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

1. Introduction
This is Part 4 in the series Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max. In Part 1 and Part 2 we discussed 3-betting heads-up with the 3-bettor in position. In Part 3 we began working on the scenario where the 3-bettor is out of position (in other words, blind defense heads-up).

We then looked at the scenario where the raiser open-raises on the button, and then the 3-bettor is in the small blind, or in the big blind after the small blind has folded. We assumed that the raiser (Alice) opened our default button range, and then we defined default ranges for flatting and 3-betting for the player in the blinds (Bob). We also defined a defense strategy for Alice to use against Bob’s 3-bets.

In Part 4 we’ll continue this work and generalize the strategies from Part 3 to include openraising from other positions than the button. We’ll let Alice open-raise from UTG, MP, CO or button, and then Bob 3-bets heads-up from one of the blinds. In Part 3 we used a simple model where we assumed that the two players in the blind shared equally the responsibility of defending the blinds against Alice’s raise. Since Alice there raised from the button, the two players in the blinds had to do the whole job of defending the blinds enough to prevent Alice from raising any two cards profitably. We found that they had to defend a total of 30% to achieve this. And when this job is shared equally between them, they have to defend about 16% each.

In Part 4 we’ll let Alice open-raise from other positions. Then there will be some player(s) between her and the blinds. The total job of defending the blind 30% is then shared between the blinds and the player(s) between Alice and the blinds. This means the players in the blinds don’t have to defend as much as they had to against Alice’s button raises. Furthermore, it’s reasonable that most of the blind defense should be done by the players with position on Alice, especially the player on the button.

We’ll use a simple model to study the distribution of blind defense responsibility between the players left to act after Alice’s open-raise. When Alice raises, the remaining players have to defend at least 30% to prevent her from having a profitable raise with any two cards. The more players left to act, the less each of them have to defend for this to be achieved.

Our starting point will be the default ranges we have defined in the first 3 parts of this article series plus simple mathematical modeling. We want to study how often the blinds minimum have to defend heads-up against raises from various positions to prevent the raiser from having a profitable raise with any two cards. We’re mostly interested in qualitative trends, but we’ll also use the results to estimate reasonable blind defense ranges to use against raises from all positions.

When we have generalized Bob’s blind defense strategies to defense against raises from all positions, we’ll turn to Alice and generalize here defense strategies in all positions against Bob’s 3-bets from the blinds.

The structure for Part 4 is thus:

• A generalization of heads-up 3-betting out of position (i.e. blind defense) against an openraise from any position
• A generalization of the raiser’s defense heads-up in position against a 3-bet from the blinds

The work in this article will be somewhat abstract and mathematical in nature, and our purpose is first and foremost to learn how to think correctly about these topics. For example, we’ll learn that there is a huge difference in blind defense strategies against a button raiser and against an UTG raiser, and we’ll use mathematical modeling to quantify this difference.

We always want to play hands that are profitable and fold those that are not, but in practice we don’t know for sure which hands are the profitable ones in a given scenario. We know which hands are clearly profitable (e.g. the big pairs) and which hands are clearly unprofitable (e.g. 72o), but in all preflop scenarios there is a wide range of hands that are not clearly profitable or unprofitable (e.g 99, AJ, KQ, JTs and other medium strong hands).

Instead of thinking about how profitable a hand is, we can complement our understanding of the situation by attacking it theoretically from a different angle. Instead of asking “which hands are profitable?“we can ask “How many hands do I have to play to prevent my opponent from having a profitable bluffing opportunity with any two cards?“. Working along this line we can paint a picture of which hands we should be able to play profitably in a given scenario to prevent our opponent(s) from exploiting us. This is the kind of thinking we’ll use in the modeling work done in this article.

The original plan for Part 4 was to also talk about squeezing (3-betting in a multiway pot after the raise has been called before it’s our turn to act) and small blind vs big blind scenarios, but we’ll move these topics to future articles.

