Proof of fundamental theorem of poker?
In heads up (no limit holdem?), 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 their cards, they lose.
Is it really a mathematical theorem, or is the word "theorem" used in an informal sense? I couldn't find any proof or even a formal statement of it. Is there any of those?
4 Replies
If you update the theorem to include your opponents cards AND their strategy, then it holds.
Every time you play a hand differently from the way you would have played it if you knew their exact cards AND their complete strategy for playing those cards, they gain (or you fail to maximize your gain)
Of course, even Nash equilibrium strategy makes a ton of "mistakes" in the hand-vs-hand sense.
It seems wrong to have something as big and important-sounding as "The fundamental theorem of poker" be incomplete. So I took it upon myself to update the wiki page. I hope David won't mind.
This whole conversation strikes me as bizarre. It is a conversation that misses the forest for the trees.
To get to the OP's question, is it mathematically proven? Yes and no.
On the flop if your opponent goes all in and explicitly shows you his cards, then yes, it is mathematically proven. You can literally decide based on mathematics whether you should call or not.
If it involves you going all in and your opponent reacting, then it depends. It depends upon your opponents understanding of the math involved and it depends upon their other tendencies reacting to the math. If they are willing to throw the math out the windows and go off of gut, then no, it isn't a given.
All of this misses the whole point of the lesson.
The point is, if you knew your opponents hand and could therefore play perfectly against them (including when you could get them to fold weak, but better hands), any deviation from that is a negative.
I think too many people are trying to tie down a very specific answer and missing the whole point of the Fundamental Theroem of Poker.
Sklansky's answer is quite bizarre. Here he has encapsulated a very important lesson about poker and haughtingly got caught up in a very technical answer that ruined the whole point.
The forest for the trees throws so many people off.
This thread is asking for a formal proof, not a discussion of the pedagogical vibe behind the theorem. My earlier post was just an attempt to make the statement rigorous by spelling out the missing assumptions.
Secondly, the real lesson isn’t literally “play as if you can see cards.” It’s that good strategy tends to maximize your opponent's "mistakes" (in the hand-vs-hand sense). For example, going all-in with a polarized range will maximize the "mistakes" of their bluffcatchers, and this is ultimately where the polarized player's EV comes from.
