GTO Characteristics

I'm fairly sure the books on this subject all say exactly this.
Players who think GTO play is the optimal play in all circumstances are simply incorrect. GTO play doesn't claim to be optimal; just unexploitable. A couple of books describe it as a "strong defensive baseline", which seems to be an accurate description to me.

I think what this probably DOES highlight though is that the marketing around solvers/certain sites would certainly suggest that all you need is GTO. Wonder why they'd do that...

Explain.

Nash is the theoretical end-state calculation itself.

GTO is the strategy (approximating Nash) used by one single player to remain unexploitable.

But again, my main point being poker game theory terminology needing a thorough revamp to sort out the perpetual confusions.

Wow fiery thread, nice one Statmanhal. Good to see this forum active once in a while!

Does anyone other than you use this definition?

Congratulations. I'd knew you'd get there eventually

1. Yes. There are cases where you can lower the EV of GTO in multiway and non-zero-sum spots though (rake/icm). But I guess you're only referring to HU play.
2. Yes.
3. Yes.
4. Only pure mistakes, not mixing mistakes.
5. Yes.
6. Yes.
7. Yes, hence the "mixing mistakes" quip above.
8. A stronger definition is to say GTO will always continue a hand that has EV/pot >= pot odds.
9. Yes.
10. Yes.

Stremba, I'm very confused about your point here.

On the one hand you seem to believe it is trivially obvious that mixing mistakes do not lose EV to the GTO player:

Then here you say that it's an open question?

I mean it's not an open question, it follows from the definition of Nash that mixing mistakes do not lose EV to a GTO strategy.

I think you'd be interested in this post. I ran experiments showing that:

1) Mixing mistakes do not lose EV vs GTO
2) If both players are making unbiased mixing mistakes neither of their EVs changes from GTO, so long as the tree is big.

https://forumserver.twoplustwo.com/15/po...

I think the confusion is that GTO is not really a mathematical term. ItÂ’s a term prevalent in the poker community, but nowhere found in actual game theory. For some reason poker players donÂ’t seem to like Nash Equilibrium as much, but that is the correct mathematical term. And, no, there is absolutely zero difference between game theory optimal and Nash Equilibrium. They are one and the same. No poker player and no solver plays GTO. True GTO strategy is not known, only approximations to it.

This is necessarily true, if for no other reason, due to the fact that bet values only have finite resolution. In a given spot, the GTO strategy might well be “bet $13.45736”. Obviously, depending on what game we are in, we could never follow that strategy exactly. In an online 10NL or 25NL, we could bet $13.46. In a live game we probably could at best bet $13 or maybe even $15 depending on what the smallest chip in play is.

In practice it is impractical to give solvers more than a certain number of sizing options to choose from in a given spot. A typical solve might give options like check, 25% pot, 50, 100, 200 and all in. The true GTO sizing might not be exactly one of these options. Practically this usually would make little difference, but this is a theoretical discussion. Theroretically speaking, it is not possible to determine exact GTO strategy. Exact GTO strategy is indeed a purely theoretical notion (and it is the strategy that makes us indifferent to what our opponent does, I.e., by dedinition, Nash Equilibrium)

It's hard to have a discussion when you are the only one speaking the language.

No I said it’s trivially obvious that if a strategy contains non-mixing mistakes then it will lose EV to a GTO strategy. Equivalently, the contrapositive is true — If a strategy doesn’t lose EV to GTO it cannot contain non-mixing errors. That does not imply the converse of the contrapositive, namely that if a strategy does not contain nonmixing errors it cannot lose EV to a GTO strategy. That statement is independent of the trivial one and must be proven seperately.

What I maintain as an open question is whether or not mixing mistakes can increase the EV of a GTO strategy. Your experimental results suggest that the answer is no, but that isn’t mathematical proof. I’m also thinking beyond poker to game theory in general here and thinking about arbitrary strategies in arbitrary games. It is clear that a mixing mistake can lose EV to a non-GTO strategy. A simple example is rock-paper-scissors. For convenience, I will designate strategies as (R,P,S) where the numbers in the ordered triple are the frequencies at which we play each option. GTO is (1/3,1/3,1/3). This strategy is zero EV vs ANY possible strategy. Suppose we play an opponent who uses (0.9,0.05,0.05). GTO still gives zero EV -1/3 wins, losses and ties. However if we were playing a non-GTO strategy of say (0.05,0.9,0.05), our opponents mixing error would lose EV against our strategy. Thus it is not true that a mixing error does not lose EV against an arbitrary strategy.

You indeed may be right in asserting that mixing errors cannot lead to EV loss vs a GTO strategy, but I don’t believe that is inherent to the definition of GTO and I’m not sure it’s ever been mathematically proven. The definition of GTO is that the GTO cannot lose EV to any opposing strategy. It does not say it cannot gain EV against one with mixing errors.

Hence my statement that it’s an open question.

There is a simple proof from first principles:

1) The definition of NE is that no player can increase their payoff by unilaterally changing their strategy.

2) Therefore, at equilibrium, every action within a player’s mixed strategy must yield the same EV against the opponent’s fixed strategy. If any action had higher EV, the player could change their strategy to increase their payoff, meaning they weren't at equilibrium.

3) Therefore, by linearity of expectation, any mixture between those indifferent actions must also have the same EV.

For example, if a hand is mixed between checking 33% and folding 67% at some node in GTO, then any mixture of calling and folding has the same payoff vs GTO. They could always call or always fold or flip a coin.. This is the law of indifference.

If they are exactly the same then why are they called different things at all? 🤔

You know what they say, sometimes the answers in life are a google away ..

