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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?

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    $\begingroup$ I think that this must be related to game theory and, in particular, mixed strategies (i.e. bluffing with some probability). Not necessarily to probability. Also, for interest, you could look at the other theorem of poker; Zeebo's theorem which states that a weak player, no matter what, is incapable of folding a hand better or equal to a full house. :) $\endgroup$ – stuart stevenson Jun 12 '18 at 11:10
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    $\begingroup$ It certainly looks interesting, I'll definitely check it out! $\endgroup$ – Mildwood Jun 12 '18 at 12:20
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    $\begingroup$ It's kind of tongue in cheek but I find it to be fairly accurate, especially in cash games. $\endgroup$ – stuart stevenson Jun 12 '18 at 13:13
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I like Sklansky but I am having trouble calling 'The Fundamental Theorem of Poker' a theorem in a mathematical sense.

If each player could see the all the cards heck yes they could all play optimal poker. It would be a very boring game in which all the chips would go in pre flop and every one would fold unless you had identical hands (JJ versus JJ). The are some other near 1:1 like 89s versus 44 that you would play for the blind/ante overlay.

Sklansky is making the point that hand reading (what your opponent has) and deceit (representing a different hand than you have) is critical. He is saying poker is more than math.

Game theory optimization is the closets thing to a mathematical model. It can be solved with just one card to come or pre flop head up all in. GTO is not optimal. It has an EV (Expected Value) of 0 if both are playing GTO. You go GTO when you don't really know what to do and just want to put your opponent in a tough spot. Or GTO says I should be bluffing here x%.

The problem you have with poker is with betting soooooo many degrees of freedom.

Bots rely on AI. Bots are not allowed in any money game to my knowledge. Bots will take some betting lines you think what? A bot will identify what is called a hole and just exploit it until you plug the hole. A mortal cannot compete if you let the (good) bot play a 1000 hands against you or the bot has a hand history on you.

Poker is not about winning hands it is about winning money. EV is not the same as return. I could put 3000 at risk and win 2000 or maybe put 200 at risk and win 200.

In a tournament when out is go home it changes model versus a cash game where you can re-buy.

May be optimized if your opponent plays properly is a different dynamic. A speculative hand like 89s wants multiple players. If you hit (straight or flush) you are likely to win. You want someone to hit a set so you can get a lot of chips from them. Mathematically they should put it all in.

It is not an easy game. Bots don't really know how to play poker they just know how to run a lot of lines.

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I think I can explain, in a purely hypothetical way, why Morton's Theorem needn't contradict the Fundamental Theorem. Suppose I have a choice between two strategies. If I choose the first strategy, I win 30% of the time, you win 30% of the time, and she wins 40% of the time. If I choose the second strategy, I win 40% of the time, you win 40% of the time, and she wins 20% of the time. So I correctly choose the second strategy, and that maximizes your chance of winning.

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    $\begingroup$ Change of winning is not the goal. You could lose 2 for 100 hands and then make 2K in 1 hand. $\endgroup$ – paparazzo Jun 14 '18 at 18:31

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