I think Durrrr was sent here from the future and his mission is to destroy SlumBot before it becomes an unstoppable force.

what if....

I don't think maximize EV is used correctly here, might be wrong though. I think we should rather say the strategy that achieves your MiniMax value for the game. EV is misleading in this case.

GTO doesn't maximize EV.
GTO means playing an unexploitable game. In order to achieve maximum EV you would have to exploit all your opponents' mistakes. Which is not what GTO is about, it is about making sure that you have no mistakes that could be exploited. It doesn't even have to be +EV. GTO could well be 0EV if you're playing against someone who is using the same GTO strategy. And -EV if you take the rake into account in this case.

Look at my wording. It is the BALANCED strategy with the highest EV. Not the overall strategy with the highest EV.

Misleading sentence. Wouldn't a perfectly balanced strategy equal GTO by defenition anyway?

Meh, not sure I agree. I think EV is somethig that most poker players will understand. I am talking about in the way that the MoP do it in Chapter 18 - Value!.

So you come up with a balanced strategy. As an example:
Say in LHE you know that cbetting 100% is balanced and cbetting 0% is balanced.
So then you start thinking about which of those strategies has the most value. If you think of most of the valuable balanced counterstrategies, then it is quite natural to see that cbetting 100% has more value than cbetting 0%. Now is 100% the optimal number. I don't know and probably not, but it"s much closer to the optimal value than 0%.

But you"re right, in purely mathematical terms, I am simplifying it or mangling the terminology. And again, this approach assumes linearity in the solutions which is not a given in multi-street games.

No, it wouldn"t!

Say on the river you know that your opponent holds a bluffcatcher. You will valuebet 100% of value hands and bluff alpha of your bluff hands - the GTO solution.
But if you only valuebet 50% of your value hands and bluff alpha/2 of your bluff hands, you are still perfectly balanced, yet you are losing value.

That reminds me of a time I was betting ε value-hands and θ² bluffs on the river, but villain owned me by jamming a quadratic equation.

What's a precise definition of "balanced"?

True, stupid of me..

That's actually a problem IMO if you apply it to a complete strategy. I don't have my copy of MoP here, but it says something like "balanced means indifferent to opponent strategies". Under this definition, a balanced strategy cannot win against anybody (GTO in RockPaperScissors has that property).

Early on, they call it, "the idea that each action sequence contains a mixture of hands that prevents the opponent from exploiting the strategy."

I think another way to say this is to say that our strategy is balanced iff Villain's best response to it is his unexploitable strategy. Notice that a balanced strategy could still be suboptimal since we could be giving up value in a way that Villain benefits from without needing to adjust.

But then, it looks like the problem of finding a balanced strategy is just as hard as the problem of finding an unexploitable strategy. I mean, sure we can write it down in spots like the example bellatrix gave, but we know the GTO strategies there, too.

As far as I can tell, saying something like, "Playing GTO is too hard but I can at least try to play balanced," is pretty nonsensical.

Not exactly. It describes a balanced strategy as one which cannot be exploited by deviation from GTO. This doesn't mean it's indifferent to any deviation at all.

It's just a made-up term to describe strategies which are hard to exploit.

How could we try to formalize it? (for a two-player game)

Let T be an acyclic extensive-form game tree. Let S be some information set in that tree. Let (x*,y*) be an equilibrium strategy pair for the entire game and let y* contain no dominated strategies. If x is a strategy, let x_S be the substrategy in x for the subgame beginning with any node in S. Then x_S is balanced if y*_S is a best response in the subgame to x_S.

Right, so balanced strategies aren't necessarily any good, they aren't any easier to find than GTO strategies, nor do I see how it's easier to to find "near-balanced" strategies than near-optimal ones. So why does everyone talk about them or about wanting to play them?

Well if you don't try to play in a balanced manner, you are committing yourself to the game of leveling exploitation, because otherwise the adaptive other guy is going to kill you by exploiting your play.

A lot of people like to a have a decent base strategy that doesn't make them broke against adaptive opponents. Balance or near-balance is the only way to achieve this. Who cares if the strategies aren't good against their nemesis? They're at least something you can just play against your opposition when you don't know that much about them.

When most people talk about "playing GTO," they have no idea what they are talking about and their strategies are miles away from GTO. You need computers to even get a sense of what GTO strategies might look like, and I'm not talking about PokerStove.

Informally, when talking about actual poker situations, when I say that someone is playing in a balanced way, I mean that they are giving me a tough decision with a lot of my hands.

If I say "I think his river betting range here is pretty balanced," I mean that I think I have a close decision if I'm holding a hand that beats all of his bluffs and none of his valuebets and my range contains a lot of hands like that.

So you do think it's easier to find a near-balanced strategy than a near-GTO one? It's not clear to me how, especially since we defined a balanced strategy for a game in terms of its equilibrium.

Balance can still be +EV. Imagine you're pot-size jamming a 2:1 mixture of hands—stronger hands (value) and weaker hands (self-owned value and bluffs), respectively—into villain's all-bluffcatcher range.

S/he says, "Well **** you," and uses a random number generator to determine what % of the bluff-catchers to call with.

Whatever % s/he gets, your EV is +(pot size), which wins against everybody.

That makes sense, but that's still an example in terms of a simple river situation which we pretty much know the GTO solution for, and I suspect that your being able to say that Villain is balanced there comes from your understanding of the solutions to related toy games.

Do people approach poker by trying to design strategies such that all their lines give Villain tough decisions with a lot of his hands on early streets? And if so, would that be easier or better than just trying to play near-optimally?

It appears a lot of peoples confusion in this thread relates to what poker players use terms like balance for and what they mean when used in an academic game theory setting.

Just because some trivial balanced strategies are bad isn't an indication that balanced strategies generally tend to suck. It is true, silly "fold all the time" balanced lines are bad. So what? The default LHE balanced lines tend to be pretty good. See Bellatrix's example of cbetting 100% of flops when HU on the flop in LHE. That will get you into the mid/high stakes just fine.
Uh GTO is the best balanced strategy -- thus your statement here is non-nonsensical. Of course finding GTO is harder than finding merely decent balanced strategies. It is the best of the whole family.

In some ways it seems like the bon mot about the best exploitative strategy making more money against fish convinces people to try to sound smart by cleverly saying that GTO doesn't make the most money. You have the suggestion that GTO leaning bots tend to do great in fishy games. You see a number of high stakes LHE players who crush live and online talking about starting from a balanced strategy before looking for ways to deviate. If you're looking for where conventional wisdom being wrong, you should start with the idea that GTO (in practiced balanced) strategies not making much money.

But it doesn't maximize EV.

Yes, because it is easier to figure out that you don't have any bluffs here than it is to figure out how many hands you can profitably bet for value. And getting the marginal hands you can profitably bet for value right is only worth a little, because they are marginal hands. But correcting the fact that you aren't bluffing in a particular spot could be worth a lot, if your opponents exploit you by folding all the time. And frankly there are a lot of spots in the poker tree.