zero sum game). Given rake, GTO can actually have a negative expectation.

Thanks, that was a good read.

I guess the one thing that stands out for me is that their are a lot of do called GTO experts who claim that if they play GTO and their opponent doesn't they will show a profit.

I have disagreed with this statement and I am glad you feel the same way and explained it in a very simple way.

I am a golfer and every year there's so new top revolutionary technology that is supposed to change golf het the average handicap never goes down. Why - because its all hype with no substance.

GTO for poker is the most over rated concept IMO.

For one I think knowing GTO is only useful to know what mistakes people are making and then exploit those mistakes by playing exploative.

And two these experts books usually they play all their combos of hands the same way. Eg 3 bet AK from BB vs MP when in reality there would have to call some of those combos so they aren't unbalanced and certain board textures.

Interesting thread. I'm not much of a math/theory guy -- would the following be fair take-aways from this thread?:

a.) GTO isn't necessarily "optimal" in the sense of being optimally +EV, but "optimal" in that it no strategy dominates it.

b.) GTO for NLHE is wildly complicated and almost certainly unsolvable, but almost as certainly involves concepts like balancing ranges, optimal bluffing frequency, and randomizing play. Most of the time, we don't need to worry about this too much because...

c.) We're not playing against EV maximizing computers, but against people who, for whatever reason, tend to play simple strategies that are dominated. Therefore, we play a set of dominated strategies against individual villains designed to exploit how our villains playy -- i.e., we bluff MUBSY nits, station bluffy aggros, value bet calling stations relentlessly, etc. The better "read" we get on how our opponents play in certain situations, the more closely we can tailor our strategy to crush theirs.

d.) That having been said, the better our opponents are, the more we need to consider randomizing/balancing our play, but our focus should still remain on identifying how their strategy is dominated and playing to exploit their leaks.

E: One basic reason for d.) is that better opponents almost by definition aren't playing a static strategy, so, depending on the situation, we might be able to get them to fold, we might catch them bluffing, or we might get them to station us when we have a value hand.

Sound about right?

There's been an argument going through the thread for awhile now and I think I have time to chime in and hopefully clear it up:

I don't know exactly where this quote that thetruewheel found came from, but what I'd say here is that it is unfortunate that the term being defined was "theoretically optimally" and not "game theoretically optimally", because they don't mean the same thing.

This confusion leads to things like this:

(The calculations are correct, but it is not quite right to call the GTO strategy "theoretically correct". More below.)

There seems to be some disconnect over the idea that "optimal" and "game theory optimal" are not the same thing, so let me try to clear this up.

Mathematicians (and economists, I guess, and everyone in similar fields) tend to use the word "optimal" to mean a solution to a problem with fixed parameters or constraints, that is the best solution given those constraints. For example, earlier in this thread, if we're considering RPS and our opponent is playing a strategy that is unbalanced toward S...we can ask, "What is our optimal strategy?" In other words, "What's the best way to play once we know the conditions we're going to be playing in?" The idea is that we are given some set of assumptions and we optimize for them.

Game theory optimal is a much more specific term that refers to a specific set of assumptions that we are trying to optimize for. Namely, we are assuming that once we fix our strategy (whether it is static or mixed), our opponent can always figure out our strategy and play his optimal counter-strategy. In terms more familiar to poker players, game theory basically assumes that our opponent is a master of exploitation!

This is in line with the idea that game theorists want to find the maximin strategy (a term I brought up in my OP). If our opponent is so good at playing his optimal strategy against us, then as a corollary of that, for any strategy that we pick, we are expecting the worst-case scenario (i.e. our opponent's best-case scenario). So *our* optimal strategy against this "ideal" opponent will be the one that has the best worst-case scenario. Another way of saying this is that under the assumptions, our optimal strategy will be one that stops our opponent from having a unique optimal counter-strategy and instead forces all his best counter-strategies to have the same EV.

In a zero-sum fair game, we can re-state this in a less confusing way as follows: we are assuming that no matter what we do, our opponent is capable of adjusting so that his EV is never negative--it's either 0 or positive. In that scenario, the best we can ever do is guarantee ourselves and our opponent an EV of zero.

That is what game theory optimal really means. It means you are maximizing your EV under a set of assumptions that imply that you cannot beat your opponent and the best you can do is tie.

