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