Morton's theorem

Morton's theorem is a poker principle articulated by Andy Morton. It states that in multiway pots, a player's expectation may be maximized by an opponent making a correct decision.

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 a player wants his opponents to make decisions which minimize 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 expressed the belief that his theorem is generically applicable in multiway pots, so that the fundamental theorem rarely applies except for headsup 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 limit holdem a player holds  A♦ K♣ and the flop is K♠ 9♥3♥ , giving the player top pair with best kicker. When the betting on the flop is complete, the player has two opponents remaining, one of whom he knows has the nut flush draw (for example,  A♥T♥ , giving him 9 outs) and one of whom the player believes holds second pair with random kicker (for example Q♣9♣, 5 outs), leaving the player with all the remaining cards in the deck as his outs. The turn card is an apparent blank (for example  6♦ ) and the pot size at that point is P, expressed in big bets.

When the player bets the turn, opponent A, holding the flush draw, is sure to call and is almost certainly getting the correct pot odds to call the player's bet (note that, due to large reverse implied pot odds, this would not be true in a no limit game). Once opponent A calls, opponent B must decide whether to call or fold. To figure out which action opponent 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.)


 * $$E(\mbox{ opponent B }|\mbox{ folding }) = 0$$


 * $$E(\mbox{ opponent B }|\mbox{ calling }) = (4/42) \cdot (P+2) - (38/42) \cdot (1)$$

Opponent 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:


 * $$E(\mbox{ opponent B }|\mbox{ folding }) = E(\mbox{ player B }|\mbox{ calling })$$
 * $$\Rightarrow P = 7.5 \mbox{ big bets }$$

When the pot is larger than this, opponent B should continue; otherwise, it's in B's best interest to fold.

To figure out which action on opponent B's part the player would prefer, calculate the player's expectation the same way


 * $$E(\mbox{ the player }|\mbox{ B folds }) = (33/42) \cdot (P+2)$$


 * $$E(\mbox{ the player }|\mbox{ B calls }) = (29/42) \cdot (P+3)$$

The player's 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 the player is indifferent whether B calls or folds:


 * $$E(\mbox{ the player }|\mbox{ B calls }) = E(\mbox{ the player }|\mbox{ B folds })$$
 * $$\Rightarrow P = 5.25 \mbox{ big bets }$$

When the pot is smaller than this, the player profits when opponent B is chasing, but when the pot is larger than this, the player's 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 the player makes more money when he does so than when he incorrectly chases. This can be seen graphically below

|                B SHOULD FOLD | B SHOULD CALL |                              v                      | WANTS B TO CALL | WANTS 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 the player wants his opponent to fold correctly, because the player loses expectation when his opponent calls incorrectly.

Analysis
In essence, in the above example, when opponent B calls in the "paradoxical region", he is paying too high a price for his weak draw, but the player is no longer the sole benefactor of that high price &mdash; opponent A is now taking B 's money those times that A makes his flush draw. Compared to the case where the player is heads up with opponent B, the player still stands the risk of losing the whole pot, but is no longer getting 100% of the compensation from B 's loose calls.

It is the existence of this middle region of pot sizes, where a player wants at least some of his opponents to fold correctly, that explains the standard poker strategy of thinning the field as much as possible when a player thinks he holds the best hand. Even opponents with incorrect draws cost a player money when they call his bets, because part of their calls end up in the stacks of other opponents drawing against you.

Because the player is losing expectation from B 's call, it follows that the aggregate of all other opponents (i.e., A and B) must be gaining from B 's call. In other words, if A and B were to meet in the parking lot after the game and split their profits, they would have been colluding against the player. This is sometimes referred to as implicit collusion. It should be contrasted with what is sometimes called schooling. Schooling occurs when many opponents correctly call against a player with the best hand, whereas implicit collusion occurs when an opponent incorrectly calls against a player with the best hand.

One conclusion of Morton's theorem is that, in a loose hold'em game, the value of suited hands goes up because they are precisely the types of hands which will benefit from implicit collusion.