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Poker players results very much depend on their knowledge and skills.
So lets make it difficult ;-) and lets start speaking of:
Fundamental Theorem of Poker and Mortons theorem
Recently on Poker Engineering I wrote about the River Rock Casino Online 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 Id like to show the limits of the theorem and the support given by what is nowadays known as the Mortons Theorem with some maths.
Against Fundamental Theorem of Poker, Mortons Theorem states that in multi-way pots, a players 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 Mortons theorem occurs when one Monzo Casino 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 holdem game the following situation:
Flop > K 9 3
Player A > A K (top pair and best kicker)
Opponent B > AT (9 outs for the flush draw)
Opponent C > Q9 (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 blinds.
When the player A bets the turn, opponent B, holding the flush draw, will call having the correct pot odds to call the players bet.
Once opponent B calls, opponent C must decide whether to call or fold. To understand what is the right path for her, lets 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 blinds with probability 4/42 and will lose what she has put into the pot so far + 1 new big blind 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 blinds 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 lets try to understand what is the best move of opponent C from the point of view of the player A.
Lets calculate the player As 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 blind with probability 33/42 and will lose 1/3 of the pot + 1 big blind 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 blinds with probability 29/42 and will lose 1/3 of the pot + 1 big blind with probability 13/42
Once again setting these two equal we find that for a pot size (P) equal to 5,25 big blinds, 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 Bs 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
Mortons theorem From Wikipedia, the free encyclopedia
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