Morton's theorem
Encyclopedia
Morton's theorem is a poker
principle articulated by Andy Morton in a Usenet
poker newsgroup
. It states that in multi-way pot
s, 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 their 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 multi-way situations is a subject of controversy. Morton expressed the belief that his theorem is generically applicable in multi-way pots, so that the fundamental theorem rarely applies except for heads-up situations.
newsgroup rec.gambling.poker.
Suppose in limit hold'em
a player named Arnold holds A♦K♣ and the flop is K♠9♥3♥, giving him top pair with best kicker
. When the betting on the flop is complete, Arnold has two opponents remaining, named Brenda and Charles. Arnold is certain that Brenda has the nut
flush draw
(for example A♥T♥, giving her 9 outs
), and he believes that Charles holds second pair with a random kicker (for example Q♣9♣, 4 outs — not the Q♥). The rest of the deck results in a win for Arnold. The turn card is an apparent blank (for example 6♦) and the pot
size at this point is P, expressed in big bets.
When Arnold bets the turn, Brenda, holding the flush draw, is sure to call and is almost certainly getting the correct pot odds
to do so (note that, due to large reverse implied pot odds, this would not be true in a no limit game
). Once Brenda calls, Charles must decide whether to call or fold. To figure out which action he should choose, we 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 current size of the pot. (Here, as in arguments involving the fundamental theorem, we assume that each player has complete information
of their opponents' cards.)
Charles 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 and solving for P lets us determine the pot size at which he is indifferent to calling or folding:
When the pot is larger than this, Charles should continue; otherwise, it's in his best interest to fold.
To figure out which action on Charles' part Arnold would prefer, we calculate Arnold's expectation the same way:
Arnold's expectation depends in each case on the size of the pot (in other words, the pot odds Charles is getting when considering his call). Setting these two equal lets us calculate the pot size P where Arnold is indifferent whether Charles calls or folds:
When the pot is smaller than this, Arnold profits when Charles is chasing, but when the pot is larger than this, Arnold's expectation is higher when Charles folds instead of chasing.
Hence, there is a range of pot sizes where both:
(a) it's correct for Charles to fold, and
(b) Arnold makes more money when Charles (correctly) folds, than when he (incorrectly) chases.
This can be seen graphically below.
|
C SHOULD FOLD | C SHOULD CALL
|
v
|
WANTS C TO CALL | WANTS C 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 Arnold wants Charles (C) to fold correctly, because he loses expectation when Charles calls incorrectly.
' money those times when Brenda makes her flush draw. Compared to the case where Arnold is heads up with Charles, Arnold still stands the risk of losing the whole pot, but he is no longer getting 100% of the compensation from Charles' loose calls.
It is the existence of this middle region of pot sizes, where a player wants at least some of their opponents to fold correctly, that explains the standard poker strategy of thinning the field as much as possible whenever a player thinks they hold the best hand. Even opponents with incorrect draws cost a player money when they call their bets, because part of these calls end up in the stacks of other opponents drawing against them.
Because Arnold is losing expectation from Charles' call, it follows that the aggregate of all other opponents (i.e., Brenda and Charles) must be gaining from Charles' call. In other words, if Brenda and Charles were to meet in the parking lot after the game and split their profits, they would have been colluding against Arnold. 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 type of hand that will benefit from implicit collusion.
Poker
Poker is a family of card games that share betting rules and usually hand rankings. Poker games differ in how the cards are dealt, how hands may be formed, whether the high or low hand wins the pot in a showdown , limits on bet sizes, and how many rounds of betting are allowed.In most modern poker...
principle articulated by Andy Morton in a Usenet
Usenet
Usenet is a worldwide distributed Internet discussion system. It developed from the general purpose UUCP architecture of the same name.Duke University graduate students Tom Truscott and Jim Ellis conceived the idea in 1979 and it was established in 1980...
poker newsgroup
Newsgroup
A usenet newsgroup is a repository usually within the Usenet system, for messages posted from many users in different locations. The term may be confusing to some, because it is usually a discussion group. Newsgroups are technically distinct from, but functionally similar to, discussion forums on...
. It states that in multi-way pot
Pot (poker)
The pot in poker refers to the sum of money that players wager during a single hand or game, according to the betting rules of the variant being played...
s, a player's expectation
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
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
Draw (poker)
A poker player is drawing if they have a hand that is incomplete and needs further cards to become valuable. The hand itself is called a draw or drawing hand. For example, in seven-card stud, if four of a player's first five cards are all spades, but the hand is otherwise weak, they are drawing to...
. 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
Fundamental theorem of poker
The fundamental theorem of poker is a principle first articulated by David Sklansky that he believes expresses the essential nature of poker as a game of decision-making in the face of incomplete information....
