I'm not an expert, but I believe the difficulty with Texas Hold'em is modeling the other human players. "Heads up", where there is an optimal strategy, is essentially solved.
Firstly, although it is proven that there is an optimal (game theoretic) strategy to play (Nash), we are not close to 'solving' it. There are attempts to solve limit hold'em, and the current systems beat human players, but they are not close to fully solving the game tree. Attempts to solve big-bet games (NL, PLO) are not even "solved" at a rudimentary level.
Modelling human players is the much easier part. There are winning cash game bots in both FL and NL up to midstakes games, and they are a serious issue. Many use default exploitative strategy that perform well against common opponents.
But the main thrust of my post was that heads up (two players) is mostly solved and you don't seem to address that point. Is that not true? It might not have a closed form solution, but I thought it was approximately solved in that there were algorithms that did no worse than epsilon off ideal play.
I'm not sure if we're talking in the same language here; you cannot lose (beyond your share of the rake) playing a GT strategy, regardless of what your opponent is doing. This is the basis of what GT is.
> But the main thrust of my post was that heads up (two players) is mostly solved and you don't seem to address that point. Is that not true?
No, it is not true. Even for FL hold'em, which has a significantly simpler game tree, the best attempts are still far from a complete solution. Large simplifications/assumptions to the game tree need to be made to make it a manageable size computationally. Much simpler representations for HUFLHE have been solved.
For all other heads up games, researchers are barely scratching the service; I don't even think there's a clear understanding of how to even go about simplifying the game tree to reduce it to a manageable size to even consider solving it.
[I put in the caveat that i (a) haven't read up on the latest in the last couple of years and (b) have not been involved directly with any research projects. I'd love to be corrected from someone who's involved in the latest research.]
What does GT mean here? Not game theoretic, since general Nash equilbria don't have this property. Nash is a strategy where no individual has a motivation to change from the equilibrium.
There are multi-player games that have this property that optimal play always results in statistical winnings, but I doubt that poker is one of them. For example, that would imply that collusion isn't effective against an optimal player.
Heads up is a different matter. There it's fairly elementary game theory that there exists an unexploitable strategy. A brief search says you're probably right, though, about heads up not being solved.
So I maintain everything i said before; HU games are far from solved. Much the same that chess is far from solved.]
The game of poker is symmetrical and zero-sum (ignoring rake).
As you say, "Nash is a strategy where no individual has a motivation to change from the equilibrium." If I were playing a GT optimal strategy, the best you can do yourself is play the same strategy - you are not motivated to deviate. This will be EV neutral to both of us in a symmetric, zero-sum game.
Any deviation you make will either be EV neutral and therefore indifferent, or EV negative. If it is EV negative to you, I gain.
Of course none of this applies to multiway games, we're talking heads up.
> Firstly, although it is proven that there is an optimal (game theoretic) strategy to play (Nash) [...]
The reference to Nash here made me think you were claiming there to be an optimal strategy in the multiway case. I've been talking mostly about multiway ever since. I associate Nash equilibrium with multiway games and wouldn't use the term "Nash" to describe GTO play in a heads up game, even though a Nash equilibrium would be GTO. But maybe this is standard lingo?
Question: So is it the case that there are human players that are measurably better (in a statistically significant way) than the best AI players, heads up?