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Say, for example, you wanted to know what percentage of the time a player plays A9 UTG. Simply look at all hands where this player was UTG, one of the blinds was forced all-in due to low chips, there is no preflop raise and the flop was AA9. Every time the player has A9 in this situation he will show it down (assuming he doesn't fold flopped boats). If the player plays A9 100% of the time then on 1.2% of such instances (blind all-in, no raise, flop AA9) the player will show down his AAA99 boat. If he plays A9 UTG 1/2 the time, he will show down AAA99 0.6% of the time, and so on.
Once you determine the percentage of time he enters utg with A9, you could also figure out how often he folds it to a raise preflop. Again look at AA9 flops after a preflop raise. If he shows down AAA99 1/2 as often as when there was no preflop raise then obviously he folds 1/2 the time to this raise.
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Yes, I think this idea is the 'transform' I was looking for in my original post (ty!) and I think in practice, some kind of abstract version of this idea could be helpful (for NL maybee [in practice, with finite data] a model will converge quicker?).
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The problem is that often everyone will fold after he bets so no showdown.
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Every time I think about this, it seems to come back to this question (again I think more in terms of limit, and trying to ignore stack size as a variable...).
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Your main question, though, I think, is if you can extrapolate an opponent's entire preflop/postflop strategy based only on the cards they show down. And knowing that they only call with nut hands, if you can somehow use EV weighting to check your assumptions about what they're folding.
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Yes, this is my goal (to start with thinking in terms of inifinte resources, but ultimately wondering if this can be abstracted to some real level), but not just in the context of nut-hands (there were just very extreme examples to show the non-linearity of the problem for differnt opponent stratergys showing similar stats on observation).
Sry, I forget to post here too (this discusion going in two threads, sry!), but i posted this in the poki-forums (see
here):
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Player Z
======
Pre-flop: Only calls with AA, KK, 22 and 77.
Post-flop: Always calls down to showdown with AA or KK, always folds 22 on the flop and then folds 77 on the turn.
(Assumes that the flop and turn are always bet)
Our prediction about this player BEFORE we see him take an action:
P(call)=4/220 {this from memory, so correct me if 220 a bit out!}
P(fold)=216/220
Our pre-flop distribution for this player AFTER we see him call:
P(AA)=1/4
P(KK)=1/4
P(<unknown>
=1/2
On the flop we now predict, BEFORE we see him take an action:
P(call)=P(AA)+P(KK)+P(<unknown>
=1/4+1/4+1/4=3/4
P(fold)=1-P(call)
Our flop distribution for this player AFTER we see him call:
P(AA)=1/3
P(KK)=1/3
P(<unknown>
=1/3
On the turn we now predict, BEFORE we see him take an action:
P(call)=P(AA)+P(KK)+P(<unknown>
=1/3+1/3+0=2/3
P(fold)=1-P(call)
Our turn distribution for this player AFTER we see him call:
P(AA)=1/2
P(KK)=1/2
NOTE: Addition of P(raise) doesn't really complicate the calculations, but makes a simple example harder to understand.
NOTE: <unknown> is a hand (hole cards) we have never seen goto showdown. We see player call/raise with this 'unknown' hand, only to later see them fold it before showdown (ie: This is not really a hand, but a range of 'unknown' hands that they could have, but will fold later [most importantly = THE ACTUALY HAND RANGE DOES NOT SEEM TO MATTER...]). - Perhaps a better name is <never-known> or <never-seen>.
We now have a seperate hand-independant prediction model for use in search capable of predicting P(f)/P(c)/P(r) based purely on all the available 'state' information (appart from the actual hole cards a player holds...), along with an extra <unknown> hand classification added to the distribution.
Thinking back to all of the hypothetical players (A,B,C & D), this idea now seems to work well at countering their stratergys (even the non-linear) in a game where ceratin folded hands we will never see (ie: Game C - Real Poker, see previous post(s)).
Is this maybee the best that can be done (this does not require a 'transformation' and it ignores the "intermediate distributions" to some extent [see terence's reply] - at least for the <unknown> hands)?
Is their anything fundamentally wrong with this idea in the context of my problem outlined in the previous posts?
Juk [img]/images/graemlins/smile.gif[/img]
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But after consideration, I THINK that this idea is now almost equivalent to a "best-response strategy --terence" (again see the poki-thread):
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I hardly read any of these last three posts (just sped through them partially absorbing things) since they are so long and I unfortunately don't have any spare time these days. Anyway, I think in the last post you gave an example of what I was saying. Basically, the whole idea is fundamental idea that Vexbot uses for its modelliing (if I understood any of what you were saying - but again I didn't get much of a chance to read it). You can read about it in Aaron's thesis where he talks about Miximax, or in the UofA paper about search in poker which I can't remember the title of. Unfortunately, those were written with a focus on the search part and not the modelling part (at least, not the modelling part that you are interested in). My thesis will have a bit more on the modelling part, but I am finishing it up so there is nothing to read yet.
Hope any of what I said helps,
Terence.
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Why I think this now, is bc my 'Player Z' calculations actually ignored the fact that sometimes (in general/reality) a player would play say AA or KK, but fold it later in the hand [sometimes!] (so now the <unknown> class contains some instances of AA or KK, and the classed I called AA and KK actually hold P(AA at showdown) or P(KK at showdown) - so i think it come back to purely a "best-response strategy" (which does seem to model enough to counter all the abstract modeels A,B,C & Z without the non-linear descisions causing any problems...).
I guess this gona confuse me for some time to come, lol. But eventually it would be nice to come to some conclusions about a 'maximal' counter-stratergy. In chess it is possible, if tried, to attempt to extrapolate (ingnoreing potential future botlenecks) the point in the future where computers will have advanced enough to actaully know the full search tree (at this point chess will be 'solved' and enter 'the hall of fame', leaving only 'maximal' stratergy vs imperfect humans, open for research). The same idea must also apply to poker to some extent (see UofO research on HU optimnal players), but of more interest to me is can any of these ideas be abstracted enough to be of any help with current technology and finite data... [img]/images/graemlins/smile.gif[/img]
Juk [img]/images/graemlins/smile.gif[/img]