, which states that a player wants their 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 multi-way situations is a subject of controversy. Morton expressed the belief that his theorem is generically applicable in multi-way pots, so that the fundamental theorem rarely applies except for heads-up situations.
An example
The following example is credited to Morton, who first posted a version of it on the UsenetUsenet
Usenet is a worldwide distributed Internet discussion system. It developed from the general purpose UUCP architecture of the same name.Duke University graduate students Tom Truscott and Jim Ellis conceived the idea in 1979 and it was established in 1980...
newsgroup rec.gambling.poker.
Suppose in limit hold'em
Texas hold 'em
Texas Hold 'em is a variation of the standard card game of poker. The game consists of two cards being dealt face down to each player and then five community cards being placed face-up by the dealer—a series of three then two additional single cards , with...
a player named Arnold holds A♦K♣ and the flop is K♠9♥3♥, giving him top pair with best kicker
Kicker (poker)
A kicker, also called a side card, is a card in a poker hand that does not itself take part in determining the rank of the hand, but that may be used to break ties between hands of the same rank. For example, the hand Q-Q-10-5-2 is ranked as a pair of queens. The 10, 5, and 2 are kickers...
. When the betting on the flop is complete, Arnold has two opponents remaining, named Brenda and Charles. Arnold is certain that Brenda has the nut
Nut hand
In poker, the nut hand, is the strongest possible hand in a given situation. The second-nut hand or third-nut hand may refer to the second and third best possible hands...
flush draw
Draw (poker)
A poker player is drawing if they have a hand that is incomplete and needs further cards to become valuable. The hand itself is called a draw or drawing hand. For example, in seven-card stud, if four of a player's first five cards are all spades, but the hand is otherwise weak, they are drawing to...
(for example A♥T♥, giving her 9 outs
Out (poker)
In a poker game with more than one betting round, an out is any unseen card that, if drawn, will improve a player's hand to one that is likely to win. Knowing the number of outs a player has is an important part of poker strategy. For example in draw poker, a hand with four diamonds has nine outs...
), and he believes that Charles holds second pair with a random kicker (for example Q♣9♣, 4 outs — not the Q♥). The rest of the deck results in a win for Arnold. The turn card is an apparent blank (for example 6♦) and the pot
Pot (poker)
The pot in poker refers to the sum of money that players wager during a single hand or game, according to the betting rules of the variant being played...
size at this point is P, expressed in big bets.
When Arnold bets the turn, Brenda, holding the flush draw, is sure to call and is almost certainly getting the correct pot odds
Pot odds
In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call. Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value....
to do so (note that, due to large reverse implied pot odds, this would not be true in a no limit game
Betting (poker)
In the game of poker, the play largely centers on the act of betting, and as such, a protocol has been developed to speed up play, lessen confusion, and increase security while playing...
). Once Brenda calls, Charles must decide whether to call or fold. To figure out which action he should choose, we 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 current size of the pot. (Here, as in arguments involving the fundamental theorem, we assume that each player has complete information
Complete information
Complete information is a term used in economics and game theory to describe an economic situation or game in which knowledge about other market participants or players is available to all participants. Every player knows the payoffs and strategies available to other players.Complete information...
of their opponents' cards.)
Charles 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 and solving for P lets us determine the pot size at which he is indifferent to calling or folding:
When the pot is larger than this, Charles should continue; otherwise, it's in his best interest to fold.
To figure out which action on Charles' part Arnold would prefer, we calculate Arnold's expectation the same way:
Arnold's expectation depends in each case on the size of the pot (in other words, the pot odds Charles is getting when considering his call). Setting these two equal lets us calculate the pot size P where Arnold is indifferent whether Charles calls or folds:
When the pot is smaller than this, Arnold profits when Charles is chasing, but when the pot is larger than this, Arnold's expectation is higher when Charles folds instead of chasing.
Hence, there is a range of pot sizes where both:
(a) it's correct for Charles to fold, and
(b) Arnold makes more money when Charles (correctly) folds, than when he (incorrectly) chases.
This can be seen graphically below.
|
C SHOULD FOLD | C SHOULD CALL
|
v
|
WANTS C TO CALL | WANTS C 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 Arnold wants Charles (C) to fold correctly, because he loses expectation when Charles calls incorrectly.
Analysis
In essence, in the above example, when Charles calls in the "paradoxical region", he is paying too high a price for his weak draw, but Arnold is no longer the sole benefactor of that high price — Brenda is now taking CharlesIt is the existence of this middle region of pot sizes, where a player wants at least some of their opponents to fold correctly, that explains the standard poker strategy of thinning the field as much as possible whenever a player thinks they hold the best hand. Even opponents with incorrect draws cost a player money when they call their bets, because part of these calls end up in the stacks of other opponents drawing against them.
Because Arnold is losing expectation from Charles
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 type of hand that will benefit from implicit collusion.