Poker Strategy Pot Odds

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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. Use my free pot odds & implied odds tool today: video was created by for htt.

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:

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]

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.

Pot odds

Pot odds are defined as the ratio between the size of the pot and the bet facing you. For example, if there is $4 in the pot and your opponent bets $1, you are being asked to pay one-fifth of the pot in order to have a chance of winning it.

A call of $1 to win $5 represents pot odds of 5:1.

If you are asked to pay $1 to win $10, you have odds of 10:1. If you need to find $3 to win $9, you have 3:1 and so on.

(Note: The size of the pot refers to the chips that are already in the pot, as well as all the bets made in the current betting round.)

Poker Pot Odds

Once you have determined the pot odds, you need to determine the odds of hitting your draw.

Odds of hitting your draw

In the basics course we introduced the Rule of Two and Four, which offered an easy way of calculating your odds when holding a drawing hand on the flop.

In that lesson, we calculated your odds of winning a hand in a percentage, but it can also be displayed as a ratio between winning and losing. A 20% winning probability can be translated as 4:1 odds – you will lose four in five times.

The precise mathematics behind this is not crucial at this stage. But the chart below shows a list of the most common draws you face in Texas Hold’em and the approximate chance you have of hitting them.

The first column (“Outs”) shows the number of outs you have; the second column (“Odds flop to turn”) shows the chance of hitting the draw on the next card; the next column (“Odds flop to river”) shows the odds of hitting on turn or river, ie, on either of next two cards.

Comparing ratios to determine expected value

After you have found the two ratios, you must compare them against each other – the odds of you winning the hand (based on your outs) compared with the pots odds offered on your call.

If the pot odds are higher than your odds of winning, you should call (or raise, in exceptional circumstances). If your pot odds are lower than your chances of winning, you should fold.

Here are a couple of solid examples:

Example with the nut flush draw:

You have the nut flush draw (nine outs) on the turn and the pot is $6. Your opponent bets $1. There is now $7 in the pot ($6 + $1), and it is $1 to call. The pot odds are therefore 7:1.

Poker Pot Odds Chart

your odds are 4:1 to hit your flush draw. The pot odds are higher. You should therefore call.

You can see why this call is correct by looking at the long-term picture. If you make this call 5 times, the odds says that you will hit your draw once on average. That means you stand to win $7 for every $5 (5 * $1) you invest. That is good business.’

Example of pot odds with a straight draw:

You have a gutshot straight draw (four outs) on the flop and there is $25 in the pot. Your opponent bets $5. There is now $30 in the pot ($25 + $5), and it is $5 to call. Your pot odds are therefore 6:1.

However, according to the table the odds of winning the hand are 11:1. You don’t have the right pot odds to call here and should therefore fold.

Again, a glance at the long-term picture reveals why this is so. In this instance, you would need to play twelve times in order to win $30. But those twelve calls would cost you $60 ($5 * 12) and so this is not profitable.

How to play against an all-in

If an opponent moves all in on the flop, you can make the same calculations as described above, but this time look at the “Odds Flop to River” column. If your opponent is all in, you have the advantage that no further bets are possible.

If you call, you therefore get to see not only the turn, but also the river without having to risk more chips.

Example of odds with a straight draw against an all-in:

You have an open-ended straight draw (eights outs) on the flop. There is $50 in the pot and your opponent moves all-in for $25. You therefore have pot odds of 75 to 25 ($50 plus the $25), and it’s $25 to call.

When simplified, the pot odds are 3:1, and if you call you get to see both the turn and the river. According to the column “Odds Flop to River” in the odds table, the odds of winning the hand are 2:1, and because the pot odds are higher, you should make the call.

Conclusion

Calculating odds and outs can seem difficult and time-consuming, especially if you are a beginner. But this process is critical to make the right decisions. If you continually play draws without getting the right odds, you will lose money in the long run.

There will always be players who don’t care about odds and call too often. These players will occasionally get lucky and win a pot, but mostly they will lose and pay for it.

On the other hand, you might be folding draws in situations where the odds are favorable. If you use the strategies in this article consistently, you can avoid mistakes and gain an edge over your opponents.

Avoiding results oriented thinking

Poker Pot

Even if you have made a correct calculation of your expected value, the fact remains that you will often make a correct call yet still lose the pot. We have factored into the calculation that, for example, you will not hit a flush draw on three out of four occasions.

But you must remember that the key determining factor in these calculations is whether or not you are getting good “value” on your call in the long term. Cash games are essentially endless and you can re-buy if you lose your chips. We are therefore looking at the decision in the abstract and determining whether this would be a profitable play if you made it time and time again.

It is a mistake in cash game poker to base your decisions only on the results of one particular hand – or even one particular session. Sometimes you might make a good call and lose; sometimes you will make a bad call and win. But don’t allow the specific result alter your decision making. You should base it in mathematics.

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