Morton's theorem is a
poker
Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game w ...
principle articulated by Andy Morton in a
Usenet
Usenet () is a worldwide distributed discussion system available on computers. It was developed from the general-purpose Unix-to-Unix Copy (UUCP) dial-up network architecture. Tom Truscott and Jim Ellis conceived the idea in 1979, and it was ...
poker
newsgroup
A Usenet newsgroup is a repository usually within the Usenet system, for messages posted from users in different locations using the Internet. They are discussion groups and are not devoted to publishing news. Newsgroups are technically distinct ...
. It states that in multi-way
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 contrasts with the
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.
The fundamental theorem is stated ...
, which states that a player wants their opponents to make decisions which minimize their own expectation. The two theorems differ in 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
[Some numbers have been changed to allow for ]complete information
In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...
. on the
Usenet
Usenet () is a worldwide distributed discussion system available on computers. It was developed from the general-purpose Unix-to-Unix Copy (UUCP) dial-up network architecture. Tom Truscott and Jim Ellis conceived the idea in 1979, and it was ...
newsgroup rec.gambling.poker.
Suppose in
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
hold'em
Texas hold 'em (also known as Texas holdem, hold 'em, and holdem) is one of the most popular variants of the card game of poker. Two cards, known as hole cards, are dealt face down to each player, and then five community cards are dealt fac ...
a player named Arnold holds
A♦K♣ and the flop is K♠
9♥3♥, giving him top pair with best
kicker
Kicker or The Kicker may refer to:
Sports
* Placekicker, a position in American and Canadian football
* ''Kicker'' (sports magazine), in Germany
* Kicker, the German colloquial term for an association football player
* Kicker, the word used i ...
. When the betting on the
flop
In computing, floating point operations per second (FLOPS, flops or flop/s) is a measure of computer performance, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate meas ...
is complete, Arnold has two opponents remaining, named Brenda and Charles. Arnold is certain that Brenda has the
nut
Nut often refers to:
* Nut (fruit), fruit composed of a hard shell and a seed, or a collective noun for dry and edible fruits or seeds
* Nut (hardware), fastener used with a bolt
Nut or Nuts may also refer to:
Arts, entertainment, and media Co ...
flush draw (for example
A♥J♥, giving her 9
outs
In baseball, an out occurs when the umpire rules a batter or baserunner out. When a batter or runner is out, they lose their ability to score a run and must return to the dugout until their next turn at bat. When three outs are recorded in a ha ...
), 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
In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call. Pot odds are compared to the odds of winning a hand with a future card in order to estimate the call's expected value. The purpose of this is to st ...
to do so. 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
In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...
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:
: