The Fundamental Flaw of Poker
In The Theory of Poker
Every time you play a hand differently from the way you would have played it if you could see all your opponents cards, they gain; and every time you play your hand the same way you would have played it if you could see all thier cards, they lose. Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose.The primary problem with this formulation of poker strategy is that it requires you to step outside the game. Specifially, it measures the appropriate response in terms of parameters which are not available during the game. Once you know what hand your opponent holds, it’s not poker anymore. It might be an interesting game, but by removing the element of the unknown, you’ve removed an essential aspect of poker.
This leads to a vivid contradiction between the FTOP and the practice of playing poker. When you are actually faced with a decision in poker, you must take all the information that you have at hand, and make a decision. There are many ways of formulating the correct action, but none of them allow you access to unknown information (without cheating). In the absense of specific player knowledge, the ideal tool to use is game theory.
Game theory is a branch of mathematical analysis developed to study the decision making in conflict situations. One unique feature of game theory is that when solving a problem, the agents in the conflict are assumed to be fully rational. In other words, your poker opponents play as well as you do. There are no fish in the game theory model, only sharks.
Specifically, once a game theory solution has been found, if both players play according to the strategy, they will break even in the long run, assuming there is no rake. Any deviation from that strategy by either player will result in that players expectation dropping to a negative value.
Using this sort of framework, there are certain classes of poker problems which become solvable. For example, heads up no limit hold’em can be solved in an optimal game theoric manner if you assume the players are restricted to either folding, or going all-in as their first option. Aproximate solutions have been published by several people, including the Jones/Kittock system, and Tysen Streib. And a complete solution has been published by Alex Selby.
As an example of how to use the system, let’s consider the specific scenario of what each player should do when they each have 20 chips. The blinds are 1/2, and the small blind looks down to find 98s. In accordance with the optimal solution, he jams. The big blind looks down to find T8o. Facing the all-in jam by the small blind he consults the optimal solution, and finds that he should fold, and does so correctly, despite the fact that he is better than a 2:1 favorite to win the pot.
- Is this a violation of the Fundamental Theory of Poker? Yes.
- Is this the optimal play for both players? Yes.
The seeming contradiction comes from the fact that the Fundamental Theory of Poker is a theory which is not about poker. Instead it is a theory about the results of poker. In other words, you cannot use the Fundamental Theorem of Poker to solve any actual poker problems. Instead you can only use it to determine that you were, or were not, unlucky to have been involved in a confrontation where your opponent just happened to be holding a specific hand.
May 16th, 2006 at 10:12 am
I wonder sometimes if, at the highest level, your margins are made by adjustment in such a way as to prohibit perfect subgames, if skilled players are so astute as to have your history nailed down so well. Maybe online where the database comes into play.
May 17th, 2006 at 11:22 am
I thought about what I wrote a little more. In the end, the only way to prohibit perfect subgames is to stop playing. Either you run out of hands to play or run out of money. Maybe the next best idea is to discover the perfect strategy and then never play to it. Sounds counterintuitive, but you have to create an inefficiency, in order to feign exploitability, then compensate. Just can’t create the same inefficiency over and over again.
May 21st, 2006 at 5:25 pm
The more detailed and relevant a post is, the fewer the comments.
May 21st, 2006 at 8:56 pm
Good thing I don’t write this stuff for a living.
:)
May 22nd, 2006 at 9:38 am
Oh no You didn’t.
May 23rd, 2006 at 8:05 am
Lol.
May 24th, 2006 at 11:14 am
Just came by your blog. It’s nice. I think you are right claiming that the FTOP is only concerned with results but I don’t think this paradox is good enough to abandon the FTOP. The FTOP essentially shows that poker is a game of incomplete information and that it is your primary objective to overcome this as much as possible. It is (almost) impossible to make the ‘perfect play’ everytime and so while the FTOP is a theorem concerning a ‘perfect poker game’, more abstract than we can imagine (maybe you can look at it as some sort of a Platonic Idea), game theory is the more ‘down-to-earth’ approach dealing with one case at a time. I haven’t read about specific game theory solutions to poker games, do you have any interesting links?
May 27th, 2006 at 4:26 pm
Careful analyzation of The FTOP reveals that it is irrelevant during gameplay, because poker is about making +EV decisions based upon interpretation of incomplete information.
June 3rd, 2006 at 7:39 pm
wrong
how is pushing allin with 98o the ‘optimal solution’ ? it clearly isnt, even if you are heads up vs a guy and assign him the random hand Q7o
June 4th, 2006 at 11:00 am
DP, you’re pretty much exactly right. The FTOP addresses a game which is distinct from poker as it is played. It’s almost as bad as saying, “Every bet that you don’t put in the pot when you’re going to get sucked out on, is a bet that you save.” It isn’t about expected value, it’s about being results oriented.
mike, I’m not sure how to respond. I’ll just point out that Q7o is actually not a random hand, but is in fact — Q7o.
