1. Introduction
This is Part 6 of the article series "PLO From Scratch". The target audience is micro and low limit players with some experience from limit or no-limit Hold'em, but little or no PLO experience. My goal with this series is to teach basic PLO strategy in a systematic and structured manner.

Part 6 is a rather technical article where we'll study a narrow topic thoroughly:

- 4-betting
- Playing against a 4-bet

When we say "technical", we mean that the discussion will revolve around mathematics, flop equity distributions, and simple assumptions about opponent ranges. The reason is that with 100 BB starting stacks, we have 1 pot-sized bet left after a 4-bet (assuming all raises up to this point were pot-sized). So the postflop play (if there is any) comes down to making range/pot odds decisions.

This means two things for us:

1. 4-bet pots with 100 BB stacks are big pots, and big pots are important pots
2. Correct play in 4-bet pots with 100 BB stacks can be well described by (relatively) simple mathematics

Strategy that is technical and cut-and-dried is easy to learn. Therefore the topic 4-betting/playing against a 4-bet gives us plenty of "low hanging fruit", and leaks in this area (e.g. folding too easily on the flop in heads-up 4-bet pots) are often easy to fix.

We already have the necessary tools (flop equity distributions, pot odds/equity, ProPokerTools), and everything we do in this article have been discussed in previous articles. Those who need to brush up flop equity distributions, can reread Part 3.

To keep the discussions simple with focus on the most important concepts, we will assume we're playing with 100 BB stacks throughout this article, and we'll look at play heads-up outside the blinds. In other words, we'll be studying heads-up scenarios where:

- We raise and get 3-bet by a player with position on us
- We 3-bet a raiser we have position on, and he 4-bets us

We discuss 4-betting first, then playing against a 4-bet. We'll illustrate the theory with examples along the way.

2. 4-betting
As an introduction to the topic, let's look at an example scenario:

2.1 Example of getting 3-bet by a loose range

You ($10) raise to 3.5 BB with A K K J in CO, button ($15.20) 3-bets to $1.20 BB, and it's folded to you. You have only played 60 hands against button, but you have already seen him 3-bet 4 times in position with speculative hands that went to showdown. You suspect he has a wide range for speculative 3-betting and bluff 3-betting in position. What is your plan?

First, let's think about how this scenario would have played out against an opponent who only 3-bets AAxx:

Against a Villain who only 3-bets AAxx, we have at most 34% preflop-equity against a random AAxx (ProPokerTools calculation) with a hand that has both a pair and an ace in it. 4-betting is therefore out of the question, since we will get 5-bet and be forced to call off the rest of our stack as a big underdog (we have to call since we'll be getting more than 2 : 1 pot odds as less than a 2 : 1 underdog)

If Villain 3-bets all AAxx and then overplays them postflop, it will be mathematically possible to play this premium pair profitably by calling and playing fit-or-fold postflop (and we can show this using modeling with flop equity distributions), and we can consider a call in this case. But if Villain only 3-bets good AAxx and plays them well postflop, he will keep our implied odds to a minimum, and we'll have a hard job making this a profitable call for us.

So regardless of how nice our hand looks, we have good arguments for folding it against a competent player who only 3-bets good AAxx, if the alternative is to call and play fit-or-fold out of position. At any rate, if we get 3-bet by AAxx, we have a close and tricky decision with our premium hand, regardless of Villain's tendencies. But what happens when Villain 3-bets a wide range?

Against a Villain who exploits his positional advantage by 3-betting a wide range of non-AAxx hands, we clearly can not fold. We have a hand that dominates a large portion of Villain's hands pretty hard, and we will often flop either the best hand, the best draw, or both. So we should at least call.

Is calling the best choice? Not necessarily, since calling sets us up for playing postflop out of position with almost 90% of the stack intact. This gives Villain more opportunities to use his positional advantage postflop.

But what if we 4-bet pot to $3.75? Now we eliminate Villain's implied odds, since a call from him sets us up for flopping at least an overpair in a scenario where we have less than one pot-sized bet left in the stack. Note that we have to call if he 5-bets all-in, since we'll be getting pot odds $13.9 : $6.25 =2.22 : 1 (we need 31% equity, and we have 34%).

But since Villain has a wide 3-betting range, and since we have an ace in our hand (which reduces the likelihood Villain has AAxx by half), he usually won't have AAxx. In this case he folds or calls. If he folds, great! If he calls, we have the best hand preflop, and we can play the rest of the hand as if we had AAxx. In other words, we can push any flop profitably, and Villain won't be able to do anything about it.

Thus, the most important factors are:

- Villain rarely has AAxx, since he 3-bets a wide range, an we have an ace in our hand
- Our KKxx plays almost as well as AAxx against Villain's non-AAxx hands.

We 4-bet pot to $3.75 BB. Villain calls.

Flop: T 6 3 ($7.50)
You have $6.25 left in the stack. What is your plan?

You have set yourself up for pushing any flop profitably, without Villain being able to exploit it, and this is one of the better flops for you. It's uncoordinated, and you have an overpair + outs to the only better pair + two backdoor flush draws and one backdoor straight draw.

Flop: T 6 3 ($7.50)
You ($6.25) push, button ($11.45) calls.

Turn: T 6 3 7 ($20.15)

Turn: T 6 3 7 2 ($20.15)
You win with the nut flush. Button has T 9 8 6 . He flopped two pair, improved to a straight on the turn, but lost to our runner-runner flush.

