So, I have a whinery, and I have a brag factory. This is a very polarized range of blogs. In the interest of balancing, I need a blog to (pretend to) present neutral thoughts as well. (The other ones reflect my personality better

.)

Observation of today: There really is a (at least one) GTO strategy for each and every form of poker.

Another observation: I have gone to Nash himself to get the real statements and proofs. The OP is wonderful. There is nothing like original sources. I have skimmed the prof only so far. An interesting technicality is that the proof seems break down if the domain of the functions that are involved in what constitute "strategies" is not

*compact*. Unless you are a total math nerd, you aren't going to have a clue here about what I mean. Thus: If the stacks (and allowed bet sizes) of the participating players include the very last chip, then the proof holds. It may break down if allowed bets B are in the range 0<B<stack excluding endpoints, while every real number in between is allowed. This is a technicality, and the theorem may still be true, but harder to prove. [Simplical Complexes and the Brouwer Fixed Point Theorem are involved, as are compact convex subsets. Most theorems in this area need compactness for their simplest proofs, and Browers theorem isn't true without the compactness hypothesis.] I may be wrong, particularly since I have skimmed only.

So much for a "normal" blog, huh?

/Johan =