Editor’s Note: This article was first published in the May, 2013 issue.
The following excerpt is from our next book, Applications of No-Limit Hold ’em; A Guide to
Understanding Theoretical Sound Poker by Matthew Janda. Our estimated date of publication is May 20, 2013.
Although there is no way to solve for the perfect 3-bet, 4-bet, and 5-bet sizings and frequencies from every possible position, we can analyze the effects of the most commonly used sizings and see how they affect each player’s ranges and frequencies. In addition, notice that the methodology we discuss will be much more important than making sure we used the perfect sizings. In reality, small changes in sizing usually only slightly alter each player’s range. Also, keep in mind that online most players have 100 big blind stacks and a common opening size is a pot-sized open for 3.5 big blinds in most positions other than the button. In addition, good players usually 3-bet a 3.5 big blind open to between 10 and 12 big blinds.
Let’s consider how often we must defend versus a 3-bet in order to make our opponent indifferent to bluffing. When a player 3-bet bluffs a 3.5 big blind open in position to 10 big blinds, he’s risking 10 big blinds to win 5 big blinds (the opening bet and the posted blinds). Thus, this needs to work more than 66.7 percent of the time to yield an immediate profit.
0.667 = 10 / (10 + 5)
Likewise, if the 3-bet was to 12 big blinds instead of 10, it would need to work 70.6 percent of the time.
0.706 = 12 / (12 + 5)
Observe that even a 20 percent increase in 3-bet sizing only changes how often the original raiser must defend by 4 percent. Also note the original opener is not the only player who can defend against the 3- bet — there are still players yet to act who may have a strong hand or decide to bluff.
Here’s an example. The cutoff opens to 3.5 big blinds and the button 3-bets to 10 big blinds. Since the 3-bettor needs folds 66.7 percent of the time to make an immediate profit, the remaining players must defend a combined total of at least 33.3 percent. If each of the blinds 4-bets cold 3 percent of the time and otherwise folds, then the original raiser would need to defend at least 29.1 percent.
(1 – X)(1 – 0.03)2 = 0.667
X = 0.291
where
X is the probability that the original raiser defends, and
1 – 0.03 is the probability that each blind folds.
But no-limit hold ’em isn't always this simple. That’s because players usually make smaller 3-bets in position than out of position since they want to encourage their opponent to call so they get to see a flop (in position). If we raise first in and an opponent makes a small 3-bet, he’ll usually be in position and there will be multiple people left to act who can help us defend. If this player is out of position, his 3-bet will usually be larger so we won’t have to defend as much even if there are no other players left to act. In general, our strategy will usually be to defend at least 27 to 31 percent of our opens against 3- bets.1
The most common 4-bet sizings are usually around 22 to 24 big blinds to win a pot of around 15 to 16 big blinds. If the original raiser 4-bets, he effectively only risks 18.5 to 20.5 big blinds since his original 3.5 big blind open is now dead money. Therefore, a 4-bet from the original preflop raiser must succeed around 55 percent of the time to make an immediate profit, whereas a 4-bet cold from one of the blinds must succeed around 60 percent of the time to make an immediate profit.
Lastly, a player can 5-bet bluff which will almost always result in him going all in when stack sizes are 100 big blinds or less. So when someone 5-bet bluff jams, he’ll usually be using a small pocket pair or a weak suited ace since these hands have the most equity when called.
However, how much equity a specific hand has will change significantly based on the opponent’s calling range. For instance, against a calling range of JJ+/AK, a pair of treys and ace-five suited have 32.2 and 30.7 percent equity respectively. But against a calling range of aces and kings, the pair of treys has only 18.4 percent equity while the ace-five suited has 26.7 percent equity.
Another point to consider is the 5-bet bluff will usually come from the original 3-bettor, and since he has already invested around 11 big blinds, he’s only risking on average another 89 big blinds (assuming he starts the hand with 100 big blinds) by going all in. Therefore, since the final pot will be 201.5 big blinds, his expectation is just over 2 big blinds back for every 1 percent of equity. So if, for instance, his hand has 31 percent equity when called, his expectation is 62.5 big blinds when his opponent does not fold,
62.5 = (201.5)(0.31)
which is the same as an overall expectation of -26.5 big blinds.
-26.5 = 62.5 – 89
But when the opponent folds, assuming that the 4-bet (to 24 big blinds) comes from the preflop raiser, the 5-bet bluffer will win on average 36.5 big blinds.
36.5 = 24 + 11 +1.5
where
1.5 is the amount of the blinds,
11 is the amount of the 3-bet raise, and
24 is the amount of the 4-bet raise.
Therefore, in this example, the 5-bet bluff needs to work more than 42 percent of the time to yield a profit.
(X)(36.5) + (1 – X )(- 26.5) = 0
63X = 265
X = 0.42
And in general, 5-bet bluffs need to work somewhere between 40 to 50 percent of the time to be profitable. The precise percentage is dependent on the equity versus the 5-bet calling range, and the bet sizings used for opening, 3-betting, and 4-betting respectively.
Summing up, the frequencies at which 3-bet, 4-bet, and 5-bet bluffs must succeed will usually fall within the following percentages:
- 3-bet bluffs usually need folds around 67 to 70 percent of the time to yield an immediate profit.
- 4-bet bluffs usually need folds around 54 to 60 percent of the time to yield an immediate profit.
- 5-bet bluffs usually need folds around 40 to 50 percent of the time to yield an immediate profit.
So despite each additional bluff getting significantly more expensive (in terms of the number of big blinds placed into the pot), each bluff needs to be successful less frequently than the one before it to be profitable. Again, it should be made clear that the above frequencies are for how often our 3-bets and 4-bets must succeed to yield an immediate profit.
If our opponent sometimes defends by calling rather than re-raising, as he almost certainly will, then our 3-bets and 4-bets do not need to work as often to be profitable. That’s because even our weakest bluffs will sometimes get lucky on the flop. Likewise, we can also state the following:
- Preflop opens need to defend by 4-betting around 25 to 30 percent of the time if they never defend by calling. This number is lower than it would otherwise be because there are usually players left to act which can help defend.
- 3-betting ranges which never flat when facing a 4-bet should 5-bet jam around 40 to 46 percent of the time.
- 4-betting ranges should usually call between 50 to 60 percent of the time when facing a 5-bet jam.
These frequencies are important for determining the ratio of value raises to bluff raises in 3-bet, 4-bet, and 5-bet situations. While they do not tell us exactly what hands belong in each range, they do allow us to check and see if a range is too heavily skewed towards value bets or bluffs.
