Re: Mathematical models
Hey, thx! I'll check those out and see if there's anything similar.
Actually, I'd also be more than happy to post the model, but it's already 35 pages single-spaced in a Word file including LOTS of functions--and, in working on it, I just found another bug.
The short version is simply this: I define a certain partially ordered set D, which represents what I call card-situations. These are conjunctions of cards and betting actions and tells. The set is partially ordered because from one situation, there are various possible successors to it in terms of the rules of the game. So, I set up a bunch of axioms (I have 32 of them) according to which the set must "behave" in terms of various functions--which return values for the rake, for holdings, for the flop, etc., etc.
Anyhow, (and here's the trick to it) I then introduce these functions, which I call psychological state functions or PSFs, which give the probabilities that a player takes a certain course of action in card-situation d. You could also call them "strategies," and they are to some extent, but I feel like the PSF idea is a little more general and suggests also any kinds of uncertainty (as well as randomized strategies), feelings of confidence, feelings of fear, etc. into the probabilities of taking various actions. So, a PSF, while it gives a probability function specifying probabilities for all allowable actions at d, has a lot of different factors going into it.
The first actual substantive result (no idea what becomes "provable" on the basis of this model--maybe only trivialities, maybe some interesting stuff) would then be a general definition of value for all possible card-situations and all possible PSFs. Obviously, a PSF in conjunction with a card-situation only has value relative to the other players' PSFs.
Anyhow, that's the basic idea, but it's not at all easy getting to an explicit (recursive) definition of value, which begins, so to speak, at the end, namely when the pot is distributed either by default (all but one fold) or at showdown.
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