Wo-ho! It might appear like we got lucky here, but did we? We had 65% equity preflop(ProPokerTools calculation), and got most of our stack in as a big favorite.

Villain then out-flopped us with top two pair, and the rest of the stack went in on the flop with only 34% equity (ProPokerTools calculation). But our flop play was mathematically correct, and we would have gone all-in even if we had seen Villain's hand, since we got pot odds more than 2 : 1 as less than a 2 : 1 underdog.

Note that a significant portion of our flop equity (3 backdoor draws =3 outs) came as a result of us starting with a premium coordinated hand. And it was one of the backdoor draws that saved us this time. For some, this will look like luck, but this is something we expect to happen a lot when we start with premium hands. Premium starting hands often flop extra pieces of equity in addition to the hand's main strength component, and this gives us more flops where we can commit profitably and realize all our equity.

We'll take the insights from this example with us and do a more thorough study of light 4-betting (i.e. 4-betting with non-AAxx hands) against a loose 3-bettor.

2.2 What is a loose 3-betting range?
Before we decide which hands to 4-bet against a loose 3-bet, let's first agree on what a loose 3-betting range is.

We start by dissecting our own range for 3-betting heads-up in position, and then we assign percentages to the different types of hands in this range. For this task, we use the "Count" function found in the ProPokerTools Beta Version

From Part 4 we remember that the value part of our core strategy 3-betting range is:

- Premium AAxx, at least single-suited, with a pair, 2 Broadway cards, or a connector
- Premium Broadway wraps, at least single-suited, and preferably with an ace
- Premium KKxx, QQxx, JJxx, at least single-suited, and with connected side cards, or another high pair

For example:

- A A Q Q
- A A K J
- A A T 9
- A Q J T
- K Q J T
- K K Q J
- K Q Q J
- Q J J 9
- K K Q Q

While the speculative 3-betting range is:

- Good, suited rundowns
- Suited aces with good rundowns

For example:

Q J T 9
9 8 7 6
Q T 9 8
J 9 8 7
T 8 7 5

A T 9 8
A 9 8 7
A J 9 8
A 8 7 6

Let us be very specific and define a total 3-betting range, formulated in ProPokerTools notation, based on these definitions. We start by splitting the total range into non-overlapping sub-ranges (non-overlapping =each hand lies in one and only one sub-range).

For each sub-range we count the number of combinations and the percentage of all starting hands they make up (by dividing on the total number of Omaha hands, which is 270725). Finally, we calculate the total 3-bet percentage by summing the percentages for all the sub-ranges (and we can do this summation since the sub-ranges do not overlap).

2.2.1. Premium AAxx hands
Our starting point is the range of all AAxx hands (but not AAAx):

AA** ! AAA*
=6768 /270725 (2.50%)

Then we define the range of premium AAxx hands as the range of all double-suited AAxx, plus all single-suited AAxx with either two Broadway cards, a pair, or a connector/one-gapper down to 76/86:

(AA** &(*s*s*h*h,*s*s*d*d,*s*s*c*c,*h*h*d*d,*h*h*c*c,*d*d*c*c)), ((AABB,AABT,AATT,AA99,AA88,AA77,AA66,AA55,AA44,AA33,AA22,AAT9,
AAJ9,AA98,AAT8,AA87,AA97,AA76,AA86) & (*s*s**,*h*h**,*d*d**,*c*c**))
=2160 /270725 (0.80%)

under this definition, 2160/6768 =0.32 =32% of the AAxx hands are premium.

2.2.2 Premium Broadway hands
We define this as the range of any 4 cards T or higher, or any 4 cards 9 or higher with an ace (but not AAxx, TTxx, 99xx, or trips). We write this range as two sub-ranges:

Four cards T or higher:

(BBBB,BBBT) &(*s*s**,*h*h**,*d*d**,*c*c**)
=2762 /270725 (1.02%)

Four cards 9 or higher with an ace:

(A9BB,A9BT) &(*s*s**,*h*h**,*d*d**,*c*c**)
=1644 /270725 (0.61%)

Note that these ranges don't overlap, and they also don't overlap with the AAxx range defined previously. You can check this by merging the ranges. You will then see that the merged range contains exactly as many hands as the sum of the number of hands in the sub-ranges.

At any rate, the sub-range of premium Broadway hands makes up 2762 + 1644 =4406 combinations, and 1.02% + 0.61% =1.63% of all Omaha hands.

2.2.3. Speculative hands
We define the sub-range of good suited rundowns as the range of all rundowns xxx9 down to xxx5, at least single-suited and with at most single gap in the structure:

=4640 /270725 (1.71%)

The range of suited aces with good rundowns is defined as the range of suited aces with rundowns Axx8 down to Axx5 with at most one single gap in the rundown structure:

A765,A865,A875) &(As*s**,Ah*h**,Ad*d**,Ac*c**)
=1776 /270725 (0.66%)

With these definitions, the sub-range of speculative 3-betting hands makes up 4640 + 1776 =6416 combination in total, and 1.71% + 0.66% =2.37% of all Omaha hands.

2.2.4. Total range
- Premium AAxx hands: 2160 /270725 (0.80%)
- Premium Broadway hands: 4406 /270725 (1.63%)
- Speculative hands: 6416 /270725 (2.37%)
- Total: 12982 /270725 (4.80%)

So we end up with a 3-bet% of 4.80%. We round this number up to 5%, and use this as a baseline for 3-betting heads-up in position with premium high cards hands and premium speculative hands.

