Playing perfect poker with the cards face up

[t_perkin]

Following from Zeno's post on the FTOP, I began to think about playing poker with all the cards face up.

Has anyone ever created a complete list of instructions on how to play a perfect game if all the cards are face up?

Is this straightforward? I think it probably is not but I am not sure...

The resulting player would only be required to be perfect rather than optimal (i.e. you assume that all the other players were also playing perfectly)

Tim

[bigpooch]

It's not that easy for a complete list of instructions.
If the game has a very big ante, the underdog often has
a call still. In a game like draw (hi only or lowball),
there are sometimes close decisions depending on the
bet size and total pot size. To play optimally means
something altogether different I thought: the cards are
unknown but ideas such as game theory come into play for
bluffing and semibluffing. There are game theoretic
optimal calling and bluffing frequencies depending on
the ratio of the bet size to the pot size. Also, in a
game like NL, this would be quite boring in NL if the
cards were exposed! So if the cards were face up I think
Sklansky was mostly referring to limit games.

[daryn]

if the game were played face up, all the information would be complete , and using mathematics it would be rather easy to play perfectly i think

[t_perkin]

First I will clarify what I mean by the difference between perfect and optimal play (in terms of playing poker with the cards face up):

Perfect play is to play in a mathematically correct way regardless of who your opponent is. This can be considered the same as assuming all your opponents are playing mathematically perfectly as well.

Optimal play is to play a game which will maximise your winnings due to the bad play of your opponents. i.e. if you used the same strategy that you used against a bad player against a perfect player you would lose, however by playing non-perfectly you exploit the weaknesses in this particular opponents game to increase your profit above and beyond what playing against them perfectly would achieve. Although it should be noted that playing perfectly would still lead to a profit.

i.e perfect play will *at least* break even against *everyone*, and optimal play will *maximise* profit against an individual, but the strategy must be specific to the individual.
The optimal strategy against a perfect player is perfect play (and will result in break even).

I think that playing a perfect game face up is not as simple as it first appears. Although it is a purely mathematical problem, perfect play still includes perfect betting strategy. This means that the future actions of your opponents must still be predicted (although they are perfectly predictable due to their perfect play and the fact that there is complete information). Predicting their actions is not straightforward, as to do this one must also consider that they are also predicting the future actions of their opponents their opponents include you, and so predicting their actions you must also "predict" your own actions. I imagine that this ends up becoming a recursive function, but I am not quite sure exactly what it will be a function of...or what the base case would be...

I think the problems would arise around calling and betting strategy for drawing hands, one would have to predict how ones opponents are going to act to work out whether or not it is worth calling a bet with a drawing hand.

More to the point if it is so straightforward, does anyone care to have a go at doing it?

Tim

[daryn]

i still think it's very easy to play perfectly and optimally with the cards face up.. unless you are against a total idiot.

for instance, .. i have AA face up, he has KK.. do i bet preflop? or do i check? well, i bet, because i want him to fold. if i check, he could catch a K to beat me, and there's no way he would call any bets post flop if he didn't have me beat.

[t_perkin]

Ok how about this situation:

The players going round the table (TcTd in the SB) have (with their twodimes EV):

Note: Although in actual fact when playing perfectly and when ones opponents are also playing perfectly, does position (with the exception of the blinds) make no difference?


Tc Td 0.056
Kd Jd 0.160
6s 5s 0.201
7d 6d 0.145
Ts Th 0.056
Js Ad 0.100
3d Ah 0.070
Ac 2c 0.125
2d 2h 0.087


Assume a limit game.

Now if you can tell me who is the favorite in this hand, how much they will win and exactly how this hand should play out then I will concede that playing face up is trivial. (although 56 has the highest EV does not mean that it should play and everyone else should fold)

To be honest I would struggle to even tell you which hands should play before the flop, after all, the AJ has only one A to run for, the pair Ts are the current best hand but have nothing to draw to, the KdJd has pretty much had his flush chance killed by the number of other diamonds already in play, and so it continues. I would not like to try and guess who should be betting, folding or raising PF. Things probably get much simpler on the flop, but there are still other issues to look at.

