[DcifrThs]

i hope you have a cray...too many variables...too many unknowns to optimize.

but you asked a question so i'll give you an answer.

first step is to catagoize all 169 distinct starting hands by overall value. one method is the HU equity vs. a random hand. let HVi be {AA,KK... 23o} where i=169....1 so 1 is the worst hand and 169 is the best (since we're trying to maximize the overall value of a range.

next we have to assign ranges to the oppoennt say he opens with the best X% hands equiprobably. this range can be modeled by a modified normal distribution where the highest valued hands sit where the most mass is. it would also be truncated at that value b/c no two hands are given equal weights so this final density looks like a reverse "s." then we add other variables similarly (which is where this becomes extremely subjective and difficult since the outcome is so highly sensitive to these variables slight changes will likely yield wide movements in optimal 3betting ranges) starting with the value of your position - his position. so if UTG is worth .05bb/100 and the button is worth .11bb/100 then this variable is worth .11-.05=.06 ...for modeling purposes this difference should be given a negative value and mapped to an increase or decrease in the size of a range. it should be negative b/c the farther away you are from the person (you hold YOUR position constant so lets say you're the button here), the tighter you have to be.

this goes on and on, thinking about the inputs, and affects on a player's hand range (pfr and vpip..these are tricky and id make some adjustment based on the ratio of pfr/vpip in addition to just a raw pfr figure in there...but a raw vpip does nothing imo.) then you quantify them, look at it and assign a model to (multivariate probit would be my guess, but to do that you'd actually have to break the dependent variable-your hand 3bet range- into ranges themselves b/c the outcome would have to be a scalar as a PART of a vector. the dependent variables are then assigned like 12=Truncated Normal ~ (HV{169 to 130)/#hand values from 169-139, variance-which you'd also have to calculate from this odd distribution)....that all reduces to a # using a MVProbit model like 12 in this case.

clearly its a tough model to make but imo it can be theoretically done...the problem is there are too many subjective steps and reductions and such that the model loses its power completely.

anyways, i hope that reading this monstrosity has assisted you in some way in your hopes to conquer the world....er...limit holdem.

PS- im not rereading this or editing it and thus im SURE ive made some mistakes/omissions...but the general gist should remain in tact)

Barron