The AAxx/Broadway part of this 3-betting range is 0.80% + 1.63% =2.43% of all hands. These hands make up 2160/12982 + 4406/12982 =0.166 + 0.339 =16.6% + 33.9% =50.5% of the total 3-betting range. So the 3-betting range is approximately divided 50-50 between premium high cards hands and premium speculative hands (the way we chose to define these categories) and 1/6 of the range is AAxx hands.

Note that the number of premium AAxx + premium Broadway hands (2.43% of all Omaha hands) is almost equal to the percentage of all AAxx hands (2.50% of all Omaha hands). So if you get 3-bet by a player with a 3-bet% around 2.5%, this does not necessarily mean he is only 3-betting AAxx hands. It can also mean a tight range of premium AAxx and premium Broadway hands. At any rate, this type of tight 3-betting range is not something you want to attack with light 4-betting, even if Villain does not always have AAxx.

Another thing to comment on is this: The 3-betting range we have put together here is approximately the core strategy range we use for 3-betting heads-up in position. But this does not mean we 3-bet all these speculative hands every time the situation comes up (we also have to assess the circumstances, and not only our hand). This means that the distribution of hand types in our range after we have 3-bet is not necessarily the same as the distribution of hand types in the range of possible 3-betting hands.

At any rate, this range consists of premium suited and coordinated high card hands and speculative hands, that all play well in a 3-bet pot heads-up against the raiser. And we will use the associated 3-bet% of ~5% as a baseline to assess other 3-betting ranges.

We now ask:

How large does the 3-bet% have to be for us to be sure that Villain is 3-betting a wide range of speculative hands?

We start by assuming that an aggressive 3-bettor with position on us first will loosen up his 3-betting with AAxx hands. So let's include all AAxx hands in the range and see what we get:

- All AAxx hands: 6768 /270725 (2.50%)
- Premium Broadway hands: 4406 /270725 (1.63%)
- Speculative hands: 6416 /270725 (2.37%)
- Total: 17590 /270725 (6.50%)

The 3-bet% increases to 6.5%. Note that the relative percentage of AAxx hands in the range increases to 6768/17590 =0.385 =38.5%, so this isn't necessarily a range we want to attack with light 4-betting.

But what we can conclude is this: When we have counted all AAxx hands plus the best of the rest from the other hand categories, the 3-bet% is still less than 7%.

Now, let's say we encounter a very loose-aggressive player with a 3-bet% of 12% on the button. Based on the range analysis above, we know that his range has to contain lots of medium/low so-so hands. For example, rough rundowns like T 8 6 5 .

So the thing you can take away from this range analysis work is that the threshold for value 3-betting and 3-betting with good speculative hands is somewhere in the region 5-7% (depending on how many AAxx hands we include). We also saw that the threshold for 3-betting with only premium AAxx hands and premium Broadway hands is ~2.5%. Finally, a player who 3-bets all AAxx hands and only AAxx hands, will have a 3-bet% of 2.5%.

Armed with these percentage you can now use player stats + reads + hands seen at showdown to decide whether or not you should counter a 3-bettor with light 4-betting.

"Light 4-betting" here means 4-betting a strong range of all AAxx + selected premium non-AAxx hands. Determining which non-AAxx hands we can 4-bet is the next step of the process:

2.3 What is a reasonable core strategy 4-betting range against a loose 3-bettor?
When we go from 4-betting only our AAxx hands to 4-betting a range of AAxx hands and premium non-AAxx hands in a heads-up 3-bet scenario, we base this on the fact that Villain is 3-betting a wide range. Therefore, he rarely has AAxx, and many of his non-AAxx hands are of dubious quality.

In this case, the equity for premium high cards hands (e.g. K K Q Q , A K K J , A K Q J ) will increase for two reasons:

- We're less often up against AAxx
- We have good equity against Villain's non-AAxx hands

The latter is based on the fact that when Villain has some medium/low hand (e.g. 9 8 6 4 ), our high pairs will perform almost as well as AAxx.

For example, against 9 8 6 4 we have 40.32% equity with AAxx (ProPokerTools calculation) and 39.84% with KKxx (ProPokerTools calculation), so it doesn't matter much which hand we hold when we 4-bet.

To further decrease the probability that Villain holds AAxx, we can also demand that we hold an ace ourselves. This reduces the chance Villain has AAxx with 50%, as shown below:

We first count all AAxx combinations, then we count all AAxx after removing one ace from the deck. The number of possible AAxx hands is then reduced from 6768 to 3384, which is exactly half.

We now define our premium core strategy 4-betting range as:

- AKKx + AQQx, at least single-suited to the ace and with a Broadway kicker
- AKxx, at least single-suited to the ace and with two Broadway kickers

In other words, hands like these:

- A K K Q
- A K Q Q
- A K Q T

This range is of course not carved in stone, and if we want to, we can include more premium AKKx/AQQx/AAKxx hands. For example, we can lift the criterion about suitedness to the ace, and accept any suited combination. We can also add some deception to the range by including the best rundown hands, for example double-suited KQJT, QJT9 and JT98 (these hands perform well against the wide range of rough rundown hands like J976 in a loose 3-bettor's range).

So you can experiment with an even wider 4-betting range against very loose 3-bettors as you gain more experience. But use the range we have defined here as a core strategy component to fall back on when in doubt.

