[jukofyork]

I think maybee here my fault... I only posted about certain abstract situations in relation to a bigger idea that I am stuck with [img]/images/graemlins/smile.gif[/img]

Let me try to re-formulate my exact problem I am considering:

1. We have infinite computational resources.
2. We wish to play 'maximally' rather than 'optimally' against a set of KNOWN opponent(s) where for each: we have previously observed playing and have collected an INFINTE amount of data on each opponent, where each opponent has played against an INFINITE number of opponents, with an infinite range of other playing styles.

The question (in relation to all forms of poker [with some degree of hidden information]) I ask then is:

In any given STATE where YOU have a decision to make, then assuming infinite computational resources (both space and time), can we theoretically compute via Minimax algorithm the EXACT EV of call or raise?

Now consider these abstract versions of poker:

A: Poker played without hidden information (ie: with all hands played face up)
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This abstact game can be solved (with infinite computational resources), even for N-players using Minimax algorithm and their are no ditributions of possible player holding to consider (eg: No harder than backgammon, just using chance nodes, but still need opponent modeling, etc).
NOTE: The question here is: If each of our opponents also thinks beyond the 1st level and they themselves use the models of others players when making their decisions (2nd level), and then they also consider models of other players (3rd level)... then will their actually be an Nth level where the tree stops growing (is the tree finite?) - I think for poker yes, assuming their are actually a finite number of actions a player can make in a hand (the 4 bet capp), because this will limit the amount of possible Nth-level thinking making the tree finite.

B: Poker played where every time you fold you are forced to show your hand:
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I think that theoretically this abstract game can also be solved, as in actual fact it is just a more complex version of Game A. Even though we do not know the exact hand our opponent hold, we should at any point in the game be able to say the exact probability distribution of his holdings. This now means we have an extra layer of chance nodes each time an opponent takes an action, with each chance node corresponding to the probability of this player having a certain hand. Again thinking about the point made about Game A, I think that again because of the limit on the number of possible action, this tree is also finite and fully observable, and we can thus search and find exact EV of call and raise.

C: Real poker:
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Now if you read my original post about my three abstract games, it should make more sense why I am considering them.
The problem now is that even with inifinite data on opponents, I think it is impossible to create the whole tree, and thus cant find exact EV of call and raise, because now we cannot either be sure to have the exact hand distribution of each opponent nor can we be sure that our future predictions (via the mini-max search) will be correct due to the fact that a opponents fold descion could be non-linear (see my 3 original examples / counter-examples).

Is my reasoning flawed for Game A or Game B, is the tree really fully observable (thus exact) and finite?

"In general, determining an exact opponent model from showdown observations is impossible -- it is strictly unknowable" - Darse, sums up my problem with Game C... [img]/images/graemlins/smile.gif[/img]
Juk [img]/images/graemlins/smile.gif[/img]