2. Generalization of the theory for 3-betting/blind defense heads-up and out of position
Let’s first quickly repeat the theory for blind defense against a button open-raise defined in Part 3:

We used the following model:

• Both players start with 100 bb stacks
• Alice open-raises pot (3.5bb) on the button
• Bob defends against Alice’s raise by 3-betting pot (12 bb) with an optimal 40/60 ratio of value hands and 3-bet bluffs, plus flatting some medium strong hands
• Alice defends against Bob’s 3-bets by 4-betting to 25 bb (a little less than pot) with an optimal 60/40 ratio of value hands and 3-bet bluffs, plus flatting some medium strong hands
• Bob defends against Alice’s 4-betting by 5-betting his value hands and folding everything else

Alice’s open-range from the button was defined in Part 2:

Default button-range

22+
A2s+ A7o+
K2s+ K9o+
Q6s+ Q9o+
J7s+ J9o+
T7s+ T9o+
96s+
86s+
76s
65s

458 combos
35%

Since Alice risks 3.5 bb to win 1.50 bb when she open-raises, she can steal profitably with any two cards if she succeeds more than (1.5 + 3.5) =70% of the time. The blinds can’t allow this, so they have to defend 30% of the time to make Alice’s weakest raising hands break even. We’ll assume they share this responsibility equally, and that they both defend some percentage x.

The probability that both of them fold is then (1-x)(1-x), so the chance that at least one of them defends is 1 – (1-x)(1-x). This expression should be 30% to prevent Alice from raising any two cards profitably, so we get:

1 - (1-x)(1-x) =0.30
1 - (1 -2x +x^2) =0.30
1 - 1 + 2x - x^2 =0.30
x^2 - 2x + 0.30 =0

This quadratic equation has the solutions x =1.84 and x =0.16 (you can use the online Quadratic Equation Solver), and we choose the solution x =0.16 =16%, since x is a probability (a number between 0 and 1). We then defined a 3-bet value range, a 3-bet bluffing range, and a flatting range for Bob so that his total blind defense was approximately 16%. The ratio of value hands to bluffs in his 3-betting range was the optimal 40/60 ratio that we have used throughout this article series. In addition Bob flats a range of medium strong hands that are not good enough to 3-bet for value, but too strong to fold or turn into 3-bet bluffs.

2.1 Minimum default blind defense heads-up against a button open-raiser
We estimated the following defense ranges for Bob against Alice’s button open-raise:

Value 3-bet-range OOP against a button open-raise

TT+
AQ+

62 combos

We remember that the weakest hands in this value range work as a “hybrid” between value hand and 5-bet-bluff. Alice will often flat Bob’s 3-bet with position, and TT/AQ have good equity against her flatting range (medium hands like 99, AJ, KQ, etc). When she 4-bets, we don’t expect TT/AQ to be favorites against her value-range (i.e. the hands she plans to call an all-in 5-bet with), so when we 5-bet these hands it makes more sense to think of them as 5-bet bluffs (that profit from folding out Alice’s 4-bet bluffs, but are underdogs when she calls). See Part 3 for a more thorough discussion of this topic.

In addition to the value hands Bob 3-bets, planning to 5-bet all-in, he uses a range of 3-bet bluffs (“OOP 3-bet air list”) and a range of medium strong hands that he flats (“OOP flat list”):

OOP 3-bet air list

66-22
A9s-A6s
K9s-K8s
QTs-Q9s
J9s-J8s
97s+
87s
76s
65s

98 combos

OOP flat list

99-77
AJs-ATs, AJo
KTs+ KQo
QJs
JTs

70 combos

Bob’s list of 3-bet bluffs to use out of position is stronger than the list of hands he 3-bet bluffed with in position (see Part 1 and Part 2). This is because Alice will sometimes flat the 3-bet in position, and then Bob will be forced to play postflop out of position. To make the most out of these scenarios it’s important for Bob to 3-bet bluff with the best of his worst hands, i.e. the range of hands just below his flatting range.

Bob defends a total of 62 + 70 + 98 =230 combos against a button steal raise, or 230/1326 =17% of the time (a bit more than the minimum 16% that we need, which is fine). We’ll now place Alice in CO, MP and UTG and estimate how often Bob minimum has to defend to prevent her from having a profitable raise with any two cards.

With players between Alice and Bob we also have to take into account the blind defense done by these players, and less of the total blind defense responsibility falls on the two players in the blinds. We’ll account for this by using a simple mathematical model.

Note that when Bob tightens up his blind defense against open-raises from earlier positions, hands should in theory move between ranges. When we drop some hands from our value range, these hands should be moved down to the flatting range, and some flatting hands should be demoted to 3-bet bluffs. This follows from the strength principle. But in practice we’ll simplify things by keeping our “OOP 3-bet air list” constant, so that we won’t have to memorize a range of 3-bet bluffing hands for each of the raiser’s positions. This is not quite optimal, but we accept this simplification in order to make it easy to build and memorize sound default ranges. When this work is done, we can always fine-tune our ranges later.