Tim

[daryn]

ok.. if i have access to all of this information , then everyone else does too. now i look up on twodimes and i see that 6s 5s has the biggest chance to win, 18.74%, and is thus the favorite if all the cards were to be dealt out right now.

the 7d 6d has a 13.08% chance of winning in the end.


the two pairs of tens are looking bad here. 11.3% of the time they will tie and split the pot. .04% of the time, one of them will win, and .04% of the time, the other will win. so their odds are not as good as they seem because most of the time they will split the pot, so they both fold.

the pair of 2's has an 8.68% chance to win, and will fold also because he doesn't have the odds to call, especially with one of his 2's dead, he's basically drawing to one card in the deck.

Ah 3d has a 6.60% chance here, and will fold.

Ad Js has a 9.39% chance, and will fold.


the concept of implied odds goes out the window, since everyone knows what everyone else has at all times..

this particular hand basically comes down to these hands...

Kd Jd
6s 5s
7d 6d
Ac 2c

the people who hold these hands know the others will fold, because the cards are all played face up and the game has perfect information.


say this is a 10-20 game, with a small blind = 1/2 big blind structure. in the pot to start is $15. first action is on the 6s 5s. does he raise? well if he does, he can expect to lose the 7d 6d and the Ac 2c, because they can't possibly call a raise with worse hands. they're not getting the correct pot odds, and also their implied odds are zero, so they both fold. it comes back to the Kd Jd, and now there is $35 in the pot, and he has to call $10 to play. he's getting 3.5-1 on this call, does he call? well it's heads up, and if we check twodimes for the stats on these two hands heads up, with all the dead cards accounted for , we get:

Kd Jd wins 50.80% of the time
6s 5s wins 48.44% of the time

so the Kd Jd obviously has to make the call. and after this, the hand plays out rather simply and would be dictated by whatever cards fall on the flop, and the probabilities thereafter.

now, the 6s 5s knows that if he raises preflop, it will play out like this, so the question is, is raising preflop the best play for the 6s 5s? if the answer is yes, then this hand analysis is done. if the answer is no, then we continue on...

clearly the 6s 5s can't fold preflop, so what if he just calls? what changes?

say he just calls for the sake of argument. now there's $25 in the pot, and it's on the 7d 6d. the 7d 6d is getting 2.5-1 on his call, but if he looks ahead, can he expect the Ac 2c to call? well.. say 7d 6d calls, then there would be $35 in the pot and Ac 2c would be getting 3.5-1 on his call.. would it be correct for Ac 2c to call here? well let's run twodimes with these 4 hands and dead cards included...., we get:

Kd Jd wins 27.89%
6s 5s wins 24.85%
7d 6d wins 19.17%
Ac 2c wins 22.71%

so if 7d 6d called, Ac 2c would be getting 3.5-1 on his call, but his odds to win the hand would be about 3.43-1 against... so he would make the call, and be correct. now there would be $45 in the pot. now if we go back, the 7d 6d knows that if he calls, the Ac 2c will correctly call, so the 7d 6d is actually getting 3.5-1 on his call as well. but since he is only 19.17% to win the hand, or around 4.22-1 against to win, he still can't call, even if the Ac 2c comes along! so 7d 6d folds, even if 6s 5s just calls, and even if he knows the Ac 2c will come along.

now that we know the 7d 6d will fold regardless, we have to rethink Ac 2c's call. he will be getting 2.5-1 on his call, since the 7d 6d won't be in, and if we run twodimes with all the dead cards, we see that :

Kd Jd wins 32.02%
6s 5s wins 39.16%
Ac 2c wins 28.31%

so the Ac 2c is getting 2.5-1 on the call, and he is 2.53-1 against winning the hand, so he folds correctly!

alright.... so even if the 6s 5s flat calls the big blind, we lose both the 7d 6d and the Ac 2c anyways...

