Morton’s theorem

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 make more money in the long run when an opponent folds to a bet, even if that opponent is folding correctly and would be making a personal mistake to call the bet. This type of situation is sometimes referred to as implicit collusion.

Morton’s theorem should be contrasted with the fundamental theorem of poker, which states that you want your opponents to make decisions which minimise their own expectation. The discrepancy between the two “theorems” occurs because of the presence of more than one opponent. Whereas the fundamental theorem always applies heads-up (one opponent), it does not always apply in multiway pots. The scope of Morton’s theorem in multiway situations is a subject of controversy. For example, Morton himself expresses the belief that the fundamental theorem rarely applies to multiway situations.

An example

The following example is credited to Morton, who first posted on rec.gambling.poker. (Some numbers have been changed to allow for complete information, see below.)

Suppose in holdem you hold A♦K♣ and the flop is K♠9♥3♥, giving you top pair with best kicker. When the betting on the flop is complete, you have two opponents remaining, one of whom you know has the nut flush draw (say A♥T♥, giving him 9 outs) and one of whom you believe holds second pair with random kicker (say Q♣9♣, 4 outs), leaving you with all the remaining cards in the deck as your outs. The turn card is an apparent blank (say 6♦) and say the pot size at that point is P, expressed in big bets.

When you bet the turn player A, holding the flush draw, is sure to call and is almost certainly getting the correct pot odds to call your bet. Once player A calls, player B must decide whether to call or fold. To figure out which action player B should choose, calculate his expectation in each case. This depends on the number of cards among the remaining 42 that will give him the best hand, and the size of the pot when he is deciding. (Here, as in arguments involving the fundamental theorem, we assume that each player has complete information of their opponents’ cards.)

E( player B | folding ) = 0

Player B doesn’t win or lose anything by folding. When calling, he wins the pot 4/42 of the time, and loses one big bet the remainder of the time. Setting these two expectations equal to each other and solving for P lets us determine the pot-size at which he is indifferent to calling or folding:

E( player B | folding ) = E( player B | calling )

When the pot is larger than this, player B should chase you; otherwise, it’s in B’s best interest to fold.

To figure out which action on player B’s part you would prefer, calculate your expectation the same way

Your expectation depends in each case on the size of the pot (in other words, the pot odds B is getting when considering his call.) Setting these two equal lets us calculate the pot-size P where you are indifferent whether B calls or folds:

E( you | B calls ) = E( you | B folds )

When the pot is smaller than this, you profit when player B is chasing, but when the pot is larger than this, your expectation is higher when B folds instead of chasing.

In this case, there is a range of pot-sizes where it’s correct for B to fold, and you make more money when he does so than when he incorrectly chases. You can see this graphically below

                              |
                B SHOULD FOLD | B SHOULD CALL
                              |
                              v
                     |
   YOU WANT B TO CALL| YOU WANT B TO FOLD
                     |
                     v
+---+---+---+---+---+---+---+---+---> pot-size P in big bets
0   1   2   3   4   5   6   7   8
                     XXXXXXXXXX
                         ^
                "PARADOXICAL REGION"

The range of pot sizes marked with the X’s is where you want your opponent to fold correctly, because you lose expectation when he calls incorrectly.

Link

Original discussion of Morton’s theorem

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.