In a recent post 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.”

Today 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. This situation may happen many times during a poker tournament. 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 hold’em game the following situation:

Flop       –> KS9H3H
Player A   –> ADKC (top pair and best kicker)
Opponent B –> AHTH (9 outs for the flush draw)
Opponent C –> QC9C (4 outs — not the QH which gives the flush to the opponent B)

Turn       –> 6D

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