To draw a Hold'em analogue, this 4-betting range corresponds roughly to 4-betting KK, QQ and AK instead of only AA. To find the number of combinations of non-AAxx 4-betting hands, we formulate the range in ProPokerTools notation and count it:

& (As*s**,Ah*h**,Ad*d**,Ac*c**)
=804 /270725 (0.30%)

Our total 4-betting range made up of any AAxx plus the AKKx/AQQx hands above then becomes:

(AA** ! AAA*),
& (As*s**,Ah*h**,Ad*d**,Ac*c**))
=7572 /270725 (2.80%)

This gives us a 4-betting range of 2.80% of all hands. The range is heavily weighted towards AAxx, but with 804/7572 =0.11 =11% premium non-AAxx-hands. However, before we can start using this range, we need to answer one last question:

How do we play the non-AAxx hands against a 5-bet?

This is a simple pot-odds question (remember: we're discussing play with 100 BB stacks). Assume we raise to 3.5 BB from CO, Villain 3-bets to 12 BB from the button, we 4-bet to 37.5 BB, and Villain pushes. The pot is now 139 BB with 62.55 BB to us. The pot odds are then 139 : 62.5 =2.22 : 1, and we need 1/(2.22 + 1) =0.31 =31% equity to call profitably all-in.

Below are the equities against random AAxx for 3 double-suited versions of AKKx, AQQk, and AKxx from our 4-betting range. The equities for the single-suited versions (we turn the last card into a club) are in parentheses:

- A K K Q : 33.99% (29.80%)
- A Q Q J : 35.69% (31.62%)
- A K Q J : 34.49% (30.64%)

We conclude that the single-suited hands are break even or slightly -EV, while the double-suited hands are slightly +EV. This is a situation where we can't make a big mistake, no matter what we do if Villain only 5-bets AAxx.

But note that against a Villain who only 3-bets premium AAxx hands, we have more reason to fold when he 5-bets. The reason is we're now up against a better range of AAxx hands that are also suited and coordinated. To be sure, let's test this by repeating the calculations above where we substitute any AAxx with the premium AAxx range defined previously::

(AA** &(*s*s*h*h,*s*s*d*d,*s*s*c*c,*h*h*d*d,*h*h*c*c,
*d*d*c*c)), ((AABB,AABT,AATT,AA99,AA88,AA77,AA66,AA55,
AA44,AA33,AA22,AAT9, AAJ9,AA98,AAT8,AA87,AA97,AA76,AA86)
& (*s*s**,*h*h**,*d*d**,*c*c**))!AAA*
=2160 /270725 (0.80%)

- A K K Q : 32.36% (28.87%)
- A Q Q J : 33.81% (30.37%)
- A K Q J : 32.24% (29.00%

Not a large effect, but all the single-suited hands are now -EV while the double-suited hands are even more marginal.

But it still isn't possible to make a big mistake against a 5-bet from AAxx, no matter what we choose. So if you have reasons to believe Villain can 5-bet non-AAxx hands, by all means call with all of these hands. Making a slightly -EV call can also have a positive effect on your table image. We tell our opponents that we're willing to fight hard for pots where we have made a large investment, and this might cause them to play more straightforward against us (and this is good for us).

2.4 The effect of light 4-betting
To illustrate what light 4-betting does for us, let's go through an example where we 4-bet a loose 3-bettor. We let Villain's hand be known, while our hand is unknown (but we have a known range).


You ($10) raise to $0.35 with a wide open-raising range from CO, and button ($10) 3-bets to $1.20 with J 9 7 6 , you 4-bet to $3.75. Villain now assumes he is up against AAxx, and he has an easy call with a decent suited rundown hand (and later in this article we'll see why this is an automatic call).

Flop: K 9 2 ($7.50)
You ($6.25) push the rest of the stack in. What should Villain ($6.25) do?

Villain now has a pot odds decision. He's getting 13.75 : 6.25 =2.2 : 1, so he needs 1/(2.2 + 1) =0.31 =31% equity.

He has flopped 2nd pair + backdoor flush draw + backdoor straight draw. If you have AAxx (which he assumes), he now has:

- 2 mostly clean outs to trips
- 9 dirty outs to two pair
- 2 mostly clean outs to a backdoor flush or a backdoor straight

Villain decides to count the trips and backdoor outs as clean. The 9 outs to two pair are not all clean, and should be discounted a bit. When Villain makes two pair on the turn, our presumed AAxx hand has 8 outs (2 aces, 3 longer, 3 deuces) to top set or a better two pair

8 outs on the turn is approximately a 1/5 chance, so Villain reduced his 9 two pair outs to 9(4/5) =7.2 =7 outs. Then he conservatively removes 1 more out to account for the fact that our AAxx hand also gets some equity from the side cards. Villain now estimates that he has:

- 2 clean outs to trips
- 6 clean outs to two pair
- 2 clean outs to a backdoor flush or a backdoor straight

The total is 10 clean outs, and therefore 3 x 10 + 9 =39% equity on the flop. So he has more than the required 31%, and he happily calls our all-in c-bet. To check his math, we do a ProPokerTools calculation, and we see that Villain has 38.91% equity on the flop. His equity estimate was therefore spot on against our presumed hand.

But how is Villain doing against our actual 4-betting range made up of AAxx + premium AKKx, AQQx and AKxx? Not quite as well, since he now has 37.71% equity (ProPokerTools calculation).

