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.
Pure bluff.
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%.
Semi-bluff.
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 circumstances.
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):
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.
Optimal bluffing frequency.
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.
Example (Texas Hold'em).
Here is an example for the game of Texas Hold'em, from "The Theory of Poker":
when I bet my $100, creating a $300 pot, my opponent was getting 3-to-1 odds
from the pot. Therefore my optimum strategy was ... [to make] the odds against
my bluffing 3-to-1.
Since the dealer will always bet with (nut hands) in this situation, they should bluff with (their) "Weakest hands/bluffing range" 1/3 of the time in order to make the odds 3-to-1 against a bluff.
Ex:
On the last betting round (river), Worm has been betting a "semi-bluff" drawing hand with: A♠ K♠ on the board:
10♠ 9♣ 2♠ 4♣
against Mike's A♣ 10♦ hand.
The river comes out:
2♣
The pot is currently 30 dollars, and Worm is contemplating a 30-dollar bluff on the river. If Worm does bluff in this situation, they are giving Mike 2-to-1 pot odds to call with their two pair (10's and 2's).
In these hypothetical circumstances, Worm will have the nuts 50% of the time, and be on a busted draw 50% of the time. Worm will bet the nuts 100% of the time, and bet with a bluffing hand (using mixed optimal strategies):
formula_1 
Where "s" is equal to the percentage of the pot that Worm is bluff betting with and "x" is equal to the percentage of busted draws Worm should be bluffing with to bluff optimally.
Pot = 30 dollars.
Bluff bet = 30 dollars.
"s" = 30(pot) / 30(bluff bet) = 1.
Worm should be bluffing with their busted draws:
formula_2 Where "s" = 1
"Assuming four trials", Worm has the nuts two times, and has a busted draw two times. (EV = expected value)
Under the circumstances of this example: Worm will bet their nut hand two times, for every one time they bluff against Mike's hand (assuming Mike's hand would lose to the nuts and beat a bluff). This means that (if Mike called all three bets) Mike would win one time, and lose two times, and would break even against 2-to-1 pot odds. This also means that Worm's odds against bluffing is also 2-to-1 (since they will value bet twice, and bluff once).
Say in this example, Worm decides to use the second hand of their watch to determine when to bluff (50% of the time). If the second hand of the watch is between 1 and 30 seconds, Worm will check their hand down (not bluff). If the second hand of the watch is between 31 and 60 seconds, Worm will bluff their hand. Worm looks down at their watch, and the second hand is at 45 seconds, so Worm decides to bluff. Mike folds his two pair saying, "the way you've been betting your hand, I don't think my two pair on the board will hold up against your hand." Worm takes the pot by using optimal bluffing frequencies.
This example is meant to illustrate how optimal bluffing frequencies work. Because it was an example, we assumed that Worm had the nuts 50% of the time, and a busted draw 50% of the time. In real game situations, this is not usually the case.
The purpose of optimal bluffing frequencies is to make the opponent (mathematically) indifferent between calling and folding. Optimal bluffing frequencies are based upon game theory and the Nash equilibrium, and "assist" the player using these strategies to become unexploitable. By bluffing in optimal frequencies, you will typically end up breaking even on your bluffs (in other words, optimal bluffing frequencies "are not" meant to generate positive expected value from the bluffs alone). Rather, optimal bluffing frequencies allow you to gain "more" value from your value bets, because your opponent is indifferent between calling or folding when you bet (regardless of whether it's a value bet or a bluff bet).
Bluffing in other games.
Although bluffing is most often considered a poker term, similar tactics are useful in other games as well. In these situations, a player makes a play that should not be profitable unless an opponent misjudges it as being made from a position capable of justifying it. Since a successful bluff requires deceiving one's opponent, it occurs only in games in which the players conceal information from each other. In games like chess and backgammon, both players can see the same board and so should simply make the best legal move available. Examples include:
Artificial intelligence.
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.
Economic theory.
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.
References.
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