it looks like this hand is gonna come down to Kd Jd vs. 6s 5s. the only real question now is whether either of them wants to raise.

the probabilities are:

Kd Jd 50.80%
6s 5s 48.44%

the 6s 5s would be getting 2.5-1 on the call, and is about 1.06-1 against winning. so he would call, correctly.

but what if he raised?

well he knows it's gonna come down to just the two of them, why would he raise heads up with the worst hand? the answer, he wouldn't. but the Kd Jd would. so 6s 5s calls, then it comes around to the Kd Jd's option from the big blind.

there's $35 in the pot at this point, and the Kd Jd knows he has the best hand, so he raises. now there's $45 in the pot. the 6s 5s has to call $10 to win $45, but really, he would have known that the hand would play out like this... so before when he called, he was really only getting 25-20 or 1.25-1 on his call, since he knew everyone would fold to the Kd Jd , who would then raise.

so now we have to look at the original call by 6s 5s. he's getting 1.25-1 on his call, and he knows he will be 1.06-1 against winning when it all plays out, so he in fact does call.





and there you have it i think. that is how this hand will play out.

[t_perkin]

[ QUOTE ]
the two pairs of tens are looking bad here. 11.3% of the time they will tie and split the pot. .04% of the time, one of them will win, and .04% of the time, the other will win. so their odds are not as good as they seem because most of the time they will split the pot, so they both fold.


[/ QUOTE ]

but if only one of them folds then the other pair of tens looks much healthier. but this is a minor mistake.

Looking at twodimes is not sufficient, this only gives you the odds if the players see all five cards.
In reality for example if one of the TT hands stays in with Kd Jd, 6s 5s and the flop comes 8h Qd 4s then the TT is now 50% to win the hand. The two drawing hands must stay in the hand to the end if they want to justify their preflop percentages. Whereas if TT gets beat on the flop then it is beat for ever (e.g if a K or J comes on the flop in this example) and its twodimes results reflect this so it can happily fold on the flop and not affect its twodimes prediction. This means it can save money (the bets it would make on the flop and turn) when it is beat, thus changing the PF calling requirements.

Although this is a rather extreme example it shows what we all know: betting strategy is important and not just for deception, and players can make the decision to fold on all streets, not just preflop. And this would be an important part of a perfect player.

But even ignoring this, you are still using your own domain knowledge to decide which combinations of hands to look at, which options are obvious folds, which are obvious calls etc.
So even if we simplified the game even further such that there was just preflop betting and then the 5 cards came out and everyone showed down you need to do a lot of twodimes calculations. For the first player you need to simulate what would happen for each of their three options (fold, call, raise) and in order to do this you would need to simulate what each of the subsequant players would do when responding to each of the three actions, and to do this you would need to simulate what each of the players would do when responding to this... and so on until you reach the last player to act in the betting round who would know all of the actions of all players, which would become the base case of this recursive problem.

Tim

[daryn]

right.. but you have to go on percentages, because you're assuming that everyone is playing perfectly.. and everyone can see every card. the TT shouldn't think.. well i should call, because if the flop comes as i like it, i could have a good chance to win here..


there are no implied odds, so you have to look at the pot odds at hand, and the current odds of each hand as is, not after the flop or anything like that

[AmericanAirlines]

Hi T_Perkin,
It seems to me the answer is: "If you have an edge, try to get the more money in the pot, if not try to draw as cheaply as possible. If you must pay to draw, the pot odds must support it." (For the cards up game. For a real game it becomes, "What move is most profitable against this opponent or this group of opponents.)

The real problem is, in real time how are you going to figure the numbers fast enough. Or perhaps, if you had tables of every possible set of matchups from heads up to full table, can you condense it to a tractable number of rules.

Then from there, going back to the real game, do your best to read you opponents and apply the aforementioned rules.

So clearly:

1. Monte Carlo tools can give you the absolute numbers for a given situation, and all situations

2. The problem is to condense the rules to a tractable number.

3. Assuming (2) is possible, from there hand reading is a key skill to success. The second key skill is knowing what move is profitable against your opponent(s). I.E. a raise here will get them to fold... a check here will induce a bet so I can check raise, etc.