Not a dramatic difference, but we definitely have succeeded in reducing the profitability of his call of our 4-bet. We have also "thrown some sand" into the eyes of the opposition by demonstrating a willingness to 4-bet non-AAxx hands. What will come out of this (if anything) in future hands is hard to tell, but whatever adjustments our opponents make will probably be to our advantage.

For example, of they start 5-betting light they will get severely punished, since we have AAxx 8 out of 9 times when we 4-bet (remember: only 11% of our 4-betting hands are non-AAxx hands). Or if they they adjust by 3-betting less (since we now 4-bet them more often) this will also be good for us.

Another advantage we got from the 4-bet was making the hand easier to play. Instead of calling and sitting out of position with $8.80 left in the stack and important postflop decisions to make, we set ourselves up for auto-pushing any flop and thereby making postflop play a formality. This reduced Villain's positional advantage significantly, and this is a factor that should not be underestimated.

Let us end this example by playing out the hand. The turn and river cards are unimportant, since we got all-in on the flop with huge +EV, but let's be nasty and let the turn and river be:

Turn: K 9 2 Q ($20.15)

River: K 9 2 Q 2 ($20.15)

Villain with his J 9 7 6 rejoices over the turn and river cards until he sees your superior A K K J . Villain now screams in agony and 3-bets the next 7 hands in rage.

3. Defending against a 4-bet from likely AAxx
The last half of this article is about a scenario that occurs more frequently than light 4-betting, namely defending against a 4-bet from likely AAxx.

This is a part of the preflop game that is very important to get right for a player who often exploits his position by making light 3-bets heads-up (which we have discussed thoroughly in Part 4 and Part 5). The more we 3-bet, the more we give AAxx the opportunity to 4-bet, and it's important that we know how to correctly defend against these 4-bets.

3.1 Core strategy for defending against a 4-bet
As briefly discussed in Part 4, our core strategy for defending against a 4-bet heads-up with 100 BB stack consists of:

1. Assume Villain has AAxx and play accordingly
2. 5-bet AAxx all-in
3. Call the 4-bet with hands that play sufficiently well against AAxx, and where your postflop plan is to go all-in on flops where you have the minimum required equity

Point 1 usually holds, especially at lower limits, and if you're mistaken, it's usually not by much on average.

Point 2 is obviously correct, since we're now usually coinflipping against another AAxx. We could be an underdog with trashy AAxx (e.g. the trashy A A 7 2 is a 41.5% underdog against the premium A A K Q as shown by this ProPokerTools calculation), but the pot is big enough that we don't have to worry about that. Also, note that we rarely 3-bet trashy AAxx anyway, so this scenario is not very likely.

The point that needs work is point 3. We now want to find the answer to the following question:

Which hands perform well enough against AAxx to call a 4-bet from likely AAxx with 100 BB stacks?

We made some general comments about this topic in Part 4, and we postulated that we want suited and coordinated hands that often hit a piece of the flop. Hitting lots of flops allows us to often go all-in with +EV on the flop, and thereby get a return on our big preflop investment. Suited rundowns like 9 8 7 6 are obviously suitable for this purpose, but what other hands can we play here? And which hands can absolutely not call a 4-bet?

To get to the bottom of this, we need to experiment a bit. In previous articles we have done a lot of mathematical modeling using flop equity distributions, and we'll use this "work horse" one more time.

3.2 A model for defending against a 4-bet from AAxx with 100 BB stacks
Below is our model:
  • Villain raises to 3.5 BB in CO
  • We 3-bet to 12 BB on the button
  • Villain 4-bets to 37.5 BB with AAxx, and we call
  • Villain c-bets all flops all-in
  • We call when we have sufficient equity (31%) on the flop

Calling the 4-bet (37.5 BB) costs us 25.5 BB. The pot is now 75 BB on the flop with 62.5 BB remaining stack. Villain now c-bets the last 62.5 BB all-in. Our pot-odds are (75 + 62.5) : 62.5 =137.5 : 62.5 =2.2 : 1. We need 1/(2.2 + 1) =0.31 =31% equity to call profitably.

Now we determine the EV for calling the preflop 4-bet:

We call Villain's all-in c-bet on some percentage top_x% of flops where we have at least 31% equity. Otherwise we fold and lose the amount we called (25.5 BB). When we call all-in, we have flop equity av_equity on average in a 201.5 BB pot where we have invested 88 BB as of our call of the 4-bet.

We find top_x and av_equity from flop equity distribution graphs, and then we calculate the EV for calling the 4-bet from the following equation:

EV (call 4-bet)
=(1 - top_x)(-25.5 BB) + top_x{av_equity(201.5 BB) - 88 BB}

We now pick some representative 3-betting hands to use in our model. We remember that our core strategy for 3-betting distinguishes (conceptually) between 3 types of 3-bets:

- Value 3-betting
- Speculative 3-betting
- Bluff 3-betting

3.3 When we have 3-bet for for value and get 4-bet by AAxx
Our core strategy for 3-betting for value is:

- Premium AAxx, at least single-suited, with a pair, 2 Broadway cards, or a connector
- Premium Broadway wraps, at least single-suited, and preferably with an ace
- Premium KKxx, QQxx, JJxx, at least single-suited, and with connected side cards, or another high pair

With AAxx we 5-bet all-in, so the interesting cases are Broadway wraps and premium pairs. We begin by picking 5 ultra-premium double-suited hands from this category (i.e. a best case scenario for us):

