Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max – Part 5

1. Introduction

This is Part 5 in the series Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max. In Part 1, Part 2, Part 3 and Part 4 we studied 3/4/5-betting heads-up, with the raiser either in position or out of position. In Part 5 we’ll look at two cases of 3/4/5-betting in multiway pots, namely squeezing (3-betting after the raise has been called) and cold 4-betting (4-betting when the pot has been raised and 3-bet before it’s our turn to act). In this work we’ll use the poker simulation software Pokerazor to estimate the EV for cold 4-betting.

Multiway scenarios are far more complex to model than heads-up scenarios, so the work done for squeezing and cold 4-betting will be less exact than what we have done in the previous articles. But we can use our understanding of heads-up scenarios plus simple modeling to find qualitative guidelines for multiway scenarios.

The structure for Part 5 is:

– Squeezing
– Cold 4-betting

2. Squeezing

The definition of “squeezing” is to 3-bet a raiser after the raise has already been called. The raiser now has to respond to the 3-bet with another player left to act, and we say that he is in a “squeeze” between the 3-bettor and the caller, thus the name.

Below are three examples of squeezing with the raiser in position and out of position:

Example 2.1: Squeezing with the raiser out of position

$100NL

MP ($100) raises pot to $3.5, CO ($100) calls, button ($100) 3-bets pot to $15.50, the blinds fold, and it’s MP’s turn to act.

Example 2.2: Squeezing with the raiser out of position

$100NL

CO ($100) raises pot to $3.5, button ($100) calls, SB folds, BB ($100) 3-bets pot to $14.50, and it’s CO’s turn to act.

Example 2.3: Squeezing with the raiser in position

$100NL

Button ($100) raises pot to $3.5, SB ($100) calls, BB ($100) 3-bets pot to $14, and it’s button’s turn to act.

If the raiser in these examples should choose to defend against the 3-bet by calling, he is setting himself up for difficult postflop scenarios. He will then often have a weak or marginal hand postflop, and he will often have to respond to the 3-bettor’s c-bet without closing the betting (when the preflop coldcaller is left to act). All who have played a bit of NL understand intuitively that this is a difficult situation to play well, and many therefore fold a lot to squeezes when they aren’t strong enough to 4-bet for value.

As we shall see soon, the mathematics of the situation dictates that the raiser and the cold-caller have to defend a lot against the 3-bet to prevent the 3-bettor from having a profitable bluff with any two cards. Since many players can’t (or won’t) defend as actively as they should in an optimal strategy, squeezing is generally a very profitable strategy against weak opposition.

We shall approach the theory behind squeezing using the theory for heads-up 3/4/5-betting as a starting point. We let Alice open-raise pot from some position outside of the blinds, and then she is called by a player between her and Bob. Bob now 3-bets (squeezes) pot with a polarized range made up of value hands and 3-bet bluffs with an optimal value/bluff ratio, Note that this optimal ratio will be slightly different from the corresponding heads-up scenario since the presence of the caller changes the pot size and therefore the pot odds for 3/4/5-betting.

Alice defends against the squeeze by 4-betting/folding out of position and 4-betting/flatting/folding in position. When she 4-bets, she will make her 4-bet a bit less than pot-sized (e.g. to 32 bb in example 2.1 instead of 4-betting pot to 46 bb), and she uses an optimal value/bluff ratio. Bob’s response to a 4-bet is to 5-bet his value hands all-in, and fold his 3-bet bluffs. Alice’s response to an all-in 5-bet is to call with her value hands and fold her 4-bet bluffs.

We’ll now construct a model for a squeeze scenario with the raiser out of position, and then estimate optimal strategy pairs for the raiser and the 3-bettor like we did for heads-up 3/4/5-betting in Part 1 and Part 2

We use the following model:

  • All players start with 100bb stacks
  • Alice open-raises pot (3.5bb) from EP (UTG or MP)
  • A player in CO cold-calls
  • Bob squeezes with an approximately pot-sized 3-bet (14 bb) on the button with an optimal mix of value hands and 3-bet bluffs
  • Alice defends against the squeeze by 4-betting to 32 bb (a bit less than pot) with an optimal mix of value hands and 4-bet bluffs, and otherwise folding
  • We’ll assume that CO always folds to Alice’s 4-bet
  • Bob defends against Alice’s 4-bets by 5-betting his 3-bet value hands all-in, and otherwise folding
  • Alice defends against Bob’s 5-bet by calling all-in with her 4-bet value hands and otherwise folding

This model is similar to the one we used for heads-up 3/4/5-betting with Bob in position. An important difference is that the pot is bigger because of CO’s call when it’s Bob’s turn to act. The optimal strategy pair for Alice and Bob will therefore change relative to the strategy pairs we found for the corresponding heads-up scenario. We’ll assume that CO never continues after a 4-bet from Alice, so that his chips are “dead money” when a 3/4/5-bet war arises between Alice and Bob. We can then estimate the optimal strategy pair using the same method we used heads-up.

We use 14 bb for Bob’s 3-bet size as an average of his bet sizing from various positions. From the examples above we see that Bob risks 15.5 bb when he squeezes with a pot-sized 3-bet on the button, but only 13 bb (beyond the big blind he has already posted) when he squeezes from the big blind. So we use 14 bb as a representative 3-bet size for all positions.

We also assume that Bob uses the heads-up ranges for 3-bet bluffing (“IP 3-bet air list”), 5-bet bluffing (“IP 5-bet air list”) and flatting (“IP flat list”) when he chooses his bluffing and flatting hands:

IP 3-bet air list

A9s-A6s
K9s-K6s
Q9s-Q6s
J9s-J6s
T8s-T7s
97s-96s
87s-86s
76s-75s
65s

100 combos

IP 5-bet air list

A5s-A2s

16 combos

IP flat list

22+
ATs+ AJo+
KTs+ KQo
QTs+
JTs
T9s
98s

Without {KK+}: 162 combos
Without {QQ+}: 156 combos
Without {QQ+,AK}: 140 combos
Without {JJ+,AK}: 134 combos

So Bob’s candidate hands for 3-bet bluffing are the same as when 3-betting heads-up. But since the pot now is bigger, Bob’s optimal distribution of value hands and bluff hands will change relative to the heads-up scenario. Except for this, we’re using a model identical to the heads-up scenario.

We start by asking 3 important questions:

  • How often do Alice and the coldcaller have to defend against the 3-bet squeeze to prevent Bob from profitably 3-bet buffing any two cards?
  • How is the defense against the squeeze shared between Alice and the coldcaller?
  • What is the optimal strategy pair for the heads-up 3/4/5-bet war that occurs between Alice and Bob after Alice 4-bets and the coldcaller folds?

Next we’ll find the answers to these questions:

2.1 Optimal defense frequency against a 3-bet squeeze

When Alice and Bob were heads-up, Bob 3-bet to 12 bb to win a 3.5 + 0.5 + 1 =5 bb pot. He got effective pot odds 5 : 12, and had to win at least 12/(5 + 12) =70% to have an automatic profit with any two cards. Heads-up Alice had the whole responsibility for defending sufficiently often to prevent this. So Alice had to defend 30% of the time in an optimal strategy (and a bit more in position where she sometimes defends by calling and lets Bob freeroll flops with his 3-bet bluffs).

But when Alice’s raise has gotten called by CO, the pot is 3.5 + 3.5 + 0.5 + 1 =8.5 bb when it’s Bob’s turn to act. His 14 bb 3-bet squeeze then risks 14 bb to win 8.5 bb and the effective pot odds becomes 8.5 : 14. Bob needs to win at least 14/(8.5 + 14) =62% to have a profitable 3-bet squeeze with any two cards, and Alice and the coldcaller need to defend at least 100 – 62 =38% to prevent this.

The next question is how this 38% defense job should be shared between Alice and the coldcaller. This question can not be answered exactly, but we can state some qualitative guidelines:

  • The coldcaller has signaled a range with few premium hands when he chooses not to 3-bet Alice
  • Alice must therefore expect that the coldcaller will often fold to the squeeze
  • So most of the job of defending will fall on Alice

To get further, let’s assume that Alice uses her corresponding heads-up defense strategy as a starting point for the squeeze scenario, and then she makes adjustments in the value/bluff ratio to adapt to the new pot size. In other words, she starts with a defense strategy where she defends 30% (only 4-betting and never calling, since she is out of position), and that the cold caller takes care of the rest by defending some percentage x% . The probability of both Alice and the coldcaller folding is then (1-0.30)(1-x), and the probability of at least one of them defending is 1 – (1-0.30)(1-x). This should be 38% in an optimal strategy, and we get:

1 - (1-0.30)(1-x) =0.38
1 - 0.70(1-x) =0.38
0.70(1-x) =0.62
1-x =0.62/0.70
x =1 - 0.62/0.70 =0.11 =11%

So to make the total defense percentage 38%, the coldcaller needs to defend 11% of his range if Alice defends 30% of her range by 4-betting or folding. Furthermore, if the coldcaller defends partly by flatting, he should defend a bit more than 11%, since flatting lets Bob freeroll flops with his 3-bet bluffs instead of having to fold them to a 4-bet. But here we’ll focus on Alice’s strategy, and simply assume that the coldcaller defends enough.

We’ll see later that Alice ends up defending a bit less than 30% after adjusting her strategy to the new pot size, so CO has to defend a bit more than 11%. But we’ll assume that the distribution of the defense responsibility is 30% and 11% before Alice begins adjusting her strategies.

After choosing this starting point for her defense strategy, Alice needs to find the value/bluff ratio for 4-betting that corresponds to the actual pot size. We make a new simplifying assumption and let Alice use the same value range she would have used in the heads-up scenario. Then we only have to adjust the number of 4-bet bluffs to the new optimal ratio, which follows from the new pot size.

We remember that Alice’s ~15% EP opening range is:

22+
A9s+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s
65s

194 combos
15%

And when working with the corresponding heads-up scenario we found that Alice used the value range {QQ+,AK} when defending her EP opening range optimally out of position against Bob’s heads-up 3-bets. So we have simplified our way down to this:

  • Alice uses the corresponding heads-up strategy as a starting point for her defense against the squeeze, and then she adjusts it to match the new pot size
  • Alice uses the same value hands she would have used in a heads-up scenario, so that her only adjustment is to change the number of 4-bet bluffs to get to the new optimal ratio (which follows from the new pot size)
  • Alice assumes the coldcaller will take care of the remaining defense, so that the total defense adds up to 38%

What remains is to estimate how many 4-bet bluffs Alice needs to get to the new optimal value/bluff ratio for her 4-betting range. Heads-up this ratio was 60/40, and next we’ll recalculate this ratio as a function of the new pot size.

2.2 Bob’s value/bluff-ratio for 3-bet squeezing

Bob knows that Alice and the coldcaller will defend a total of 38% against his squeeze 3-bet (a bit more when the coldcaller defends partly by flatting). When Alice re-squeezes with a 4-bet to 32 bb, she risks 28.5bb more (32 bb minus the original raise to 3.5 bb) to win a 3.5 + 3.5 + 14 + 0.5 + 1 =22.5 bb pot.

Alice then gets effective pot odds 22.5 : 28.5, and she needs to succeed 28.5/(22.5 + 28.5) =56%. So Bob needs to defend against a 4-bet by 5-betting 100- 56 =44% of his 3-betting range to prevent Alice from having a profitable 4-bet with any two cards. Therefore, 44% of Bob’s hands need to be value hands. We can round this to the nearest 5% to keep things simple, and we find that the optimal value/bluff ratio for Bob’s 3-bet squeezing range is 45/55 (compared to 40/60 for the heads-up scenario).

2.3 Alice’s value/bluff ratio for 4-betting

When Alice re-squeezes Bob’s 14 bb squeeze by 4-betting to 32 bb, and the coldcaller between them folds, the pot grows to 32 + 3.5 + 14 + 0.5 + 1 =51 bb. When Bob shoves his remaining 86 bb, he’s getting effective pot odds 51: 86.

Bob always has some equity when his 5-bet bluffs get called, and we’ll make the same assumption we made in Part 1. There we showed that Bob’s weakest 5-betting hands (the Axs hands he used as 5-bet bluffs) had about 30% equity when they got called by Alice’s value 4-betting hands. So Bob’s 5-bet bluffs win back about 30% of a 100 + 3.5 + 100 + 0.5 + 1 =205 bb pot, or 0.30 x 205 =61.5bb. So Bob effectively risks 86 – 61.5 =24.5 with his 5-bet bluffs and not 86 bb.

The effective pot odds for Bob’s 5-bet bluffs is then 51 : 24.5, and he needs to win 24.5/(51 + 24.5) =32%. To make Bob’s 5-bet bluffs break-even, Alice needs to defend 100 – 32 =68% against Bob’s 3-bets, which we round to 70%.

It follows that Alice’s 4-betting range needs to contain 70% value hands (compared to 60% in the heads-up scenario). Alice’s optimal value/bluff ratio for 4-betting is then 70/30.

2.4 Adjusting to squeeze scenarios in practice

We have now established that Bob should change his value/bluff ratio for 3-betting from 40/60 to 45/55, which means his 3-betting range should be more weighted towards value hands. Alice’s value/bluff ratio for 4-betting should change from 60/40 to 70/30, so range also becomes more weighted towards value.

Do these changes make sense intuitively? Yes, since both players should be less willing to fold when the cold caller’s dead money has made the pot bigger, giving them a better risk/reward ratio when continuing in the hand. So bluffing becomes less effective, and both players adjust by reducing their bluffing frequency.

We have already done a systematic discussion of Alice’s and Bob’s 3/4/5-bet strategy pairs in previous articles. In Part 2 and Part 3 we estimated specific ranges for both of them when Alice raises out of position and Bob 3-bets her in position. In Part 4 we did the same for the scenario where Alice has position on Bob after he has 3-bet from the blinds.

So instead of going through these scenarios one more time with the new value/bluff ratios, we’ll instead look at an example that illustrates how we can adjust in practice. We’ll then use the previously established heads-up optimal strategy pairs as our starting point.

When we’re in a potential squeeze situation, there are two different ways to approach it:

  • We can used precisely defined ranges based on a value range + “IP 3-bet air list” and “OOP 3-bet air list” together with a randomizer. In other words, we’re trying to squeeze 3-bet optimally (the topic for this article)
  • We can realize that we’re in a squeeze and squeeze with whatever cards we have, if we think the situation is good for it (but we’re rarely squeezing with pure trash hands). We’re now playing exploitatively, probably with an unbalanced range (weighted towards an excess of 3-bet bluffs) in selected spots

For example, let’s say button open-raises and SB flats. Button folds often against 3-bets, and SB is loose-passive with a wide flatting range, and he also folds often to 3-bets. You have K7s in the big blind. K7s is to weak for flatting, and it’s not a member of the 3-bet bluff candidate list (“OOP 3-bet air list”) that we use out of position in the blinds.

So if you’re using a strictly defined optimal strategy based on lists + a randomizer, you fold. You know that in the long run you’ll squeeze an optimal amount (which is pretty aggressive) by sticking to your strategy, and you don’t have to add more bluffing hands to get there (and if you do add more bluffing hands, your strategy will become unbalanced, which isn’t necessarily what you want).

Another approach is to exploit whatever good squeezing opportunities that come your way, without worrying about moving away from an optimally balanced 3-betting range. If you want to play this way (deviating from optimal strategy whenever you see an opportunity to exploit a profitable scenario), you’ll 3-bet K7s and similar hands in the scenario described above. You do this because you expect to make a good profit from picking up the pot preflop against two players who fold too much to 3-bets (and when the loosest player calls, you will have position on him postflop). This is obviously a fine way to play these scenarios.

But if you choose the exploitative approach, be aware that you might have to tighten up your 3-betting if your opponents realize you are 3-betting too loosely and decide to fight back (for example by 4-bet bluffing you more). On the other hand, if you choose an optimal strategy, your opponents’ strategy adjustments will have less impact. If you use an optimal value/bluff ratio for 3-betting, they can’t exploit you with any change they make. So you don’ have to make any changes in your strategy, unless you want to deviate from optimal play in order to exploit your opponents new tendencies.

Below are adjustments (based on optimal heads-up strategy pairs) for Alice and Bob in a squeeze scenario where Alice open-raises 35% on the button, small blind calls, and Bob sits in the big blind. This is a common squeeze spot, and you will profit from training solid default strategies for it (both as the raiser and as the squeezer) so that you both can squeeze and defend against squeezes with strong control preflop.

Example 2.4: Squeezing from the blinds against a button steal-raise

$100NL

Alice ($100) raises pot to $3.5 from the button, small blind ($100) calls, Bob ($100) is in the big blind.

Alice uses her default 35% button range defined in Part 2:

22+

A2s+ A7o+

K2s+ K9o+

Q6s+ Q9o+

J7s+ J9o+

T7s+ T9o+

96s+

86s+

75s+

65s

458 combos

35%

Bob’s strategy
Let’s start with Bob’s 3-betting range against Alice. We have assumed that Alice uses the same value range {QQ+,AK} that she would use heads-up on the button against a 3-bet from the blinds. So Bob’s response is to use the same value range for squeezing that he would have used heads-up. Then he adds 3-bet bluffs until he has a 45/55 value/bluff ratio for his 3-betting range.