So, seems to me the Rosetta Stone of poker is (2). As you suggest, a tractable set of rules or perhaps even just a tractable table of odds for the common "hand v. hand" and "hand v. hands" matchups.

As I've ranted in the past, what is needed is a fast way at the table to calculate your "probability of winning the hand" not the "probability of catching an out".

Sincerely
AA

[t_perkin]

But a perfect player DOES look ahead, there is no reason why they should only think in terms of twodimes odds. Twodimes is just one way of looking at which hand is best (all players see all the cards). It does not adjust the win probability for them making sensible decisions later in the hand.
A perfect player should do this. They must also make decisions based on what opponents are going to do.

Tim

[daryn]

i don't think you're correct here. it's all about pot odds and probability (if playing cards face up). for instance, if we are heads up, playing with cards face up, i have AA and you have 22, i raise, do you call? you can't think to yourself, "well, if i call, and a 2 hits the flop, i'm a huge favorite", this is because if the 2 does hit, i will see it and fold to your bet! you have no implied odds! and you must know that the 2 does not come often enough to warrant your call preflop, in other words, you will lose money in the long run by calling.

this is not poker, where we constantly make decisions about other players based on post flop play.. this is a complete information game we're talking about, where all cards are seen and all perfect decisions are known to all!

[t_perkin]

Yes you are absolutely right

BUT

You have too much faith in those wonderful twodimes figures.

Those figures assume that all the hands which are given as input are seen all the way through to the showdown, this will not happen in a real game of poker regardless of whether the cards are face up or face down.

As you say, when that winning card for the flush draw comes on the turn the AA will fold. This is not accounted for in twodimes stats. This is to the detriment of made hands which are likely to win pots that would be lost if all the board cards came out.

Tim

[daryn]

ok, i understand your point and you are correct about that. however i do still feel that playing perfectly could be done easily using simple math and poker theory.

[karlson]

I have something that will calculate optimal heads up play with hole cards exposed (for any given pot size and board).

Doing this for multiway is slightly trickier, but, as Daryn points out, very much doable.

At some point this type of analysis becomes not particularly worthwhile.

[AmericanAirlines]

A second thought on this. Perhaps Petriv's "Holdem Odds"
or the other book "Percentage Poker" would give you some math based ideas based on hand v. hand matchups?

Sincerely,
AA

[AmericanAirlines]

Hi Daryn,
I'd like to add to that, "Knowing your Opponents".

Granted, not required if we assume in the face up game that all the opponents have exactly the same knowledge base.

Assuming they don't have the same base, seems to me the real issues are:

1. Where do I stand?
Here I see a 4 part table assuming the following two attributes:

A. Where am I now - "Leader, Follower"
B. What's the outlook - "Favorite, Dog"

So you have :

Leader, Favorite
Leader, Dog
Follower, Favorite
Follower, Dog

Knowing this will dictate what you *want* to happen. And determining this is the math part.

For the face up game, this could be tabularized (a large table indeed for all matchups from heads up to full table!)

Perhaps this table could be condensed to a set of useful rules so you don't have to clog your head with thoughts like, "I have 10 outs but he has 5 and two of his are mine and will make him win so I really have 8 outs..."

2. Given my opponents playing habits what's the most profitable move?

Once you know (1) now you have to figure out what to do with that information in light of the

For example, you may be Follower,Dog... but you know your opponent will fold to a raise. So you raise rather than fold.

On the other hand, given the same opponent, and you are Leader,Favorite... you might just bet or call. I.E. slow play the opponent.

Now moving back face down game... you have to add in that you're going to estimate your opponents' holdings. I guess therein lies the art.

But it would be nice to have that tractible set of "rules" or magic odds of winning.