2.2 Blind defense heads-up against a CO open-raise
We assume Alice opens our default 25% CO range:

22+
A2s+ A9o+
K9s+ KQo
Q9s+ QTo+
J8s+ JTo
T8s+
97s+
87s
76s
65s

326 combos
25%

Button and the two players in the blinds now have a collective responsibility of defending the blinds at least 30%. We start by assuming button uses our optimal 3/4/5-bet strategy against a 25% CO open-raise, and also our default flatting range in position. So we start by estimating how often button defends against Alice’s CO raise.

In Part 2 we defined the following ranges for the 3-bettor in position:

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

Button first defines a value range. Then he finds the percentage of the 3-bet bluff list he needs to use to get an optimal 40/60 value/bluff ratio. And then he chooses his flatting range. We’ll assume that a player on the button will flat all hands from “IP flat list” regardless of the raiser’s position.

In Part 2 we found the following optimal strategy to use in position behind a 25% CO raiser:

• Flat the whole “IP flat list”: {22+,ATs+,AJo+,KTs+,KQo,QTs+,JTs,T9s,98s} =140 combos when {QQ+,AK} are 3-bet for value
• 3-bet {QQ+,AK, 12 air} for value, planning to 5-bet all-in after a 4-bet
• 3-bet 70% of “IP 3-bet air list”, planning to fold to a 4-bet

So button 3-bets {QQ+,AK} =34 combos for value, together with 12 Axs-combos that he 5-bet bluffs all-in if Alice 4-bets. This gives button a total value range (more correctly: all-in range) of 34 + 12 =46 combos. Button then needs 1.5 x 36 =69 3-bet bluff combos to get an optimal 40/60 value/bluff ratio.

So we use 69% of the “IP 3-bet air list”, which we round to 70% to keep things simple. To achieve this we use a randomizer every time we have one of the 3-bet bluff candidates from the list. We 3-bet bluff when the randomizer returns a number between 0 and 70, and otherwise we fold. Since there are are 100 combos total in “IP 3-bet air list”, this corresponds to 3-bet-bluffing 70 combos on average, which is what we want. Finally, we flat the 140 combos from “IP Flat List” that remain when {QQ+,AK} get 3-bet for value.

Button then 3-bets 34 + 12 + 70 =116 combos total (i.e. 116/1326 =8.7%) and flats 140 combos (140/1326 =10.6%). This means button defends 8.7 + 10.6 =19.3% total after a CO open-raise. Since button, small blind and big blind need to defend 30% combined, this means that button does most of job of defending the blinds. The two players in the blinds can therefore tighten up considerably compared to the ranges they had to defend with against a button open-raise.

We use the same mathematical model as before and assume button defends 19.3% as estimated above. Then the rest of the blind defense responsibility is shared equally between small blind and big blind, and both of them defend some percentage x those times button folds. The probability all 3 players fold is then (1-0.193)(1-x)(1-x), so the probability at least one of them defends is 1 – (1-0.193)(1-x)(1-x). This expression should be equal to 30%, so we get:

1 - (1-0.193)(1-x)(1-x) =0.30
1 - 0.807(1 -2x +x^2) =0.30
1 - 0.807 + 1.614x - 0.807x^2 =0.30
-0.807x^2 + 1.614x - 0.107 =0

We solve this expression with Quadratic Equation Solver, and get the solutions x =0.068 and x =1.93. We choose the solution between 0 and 1, and find that each of the players in the blinds need to defend x =0.068 =6.8%. We round this to 7%.

This is a very interesting result compared to the defense percentage of 16% against a button open-raise. When we get one player (button) between the raiser and the blinds, the minimum defense percentage for the two players in the blinds is reduced from 16% to only 7%!

So what does a ~7% defense range look like? As always, we start with a value range for 3-betting. Then we add 3-bet bluffs to get an optimal 40/60 value/bluff ratio. Finally we flat with the best hands not good enough to 3-bet for value.

We have some flexibility here. As discussed in Part 2, we are trying to do the big and important things correctly, and we don’t worry about the grey areas where the differences between the alternatives are small (for example, whether we should 3-bet for value or flat with a good-but-not-great hand like JJ). Our starting point is the optimal value/bluff ratio for our 3-betting range, and then we try to design a solid and reasonable total defense strategy.

Against a button open-range we used {TT+,AQ} as our value range. Against a default CO open-range we should tighten up our value range somewhat, let’s say to {JJ+,AK} or {QQ+,AK}. Let’s choose {QQ+,AK} =34 combos and see where this takes us. We now need 1.5 x 34 =51 3-bet bluff combos for an optimal 40/60 value/bluff ratio. Since there are approximately 100 combos in our “OOP 3-bet air list”, this corresponds to 3-bet bluffing all hands on the list 51% of the time, using a randomizer. We round this number to 50% to keep things simple.