- K K Q Q

- K K Q J
- Q Q J T
- J J T 9

- A K Q J

Below are the flop equity distributions for each of these hands, plus the calculated EV for calling the 4-bet:

3.3.1 Premium double-suited double pair (K K Q Q ) vs AAxx:

EV(call 4-bet)
=(1 - 0.46)(-25.5 BB) + 0.46{0.657(201.5 BB) - 88BB}
=+6.66 BB

3.3.2 Premium double-suited + connected KKxx (K K Q J ) vs AAxx:

EV(call 4-bet)
=(1 - 0.48)(-25.5 BB) + 0.48{0.591(201.5 BB) - 88BB}
=+1.61 BB

3.3.3 Premium double-suited + connected QQxx (Q Q J T ) vs AAxx:

EV(call 4-bet)
=(1 - 0.54)(-25.5 BB) + 0.54{0.583(201.5 BB) - 88BB}
=+4.19 BB

3.3.4 Premium double-suited + connected JJxx (J J T 9 ) vs AAxx:

EV(call 4-bet)
=(1 - 0.56)(-25.5 BB) + 0.56{0.582(201.5 BB) - 88BB}
=+5.20 BB

3.3.5 Premium double-suited ace high Broadway wrap (A K Q J ) vs AAxx:

EV(call 4-bet)
=(1 - 0.52)(-25.5 BB) + 0.52{0.541(201.5 BB) - 88BB}
=-1.34 BB

3.3.6 Summary of EV for premium double-suited hands from the value 3-betting range:

- K K Q Q : +6.66 BB

- K K Q J : +1.61 BB
- Q Q J T : +4.19 BB
- J J T 9 : +5.20 BB

- A K Q J : -1.34 BB

We observe that we can call profitably with high pairs that have double-suited and connected side cards, but the ace high Broadway wraps should be folded, even double-suited.

We note that QQxx and JJxx perform better than KKxx. The reason for this is that Villain's AAxx hand blocks many of the straights KKxx can make, while the lower pairs make more straights that do not involve an ace. To investigate whether this trend continues for even lower pairs, we will also do a simulation for a low pair with double-suited and connected side cards:

- 7 7 6 5

We'll also do some more research on the Axxx hands from the value 3-betting range, and we do this by also including double-suited high pairs with an ace:

- A K K J
- A Q Q J
- A J J T

Finally, we'll study the effect of having two suits by also calculating EV for the single-suited versions of all hands above. We then turn one of the cards into a club (K K Q Q , K K Q J etc.) and perform another round of flop equity distribution simulations.

We will not include all graphs and calculations here, and we simply list the results:

3.3.7 Final data set for the hands from our value 3-betting range:
Below are the EV for calling a 4-bet for all hands discussed so far. We list the double-suited versions, sorted by type (paired/unpaired and with or without an ace). The EVs for the single-suited hands are in parentheses:

- K K Q Q : +6.66 BB (+2.63 BB)

- K K Q J : +1.61 BB (-2.74 BB)
- Q Q J T : +4.19 BB (+0.03 BB)
- J J T 9 : +5.20 BB (+1.14 BB)

- 7 7 6 5 : +6.01 BB (+2.02 BB)

- A K K J : -1.04 BB (-3.83 BB)
- A Q Q J : +0.21 BB (-3.37 BB)
- A J J T : +1.31 BB (-2.61 BB)

- A K Q J : -1.34 BB (-5.79 BB)

3.3.8 Conclusion for calling a 4-bet with hands from the value 3-betting range:
We can draws some conclusion from the data set above:

High double pairs can call profitably against AAxx, both single- and double-suited. If we had done the simulations, we would probably have seen JJTT perform better than KKQQ due to better straight potential (AAxx blocks more of KKQQ's straights). We can also call with the double-suited and connected single pairs, and we fold all unpaired ace high hands.

Having 2 suits is very important for the high pairs, and you can fold the single-suited versions of KKxx and QQxx. But you can call with single-suited and connected JJxx and lower. We don't 3-bet low pairs like 7 7 6 5 as a default, but it's interesting to note that these pairs performs better against AAxx than KKxx and QQxx (because of the straight blocker effect).

We note that all Axxx hands struggle hard against AAxx. This is obvious, since these hands effectively only have 3 cards in play. Unpaired ace high Broadway wraps should always be folded, regardless of suits. The same goes for AKKx, while double-suited AQQx and AJJx are marginal calls (again because of straight blocker effects).

3.4 When we make a speculative 3-bet and get 4-bet by AAxx
Our core strategy for speculative 3-betting is:

- Good, suited rundowns
- Suited aces with good rundowns

We begin with two double-suited best case candidates:

- 9 8 7 6

- A 9 8 7

Below are the flop equity distribution graphs for both hands, and the calculation of the EV for calling the 4-bet:

3.4.1 Premium double-suited rundown (9 8 7 6 ) vs AAxx:

EV(call 4-bet)
=(1 - 0.64)(-25.5 BB) + 0.64{0.587(201.5 BB) - 88BB}
=+10.25 BB

3.4.2 Premium double-suited ace + rundown (A 9 8 7 ) vs AAxx:

EV(call 4-bet)
=(1 - 0.59)(-25.5 BB) + 0.59{0.565(201.5 BB) - 88BB}
=+4.74 BB

3.4.3 Summary of EV for premium double-suited hands from the speculative 3-betting range:

- 9 8 7 6 : +10.25 BB

- A 9 8 7 : +4.74 BB

We have already seen (Part 3 and Part 4) )in that perfect double-suited rundowns perform very well against AAxx in 4-bet pots. However, it's interesting to see that a double-suited ace + rundown performs well too, even if it's effectively a 3-card hand against AAxx. As noted previously, it's important to have straight potential that isn't blocked by AAxx.

