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Let’s make it difficult ;-) and let’s start speaking of:
Fundamental Theorem of Poker and Morton’s theorem
Recently on Poker Engineering I wrote about the benefits of playing a non-optimal strategy in poker. I also briefly illustrated the Fundamental Theorem of Poker, introduced by David Slansky, the father of modern poker:
“Anytime you are playing an opponent who makes a mistake by playing his hand incorrectly based on what you have, you have gained. Anytime he plays his hand correctly based on what you have, you have lost.”
Here I’d like to show the limits of the theorem and the support given by what is nowadays known as the Morton’s Theorem with some maths.
Against Fundamental Theorem of Poker, Morton’s Theorem states that in multi-way pots, a player’s expectation may be maximized by an opponent making a correct decision.
But actually David himself intended to apply his theory to head-to-head situations, which involve only two players. So when one theorem falls, another comes in support.
The most common application of Morton’s theorem occurs when one player holds the best hand, but there are two or more opponents on draws. In this case, the player with the best hand might benefit from the absolutely “correct” decision of her opponent to fold to a bet.
Morton proposed an example very similar to the following one to prove his thesis.
Consider in a limit texas hold’em game the following situation:
Flop –> K 9 3
Player A –> A KC (top pair and best kicker)
Opponent B –> AT (9 outs for the flush draw)
Opponent C –> Q 9 (4 outs — not the Q which gives the flush to the opponent B)
Turn –> 6
The pot size at that point is P, expressed in big bets.
When the player A bets the turn, opponent B, holding the flush draw, will call having the correct pot odds to call the player’s bet.
Once opponent B calls, opponent C must decide whether to call or fold. To understand what is the right path for her, let’s calculate her expectation in the two cases. Mathematical expectation is the amount a bet will on average win or lose. In our case the expectation depends on 2 factors: the number of cards among the remaining 42 that will give her the best hand and the size of the pot when she is deciding.
Expectation of C when she decides to fold = E( opponent C | folding ) = - (1/3) * P that means that C, folding, will lose what she has put into the pot so far.
Expectation of C when she decides to call = E( opponent C | calling ) = (4/42) * (2/3*P+2) - (38/42) * (1/3*P+1) that means that C, calling, will win 2/3 of P + 2 big bets with probability 4/42 and will lose what she has put into the pot so far + 1 new big bet with probability 38/42
Setting these two expectations equal to each other and solving for P we may easily discover that for a pot size (P) = 7,5 big bets it is indifferent for the opponent C calling or folding: that means that when the pot is larger than this, opponent C should call, otherwise fold.
Now let’s try to understand what is the best move of opponent C from the point of view of the player A.
Let’s calculate the player A’s expectations:
Expectation of A when C decides to fold = E( player A | C folds) = (33/42) * (2/3*P+1) - (9/42) * (1/3*P+1) that means that the player A will win 2/3 of the pot + 1 big bet with probability 33/42 and will lose 1/3 of the pot + 1 big bet with probability 9/42
Expectation of A when C decides to call = E( player A | C calls) = (29/42) * (2/3*P+2) - (13/42) * (1/3*P+1) that means that the player A will win 2/3 of the pot + 2 big bets with probability 29/42 and will lose 1/3 of the pot + 1 big bet with probability 13/42
Once again setting these two equal we find that for a pot size (P) equal to 5,25 big bets, the player A is indifferent to any decision of the opponent C: that means that when the pot is smaller than this, the player profits when opponent C calls instead of “correctly” folding, but when the pot is larger than this, the player A would benefit from B’s folding.
It results clear that there is a range of pot sizes included between 5,25 and 7,5 in our example where it is “correct” for C to fold and at the same time the player A would benefit from the “correct” play of C. And hence there is a range, a blindside, where the Fundamental Theorem of Poker is no more applicable.
References
Morton’s theorem From Wikipedia, the free encyclopedia
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