Sincerely,
AA

[Copernicus]

I think AA's post is as good an analysis as any, and highlights why AI poker has achieved its strongest successes using Monte Carlo simulations rather than rule based decisions post flop, and bases those decisions on hand reading (though putting the opponent on a broader range of hands than live players generally would, and weighting them probabilistically) and "opponent reading".

What an opponent is likely to do is only partially based on the mathematics of the hands, due to incomplete information. There are often "tells" in his hand histories that are more reliable than knowing what his pockets are and doing the math assuming rational play.

[t_perkin]

OK I am going to clarify a few things - the original question seems to have got a bit lost...

first of all we are talking about a perfect player not an optimal one. from my first post:

[ QUOTE ]

Perfect play is to play in a mathematically correct way regardless of who your opponent is. This can be considered the same as assuming all your opponents are playing mathematically perfectly as well.

Optimal play is to play a game which will maximise your winnings due to the bad play of your opponents. i.e. if you used the same strategy that you used against a bad player against a perfect player you would lose, however by playing non-perfectly you exploit the weaknesses in this particular opponents game to increase your profit above and beyond what playing against them perfectly would achieve. Although it should be noted that playing perfectly would still lead to a profit.

i.e perfect play will *at least* break even against *everyone*, and optimal play will *maximise* profit against an individual, but the strategy must be specific to the individual.
The optimal strategy against a perfect player is perfect play (and will result in break even).


[/ QUOTE ]

this means that:

[ QUOTE ]

I have something that will calculate optimal heads up play with hole cards exposed (for any given pot size and board).

Doing this for multiway is slightly trickier, but, as Daryn points out, very much doable.


[/ QUOTE ] (from Karlson)

is not correct, I think you mean perfect heads up play, rather than optimal, in order to play optimally you must know everything that the player is going to do, a computer program certainly does not.
Can you tell us what this program is? do you mean poki/SparBot? I don't think that either of these can play perfectly.

next:
[ QUOTE ]

1. Monte Carlo tools can give you the absolute numbers for a given situation, and all situations


[/ QUOTE ]

Monte Carlo tools can give a good approximation for a given situation and all situations, but by their very definition (they only explore a proportion of the search space) they do not give absolute numbers. A monte carlo estimate will be different every time you run the algorithm. Even then Monte Carlo must still search a significant part of the search space, working out exactly how much of the search space needs to be searched to give an acceptable level of accuarcy is beyond the scope of this post (but I will dig out some papers if you like!).

[ QUOTE ]

For the face up game, this could be tabularized (a large table indeed for all matchups from heads up to full table!)


[/ QUOTE ]

These tables would indeed be very large! :

There are 52C20 = 1.25*10^14 combinations of whole cards for a 10 seater table alone. (actually it is less than this, because this considers JT to be a differnet hand from TJ, but it is not significantly different)
And for each of these we have to evaluate every single board combination:
so that is 52C25 + 52C24 + 52C23 + 52C20 = 1.4 * 10^15 evaluations. (once again this is a overestimation because it considers TJQ to be a different board to JQT, but it is not hugely significant to this argument)

So that give us a very rough approximation of how many different games of poker can be played with 10 players.

So we have a pretty big table...

And this is before we consider betting!

So now for each of these 1.4*10^15 different situations we must now consider *all* the betting patterns that could lead to the sitaution.. this is more complicated stats and I don't want to try and do it without some help.

(does anyone want to check my maths? I have done it very quickly..)

So now we have a search space which ( I have read in a paper - again I will dig it out if people want to read) is considerably larger than that of chess. (in fact if I remember right just the preflop play gives a larger search space than the whole of the chess search space).


Now THIS is what a perfect player must look at when deciding what to do. He must evaluate every possible outcome for his given situation and work out what the most profitable move is on average.
BUT in order to do this he cannot simply say well I make on average 15bets if I raise here and 10bets if I fold and -2bets if I call. Because not all of the search space is equally likely to take place, this is because some branches of the search space will never take place because they involve betting actions which are not perfect plays for the opponents to make and thus they will never take place (because all our opponents are perfect as well remember!). This means that we must perform some sort of 10-way minimax search to approach our final solution to the given situation.

