Official subreddit for all things poker. Theory Optimal bluffing frequency. Archived Theory Optimal bluffing. There is no set optimum frequency for bluffing that covers every situation. It would be awesome if I could tell you that you should bluff once every 10 hands or something to that effect, but there are no hard or fast rules in poker. November 7, 2019 Beginner Poker Strategy, Complete Guide, Live Poker, Online Poker, Poker Articles Comments off 10538 Views 3 Bluffing is one of the most important skills a poker player can have. Anybody can wait around for a big hand and hope to get paid off – that takes almost zero skill.

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.^[1]

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 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 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 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[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 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)[edit]

Here is an example for the game of Texas Hold'em, from The Theory of Poker:

Poker Optimal Bluffing Frequency Definition

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, 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.^[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, 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):

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

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

${\displaystyle x=1/(1+s)=50\mathrm{\%}}$ 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).^[3]

Poker Optimal Bluffing Frequency Analyzer

Bluffing in other games[edit]

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:

• Contract Bridge: Psychic bids and falsecards are attempts to mislead the opponents about the distribution of the cards. A risk (common to all bluffing in partnership games) is that a bluff may also confuse the bluffer's partner. Psychic bids serve to make it harder for the opponents to find a good contract or to accurately place the key missing cards with a defender. Falsecarding (a tactic available in most trick taking card games) is playing a card that would naturally be played from a different hand distribution in hopes that an opponent will wrongly assume that the falsecarder made a natural play from a different hand and misplay a later trick on that assumption.
• Stratego: Much of the strategy in Stratego revolves around identifying the ranks of the opposing pieces. Therefore depriving your opponent of this information is valuable. In particular, the 'Shoreline Bluff' involves placing the flag in an unnecessarily-vulnerable location in the hope that the opponent will not look for it there. It is also common to bluff an attack that one would never actually make by initiating pursuit of a piece known to be strong, with an as-yet unidentified but weaker piece. Until the true rank of the pursuing piece is revealed, the player with the stronger piece might retreat if their opponent does not pursue them with a weaker piece. That might buy time for the bluffer to bring in a faraway piece that can actually defend against the bluffed piece.
• Spades: In late game situations, it is useful to bid a nil even if it cannot succeed.^[4] If the third seat bidder sees that making a natural bid would allow the fourth seat bidder to make an uncontestable bid for game, they may bid nil even if it has no chance of success. The last bidder then must choose whether to make their natural bid (and lose the game if the nil succeeds) or to respect the nil by making a riskier bid that allows their side to win even if the doomed nil is successful. If the player chooses wrong and both teams miss their bids, the game continues.
• Scrabble: Scrabble players will sometimes deliberately play a phony word in the hope the opponent does not challenge it. Bluffing in Scrabble is a bit different from the other examples. Scrabble players conceal their tiles but have little opportunity to make significant deductions about their opponent's tiles (except in the endgame) and even less opportunity to spread disinformation about them. Bluffing by playing a phony is instead based on assuming players have imperfect knowledge of the acceptable word list.^{[citation needed]}

Artificial intelligence[edit]

Evan Hurwitz and Tshilidzi Marwala developed a software agent that bluffed while playing a poker-like game.^[5][6] 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[edit]

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.^[7]

See also[edit]

References[edit]

1. ^'call bluff'. The Free Dictionary by Farlex. Retrieved October 22, 2020.
2. ^Game Theory and Poker
3. ^ ^abThe Mathematics of Poker, Bill Chen and Jerrod Ankenman
4. ^[1]Archived December 28, 2009, at the Wayback Machine
5. ^Marwala, Tshilidzi; Hurwitz, Evan (May 7, 2007). 'Learning to bluff'. arXiv:0705.0693 [cs.AI].
6. ^'Software learns when it pays to deceive'. New Scientist. May 30, 2007.
7. ^Goldlücke, Susanne; Schmitz, Patrick W. (2014). 'Investments as signals of outside options'. Journal of Economic Theory. 150: 683–708. doi:10.1016/j.jet.2013.12.001. ISSN0022-0531.

General references[edit]

• David Sklansky (1987). The Theory of Poker. Two Plus Two Publications. ISBN1-880685-00-0.
• David Sklansky (2001). Tournament Poker for Advanced Players. Two Plus Two Publications. ISBN1-880685-28-0.
• David Sklansky and Mason Malmuth (1988). Hold 'em Poker for Advanced Players. Two Plus Two Publications. ISBN1-880685-22-1.
• Dan Harrington and Bill Robertie (2004). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume I: Strategic Play. Two Plus Two Publications. ISBN1-880685-33-7.
• Dan Harrington and Bill Robertie (2005). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume II: The Endgame. Two Plus Two Publications. ISBN1-880685-35-3.
• Bill Chen, Jerrod Ankenman. The Mathematics of Poker.

Poker Game Theory And Math Are Actually Very Easy To Use, Here We Look At
Using Game Theory to Improve Your Online Tournament Results.

Game theory is a branch of applied math which looks at competitive situations where 2 or more people have conflicting interests. This has been applied to global politics, economics and of course to poker. David Sklansky covered poker game theory in his seminal work ‘The Theory of Poker’ and Chris ‘Jesus’ Ferguson made this famous by using poker game theory rather than experience to achieve his phenomenal success.

The key to application of poker game theory is that there are ways to vary your poker strategy depending on your opponents and the individual situation. After all, any poker strategy which does exactly the same thing each time is very easy to exploit.

