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Tactic in poker and other card games

In the card game of poker, a bluff is a bet or raise made with a hand which is not thought to be the best hand. To bluff is to make such a bet. The objective of a bluff is to induce a fold by at least one opponent who holds a better hand. The size and frequency of a bluff determines its profitability to the bluffer. By extension, the phrase "calling somebody's bluff" is often used outside the context of poker to describe situations where one person demands that another proves a claim, or proves that they are not being deceptive.

A pure bluff, or stone cold bluff, is a bet or raise with an inferior hand that has little or no chance of improving. A player making a pure bluff believes they can win the pot only if all opponents fold. The pot odds for a bluff are the ratio of the size of the bluff to the pot. A pure bluff has a positive expectation (will be profitable in the long run) when the probability of being called by an opponent is lower than the pot odds for the bluff. For example, suppose that after all the cards are out, a player holding a busted drawing hand decides that the only way to win the pot is to make a pure bluff. If the player bets the size of the pot on a pure bluff, the bluff will have a positive expectation if the probability of being called is less than 50%. Note, however, that the opponent may also consider the pot odds when deciding whether to call. In this example, the opponent will be facing 2 to 1 pot odds for the call. The opponent will have a positive expectation for calling the bluff if the opponent believes the probability the player is bluffing is at least 33%.

In games with multiple betting rounds, to bluff on one round with an inferior or drawing hand that might improve in a later round is called a semi bluff. A player making a semi bluff can win the pot two different ways: by all opponents folding immediately or by catching a card to improve the player's hand. In some cases a player may be on a draw but with odds strong enough that they are favored to win the hand. In this case their bet is not classified as a semi bluff even though their bet may force opponents to fold hands with better current strength. For example, a player in a stud poker game with four spade suited cards showing (but none among their downcards) on the penultimate round might raise, hoping that their opponents believe the player already has a flush. If their bluff fails and they are called, the player still might be dealt a spade on the final card and win the showdown (or they might be dealt another non spade and try to bluff again, in which case it is a pure bluff on the final round rather than a semi bluff).

Bluffing may be more effective in some circumstances than others. Bluffs have a higher expectation when the probability of being called decreases. Several game circumstances may decrease the probability of being called (and increase the profitability of the bluff):
Fewer opponents who must fold to the bluff. The bluff provides less favorable pot odds to opponents for a call. A scare card comes that increases the number of superior hands that the player may be perceived to have. The player's betting pattern in the hand has been consistent with the superior hand they are representing with the bluff. The opponent's betting pattern suggests the opponent may have a marginal hand that is vulnerable to a greater number of potential superior hands. The opponent's betting pattern suggests the opponent may have a drawing hand and the bluff provides unfavorable pot odds to the opponent for chasing the draw. Opponents are not irrationally committed to the pot (see sunk cost fallacy). Opponents are sufficiently skilled and paying sufficient attention. The opponent's current state of mind should be taken into consideration when bluffing. Under certain circumstances external pressures or events can significantly impact an opponent's decision making skills.

If a player bluffs too infrequently, observant opponents will recognize that the player is betting for value and will call with very strong hands or with drawing hands only when they are receiving favorable pot odds. If a player bluffs too frequently, observant opponents snap off their bluffs by calling or re raising. Occasional bluffing disguises not just the hands a player is bluffing with, but also their legitimate hands that opponents may think they may be bluffing with. David Sklansky, in his book The Theory of Poker, states "Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting." Optimal bluffing also requires that the bluffs must be performed in such a manner that opponents cannot tell when a player is bluffing or not. To prevent bluffs from occurring in a predictable pattern, game theory suggests the use of a randomizing agent to determine whether to bluff. For example, a player might use the colors of their hidden cards, the second hand on their watch, or some other unpredictable mechanism to determine whether to bluff.

Evan Hurwitz and Tshilidzi Marwala developed a software agent that bluffed while playing a poker like game. They used intelligent agents to design agent outlooks. The agent was able to learn to predict its opponents' reactions based on its own cards and the actions of others. By using reinforcement neural networks, the agents were able to learn to bluff without prompting.

In economics, bluffing has been explained as rational equilibrium behavior in games with information asymmetries. For instance, consider the hold up problem, a central ingredient of the theory of incomplete contracts. There are two players. Today player A can make an investment; tomorrow player B offers how to divide the returns of the investment. If player A rejects the offer, they can realize only a fraction x<1 of these returns on their own. Suppose player A has private information about x. Goldlücke and Schmitz (2014) have shown that player A might make a large investment even if player A is weak (i. e., when they know that x is small). The reason is that a large investment may lead player B to believe that player A is strong (i. e., x is large), so that player B will make a generous offer. Hence, bluffing can be a profitable strategy for player A.