Morton's Theorem


The theorem, named for Andy Morton, says that the expected value for a player in a multiway pot increases whenever his opponent makes a correct decision. Sklansky's fundamental theorem of poker says the opposite, namely that the opposition's mistakes increase a players EV.


For example, if a player has the best hand at the time against multiple opponents who hold draws, then it might happen that his EV grows more if an opponent correctly folds a draw than if he incorrectly calls. The opposing draws interfere with one another.

Example (Fixed Limit Texas Hold'em):

Player A Player B Player C Board

The chances for each player to win on the turn are:
 A B  C
69.05% 21.43% 9.52%

Let us assume that the players know the cards of their opponents, and hence always know the correct move.

The pot on the turn is x big bets. Player A bets and B calls. C's decision hinges on how large the pot p is. 

EV(fold) = 0 BB
EV(call) = 9.52% * p + 90.48%*(-1 BB)
EV(fold) = EV(call)

0 BB= 9.52% * p - 0,91 BB
p = 0.91 BB / 9.52%
p = 9.56 BB

Player C can call correctly if the pot is larger than 9.6 BB. The starting value for the pot x must then be x = p - 2 = 7.6 BB, since both wagers after the turn must be deducted.

Now we must examine what player A would like to see player C do. If C folds, then A will win in 79.55% of cases, otherwise he will win 69.05% of the time. The question is which pot is better for A, when C folds or when C calls.

EV(C folds) = 79.55% * (x + 1 BB) + 20.45% * (-1 BB)
EV(C calls) = 69.05% * (x + 2 BB) + 30.95%*(-1 BB)
EV(C folds) = EV(C calls)

79.55% * (x + 1 BB) + 20.45% * (-1 BB) = 69.05% * (x + 2 BB) + 30.95%*(-1 BB)
(79.55% - 69.05%) * x = 0.48 BB
x = 0.48 BB / 10.5%
x = 4.58 BB

It would be better for player A if C folds correctly, so long as the starting value of the pot x is larger than 4.6 BB. C can only call correctly after an x of 7.6 BB. If x were less than 4.6 BB, it would best for A if C stayed in the hand. Between 4.6 and 7.6, though, A's EV will be maximized if C makes a correct decision.

Related Topics:

Fundamental Theorem of Poker, Expected Value