Using the optimal heads-up strategy pair from Part 3 as our starting point, we get:

  • Bob 3-bets {TT+,AQ+} =62 combos for value from the blinds against a 35% button open-raise, planning to 5-bet all-in against a 4-bet
  • Bob then needs (55/45) x 62 =76 3-bet bluff combos to get a 45/55 value/bluff ratio. So he 3-bets 76% of “OOP 3-bet air list” as a bluff. We can round this to 75%.

We remember that “OOP 3-bet Air list” is:

OOP 3-bet air list
66-22

A9s-A6s

K9s-K8s

QTs-Q9s

J9s-J8s

97s+

87s

76s

65s

98 combos

Since the list has about 100 combos, we can convert directly between number of combos and percentages to use with a randomizer. So Bob 3-bets {TT+, AQ+} for value, and when he has a hand from “OOP 3-bet air list” he uses the randomizer. He 3-bet bluffs if the randomizer returns a number between 0 and 75, and otherwise he folds. This gives him the optimal 45/55 value/bluff ratio for squeeze 3-bets in a 3-way pot.

Alice’s strategy
From Part 3 we remember that Alice’s value range after opening her 35% button range and getting 3-bet heads-up was {QQ+,AK} =34 combos. Then she added the 4-bet bluffs {ATo,A9s-A7s} =24 combos to get a heads-up optimal 60/40 ratio between 4-bet value hands and 4-bet bluffs.

We have chosen a model where Alice uses the same value range in squeeze scenarios, but now with a 70/30 value bluff ratio instead of 60/40. So Alice needs 30/70 bluff combos for every value combo, She therefore 4-bet bluffs with (30/70) x 34 =15 combos. For example, we can drop A8s/A7s from the heads-up 4-bet bluffing range and use {ATo,A9s} =16 combos. The value/bluff ratio then becomes 34/16 =68/32 which is close to the 70/30 that we want.

In addition, Alice defends by flatting a wide range in position, also when there is a cold-caller between her and Bob. Heads-up in position we gave Alice the flatting range {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs} =120 combos, and we can use this as a starting point also in a squeeze scenario. We can adjust as needed, for example by calling tighter if the cold-caller is tight and plays well postflop.

2.5 Summary of the theory for squeezing

We used a model to estimate the new optimal value/bluff ratios that arose in a 3-way squeeze scenario. We found that these were 45/55 for Bob’s 3-betting and 70/30 for Alice’s 4-betting.

Then we looked at an example with Alice on the button, the coldcaller in the small blind and Bob in the big blind to illustrate how we can adjust to these new optimal ratios. We made some simplifications along the way. For example by assuming that Alice uses the same defense frequency (30%) as in a heads-up 3-bet scenario. We also assumed she uses the same value range. Adjusting the ranges to the new value/bluff ratios then simply becomes an adjustment of the number of 3-bet/4-bet bluffs, while the value ranges are the same as in the heads-up scenario.

This method is of course only an approximation, but it captures the essence of the difference between heads-up pots and multiway pots, namely that both the raiser and the 3-bettor should bluff less and 3/4/5-bet more for value. We can make more accurate adjustments, but I recommend you keep things simple and stick with the simple model we have used here when you find yourselves in a squeeze scenario. Use the corresponding heads-up strategy pair for Alice and Bob as a starting point, and tighten up the bluffing ranges somewhat.

The most important points to take with us from this discussion are:

  • The raiser and the cold-caller have to defend a lot (38%) against the squeeze to prevent the squeezer to have a profitable 3-bet with any two cards
  • A bigger pot before it’s the 3-bettors turn to act means a higher value/bluff ratio for all players involved. A bigger pot means better risk/reward ratios and therefore less folding. The players adjust by bluffing less.

If you understand these things and use the model presented above to design (or at least think about) new value/bluff-ranges for 3/4/5-betting adjusted to the new pot-size, you should feel comfortable playing squeeze scenarios.

We discussed one specific example here to show how these adjustments can be done. Those of you who have trained the 3/4/5-bet strategy pairs for the heads-up scenarios can now work through any squeeze scenarios on your own and implement the necessary adjustments, based on the model used in this article.

3. Cold 4-betting

“Cold 4-betting” is 4-betting in a multiway pot after a raise and a 3-bet has occurred before it’s your turn to act. When the pot has been raised and 3-bet, we expect to often clash with a value hand, and the range we 4-bet for value should therefore be very tight. Having a default 4-betting range of only {KK+} in this scenario is reasonable and fairly standard.

Below is an example of a cold 4-bet for value:

Example 3.1: Cold 4-betting

$100NL

UTG ($100) raises to $3.5, button ($100) 3-bets to $12, you have K K in the small blind and 4-bet to $25. UTG and button folds.

The example above illustrates what often happens in these situations. A typical UTG-raiser will open a tight range. Therefore, the 3-bettor will also have a tight value range (of course mixed with a lot of 3-bet bluffs if he plays optimally). So when you 4-bet, you are telling both opponents that you have an extremely tight value-range. Two thinking opponents will put you on mostly {KK+}, fold all medium and weak hands, and only 5-bet all-in with their absolutely best hands.

Since cold 4-betting with an ultra tight value-range forces your opponents to fold a lot, it’s obvious that you should balance your value hands with some 4-bet bluffs. This does two things for you:

  • Your bluffs make an immediate profit if your opponents fold too much
  • Even if they don’t fold too much, thus making your bluffs break-even or close to it, you have now guaranteed that your opponents can’t “escape” your extremely tight value range by folding all weaker hands. If they do, your bluffs will make more money

The last points needs a bit more explaining. Let’s say that you choose to only cold 4-bet {KK+} for value in this situation. Two observant opponents can now save money by folding everything but AA! They will fold KK without hesitation if they are certain you’re only 4-betting {KK+}, since the AA/KK ratio in your range is 6 : 1 when they have KK (there are now 6 possible AA combos, but only 1 possible KK combos in your range). So the probability of their KK hands running into your AA hands is 6/(6 + 1) =86%, so they have an automatic fold with KK against your {KK} value range if you never cold 4-bet bluff.

