Playing perfect poker with the cards face up

[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).