I thought I explained that. When you learn game theory you NEVER hear the term “game theory optimal”. That is not a term that is defined in the mathematics of game theory. For some reason (and perhaps it is indeed solver developers trying to market their products) the poker community rejects the mathematical term Nash Equibrium in favor of the GTO designation. They are exactly the same thing, though.

Wasn't GTO popularized as a term in "Mathematics of Poker"?

You can't think of anything else in the world that has different ways to describe it?

ummm, Solvers and GTO? 😉

...............

The irony of this thread teetering somewhere between hilarity and parody now

Spoiler

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No, Nash equilibrium is not the same as "game theory optimal," but they are closely related concepts. A Nash equilibrium is a specific type of stable outcome where no player can improve their outcome by unilaterally changing their strategy, given that all other players' strategies remain unchanged. In contrast, a "game theory optimal" (GTO) strategy is one that is unexploitable by any opponent, but it is not guaranteed to be a Nash equilibrium unless all players are playing GTO.

Nash Equilibrium

Definition: A stable state in a game where each player's strategy is the best response to the other players' strategies.
Key feature: No single player can benefit by changing their strategy if all other players keep theirs the same.
Example: In the game of rock-paper-scissors, the Nash equilibrium is to play each option with equal probability (a mixed strategy). If a player deviates from this strategy, an opponent could gain an advantage.

Game Theory Optimal (GTO)

Definition: A strategy that is designed to be unexploitable by an opponent, regardless of the opponent's strategy.
Key feature: A GTO strategy is considered "perfect" because it generates the best possible outcome against a perfect opponent and makes the best possible outcome when playing against an imperfect opponent.
Example: In poker, a GTO strategy is one that is mathematically optimal and unexploitable by any opponent. If all players in a game are playing a GTO strategy, a Nash equilibrium will result, where no one can gain an advantage by changing their strategy.

Relationship between the two

Nash equilibrium is often a result of players playing optimally: When all players are playing an optimal, GTO strategy, the resulting state is a Nash equilibrium.
A Nash equilibrium does not always mean players are playing optimally: In games with multiple Nash equilibria, there might be a "better" or "optimal" equilibrium that is not the one that is reached. In this case, the Nash equilibrium is still a stable outcome, but it may not be the most profitable one for a given player.

A Nash equilibrium is a condition, while GTO is a strategy: Nash equilibrium describes the state of a game, while GTO describes the strategy of a single player.

Then why is it a mistake? What is mistaken about it if it results in the same EV? If, as in your example, a spot is call 1/3, fold 2/3, why is it mixed at that frequency if calling 1/2, folding 1/2, pure call, or pure fold all have identical EV?

IÂ’m not being argumentative here; IÂ’m truly trying to learn. What am I missing? Why is it a mistake to pure call instead of mixing call and fold if it doesnÂ’t make a difference to our EV? I assume that only applies at equilibrium, but isnÂ’t the solver giving us these frequencies based on equilibrium strategies being used by both players? Why does it give the specific frequencies it does for mixed strategies if those frequencies are irrelevant?

And your proof shows that if both players are playing a NE strategy then mixing mistakes cannot change the EV of either player. It does not show that if player A is playing a NE strategy and player B an arbitrary one, that player B’s mixing mistakes cannot increase player A’s EV. That was what I was getting at - whether mixing mistakes in a non-equilibrium general strategy can increase the EV of a player using a NE strategy.

Because it increases our exploitability. Our opponent could adjust and increase his EV and lower ours. But as long as he remains fixed, the EV of our mixed actions remain the same.

If we are playing paper 90%, our opponent is not making a mixing mistake by playing rock 90% -- that's a pure mistake. Why would he mix between rock and scissor when rock is very negative EV and scissor is very positive?

It's a mistake because it's exploitable. It won't be punished by a Fixed NE strategy, but would be punished by a best response strategy. The set of all unexploitable strategies (NE) is much smaller than the set of all strategies that maximize EV vs NE.

Regarding your last point; I'm not really sure what you want to prove. As long as player B's strategy doesn't contain pure mistakes, then their mixing mistakes won't lose EV vs NE, because all valid mixtures are, by definition, indifferent.

Yes, NE is more formally defined as a state where no player has an incentive to change their strategy. GTO is a less formal definition popularized in poker theory.

But this is a rather tired semantic debate Ceres. Unless you're explaining it to a beginner, everyone here understands what is meant by GTO, or NE. We understand that "optimal" means the equilibrium strategy not the best response strategy. We understand that these strategies are approximated in practice. We refer to one player's strategy as GTO or NE even if they are the only ones playing the way.

We don't need to redefine our language because we understand one another.

Analogy:
Pi is approximated. Do we need to define an entirely new definition to describe the approximation of Pi, to distinguish it from the abstract mathematical concept? No.

I was going to poke a bit and give 22/7 as the exact figure of Pi. Imagine my surprise when I thought to check it first and learned that 22/7 is in fact an approximation and not a precise value. Old dogs can learn new tricks.

The point was never people confusing those two in particular, the point was people were calling me out for cross pollinating things like solvers and GTO and the general double meanings that can be applied to all these abstract terms, whilst inevitably committing the exact same crime themselves.

Which validates my original point. The semantics are all over the place. Do I care? Not really. But it does seem to cripple every GTO debate ever because you get bogged down in ego, corrections and confusions when 90% of the time it's people (broadly) on the same page, stuck in this kind of perpetual minor clarifications la la land.

People were dunking on you because you spent half the thread making up your own random terminology, and derailing the conversation with semantics, rather than communicating in our shared language.