So we 3-bet 34 + 50 =84 combos total for 84/1326 =6.3% of all hands. This means that almost all of the estimated minimal defense responsibility of 7% can be done by 3-betting {QQ+,AK} for value, together with the optimal number of 3-bet bluffs.

In addition, we can pick the best medium strong hands to flat with. Since the 3-betting range makes up almost everything we need, we can be picky and choose for example {JJ-TT,AQ} =28 combos (32/1326 =2.1%). This gives us a total defense percentage of 6.3 + 2.1 =8.4%, which is a bit more than the minimum 7% we need according to our model.

Note that what we’re doing here is to play with a mathematical model to estimate how often we minimum have to defend in the blinds against a CO open-raise to prevent him from open-raising any two cards profitably. We’re assuming button does his part of the job by following our optimal 3/4/5-bet + flat strategy in position, and then the small blind and big blind take care of the rest.

This is not the same as estimating which hands are profitable to play from the blinds after a CO steal raise when button has folded. But is gives us a starting point to build on. For example, if CO is passive postflop and often lets you get cheaply to showdown with marginal hands, this will make it easier to play postflop out of position with your medium strong flatting hands. So it would make sense to exploit his tendencies by adding more hands to the flatting range, for example 99-88,AJ,KQs. But our model indicates that you can’t get exploited by CO if you choose to play very tight, and fold these hands heads-up out of position (assuming button defends as actively as he should).

It’s also possible to define a minimum blind defense strategy against a CO open-raise without flatting. For example, we can choose to 3-bet {JJ+,AK} =40 combos for value, and then 1.5 x 40 =60 3-bet bluff combos (i.e. all hands from “OOP 3-bet air list” 60% of the time using a randomizer). This gives us a total 3-bet% of (40 + 60)/1326 =7.5% which is slightly more than the required 7%.

You can of course also use the looser 3-betting range {JJ+,AK} + {3-bet bluffs} together with a flatting range, as you wish. As we discussed in Part 3, what’s most important for us is to use strategies and ranges that are consistent and based on optimal 3/4/5-betting. Exactly what we choose in marginal spots (e.g. should JJ be flatted or 3-bet for value?) is less interesting for us when we’re defining a reasonable default strategy. Also, note that in practice we’ll often use reads to help us choose between similar alternatives at the table.

At any rate, we can use the following strategy as a minimum default blind defense strategy against a CO openraise heads-up from the blinds:

Minimum ~7% blind defense strategy heads-up against a CO open-raise
The minimum 7% blind defense is covered by:

• 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
• Flat {JJ-TT,AQ}
• This gives ~8% total defense

Our starting point is to play minimum 34 + 50 + 28 =112 combos, i.e. 112/1326 =8.4% of all hands. Then we can add more hands when we think it’s profitable. Exactly what the range of profitable hands is for us in this scenario is partly a function of factors like CO’s postflop skills, our postflop skills, and the history/metagame between us, so this is impossible to determine exactly in practice. But we’ll talk more about this towards the end of this article series where we’ll discuss exploitative play versus optimal play, and adjustments based on reads and metagame.

We move on, and place Alice in MP. There are now two players (CO and button) between Alice and the blinds, and the blinds can now get away with defending even less.

2.3 Minimum default blind defense heads-up against an MP open-raise
We assume Alice opens our default 15% EP range from MP:

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

194 combos
15%

Button, CO and the two players in the blinds now share the collective responsibility of defending 30% against Alice’s open-raises. We start by assuming CO and button both use our optimal 3/4/5-bet + flat strategy in position against a 15% open-raise. From Part 2 we remember that the 3/4/5-bet strategy is:

• 3-bet {KK+, 7 air} for value, planning to 5-bet all-in after a 4-bet
• 3-bet 30% of “IP 3-bet air list”, planning to fold to a 4-bet

CO and button then both 3-bet {KK+} =12 combos for value, together with 7 Axs-combos as 5-bet bluffs for a total value range of 12 + 7 =19 combos. Then we need 1.5 x 19 =29 3-bet bluff combos for an optimal 40/60 value/bluff ratio, which we can round to 30. So we add 30% of the 100 combos in “IP 3-bet air list” using a randomizer. Both CO and button then 3-bets 12 + 7 + 30 =49 combos total, i.e. 49/1326 =3.7%.