So as far as I can see this is what would have to be done to create a perfect player!

And all of this ignoring the fact that it may not be perfect play to play each individual hand perfectly! If our perfect player is playng in a tournament for example then it must also consider bankrolls, and the possibility of busting players out, probability of being busted out, and the problems associated with increasing blinds, and the money payouts, this becomes fiendishly complicated!

But nonetheless this is what I am looking for! I want to find a set of functions (even if it is not practical to calculate them) that will produce a perfect action for every possible situation.

You may be thinking what is the point? if you can't calculate the answer then what is the point in knowing *how* to calcualte the answer? well I am afraid I am interested in things like this...

Tim

p.s.: one thing if you are going to reply - please remember that this thread is about playing poker with all hole cards face up. The aim of saying this was to remove all problems of hand reading and player analysis.

[karlson]

Tim,

I think you are forgetting just how much things are simplified by the cards being face up.

To answer your first question, my point was about "perfect" play, assuming your opponent also plays perfectly. The program is something I wrote myself, it enumerates all combination, and, for now, only works for heads up (I don't think that the effort required to make it work multiway is justified. I have used the damn thing exactly twice since writing it.)

So you want to put in opponent modeling. Great. But the cards are face-up! So what kinds of things do you know?

a) He will go to the river with AA every time regardless of your cards/flop? Ok, easy to add in to calculations.
b) (Multiway) He will not raise to knock out other hands even when he should? Ok, again, fairly easy to calculate in.

I did do some calculations, and you are right, for 10-handed, the complexity gets slightly out of hand. (For heads up, my program runs in a few seconds.) I don't see how MC would be particularly helpful in this case. The random boards are not the dominant term.

I still don't see why this is particularly useful...If someone points out to me why it is, then maybe we can do something to reduce the complexity (it seems like the tree should be very suitable for significant pruning).

[Henke]

My guess is that the hand would end up heads up between SB and BB. I think that most hands played (perfectly) with all the cards face up would end up heads up, because there would (almost?) allways be a preflop raise by the best hand.

This is because if all played perfectly, you could easily say that the biggest dog (if not in the blinds) would throw his hand away.

For example, the 6/images/graemlins/spade.gif 5/images/graemlins/spade.gif would have to assume that T/images/graemlins/spade.gif T/images/graemlins/diamond.gif would be thrown away, and after that A/images/graemlins/heart.gif 3/images/graemlins/diamond.gif and so on...

Disclaimer
My analysis is wrong in that it uses twodimes (and hence should require that all players that join the pot go all in), which creates a bias for KJ. A more correct analysis would require analysis of the flop vs. all possibilities of opponents instead.

Or am I far off?

[t_perkin]

[ QUOTE ]

You may be thinking what is the point? if you can't calculate the answer then what is the point in knowing *how* to calcualte the answer? well I am afraid I am interested in things like this...


[/ QUOTE ]

and you replied

[ QUOTE ]

I still don't see why this is particularly useful...If someone points out to me why it is, then maybe we can do something to reduce the complexity (it seems like the tree should be very suitable for significant pruning).


[/ QUOTE ]

so yes this is not an immediately applicable application to your friday night game. But then i did state this at the start of the post, and this forum is for poker *theory*.

But if you like I will now try and justify why *I* am interetested in this (and things like it).

First up I am an academic, and academics were put on this planet to think about things that are not immediately applicable to everyday life. You may not like it, but it pays the rent (unfortunately my poker play does not..) and I love it so that is me.

Secondly I think that by understanding the extremities of the game and the components of the game it makes me a better player (although still not a very good one).

so there we go.