So you will pick up lots of pots with your {KK+} value range, but you will never get action from worse hands, assuming your opponents are observant and play well. Of course, your opponents will not play perfectly in reality, but we should remember that playing optimally means removing opportunities for our opponents to exploit us. And if we never cold 4-bet bluff, they can exploit us by folding all hands we beat.

Therefore, by adding some cold 4-bet bluffs to our {KK+} value range, we’re forcing our opponents to choose between:

– Give our {KK+} value range action with more hands than AA
– Or lose to our 4-bet bluffs

An optimal mix of {KK+} value hands and some cold 4-bet bluffs guarantees a better average profit than only 4-betting for value, regardless of what our opponents do to defend themselves. We will not estimate what the optimal ratio is, but instead talk about qualitative guidelines for cold 4-betting, so that you can recognize the good cold 4-betting spots when they occur.

We’ll use an exploitative mindset where we’re trying to cold 4-bet bluff in situations where we expect our bluffs to make money. As a bonus we’ll balance our {KK+} value range, but we’re not necessarily trying to use an optimal value/bluff ratio for all situations. This is fine if we save our cold 4-bet bluffs for situations where expect our opponents to fold too much, so that we can exploit them by 4-bet-bluffing more than optimally.

We’ll look at how the following 4 factors influence the EV of cold 4-bet bluffing:

  • The effect of opponent ranges for openraising and 3-betting (which are functions of the openraiser’s position)
  • The ranges they choose to go all-in with against our cold 4-bet
  • Our choice of cold 4-bet bluffing hands (where we use the blocker effect to our advantage)
  • Our choice of value range. {KK+} is a sensible default, but when our opponents start out with wide ranges, we might want to also include QQ and AK

We start by defining the model we’ll use to study the situation:

3.1 Model for cold 4-betting

We’ll use the following model:

  • All players start with 100 bb stacks
  • The raiser (Alice) and the 3-bettor (Bob) use our default ranges for openraising and our estimates of optimal heads-up 3/4/5-bet strategies
  • Alice raises pot (3.5bb) from some position outside the blinds, and Bob 3-bets pot from some position between Alice and us
  • We cold 4-bet to 25 bb (a little less than pot) from the big blind with a mix of value hands and 4-bet bluffs
  • Alice and Bob defend against our 4-bet by 5-betting all-in with some value range and otherwise folding
  • If we get 5-bet all-in, we fold our bluffs and call with our value range (where {KK+} is a good default to use against unknown or tight opponents)

We will in the following only study the EV of our cold 4-bet bluffs, and not the EV of our total cold 4-betting range (remember: We’re trying to exploit our opponents in this situation, so we’re looking for the spots where cold 4-bet bluffing is most profitable)

We start by investigating the effect of opponent ranges, which is a function of the position Alice openraises from. For example, it’s reasonable to assume that the profitability of a cold 4-bet bluff will increase as Alice moves from UTG (tight openrange) to the button (loose openrange). When Alice’s openrange widens, Bob will respond by 3-betting a wider range, and both of them will have to fold more hands to a cold 4-bet.

Of course, if Alice and Bob are trying to play optimally against our 4-bet, both of them will make sure they are defending with an optimal mix of value hands and 5-bet bluffs, so that they are defending correctly against getting exploited by a cold 4-bettor who is bluffing with any two cards. But in practice most players you meet will only shove a tight valuerange and almost never 5-bet bluff. So we should be able to exploit them by 4-bet bluffing more than we should be allowed to, if we pick good spots for it.

We’ll start by assuming Alice and Bob are using the same value ranges for 5-betting all-in that they would have used in a heads-up 3/4/5-bet war against each other (but they drop all 5-bet bluffs from their strategies after we come charging in with a cold 4-bet and make the pot multiway). So if Alice openraises from UTG, and gets 3-bet by Bob on the button, their value ranges are {QQ+,AK} and {KK+}, respectively, as shown in previous articles (see Part 1, Part 2). Later we’ll study the effect of allowing them to use other value ranges against our 4-bet.

3.2 The profitability of cold 4-bet-bluffing as a function of our opponents’ positions

We’ll investigate two scenarios:

– Alice openraises from UTG and Bob 3-bets from the button
– Alice openraises from CO and Bob 3-bets from the button

In both scenarios we’ll assume that Alice and Bob are using the optimal heads-up 3/4/5-bet strategies we have defined in previous articles. We’ll assume that we are in the small blind with a hand we elect to cold 4-bet bluff. Then we use Pokerazor to calculate the EV of our cold 4-bet bluff.

For each scenario we’ll cold 4-bet bluff with 4 different hands:

– 7 2
– A T
– A K
– Q Q

7 2 is a worthless bluff with no blocker effect. A T takes advantage of the blocker effect, since an ace in our hand reduces the probability that our opponents have the value hands AA and AK. A K and Q Q block the value hands AA, KK, QQ and AK, and they also have decent equity against our opponents value ranges. They are also borderline value hands for us in this situation, so it will be interesting to see if they can be played profitably as value hands (i.e. we 4-bet them, planning to call a 5-bet), even if we start out with a default valuerange of only {KK+}.