Button flats the whole “IP flat list” as before. This is 162 combos when {KK+} is 3-bet for value, which gives 162/1326 =12.2%. For the flatting done by CO, we assume he will flat tighter than button because of poorer position (a reasonable assumption) and that he effectively flats with half the flat list. So MP flats 162/2 =81 combos, or 81/1326 =6.1%.

Under these assumptions CO defends 3.7 + 6.1 =9.8%, while button defends 3.7 + 12.2 =15.9%. As before we find that the probability of all players folding to Alice’s raise is (1-0.098)(1-0.159)(1-x)(1-x), so the chance of at least one of them defending is 1 – (1-0.098)(1-0.159)(1-x)(1-x). This should be 30%, so we get:

1 - (1-0.098)(1-0.159)(1-x)(1-x) =0.30
1 - 0.759(1 -2x +x^2) =0.30
1 - 0.759 + 1.517x - 0.759x^2 =0.30
-0.759x^2 + 1.517x - 0.059 =0

We plug this expression into Quadratic Equation Solver, and find the solutions x =0.040 and x =1.96. We choose the solution between 0 and 1 and find that each of the players in the blinds need to defend x =0.040 =4.0%.

As expected even less of the blind defense responsibility falls on the players in the blinds. We now have two players, CO and button, with position on Alice, and they do most of the defense. Button defends tighter against MP than he did against CO, since the 3-betting range becomes tighter against an MP open-range. But this is more than compensated for by the presence of CO, who also defends with 3-betting and flatting.

Defining a 4% minimal blind defense range is simple. We can use the value range {QQ+,AK} =34 combos as our starting point like we did against CO. We then used 50% of “OOP 3-bet air list” for optimal 3-bet bluffing, and landed on a total 3-betting range of 34 + 50 =84 combos, or 84/1326 =6.3%.

This means that we cover the minimum necessary blind defense (and then some) against an MP raiser by only 3/4/5-betting optimally with a {QQ+,AK} value range and no flatting. We don’t have to use a default flatting range to prevent MP from open-raising any two cards profitably when CO and the button defend optimally in position. So if you want to, you can play very tight and fold hands like JJ, AQ and KQ against a 15% open-raise from early position. The raiser can’t begin to exploit this by loosening up, even if it might feel like you’re being exploited when folding decent hands. Of course, if you think you have a profitable hand, you should play it, but the model indicates that we don’t have to play more than 4% of hands.

Note that the tighter the raiser’s range, the more 5-bet-bluff-like the hands QQ/AK become. QQ/AK have good equity against the range the raiser flats 3-bets with, but we don’t necessarily have good equity against the range of hands a tight player calls an all-in 5-bet with (for example, if he only calls a 5-bet with {QQ}). So in a sense, QQ/AK can be viewed as a value/bluff hybrid against a tight open-raising range, similar to how we played TT/AQ against a button open-raiser.

Alternatively, there’s nothing that prevents us from 3-betting an even tighter value range against MP and then adding a flatting range that includes QQ/AK. If we choose this, it’s obvious to reduce the value 3-bet range to {KK+} =12 combos, and then we add 1.5 x 12 =18 combos from “OOP 3-bet air list”, or 18% for all hands using a randomizer (and we can round this to 20%). Then we effectively 3-bet 12 + 20 =32 combos total, or 32/1326 =2.4%.

We then need a flatting range of at least 4.0 – 2.4 =1.6%, or 0.016 x 1326 =21 combos. This is covered pretty accurately by {QQ,AK} =22 combos. So we can easily defend the minimum 4.0% even with a super-tight strategy where we only 3-bet {KK+} for value together with an optimal number of 3-bet bluffs, and then we flat only {QQ,AK}.

We list both these alternatives:

Minimum ~4% blind defense strategy heads-up against an MP open-raise
The minimum 4% defense is covered by:

• 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
• This gives 6% total defense

Alternatively

• 3-bet {KK+} for value together with 20% of “OOP 3-bet air list”
• Flat {QQ,AK}
• This gives ~4% total defense

Starting with one of these minimal default strategies we can then add more flatting hands if we think it’s profitable (hands like JJ, TT, AQ, etc). We can use reads to help us here.

