I've been doing an investigation into the mathematics behind poker, and I have stumbled upon this theorem called 'The Fundamental Theorem of Poker', which is as follows:
"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."
There has been many different articles on this rule, and most state that there is 'a strong mathematical background' and 'a practical application' of the Law of Iterated Expectations/Law of Total Expectation:
E(X) = E(E(X|Y))
However, I am yet to encounter an explanation HOW those two things are related, and I'm not sure why the two are related in the first place. To my understanding one is more or less common sense and the other is about expectations.
Can someone please explain to me the Law of Iterated Expectations implicates or at least is related to the Fundamental Theorem of Poker?
P.S Upon further search, I've also found another theorem called 'Morton's Theorem'. It states that:
In multi-way pots, a player’s expectation may be maximized by an opponent making a correct decision."
It's a direct contrast to the fundamental theorem in the way that it's stating players win when the opponents make a correct decision. I'm not quite sure why the two theorems exist when they seem like they're literal polar opposites of each other. I understand that Morton's theorem is for several players while the Fundamental Theorem is only for two players, but I'm unclear on why more players would suddenly reverse what is described by the Fundamental Theorem. If you can, can you please explain why such is the case?