So we have 2 dynamics at work – on one side a mixed vs static strategy, on the other an Unexploitable vs exploitable strategy. Unexploitable strategies are defensive in nature, they will prevent you losing chips and will win small amounts. In order to win you need to play in such a way as to exploit your opponents weakness – this of course involves deviating from Unexploitable strategy yourself and is where Poker Game Theory really starts to get interesting.

Below are 4 common ways in which poker game theory can be put to use at the table.

Poker Game Theory Application #1 – Optimal Bluffing Frequency

As made famous by David Sklansky the idea is that you should bluff with a busted hand on the river at the same frequency as the odds you are offering your opponent from the pot. Thus if the pot offers 4/1 at the river bluffing 25% of the time will mean your opponent loses the same amount whether he calls or folds, all else being equal. A great way to do this is to calculate the odds you will offer on the end of the hand when you bet and then randomize your bluffing by choosing a number of extra cards which do not make your hand. For example if you are offering 4/1 and have 8 real 'outs' to make your hand then choose 2 bust cards to bluff with.

Of course – this is the Unexploitable strategy. If you feel that your opponent either calls or folds more frequently than is correct you can adjust your bluffing frequency accordingly... adaping to expliot your opponents tendencies.

Poker Game Theory Application #2 – The Independent Chip Model (ICM)

ICM simply represents your current chip stack in terms of 'dollar equity' in the prize pool of the tournament. It is most commonly used in Sit and Go play as the numbers are easier to calculate (due to fewer paying places)but also has applications at MTT final tables. When you or your opponents have short stacks in comparison to the blinds (less than 10BB is usually quoted) then it is possible to calculate a mathematically ‘correct’ range of hands to push all-in with or call an opponents all-in bet. Playing using ICM is an Unexploitable strategy, as long as you can put opponents on correct hand ranges for calling or pushing then you will profit from ICM based decisions over time. Even if you choose not to use ICM yourself, it is worth learning about this - since many of your opponents will be basing their decisions on this math, once you understand what is driving their decision making it will be easier to put them on a range of hands.

As with any Unexploitable poker strategy there is a counter. Adjusting play based on stack sizes (especially the presence of very small stacks) or making mathematically ‘bad’ calls against regular opponents to prevent them correctly using this strategy in future are just 2 examples of poker game theory in action. For more on ICM see my SNG Strategy Articles section.

Planet Mark's Rec : If you are thinking about Poker Game Theory, then you are already more advanced than the majority of players. What you need to do next is stop playing against other experienced players on the big sites, and check out sites where the majority of players are amateurs, recreational poker fans or just plain fish! Your returns will be much bigger at 888 Poker - I strongly recommend you check out the tables at 888 Poker now and see just how soft their games are for yourself!

Poker Game Theory Application #3 – Stack Sizes and The Gap Concept

The gap concept states that you need a stronger hand to call an opponent's bet than you need to raise yourself. In a no-limit Holdem tournament the ‘gap’ is usually small when stacks are shallow compared to the blinds (since implied odds are far higher) and shrinks considerably as the game approaches the bubble.

Poker Game Theory suggests that opponents understanding of the gap concept can be exploited. This happens when you strongly re-raise a player who has raised a pot ahead of you when stacks are getting shallow. If your opponent understands the gap concept he may be open-raising with a wide range of hands. The second ‘gap’ – that between the hands he will open with and those he will call a re-raise with – may be very wide. If your opponent raises with 20% of hands but will only call with 5% he will fold to your re-raise 75% of the time.

Using mathematics to exploit this concept will gain chips in many tournament situations. This is particularly true when stack sizes are balanced in such a way as to commit your opponent to the pot. The danger of using such a strategy is that yet another opponent picks up a monster hand behind you – so this is best used from late position!

More recently the 'exploitative' strategy of re-raising light is well known by regular tournament players. You might find your re-raised raised once more (4-bet) simply because your opponent knows that you will fold a good percentage of your hands (this could continue to 5-betting etc if stacks are deep enough). Remember, a novice player is unlikely to be bluffing when they make the 4th bet - since they are not considering game theory, just looking at their own cards.

Poker Game Theory Application #4 – S.A.G.E. Poker System / Nash Equilibrium

The SAGE poker system uses Nash Equilibrium math to create an unexploitable system for heads-up play at the end of a poker tournament. ICM does not work heads-up, since you are effectively playing for a single prize (the gap between the 1st and 2nd place payouts). Sage has a points system based on the number of big blinds in your stack, and tells you whether to shove all-in or call your opponents all-in with certain hands.

SAGE is best used when you are playing against an opponent who is more skilled or experienced than you are. While this system is 'unexploitable' it is not flexible enough to adjust to errors that your opponent is making. To take advantage of errors you need to step outside of the nash equilbrium 'game-theory' system, and call more or bet more depending on how you perceive the weaknesses of your opponent. For example, if your opponent folds too often (according to game theory) you can exploit this by betting more frequently.

Remember, applying Game Theory is even more effective against weak opposition – find out which is the best place to find those easy to exploit opponents in our Best Poker Tournament Sites section.

Planet Mark's Rec : If you are thinking about Poker Game Theory, then you are already more advanced than the majority of players. What you need to do next is stop playing against other experienced players on the big sites, and check out sites where the majority of players are amateurs, recreational poker fans or just plain fish! Your returns will be much bigger at 888 Poker - I strongly recommend you check out the tables at 888 Poker now and see just how soft their games are for yourself!

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