It’s obvious that a minimal heads-up blind defense strategy against an UTG open-raise will be squeaky tight, based on our model. But let’s complete the modeling by also working through this case:

2.4 Minimum default blind defense heads-up against UTG open-raise
We assume Alice opens our default 15% EP range from UTG:

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

194 combos
15%

MP, button, CO and the two players in the blinds now share the collective responsibility of defending 30% total against the raise. We assume MP, CO and button all use our optimal 3/4/5-bet + flat strategy in position against the raiser. The 3/4/5-bet part of that strategy is:

• 3-bet {KK+, 7 air} for value, planning to 5-bet all-in after a 4-bet
• 3-bet 30% of “IP 3-bet air list”, planning to fold to a 4-bet

And all players between UTG and the blinds thus 3-bet 3.7% as shown previously. In addition button flats the whole flat list of 162 combos =12.2%, while CO flats half the list and 162/2 =81 combos =6.1%. We now assume (somewhat arbitrarily) that MP uses a tight flatting range of 1/4 of the list because of his poor position. In other words 162/4 =41 combos (rounded), or 41/1326 =3.1%.

So MP defends 3.7 + 3.1 =6.8% total, while CO and button defend 9.8% and 15.9%, like they did against MP previously. We set up the same equation as before and get:

1 - (1-0.068)(1-0.098)(1-0.159)(1-x)(1-x) =0.30
1 - 0.707(1 -2x +x^2) =0.30
1 - 0.707 + 1.414x - 0.707x^2 =0.30
-0.707x^2 + 1.414x - 0.0070 =0

We find the solutions x =0.0050 and x =1.995, and choose x =0.0050 =0.5%.

0.5% corresponds to 0.05 x 1326 =7 combos, which is basically {AA}. In other words: If MP, CO and button defend in position with a combination of optimal 3/4/5-betting and flatting, the blinds don’t have to defend with anything else than {AA} to prevent UTG from exploiting them!

The minimal defense percentage 0.5% is of course only meaningful within the context of our model. The number itself is much less interesting than what it represents. What the trend in our model (16% –> 7% –> 4% –> 0.5%) tells us is that you don’t have to worry about getting exploited if you should choose to play very tight from the blinds heads-up against an early position raiser. UTG and MP are handled effectively by the players with position on them, and the players in the blinds can basically just sit back and cherry pick hands they think are clearly profitable.

This is in strong contrast to the 16% default blind defense we were forced to do against a button steal-raise. There we had to 3-bet a wide range and also flat out of position with many so-so hands like 77, ATs, KTs, QJs, etc. All this to prevent button from running over us with loose open-raising. In other words, we were forced to do a lot of “dirty work” out of position with less-than-stellar hands.

On the other hand, our attitude heads-up in the blinds against a tight early position raiser should be more like this:

So should we only 3-bet {AA} from the blinds against an UTG raiser and fold everything else? Of course not, and that’s not what our model tells us. But what it does tell us is that we can’t be exploited by a loose UTG raiser, even if we should choose to defend extremely tight when it’s folded to us. In practice, let’s use the same minimum default defense range we used against MP:

Minimum ~0.5% blind defense strategy heads-up against an MP open-raise
The minimum 0.5% defense is covered by:

• 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
• This gives 6% total defense

Alternatively

• 3-bet {KK+} for value together with 20% of “OOP 3-bet air list”
• Flat {QQ,AK}
• This gives ~4% total defense

And then we can add a range of flatting hands on top of this, when we think this is profitable.

2.5 Summary of the theory for 3-betting/blind defense heads-up and out of position
Based on simple mathematical modeling we defined the following minimum blind defense strategies heads-up from the blinds against an open-raiser from UTG, MP, CO and the button:

Minimum ~0.5% blind defense strategy heads-up against an MP open-raise
The minimum 0.5% defense is covered by:

• 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
• This gives 6% total defense

Alternatively

• 3-bet {KK+} for value together with 20% of “OOP 3-bet air list”
• Flat {QQ,AK}
• This gives ~4% total defense

Minimum ~4% blind defense strategy heads-up against an MP open-raise
The minimum 4% defense is covered by:

• 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
• This gives 6% total defense

Alternatively

• 3-bet {KK+} for value together with 20% of “OOP 3-bet air list”
• Flat {QQ,AK}
• This gives ~4% total defense

Minimum ~7% blind defense strategy heads-up against a CO open-raise
The minimum 7% blind defense is covered by:

• 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
• Flat {JJ-TT,AQ}
• This gives ~8% total defense

Minimum ~16% blind defense strategy heads-up against a button open-raise
The minimum 16% blind defense is covered by:

• 3-bet {TT+,AQ} for value together with 100% of “OOP 3-bet air list”
• Flat the whole “OOP flat list”: {99-77,AJs-ATs,AJo,KTs+,KQo,QJs,JTs} =70 combos
• This gives ~17% total defense

Understanding the trend is just as important as the ranges we have defined. There should be as dramatic change in mindset for the players in the blinds when the raiser moves from the button to UTG. Against a button raise we’re prepared to fight fiercely, but against an UTG range we’re content playing only our premium hands for value and avoiding trouble against a strong range with our medium strong hands and our weak hands. We should not try to outplay a tight UTG range, since he is effectively protected by his range (a tight range is easy to defend correctly), and he has position to boot.

