Thoughts on the following statement:

Theorem of balance:

One should balance if and only if the external EV of balancing on avewrage over the entire session (diachronic) is greater then the EV you gain by playing the single hand optimally (synchronic)

well i think obviously yes if im understanding correctly, but id suggest maybe expanding on this with some examples

I don't think you worded it properly.

"playing the single hand optimally", as in GTO or as in exploitivelly optimally?

This is actually an interesting question, in the OP i ment in a manner that yields the highest EV for that particular hand. which MAY be GTO but in most cases is probably max exploit

I am no game theory expert so hopefully I can learn something here.

questions:

1) is "GTO" the same as Nash?

2) is GTO/nash/Balance the same?

3) If the maximally exploitative or best response strategy earns the highest EV i.e most optimal, why do we in the poker community use "GTO" to refer to a strategy that is balanced/nash.

4) if our opponent is not balanced themselves i.e they are exploitable, why would we choose to balance?

Not necessarily as a nash equilibrium could be a subset of the game, same with balance. Where I'd argue GTO is an equilibrium of the entire game.

For example you may be balanced in a specific river spot, but not playing GTO.


1. not necessarily

2. no

3. GTO is not the highest EV move, it's the highest ev move against a perfect counter strategy

4. you shouldn't

I actually just learned that this is mathematical nomenclature and the term GTO stems from mathematics of poker(which I have not read) and is somewhat a unique term to poker which is not really discussed in books on general game theory(some of which I have read)

so apparently the terms "optimal" and "best" are not synonymous in game theory regarding poker

But if you don't know Villain deviates in a certain spot you best approach ist to play GTO. Often if u face opponents u want to be balanced in certain spots to avoid getting exploited

Only if you have 0 information, which is rare. Even population tendencies are information and that can give you cause to deviate.

1 and 2. Nash proved that some games (e.g. poker) always have optimal strategy for any player. When we know and play the strategy it is called GTO.
We know that optimal game (in poker) exists but dont know what it is.
Optimal game is known for some simple situations, most push/fold in SNG.
SNG players talk about Nash and mean optimal game.
Cash players talk rather about balance/GTO but we must know that it is only aproximation, game is not solved.

3 and 4.
We should always use exploit when possible. We should be aware what is roughly optimal game. In the end how could we adjust if we dont know what is normal, then we end tightening range already too tight etc...

The original problem was that "optimal" to game theory academics meant best play assuming your opponent was playing perfectly, while "optimal" to poker players meant the best play assuming your opponent was an average joe. So we have the same term meaning different things, so using the term "game theory optimal" is supposed to make things less confusing by clarifying which meaning we're using. The problem is, many poker players (including myself) starting using "optimal" as shorthand for "game theory optimal" meaning the "optimal vs average joe" usage of "optimal" was abandoned. So yes, this would mean that if we knew our opponent's strategy "optimal" and "best" would mean two different things. From a pure English language perspective this is probably wrong. When we know our opponent is non-perfect its probably better to call a game theory optimal strategy a "Strategy at a Nash Equilibrium" or something and just concede that it isn't optimal at all.

Seems like this thread got off on the wrong foot.

I think there's an interesting question mixed in there:

There are exploitive plays and exploitive strategies. The latter being much more exploitable in return. Exploitive plays may be mixed into an otherwise non exploitive strategy, but be aware that the more you use these plays, the more exploitive your strategy becomes; the more exploitable your strategy becomes.

So now how to form an exploitive strategy?

We have options:

Exploit a tendency to overfold immediately.
Exploit a tendency to overfold on a later street.

Exploit a tendency to overbluff immediately.
Exploit a tendency to overbluff on a later street.

Exploit a tendency to call too much immediately.
Exploit a tendency to call too much on a later street.

Exploit a tendency to value bet too thin immediately.
Exploit a tendency to value bet too thin on a later street.

Exploit a tendency to value bet too tight immediately.
Exploit a tendency to value bet too tight on a later street.

----

Now some of these seem obvious, such as choosing to exploitively fold immediately vs a player that value bets too tight. However, there will be exceptions, such as vs a player that value bets too tight, but fails to reduce his bluffs; he's bluff heavy because of his missed value; exploit by bluffcatching wider.

If the tendency to value bet too thin is accompanied by a tendency to bet fold too much, then it can be of great value to delay the exploitive bluff.

If the tendency to value bet too thin is accompanied by a tendency to bet call too much, then value raising thin goes up in value.

----

Be aware of going off the exploitive deep end. You might be winning tons of pots vs a nit, but then all of a sudden he has a real hand and you lose a big pot. You might be winning tons with your bluffcatchers vs a habitual bluffer, but if he changes gears and stops bluffing so much, then you gotta be ready to fold you weak bluffcatcher, particularly when facing increasing betsizes.

There are some players out there that are awesome at metagame stuff. These players are the reason why I avoid the very exploitable adjustments. Leveling wars are great when you have the upper hand, but that's not always the case.

Any purported universal theory of 'balance' is almost certainly either wrong or tautological.

I think the definition of balance should be "having the properties of a properly mixed strategy."

But I've come to accept that the term isn't going away so we may as well use it as correctly as we can.