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Jojo Poker Maison Du Bluffs
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A game of Texas hold 'em in progress. 'Hold 'em' is a popular form of poker.

In this 1904 cartoon by E. A. Bushnell, the Russian Empire (represented by a bear) and the Empire of Japan (represented by a fox) play poker, with their respective arsenals as stakes. Both wonder if the other is bluffing. The Russo-Japanese War began 17 days later.

Poker Joe lived in Missoula, Montana before his involvement with the Nez Perce War. Under the guidance of Chief Joseph, the Nez Perce avoided a run-in with soldiers and local militia in Missoula, and ended up in Yellowstone National Park-unchartered territory for the group.
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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 cases where one person 'demand[s] that someone prove a claim' or prove that he or she 'is not being deceptive.'^[1]

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Pure bluff[edit]

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 he 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 busteddrawing 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[edit]

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 he is favored to win the hand. In this case his bet is not classified as a semi-bluff even though his 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 his opponents believe he already has a flush. If his bluff fails and he is called, he still might be dealt a spade on the final card and win the showdown (or he might be dealt another non-spade and try his bluff again, in which case it is a pure bluff on the final round rather than a semi-bluff).

Bluffing circumstances[edit]

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.

Optimal bluffing frequency[edit]

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 his bluffs by calling or re-raising. Occasional bluffing disguises not just the hands a player is bluffing with, but also his legitimate hands that opponents may think he 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 his hidden cards, the second hand on his watch, or some other unpredictable mechanism to determine whether to bluff.

Example (Texas Hold'em)

Here is an example from The Theory of Poker:

when I bet my $100, creating a $300 pot, my opponent was getting 3-to-1 oddsfrom the pot. Therefore my optimum strategy was ... [to make] the odds againstmy bluffing 3-to-1.

Since the dealer will always bet with (nut hands) in this situation, he should bluff with (his) 'Weakest hands/bluffing range' 1/3 of the time in order to make the odds 3-to-1 against a bluff.^[2]

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, he is giving Mike 2-to-1 pot odds to call with his 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):

${displaystyle x=s/(1+s)}$ ^[3]

Where s is equal to the percentage of the pot that Worm is bluff betting with and x is equal to

Jojo Poker Maison Du Bluff