We now turn to Alice and generalize the theory for her defense heads-up in position against Bob’s 3-bet from the blinds.

3. Generalizing the theory for heads-up defense in position against a 3-bet
In Part 3 we studied the scenario where Alice openraises a 35% open-range on the button, and then she gets 3-bet by Bob in the blinds. Mathematics dictates that Alice defends 30% of her opening range to prevent Bob from profitably 3-bet-bluffing any two cards. If Alice only 4-bets or folds (like she did when out of position) we can stick to 30%, but when Alice has position, it will also be profitable for her to defend some hands by flatting.

But when Alice defends partly by flatting, Bob’s 3-bet bluffs will sometimes get to see a flop (instead of having to fold to a 4-bet), and then he will sometimes outflop Alice’s better hands. To compensate for the fact that Bob now gets to freeroll flops this way, Alice needs to defend more than 30% total.

We defined the concept “call multiplier” in Part 3 to take into account that Bob freerolls flops with his 3-bet bluffs those times Alice defends against his 3-bets by flatting. We start by giving Alice a 4-bet value range, and then we add the optimal amount of 4-bet bluffs to get a 60/40 value/bluff ratio. This gives us a total 4-bet%, for example 10%. In that case Alice has to flat 30 – 10 =20% by flatting g to get to 30% total defense. But since this lets Bob freeroll flops, we scale the flatting percentage with a call multiplier, which is some number > 1. We elected to use 1.5, and with these numbers Alice now has to flat 1.5 x 20 =30% of her range in addition to the 10% she 4-bets.

3.1 Default defense in position against a 35% open-raise from the button
We defined this strategy in Part 3, and we started by choosing {QQ+,AK} =34 combos as our 4-bet value range for Alice. She balances this with (2/3) x 34 =23 combos of 4-bet bluffs for an optimal 40/60 value/bluff ratio. We pick the 4-bet bluffs from the hands not quite good enough to flat, and we chose {ATo,A9s-A7s} =24 combos.

This gives us a total of 34 + 24 =58 4-bet combos, which is 48/458 =13% of Alice’s 35% button range with 458 combos in it. She now needs 30 – 13 =17% flatting to get to 30% total defense, and we scale up this percentage with the call multiplier of 1.5 and get 1.5 x 17% =26% flatting. This corresponds to 0.26 x 458 =119 combos from her opening range, and we picked {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs} =120 combos.

This gives us the following total default defense strategy on the button against a 3-bet from the blinds:

• 4-bet {QQ,AK} for value and {ATo,A9s-A7s} as bluffs
• Flat {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs}
• We then defend with 34 + 24 + 120 =178 combos total, or178/458 =39% of the button open-range. 13% by 4-betting and 26% by flatting

We now quickly repeat this process for open-raising from CO (25% open-range) and EP =UTG/MP (15% open-range for both). In all cases we choose to start with the value range {QQ+,AK}, and then we build the rest of the total defense strategy around this value range, using optimal 4-betting, 30% total defense, and a call multiplier for the flatting range.

3.2 Default minimum defense in position against a 3-bet after a 25% open-raise from CO
Alice’s default CO range is 25% with 326 combos in it. We 4-bet {QQ+,AK} =34 combos for value, and balance this with the 4-bet bluffs {AT,A9s-A8s} =24 combos. Note that this is a small change relative to the 4-bet bluffs we used on the button. As we shall see in a minute, we choose not to flat 3-bets with ATs in CO, so we demote this hand to the 4-bet bluffing range and kick out A7s.