Now as you state, this does look like a prime candidate for a pruning algorithm to reduce the search space. It may be that this becomes a very computationally feasible problem. Suddenly this would be the basis of a perfect player in real life poker. The other major component would be modelling opponents for two things - modelling the weaknesses in their game (to make you *optimise* your winnings) and model the way they play to help ascertain the cards that they hold.
SparBot - the best AI player in the world does something similar. Except that rather than searching the whole search space it uses monte carlo simulations (i think? or does it select simulations with a heuristic?) . It also plays a (nearly/maybe) world class heads up game and it *doesn´t* do *any* player modelling. They are attempting to make a perfect player, before they make it optimal. and just playing perfectly seems to be enough to make it almost world class. Just goes to show how valuable the mathematics of poker is.

Tim

[karlson]

Tim,

[ QUOTE ]
Suddenly this would be the basis of a perfect player in real life poker

[/ QUOTE ]

But it wouldn't. Because the game is so much different with the cards face down, that this research could easily be absolutely inapplicable.

I am an academic too. We still try to work on problems that will somehow in the future be applicable. Approximations and simplifications are great, assuming that we can somehow relate it to the real thing later.

I find the research on Poki and Sparbot very interesting too.

Now, actually, you may be right. It may be that the right way to go about modeling real poker is actually to look through the entire, or almost entire, space, like we we talking about for the faceup model. In that case, the same pruning algorithm could apply (however notice we've increased the space size by several orders of magnitude), and we will have achived something. Or maybe once we figure out that we can't look at the whole space for face up play, we'll reject this idea for real poker. If this is the point of view you want to approach it from, great! However, saying that the solution, as is, is a guide to perfect play, is misleading.

Now, it seems that you are most interested in opponent modeling. If you want to work on opponent modeling in a sparbot-like system, this seems like both a feasible and applicable task. I thought about doing something similar myself, but have not found the time. I still have some ideas about it.

Victor.

[t_perkin]

[ QUOTE ]

I am an academic too. We still try to work on problems that will somehow in the future be applicable.


[/ QUOTE ]

But what about all those economists!? /images/graemlins/grin.gif - no really I am only joking.

[ QUOTE ]

In that case, the same pruning algorithm could apply (however notice we've increased the space size by several orders of magnitude),


[/ QUOTE ]

yes absolutely - the search space increases considerably , but it is still the same algorithm.

[ QUOTE ]

However, saying that the solution, as is, is a guide to perfect play, is misleading.


[/ QUOTE ]

Yes I am not suggesting for a minute that it is the "magic key" to a automated player (bot). But I think nobody would deny that *if* (big if) this could be accomplished then many of the key features of a solution to this could be applied to face down (normal) p#ker
Sorry if I was misleading though.

[ QUOTE ]

Now, it seems that you are most interested in opponent modeling.


[/ QUOTE ]

yep! what I would *really* like to do is get hold of a pokerroom's database and do some serious data mining, would be great to try and train some neural nets or just generate some real life versions of the twodimes stats or a million other things. So many possibilities!

Would love to know your thoughts on player modelling, drop me a private message if you can be bothered.

Tim

[ACPlayer]

If all the cards are face up and every one played mathematically correctly (there is no psychology or mixed strategy here) the game would be a break even game (less the rake ofcourse).

[daryn]

why don't you just read my post? if every player played perfectly, the hand would play out as i described.

[Henke]

I think you put the two TT hands out of the picture too early. Yes, the T/images/graemlins/spade.gif T/images/graemlins/heart.gif wouldn't have odds to call, since it can only hope for a split pot, but the T/images/graemlins/club.gif T/images/graemlins/diamond.gif has already put money in from SB. If we include that hand in the twodimes analysis, we will see that 6/images/graemlins/spade.gif 5/images/graemlins/spade.gif would be a dog in this hand and wouldn't have odds to call. Yes, T/images/graemlins/club.gif T/images/graemlins/diamond.gif is the "worst" dog of all three, but if 6/images/graemlins/spade.gif 5:spade would call, and if we expect K/images/graemlins/diamond.gif J/images/graemlins/diamond.gif to raise, it will just need to win more than 1.5/6=25%.

If we where to analyze favorable flops instead, I think the TT would be even more likely to call, since the KJ wouldn't get two "free cards" to pair one of his cards.