June 12th, 2006 at 7:33 am
where might one find the “alex selby system”?
June 12th, 2006 at 7:52 am
daniel, you can find his work on his website:
http://www.archduke.org/simplex/index.html
June 14th, 2006 at 12:09 pm
came to you from a link on DP’s site…very thought provoking even for someone new to the game who has just recently finished TOP for the first time. you’ve hit on the great unarticulated unease I’ve had while working through it - the sense that, while admirable, it was not truly applicable. I like the analogy of a platonic ideal, but I agree with you that FTOP is actually describing something altogether different than what Harrah’s or Stars is selling.
June 22nd, 2006 at 2:20 am
In Barry Greenstein’s book, Ace on the River, he talks about this. He explains that there is a perfect play, which is like what Sklansky is talking about where you can see the cards face up, and a correct play, which is what you do based on the incomplete information that you have.
June 22nd, 2006 at 7:57 pm
No mention of the Bayesian/Probabilistic TOP Andrew? Or is that coming in a follow-up?
June 30th, 2006 at 3:41 pm
I don’t think this is that good of an article. You were unlucky to be put in a situation where making the right decision cost you money, but it did in fact cost you money. If the hand came up 10,000 more times, where your opponent pushed into you and you picked up T8o, folding on average will cost you less than calling on average. A better article would have explained that in the absense of perfect information decisions are based on formulating hand ranges and assigning probabilities to them. The results still follow from the FTOP once you assign a hand range. After you evaluate the EV of each hand in the range and then multiply by the probability, then you come up with the “correct” decision based on incomplete information. But the whole evaluation process was based on the FTOP. I know you understand this as all good poker players do, but it is a mistake to think that this is a weakness in the FTOP. While the misapplication of the FTOP can lead to results oriented thinking, it shouldn’t and the author never intended it to.
After that criticism I have the gall to ask for a favor. Pokerstove is a truly awesome achievement in poker technology. Ever since I was exposed to it, it changed the way I approached poker problems. It would be nice though if one had the option to change the preflop percentages to the Sklansky-Chubukov rankings. Small blind/big blind or blind/button preflop all-in situations arise frequently enough that the rankings using hands hot/cold in a multiway pot may have some limitations. For instance, K4o is often a better push than T9s from the SB in an unraised pot (certainly a less exploitably one). But pokertracker has T9s in the top 20% of hands and K4o just out of the top 50%. This is a good estimation for the playability of these hands, but not good in the heads up situations that so often arise in tournaments. It would be extremely helpful if one could push a button and switch to the SC rankings. Thank you for the work you’ve done.
July 1st, 2006 at 12:37 am
Wagon: I generally agree with what you’re saying about hand ranges/the functional probabilistic model we use to determine our best course of action… but I disagree that the FTOP is applied to the EV calculations you mention. The FTOP very explicitly deals with perfect information, and what you have said about how we are forced to arrive at the ‘correct’ decision based on incomplete information basically reinforces Monsieur Prock’s claims: it’s quite easy for the correct decision to be a bad one, FTOP-wise, due to the lack of information.
Playing perfectly, FTOP-wise, is unattainable.. but the closer we come, the better we do. That’s more of a testament to how important the premises are (what information we have) as opposed to how we take those premises and arrive at a conclusion(the fuzzy probability EV calcs we all know and love.) I don’t think this means that the FTOP is ‘flawed,’ necessarily, but that it isn’t very useful at all when dealing with how we should play.
Of course, this is essentially all semantic nonsense. I just can’t help but argue. Regardless, I think that’s almost all the FTOP really illustrates: as our information gets closer to perfect, and our opponent’s information gets fuzzier and fuzzier, we do better.
Andy: Isn’t it funny how people make a huge distinction between the ‘unknowns’ of the face-down cards on top of the deck, and the ‘unknowns’ of the face-down cards under our opponent’s card marker? I’m having some weird thought about how we assume the state of the cards could be anything.. but once somebody observes that state, we feel as if, despite our imperfect information, that no other outcome is possible. Something about trees falling in the woods and cats in boxes. You know.
Mike: Keep on with yo’ bad self.
July 2nd, 2006 at 3:45 am
“Instead it is a theory about the results of poker. In other words, you cannot use the Fundamental Theorem of Poker to solve any actual poker problems.”
I submit that we use the FTOP to solve all poker problems. In this example we know sb’s strategy. How do we know that it is wrong to call a push with T8o? Well, we know our equity versus every hand in sb’s range. How do we know that? Well, we pretend that we know sb’s hand and then we run it hot and cold. We know that in the absence of perfect information we will have to make FTOP mistakes, but they are mistakes. In forming our strategy we balance the mistakes to maximize equity, but in order to do that we use the FTOP to determine what a mistake is and what its cost is.