So we 4-bet a total of 34 + 24 =58 combos, or 58/326 =18% of our 25% open-range with 326 combos in it. We flat 30 – 18 =12% to get to 30% total defense, and we scale this with the 1.5 call multiplier to get to 1.5 x 12% =18% flatting. This is 0.18 x 326 =59 combos from our opening range, and we can choose {JJ-99,AQ-AJ,KQs,QJs,JTs} =62 combos (using a few combos extra doesn’t matter)

Our total minimum defense strategy in CO heads-up against a 3-bet from the blinds becomes:

• 4-bet {QQ,AK} for value and {AT,A9s-A8s} as bluffs
• Flat {JJ-99,AQ-AJ,KQs,QJs,JTs}
• We defend with 34 + 24 + 62 =120 combos total, or 120/326 =37% of our opening range. 18% by 4-betting and 19% by flatting

3.23 Default minimum defense in position against a 3-bet after a 15% open-raise from EP =UTG/MP
We treat these two positions as the same, since we open with the same default range in them. Alice’s default EP range is 15% with 194 combos in it. We 4-bet {QQ+,AK} =34 combos for value and balance this with the 4-bet bluffs {AQ,AJs-ATs} =24 combos (and we’ll see why in a minute).

So we 4-bet a total of 34 + 24 =58 combos, or 58/194 =30% of the 15% open-range with 194 combos in it. This means we don’t have to flat to defend sufficiently! We can flat medium strong hands like AQ, JJ, KQs if we want to, but we don’t have to in order to prevent Bob from exploiting us with loose 3-betting. So if we stick to a minimal strategy, we end up with the same defense against a 3-bet that we use out of position in EP.

The total minimal defense strategy in EP =UTG/MP heads-up against a 3-bet from the blinds then becomes:

• 4-bet {QQ,AK} for value and {AQ,AJs-ATs} as bluffs
• No flatting
• We defend with 34 + 24 =58 combos total, or 58/194 =30% of our opening range. 100% by 4-betting and 0% by flatting

Does this make sense intuitively? Yes, since we have to expect the 3-bettor to have a tight range when he chooses to 3-bet our tight opening range heads-up and out of position from the blinds (and we saw that very tight 3-betting was correct previously in this article). Even if it feels overly tight to fold hands like JJ-TT, AJ and KQs in this case, the mathematics of the situation ensures that we don’t have to play them to defend optimally.

When we start with a 15% opening range, our tight range protects us, and all we have to do is to defend with a tight value range {QQ+,AK} balanced with an optimal number of 4-bet bluffs. There’s nothing anyone can do to exploit this defense strategy, even if we fold everything else.

Since 3-betting from out of position against an early position open-raise usually is done with a very tight range, I recommend that you stick to this minimal defense strategy from EP and don’t try to exploit the 3-bettor by flatting lots of medium strong hands (unless you have strong reads on him). If he 3-bets very loosely and then plays poorly postflop, you can of course flat hands like JJ, TT, AQ, AJ, KQ and use position to play them profitably postflop. But as a default, protect yourselves by using the unexploitable default strategy.

3.4 Summary of the theory for defense heads-up in position against a 3-bet
We elected to use a simple approach to this problem. We used the same value range {QQ+,AK} for all positions, balanced this with an optimal amount of 4-bet bluffs (picked from the best hands not good enough to flat) and the we did the rest of the defense with a flatting range. The flatting range started out very wide on the button, tighter in CO, and it disappeared in UTG/MP where we could defend sufficiently by only 4-betting.

These strategies should be simple to memorize if you already know the strategies for defending against 3-bets out of position (where we only 4-bet or fold) and the button defense we defined in Part 3. The defense from UTG/MP is identical both in and out of position, and we have already covered defense against 3-bets on the button. So the only new strategy to memorize is the CO strategy (and it’s not complicated).

We have now defined a complete set of defense strategies to use against 3-bets heads-up after open-raising from outside the blinds, both in and out of position. Memorize all of these strategies, and you will have a strong foundation to build your preflop game on. It will also be easy for you to adjust to your opponents when you spot opportunities to improve on the default strategies by deviating from optimal play.

4. Summary
We have generalized the theory for 3-betting heads-up out of position against an open-raise from an arbitrary position, and we have also discussed the raiser’s defense against this. We used a simple model to study trends in the minimum blind defense requirements for heads-up blind defense as a function of the raiser’s position. We found that we have to defend loosely and aggressively against a button open-raise, but we can play very tight against raises from early positions (assuming the players with position on the raiser defend as actively as they should).

The reason for this trend is that all players share the responsibility of defending the blinds against a raiser. And the players with position (particularly the button) should defend more than the players out of position. So heads-up against an early position open-raiser we can play tight, without worrying about being exploited. But against an open-raise from late position we have to defend very loosely to prevent the raiser from running over us with loose open-raising. Particularly when the raiser is on the button.

In Part 5 we’ll discuss two topics:

– Squeezing (3-betting in a multiway pot after a raise + call in front of you)
– Cold 4-betting (4-betting after a raise + 3-bet in front of you)

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