Consider this

Player A knows that B only raises with big pairs.

Player B knows that A knows this.

Player A knows that B knows the above.

Player B knows that A knows the above.

Player A . . . etc.


After some number of iterations the extra knowledge becomes trivial. Does it ever become absolutely un-differentiable?

Interesting point. What's the difference here: player A is what Mason refers to as a tourist type, and so is player B. They bet and call and raise all the way through a hand with virtually nothing because, well, neither one knows any better.


Now switch to players C and D, who are world class. They put the same number of bets into a pot with the same cards as did A and B above, but their reasons are much more sophisticated: they are thinking up so many levels, that the plays make sense from that standpoint. Yet when the hand is over, C is in the same situation as A and D is in the same situation as B.


I know it doesn't help, in the long run, to make a "correct" play for the wrong reason. But when your opponent is of equal skill to you, your results may be the same as the results from a skirmish between two players who play far, far worse than you, and the same as would result from two players who play far, far better.


So I guess the answer to your question is that the player who goes one iteration farther usually comes out on top.

No - if you allow arbitrary NL-betting amounts, randomization and sufficently large stacks, that is - since optimal betting ammounts and probabilities would be real values that would react to even the slightest changes in assumtions. (This, of course, assumes that only B's initial bet is unconditional; otherwise, it would suffice that A knows B's fixed betting pattern to derive an optimal strategy.)


A more interesting question is, whether the resulting game-theoretical optimal strategy is convergent, in the sense that the EV-difference between two recursion steps goes against zero for all possible stack sizes. It seem reasonable to assume that, but I wonder if some game-theory wizard could actually come up with a prove. (It is at least possible, that there is a finite premium in thinking one step further than your opponent.)


Another interesting question is, how much it will hurt you, if you think too complicated i.e. multiple levels ahead of your opponent: Will A e.g. be better off not knowing that B raises only with big pairs, than knowing it but under the false assumtion that B is aware of that (So that A would get paranoid each time B plays back at him).


cu


Ignatius