Imagine we didn’t have pokerstove. We just have a table of every hand in sb’s range against T8o, and we are trying to determine if T8o should be in our calling range. We would say what is our equity vs. hand X? What is the probability that sb has hand X? Multiply, sum, subtract the price of the call, and we have our exact EV (which should be slightly below zero). So on balance we make less costly FTOP mistakes by folding than calling.
See how we do do the problem backwards. We pretend we know the hand (the result). That part is FTOP. Only then do we weight it and extend beyond FTOP. FTOP is very, very basic. It is how we make money at poker in a real sense. We know by FTOP that we lose money when we fold T8o when our opponent has 98s. We know by FTOP that we lose money when we call with T8o and our opponent has AK.
We base our decisions on expected values, but we arrive at those expected values by “outside of the game” results. We know we will make some FTOP mistakes that will cost us money, but the mistakes do actually cost us money, regardless of whether we played correctly or not. And it shows as much in our expectation calculations.
July 3rd, 2006 at 6:05 pm
It seems tome the FTOP is missing something. The theoretically correct action at a particular point in a poker hand is not just a function of your opponents cards, but a function of the decisions he will make in response to your action (i.e. it is a function of your opponents brain as well). For example the theoretically correct decision on whether or not to semi-bluff someone is dependent on what hands he will fold. This then creates four categories of poker skill-set:
1.) Knowledge of correct action if opponents hand and responses are completely known–knowing the odds.
2.) Ability to read oponents hand.
3.) Ability to predict oponents responses/decisions.
4.) Ability to influence opponent into making bad decisions. This could be anything from putting him on tilt to disguising your hand.
allan
July 4th, 2006 at 3:00 am
Chipp: I’m working on a post about approaches to solving poker problems, and I will definatly reference the BTOP in that.
Wagon: Thanks for your comments, they are useful, and help clarify a lot of what is actually important. It’s not really about picking nits with other people’s work — although that sure can be fun — it’s about coming to good solutions to useful problems. As far as future versions of PokerStove go, I’ll be adding more range editing tools which should satisfy your concerns. I hope to have a new beefed up version out by the end of the year.
Jake: You bring up a great point about knowing what is important to think about and understand. People have a real hard time dealing with unknown information, and that’s why poker can be so profitable. For example, one rule of thumb is “Your opponents always put you on AK.” They do it because they can’t think in terms of distributions. That one rule has made me tons of money in terms of bets won and bets saved.
July 5th, 2006 at 12:33 am
Wagon: “I don’t think this is that good of an article.”
You misunderstand. The FTOP can only be used to analyze hands afterwards; therefore, it is useless for making decisions because you must assign your opponent a probable hand range by interpretating the available information, regardless of what your opponent could have during the given hand — furthermore, even after the hand is over, the conclusion reached using the FTOP is often incorrect because it will often state that you made a bad play by folding 9 high on the river when your opponent had 8 high, even though it’s obvious that you’ve made a good play, because you will lose if you practice the act of calling with 8 high on the river.
Delving deeper:
The person with 8 high has lost (even though they won that hand), if they did not act on information which caused them to decide you would fold, because if that is the case, they would make the same bet regardless of your cards.
July 5th, 2006 at 8:45 pm
lactating mother picture
Do you mind to post some more information about this ?
July 6th, 2006 at 9:13 pm
Andrew:
Another thing that would be awesomely cool is if you could put in a range and say a minimum equity for that range and Pokerstove would find all the hands above that equity for you.
So for instance, you put a range in and say I need 45% equity, and poker stove returns a range filled with every hand exactly 45% equity and above.
Can’t wait for the new one
-Wagon
July 10th, 2006 at 5:04 pm
I suppose how useful FTOP is as a ’strategy’ depends on how it is used. To the extent that FTOP means “I was behind when my chips went in the pot, I have violated FTOP,” FTOP is worthless. It is better to think of FTOP as a generator of other principles that assist a novice player (but are, of course, old hat to someone who has played the game). Some of these thoughts include (a) don’t chase unless you actually have the odds (or, at least, the implied odds) to do so; (b) make sure that your play is sufficiently deceptive that other players will make mistakes against you; (c) think about your opponent’s holdings beyond “he is/isn’t bluffing” so that you don’t make mistakes.
In other words, FTOP is a way of life rather than a specific poker strategy. The longer you play, the more it seems like “buy low, sell high” - but to the novice investor who thinks things like “Enron in huge so it must be a good investment,” forced reflection through a simple aphorism isn’t a bad thing.
In conclusion, FTOP, FTOP, FTOP.
July 10th, 2006 at 5:07 pm
You know Ugarte, I think you are right, the FTOP is about as useful as “buy low, sell high.”
July 24th, 2006 at 10:50 pm
Andrew: “The FTOP is about as useful as ‘buy low, sell high.’”
Especially when you consider that the approach of many traders is actually “buy high, sell higher.” That is, they are momentum traders, betting that the current trend will continue, at least for a little while longer.
August 12th, 2006 at 4:00 am
Hey Andrew:
Any new posts coming soon? You always got some good stuff…let’s see something!