We’ll first find the EV for all 4 hands when we play them as bluffs (i.e. we 4-bet, planning to fold to a 5-bet), and then we’ll see if any of them can increase their EV by calling the 5-bet instead of folding. We suspect that AK and QQ might be profitable value hands for us when Alice starts with a wide 25% openrange in CO (which will cause Bob to 3-bet a wide range), but probably not when Alice starts with a tight 15% openrange in UTG (which will cause Bob to 3-bet a tight range).

These calculations will be very hard to do manually, but Pokerazor will do it for us in a few seconds. The program let us specify complete ranges and strategies for all players on all streets, and then it can find the EV for these strategies. Unfortunately, Pokerazor is no longer commercially available, but the developers seem to be working on a new and improved version to be released some time in the future. This means you don’t have the opportunity to use this fine poker software tool to verify my calculations or do similar modeling work on your own. That’s a pity, but you simply have to accept the numbers I present here, and focus on the results, not the computational method.

We won’t repeat all the optimal 3/4/5-betting ranges and strategies here, so look them up in the previous articles if you need to refresh them.

Scenario 1: Alice openraises from UTG and Bob 3-bets from the button
The complete list of strategies is:

  • Alice’s strategy in EP: Openraises to 3.5 bb with the ~15% EP default range, 5-bets the corresponding value range {QQ+,AK} all-in against our cold 4-bet and folds everything else
  • Bob’s strategy on the button: 3-bets to 12bb with an optimal {value range} + {bluff range} ={KK+, A5s,As4s,Ah4h,Ad4d} + {30% of “IP 3-bet air list”} ={KK+,A5s,As4s,Ah4h,Ad4d} + {A9s,As8s,Ah8h,K9s,Q9s,J9s,T8s,97s,65s}, 5-bets the corresponding value range {KK+} (where all 5-bet bluffs have been dropped) against a cold 4-bet from us, and folds everything else
  • Small blind: Folds a random hand
  • Our strategy in the big blind: Cold 4-bet bluff to 25 bb, folding to an all-in 5-bet

Note that we have replaced Bob’s 30% random 3-betting of bluff hands from “IP 3-bet air list” with 30 specific combos from the list (which has 100 combos). This makes it easier to construct the Pokerazor input.

We have also assumed that SB’s fold means he folds 100% of his hands, including AA and KK. To be exact, we should have taken into consideration the fact that small blind will sometimes wake up with a value 4-betting hand, but ignoring this won’t make much of a difference (since this range is very tight). Ignoring the small blind completely makes the calculations much simpler. In fact, including the small blind’s range when Alice and Bob are using wide ranges makes the calculations prohibitively complex for Pokerazor, since the number of possible hand combinations “explode”. But I checked this simplifying assumption in a set of separate calculations with tight ranges for Alice and Bob (where the calculations could be done taking into account small blind’s range), and the EV differences for our cold 4-bet bluffs were negligible (less than 0.05 bb difference between the exact and approximate calculations).

We start by computing the EV for our 4 candidate hands when we play them as pure bluffs, and always fold to a 5-bet. The EVs are given in big blinds, and computed as the difference between our final stack and our starting stack. For example, EV =+0.66 bb means our stack changed from 100 bb at the beginning of the hand (including the big blind we posted) to 100.66 bb when the hand was over.

The EV for cold 4-bet bluffing with each of the 4 candidate hands are:

– 7 2 : +0.66 bb
– A T : +2.35 bb
– A K : +5.51 bb
– Q Q : +0.83 bb

Against a tight EP openrange followed by a tight 3-betting range a random bluff with a trash hand is approximately break even under the assumptions made in our model. Using the blocker effect to our advantage increases the profitability, and A K performs best with an EV of more than +5 bb

The blocker effect for A T and A K is significant, particularly for A K which blocks both AA, KK and AK in our opponents’ value ranges, and it also has good equity against QQ. Q Q also has a small blocker effect against opponent value hands, but the hand is just barely performing better than the trash hand 7 2 . This makes sense, since Q Q only blocks other QQ in Alice’s value range {QQ+,AK} and no hands in Bob’s tight value range {KK+}.

Then we investigate how the EVs change when we play A K and Q Q as value hands heads-up those times one opponent folds and the other one 5-bets all-in. We call the 5-bet if Alice 5-bets and Bob folds, or if Alice folds and Bob 5-bets. If Alice 5-bets all-in and Bob calls, we will fold as before.

It makes sense to only call an all-in 5-bet when we are heads-up, since the probability of being up against AA or KK is huge when two players have gone all-in in front of us. In this particular case we of course know that Bob only can have {KK+} when he gets all-in (so we should fold against him heads-up as well), but we make things simple and assume we’re willing to get all-in with QQ and AK and take our chances, if we can do so heads-up.

– A K : +3.37bb
– Q Q : -6.38bb

We see that both hands perform worse as value hands than as bluffs after a tight open-raise and a correspondingly tight 3-betting range, even if the 3-bettor has a range full of 3-bet bluffs (60% of his 3-bets are bluffs in an optimal strategy). Calling a 5-bet all-in with A K when heads-up is not very bad, but we lose relative to folding (+5.51bb –> +3.37bb) and playing the hand as a 4-bet bluff. For Q Q calling an all-in is horrible, and this is due to a “double whammy” where we don’t block any of the hands AA/KK that beat us, and we have very bad equity against those hands (while AK is blocking both of those hands, and only has terrible equity against AA).

We conclude:

Against a tight openraiser from early position, followed by a 3-bet from an optimal (or near optimal) 3-betting range, cold 4-bet bluffing with a random trash hand is close to break even if the raiser and the 3-bettor use the value ranges {QQ+,AK} and {KK+}, respectively. We can increase the EV of our bluff by using the blocker effect, picking our bluffs from the best Ax hands (AK in particular). But we should not use a value range wider than {KK+} in this case. AK doesn’t suffer much from getting all-in heads-up, but QQ (and similarly all lower pairs) performs very poorly as a value hand).