To study the effect of gaps in the rundown structures, we'll also perform a simulation for

- T 8 7 5

And we'll also perform simulations for the single-suited versions of all these hands (9 8 7 6 , etc.)

3.4.4 Final data set for hands from the speculative 3-betting range:
Below are the EVs for all hands discussed so far (EV for single-suited hands in parentheses):

- 9 8 7 6 : +10.25 BB (+6.75 BB)
- T 8 7 5 : +10.19 BB (+6.05 BB)

- A 9 8 7 : +4.74 BB (+0.81 BB)

3.4.5 Conclusion for calling 4-bets with hands from the speculative 3-betting range:
We immediately conclude that suited rundowns are robust hands against AAxx. Going from a perfect rundown to a rundown with top + bottom gaps almost didn't change the EV. Rundowns benefit a lot from having 2 suits, but the single-suited hands were also clearly profitable.

When it comes to the suited ace + rundown hands, we see that these hands are very dependent on the second suit. The single-suited hands are marginally +EV, but we won't lose much by folding them.

3.5 When we make a bluff 3-bet and get 4-bet by AAxx
It's difficult to make generalizations for this category, and we have not defined any core strategy ranges here. Instead, we defined guidelines for when to consider a bluff 3-bet. We also stated that we should avoid pure trash hands, and stick to suited hands with a minimum of potential.

Now we'll look at a 4-bet scenario where we 3-bet a double-suited and very rough rundown and get 4-bet. In Example 4.11 in Part 4 we did a bluff 3-bet with Q 9 6 5 . We did not calculate the EV for calling a 4-bet, but commented that we would study this scenario in more detail in this article.

This means we're performing a simulation on the following double-suited rundown with two double gaps at the top:

- Q 9 6 5

3.5.1 Double-suited bluff 3-bet hand (Q 9 6 5 ) vs AAxx:

EV(call 4-bet)
=(1 - 0.68)(-25.5 BB) + 0.68{0.564(201.5 BB) - 88BB}
=+9.23 BB

3.5.2 Final data set for hands from the bluff 3-betting range:
We have only picked one hand to illustrate bluff 3-betting. We also run a simulation for the single-suited version (Q 9 6 5 ) and we get:

- Q 9 6 5 : +9.23 BB (+4.60 BB)

3.5.3 Conclusion for calling a 4-bet with hands from the bluff 3-betting range:
The conclusion from the one hand we studied here confirms what we found for the speculative 3-betting hands: rundowns are very robust hands against AAxx! By "robust" we mean that structural defects don't matter as long as the hand has some coordination and is suited. A gap or two changes very little, even multiple gaps at the top.

But before we fall in too much in love with these suited semitrash hands, keep in mind that we're now pitting them against AAxx only. In this scenario they perform well, but if we in some way get involved in a multiway pot with these hands, we often get in trouble because of their limited nut potential.

So here it's important to think about the relation between preflop play and postflop play, and the kind of postflop scenarios we try to set up with our 3-bet. Before we 3-bet these hands we have to assess the situation carefully. We want fold equity, and we want to steal the pot either preflop or get heads-up in position against a raiser over whom we have good control.

If we do the preflop part right, we will rarely find ourselves in tricky postflop spots. If we have assessed the situation correctly, we will usually get heads-up. If we get 4-bet, we have an easy job defending against likely AAxx. This job becomes easy when we have:

- Position
- Reliable information about Villain's hand (he only 4-bets AAxx)
- A hand that performs well against AAxx

And we have ensured these things by:

- Using position when we bluff 3-bet
- Having reads on Villain as a straightforward player
- Choosing a suited hand with a minimum of coordination

We conclude this part with two examples where we defend against a 4-bet. In the first example we have a "bread-and-butter" situation with a speculative rundown against likely AAxx, but we get a slightly trickier than usual job of counting outs on the flop when Villain probably has flopped top set.

In the second example we make a bluff 3-bet with a very rough double-suited hand, and we get 4-bet and 5-bet all-in in a 3-way pot. We shall see that this scenario is trivial (and profitable) under the assumption that both opponents are straightforward.

3.6 Example of defending against a 4-bet from likely AAxx: Counting outs against top set on the flop

CO ($10) raises to $0.35. He is straightforward and plays tight out of position after a 3-bet, so you elect to make a speculative 3-bet to $1.20 with J 9 7 5 . The blinds fold, and CO 4-bets to $3.75.

As we have seen, this is a routine call with a decent suited and coordinated hand against likely AAxx. You call, planning to go all in on all flops where you have sufficient equity. As we have seen several times already, the equity threshold is 31%. So you need to flop two pair or better, or a pair/draw combination with at least 8 clean outs (which will gives us 4 x 8 =32% equity).

Flop: A T 7 ($7.50)
CO ($6.25) pushes. Do you have enough equity to call here?

Flopping a pair + flush draw + gutshot against likely AAxx is normally worthy of rejoicing and a snap call. But here we need to be cautious. Villain probably has top set, and in that case all our two pair/trips outs are dead.