In that sense, game theory optimal play is optimal, in some sense. But it is not optimal in any real-world sense for a game like poker, where those assumptions are very clearly false!

So, I'm adding to this thread because I recently got my hands on my old game theory book that I learned a lot of my game theory from back in high school.

And I was reminded of something very important about game theory that many poker players just skate over, but is extremely important.

Here's the fundamental question we have to ask ourselves before we do any kind of "GTO"-type analysis:

How do we know whether for any given game, a Nash equilibrium exists at all?

There is a theorem in game theory that basically states that for 2-player games (even ones that are not zero-sum), there will be a Nash equilibrium.

That theorem does not apply to games with more than 2 players! And in fact it turns out that it is pretty easy to come up with a 3-player game for which a Nash equilibrium does not exist.

What does that mean? Well, it means that if we are playing 3- or more-handed poker, we actually don't know whether there exists a GTO strategy. And in my opinion, there probably isn't one (obviously still just my opinion though).

Not to derail, but...

In the RPS w/pebble example, wouldn't the correct GTO solution be 50% rock and 50% paper? Scissor and pebble are equally dominated strategies.

Not a derail at all; perfectly good question for the thread.

I think the reason you're equating the two is because both scissors and pebble lose to 2 things while only beating one thing. That is not, however, what defines dominated. (Other people have made this mistake in the thread, too--it's not just you.) Dominated means that there's a different strategy--a specific one--that outperforms it across the board, not just on average.

The reason pebble is dominated is not just because it beats 1 thing and loses to 2--it's because rock outperforms it across the board. Anytime you were thinking about throwing pebble, you would do better (or at least not worse) throwing rock. Observe:

-If your opponent plays scissors, rock and pebble both win.
-If your opponent plays paper, rock and pebble both lose.
-If your opponent plays rock, rock ties but pebble loses.
-If your opponent plays pebble, rock wins but pebble ties.

So no matter what your opponent does, there is no time when you would rather play pebble than play rock. (It beats the same other things that rock beats, and loses to the same things rock loses to, but also loses to rock itself.) That means, by definition, that pebble is dominated by rock.

This does not apply to scissors. Even though scissors beats 1 thing and loses to 2 things, it is not dominated because there is nothing else that beats paper. Scissors does better (strictly better, not just tied) against paper than any other strategy does, and that alone means it cannot be dominated.

To wit: you claim your strategy of 50% rock/50% paper should now be GTO. But it can't be, because if you play that, I can show a positive EV against you by throwing 100% paper. I'll never lose! The only way to thwart my strategy is to play scissors. That's how you know scissors can't be dominated--because you may need it to exploit a different strategy.

It is literally impossible, however, to construct a strategy where you end up saying "the only way to foil this strategy is to play pebble", and the reason is because anytime you're thinking of saying that, you could just replace it with rock. That's why pebble is dominated but scissors isn't.

^Nice explanation! And WOW, very surprising. I mean, it's 'obvious' after reading your post, but it's hard to fathom how scissors can be an equal part of the strategy when it performs so poorly on its own against 'the field'. But there's no way to 'exploit' the presence of scissors without giving up everything you would gain by losing more to paper.

Thanks!

Actually the explanation is that since pebble is dominated, you are supposed to take it out of "the field" before you consider what the GTO strategy is. The conditions in game theory are that both players are rational, so you would never expect pebble ever to be thrown, and therefore it doesn't affect the symmetry of the game. It is not really a 4th option; it is just "oh, it's like he's throwing rock, only worse".

So a strategy of 25% rock / 25% paper / 25% scissors / 25% pebble is not actually "balanced"; it's actually unbalanced toward "rock/pebble" (which you should think of as the same thing since rock is the strategy that dominates pebble), which is why you'd never throw scissors against it.

You know...

I was just thinking that in LLSNL, when it comes to pre flop:

Rock == KK+
Pebble == QQ-
Scissors == AK
Paper == everything else

The biggest misconception that I see on this subject as it related to poker is that GTO= unpredictable. Just because something is random doesn't mean it's GTO. As CMV has pointed out, GTO is often utterly predictable. He's also demonstrated that to pursue a GTO strategy in virtually all LLSNL environments is not only impossible, but (IMO) quite detrimental to your bottom line.

Instead lets focus in exploiting the crap out of everyone.