Now we move Alice to CO and let her open her standard 25% CO range, while Bob attacks her with the corresponding optimal 3-betting range

Scenario 2: Alice openraises from CO and Bob 3-bets from the button
The complete list of strategies are:

  • Alice’s strategy in CO: Openraises to 3.5 bb with the ~25% default CO range, 5-bets the corresponding value range {TT+,AQ} all-in against our 4-bet, and folds everything else
  • Bob’s strategy on the button: 3-bets to 12 bb with an optimal range {value range} + {bluff range} ={QQ+,AK,A5s,A4s,A3s} + {70% of “IP 3-bet air list”} ={QQ+,AK,A5s,A4s,A3s} + {A9s-A6s,K9s-K6s,Q9s-Q6s,J9s-J8s,T8s-T7s,97s,6s5s,6h5h}, 5-bets the corresponding value range {QQ+,AK} (where all 5-bet bluffs have been dropped) against a 4-bet by us or by Alice, and folds everything else
  • Small blind: Folds a random hand
  • Our strategy in the big blind: 4-bet-bluff to 25bb, and fold to a 5-bet

The assumptions are the same as in the previous scenario. We have here specified 70 3-bet bluff combos from “IP 3-bet air list” for Bob to use instead of a randomizer, and we have assumed small blind is folding 100% of his hands to make the Pokerazor calculations practical.

The EV for playing our 4 candidate hands as 4-bet bluffs now becomes:

– 7 2 : -1.60bb
– A T : +0.27bb
– A K : +2.00bb
– Q Q : +0.02bb

The trend between the hands is the same as in the previous case (the blocker effect is significant), but now only AK has positive EV. However, none of the hands are losing big. Is this surprising? Not really. We would intuitively expect to make more from bluffing when Alice’s and Bob’s ranges are wide, but we have to remember that both of them are using the optimal HU strategies that defend them against being exploited by a any-two-cards-bluff. These strategies/ranges are not quite optimal in multiway scenarios, but they still do a pretty good job. So it’s not really surprising that our bluffs are close to break even, no matter where Alice is opening from, when both she and Bob are using the value ranges they would have used heads-up in a 3/4/5-bet war against each other. This illustrates an important property of optimal strategies: They are robust. Small changes in the situation don’t cause large changes in the optimal strategies, and playing near-optimally for any given situation is usually good enough.

The small changes in EV for our 4-bet bluffs when Alice moves from UTG to CO probably contains some “numerical noise” as well, so we won’t draw any strong conclusions from these changes. For example, our definitions of ranges for Alice and Bob are not perfect down to the last combo, and we also did some numerical rounding along the way when we defined these strategies. The most important observation is that our cold 4-bet bluffs with random trash is close to break even when both Alice and Bob defends with something close to the optimal heads-up 3/4/5-bet strategies they would have used against each other.

This means that the defense strategies for Alice and Bob work well against random 4-bet bluffing, and defending against random bluffing is partly what they were designed to do. If we had been able to exploit Alice and Bob hard by cold 4-bet bluffing any two cards, something would have been wrong.

Like in the previous case we now move on to see if A K and Q Q work as value hands in this scenario. Like in the previous case we now call a 5-bet if we can get all-in heads-up against either Alice or Bob, but not both (we’re assuming the risk of clashing against AA/KK is too high when this happens).

– A K : +8.29bb
– Q Q : +6.91bb

Not surprisingly both hands now perform well as value hands, and with a big increase in EV compared to bluffing with them. This is obvious when we look at some of the value hands Alice is now shoving: JJ, TT and AQ. All of these are dominated hard by AK and QQ. Also, when Alice is raising from CO, Bob is also shoving QQ and AK for value, so our AK and QQ are hurt much less when they get all-in against his tight value range.

Finally we’ll run a series of calculations where we let Alice tighten up her value range from {TT+,AQ} to {QQ+,AK} out of respect for our signal of strength when we 4-bet cold from the blinds. And we let Bob continue with his {QQ+,AK} value range.

The EV for 4-bet bluffing now increases (not unexpectedly) for all hands, since Alice folds a lot more:

– 7 2 : +0.97bb
– A T : +2.56bb
– A K : +4.76bb
– Q Q : +2.12bb

And it’s here we can gain EV by 4-bet-bluffing against wide ranges. We saw previously that there wasn’t much difference between cold 4-bet bluffing against wide and tight opponent ranges, if they defended close to optimally, and had the “guts” to keep 5-bet-shoving with the value ranges they would have used heads-up against each other. For Alice this means she has to be willing to continue to shove with both TT and AQ to avoid giving us and Bob an opportunity to exploit her 25% opening range (which is what will happen when she begins folding too much). This value range is designed to use heads-up against Bob, but if she deviates drastically from it, she will make herself vulnerable against our cold 4-bet bluffs.

When Alice “chickens out” by dropping 3 of her value hands (JJ, TT, AQ), she leaves “dead money” in the pot, and creates an opening for profitable cold 4-bet-bluffing with any two cards. Bob can now decide to take some of the defense responsibility that Alice refuses to take, but remember that Bob’s 3-bet strategy against a CO raiser starts with choosing {QQ+,AK} as value range and placing the next tier of good hands (JJ, TT and AQ) in the flatting range.

So Bob can’t increase his defense frequency against our 4-bet by 5-betting more for value since he doesn’t have any more value hands to use, only 3-bet bluff hands. So if he wants to defend more, he has to start 5-bet-bluffing. But bluff-shoving all-in with hands like K 9 and A 4 after bluff 3-betting and then getting cold 4-bet, takes a better understanding of the dynamics of the situation, better reads, and more guts than most players possess.

Note that folding JJ, TT and AQ against our 4-bet isn’t unreasonable for Alice. We’re signaling great strength, and she is in a squeeze between us and Bob who has 3-bet. But the mathematics dictates that if she doesn’t defend her 25% openrange often enough, we have to make a profit with our 4-bet bluffs if Bob doesn’t do anything to prevent it (and as we saw, this is hard for him to do).

Still, there is a balancing effect at work here, since Alice now pays off less to our value hands. The EV for playing A K and Q Q as value hands is now:

– A K : +7.81bb
– Q Q : +4.61bb

But if Alice’s folds of JJ, QQ and AQ have given us an opening for 4-bet bluffing any two cards profitably, she can’t make back this loss by folding the few times we have a value hand. After all, there are only 34 combos of {QQ+,AK} in our range, and 1292 other random hands we can now cold 4-bet bluff profitably.

Finally, if Alice should be scared enough to fold even QQ and AK against our cold 4-bet (for example, of she plays too tight against 4-bets to begin with, and if our table image is good) we can print money by cold 4-bet bluffing with any two cards:

– 7 2 : +2.92bb
– A T : +4.46bb
– A K : +6.42bb
– Q Q : +3.80bb

We conclude:

Cold 4-bet bluffing is not necessarily more profitable when the raiser and 3-bettor are using wide ranges, unless they begin to deviate significantly from optimal play by folding too many value hands against our seemingly strong 4-bet. If Alice and Bob are playing wide ranges, and they defend against our 4-bet using the optimal heads-up strategies they would have used against each other, our cold 4-bet bluffs are about break even But if they begin folding value hands against us that they would have played against each other, our bluffs become much more profitable. It’s the raiser in particular who is vulnerable to this, since she is “forced” to play many hands for value to defend correctly against getting exploited. 

3.3 Summary of cold 4-betting

We learned the following:

  • Cold 4-bet bluffing with random trash against a raiser and a 3-bettor who both defend optimally (or close to it) is about break even, regardless of their positions.
  • By using the blocker effect (first and foremost hands with an ace) we can improve the EV of our bluffs significantly.
  • The raiser and the 3-bettor can give us openings for profitable cold 4-bet bluffing if they begin to fold their weakest value hands (that they would have played against each other, but now decide to fold after getting cold 4-bet). For example, if a CO raiser folds his value hands JJ, TT and AQ
  • Using {KK+} as value range for cold 4-betting is a good default against tight ranges (for example against a ~15% raiser followed by an optimal 3-betting range). But against wide ranges (for example, a ~25% CO raise followed by an optimal 3-betting range) QQ and AK also become value hands

We have discussed cold 4-bet bluffing in isolation, but it’s important to see the cold 4-bet bluffing as “twin” to the value 4-bet we make in this situation. For example, if we always have {KK+} when 4-betting cold against two tight opponent ranges, it’s easy for the opposition to adapt. They can shove {AA} and fold everything else, and never let us get our stack in as big favorites preflop. But if we cold 4-bet bluff occasionally, it’s impossible for them to avoid paying us off one way or the other. They will either fold too often and make our bluffs nicely +EV, or they will be forced to pay off our A and KK with worse hands sometimes.

In theory one can calculate how to perfectly balance a {KK+} or {QQ+,AK} value ranges in a 3-way raised and 3-bet pot with cold 4-bet bluffs, but we won’t do that here. The point of our discussion was to illustrate what makes cold 4-bet bluffing work, how different types of hand perform, how we should choose our value range, and how the raiser and the 3-bettor make themselves vulnerable if they are not willing to felt their weakest value hands.

If they refuse to get all-in with hands weaker than QQ and AK after starting with wide ranges for openraising and 3-betting, we can cold 4-bet bluff any two cards profitably. At least until they adjust, but when they begin adjusting, our 4-bet value hands make more money, and now we can dial back on the bluffing and instead exploit their looseness. Note that when they have pegged us as a loose cannon who is willing to cold 4-bet bluff often, this impression will last a long time since these situations don’t come up often.

Using bluffs to guarantee ourselves action on our good hands is one side of optimal play that we haven’t talked much about, but it’s an important part of the equation.

By mixing value hands and bluffs in an optimal (or close to it) ratio, we’re making it impossible for the opposition to “escape” our value hands by folding a lot, and we guarantee ourselves a certain minimum profit. Then our opponents can choose whether they’d like us to get this guaranteed profit from our value hands (when they call or play back at us too much) or from our bluffs (when they fold too much). We’ll talk more about this side of optimal strategies in Part 6 and Part 7.

4. Summary

We studied two cases of 3/4/5-betting in multiway pots, namely “squeezing” and “cold 4-betting”

For squeezing we started with the heads-up strategies from previous articles and adjusted them to the new multiway scenario by taking the new pot size into account and then doing some simplifying assumptions. This let us estimate new optimal value/bluff ratios, and we used an example to illustrate how we can adjust our heads-up 3/4/5-bet strategies to use in squeeze scenarios.

For cold 4-betting we assumed that our opponents started with the heads-up optimal 3/4/5-bet strategies, and that they responded to our cold 4-betting by only 5-betting their value ranges from those strategies. We used the poker analysis software “Pokerazor” to study this scenario in detail, and we saw that the profitability of a cold 4-bet bluff is very dependent on the blocker effect, and our opponents’ value ranges.

What remains to be done before we end this article series on default preflop strategies in NLHE 6-max based on principles for game theory optimal play, is to do some numerical testing of the strategies we have designed. We shall do this in Parts 6 and 7. We’ll use Pokerazor again to do numerical simulations for various preflop scenarios. We’ll also discuss exploitative play versus optimal play, and when to use one or the other. In Part 7 we’ll also look at 3/4/5-betting in a blind vs blind scenario, which is a topic we haven’t looked at so far.

Good luck!
Bugs – See more at: http://en.donkr.com/forum/optimal-3-bet-4-bet-5-bet-strategies-in-nlhe-6-max—part-5-533565#sthash.ZwlU6ch6.dpuf

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s