So let's see what we got:

- 8 outs to a good flush (3rd nut flush)
- 3 non-flush outs to a nut straight

But if we hit an out on the turn, Villain has 9 outs to quads/full house (1 ace, 3 tens, 2 sevens, plus 3 outs that pair the turn card). 9 outs on the turn correspond to ~1/5 chance (we go easy on the decimals when making these rough estimates), so we can reduce the 11 outs to 11(4/5) =~9.

In previous examples where we have counted outs, we have conservatively removed 1 additional out to account for the extra equity Villain gets from his side cards. But here all our outs are to minimum 3rd nuts (remember: all our two par and trips outs are dead), so we assume that further reduction is not necessary here.

Our 9 presumed clean outs give us 4 x 9 =36% on the flop. A check shows that the real equity against random AAxx is 34.98% (ProPokerTools calculation), and our estimate was pretty good.

So we call and let the poker gods decide our fate

Turn: A T 7 2 ($20.15)

River: A T 7 2 K ($20.15)

Not this time. Villain shows A A K 9 and takes the pot down with top set. We had 35.98% equity against his actual hand (ProPokerTools calculation) and we defended perfectly against his preflop 4-bet.

After such all-in clashes it's important to lower the shoulders and move on to the next hand calm, cool and collected. Loose-aggressive PLO play with lots of raising and 3-betting preflop unavoidably puts us in lots of marginal all-in spots like this one. And we will lose many of them. This should not bother us, and the only thing we need to be concerned with is whether or not we got our money in with sufficient equity.

If we consistently manage to get on the right side of 55/45 coinflips and marginal pot odds decisions like the one above, our job is done. The long run will take care of the rest (although we might have to wait a long time for our rightful profits if the PLO variance decides to torture us).

3.7 Example of defending against a 4-bet: Defending against 4-bet + 5-bet in a multiway pot

CO ($17.20) raises to $0.35. He is tight/solid and easy to outplay by 3-betting him in position. You assess the situation and elect to make a bluff 3-bet to $1.20 with J 8 5 4 . Small blind ($15.30), who is a solid and straightforward TAG, now 4-bets to $4.05, an CO pushes all-in. What is your plan?

Oops You have clashed with two monster hands, so you have to fold your very speculative rundown hand, right?

Not quite. Before you do anything, use the time available to you and think about what has happened up to this point in the hand. CO (tight/solid) raises, and he will have a wide range for this action. Then you 3-bet, and another tight/solid player made a 4-bet from out of position in a scenario where he is 150 BB deep against CO.

Unless the small blind is completely out of character, there is only one hand he can hold, and that is AAxx. CO probably also knows this, and when he pushes all-in, we can assume he also holds AAxx. So if we call, we expect small blind to call behind us so that we get all-in in a $30.10 3-way main pot (note that our opponents have us covered, so we're playing with a $10 effective stack).

When it comes 5-bet back to us, there is $8.80 to call in order to win a net $21.30 (assuming small blind calls behind us). Effective pot odds are 21.30 : 8.80 =2.42 : 1, and we need 1/(2.42 + 1) =29% equity to call.

Do we have this much equity? Yes, if they both have AAxx we are probably even the favorite. This is because the now block each other (i.e. they have each others outs). We punch our hand against 2 random AAxx hands into ProPokerTools and get:

Not only are we the favorite (we need more than 33% in a 3-way pot), we are a big favorite. So we're not only calling "defensively" for the pot odds, we have +EV for each additional chip that goes into the pot preflop.

By calling we invest $8.80 in an expected main pot of $30.10, where we have 41.6% equity. The EV for our call is:

EV (all-in)
=0.4116($30.10) - $8.80

Folding (EV =0) is therefore a big mistake which costs more than 1/3 of a buy-in. So we call, and we're happy to see the small blind calling behind us.

Flop: Q J 4 (Main pot: $30.10)

Turn: Q J 4 7 (Main pot: $30.10)

River: Q J 4 7 2 (Main pot: $30.10)
You take down the pot with flopped bottom two pair. CO has A A K 9 , while small blind has A A 8 8 . They both scream in agony, and CO threatens to report you to support for cheating.

You had 37% equity preflop against the actual hands, and you got your money in as a significant favorite in a 3-way pot (ProPokerTools calculation). Not bad!

What we take from this example is this: When you are up against two AAxx hands in a 3-way pot, you can go profitably all-in with an extremely wide range of speculative hand (especially when you have already invested money in the pot). And some of these hands are so ugly that we don't dare to mention them here, out of fear for the starting hand police. Note that for this all-in scenario to be profitable, you want all players to have approximately equal stacks. What you don't want is to get all-in in a big side pot heads-up against one of the AAxx hands.

4. Summary
We have discussed 4-betting/playing against a 4-bet, using a mathematical/analytical approach. Our goal has been to lay the foundations for a mathematically sound 4-bet strategy where we do the big things right.

Since this material (as usual) expanded a bit during the writing process, I chose to use a few thorough examples instead of many shorter ones. The strategies for 4-betting and playing against a 4-bet with 100 BB stacks are conceptually simple, and postflop play mostly comes down to going all-in on the flop. Playing these situations well is more a craft than an art form, and it isn't necessary to be a gifted poker player to get these things right. What you need is repetition, repetition, repetition.

With this article we have reached the end of the list of preflop concepts I planned to discuss. We have also talked about postflop play, but so far not with much subtlety and finesse.

The next stage is to talk about more advanced postflop concepts, and the plan is to make the rest of the article series be about postflop play. I haven't yet planned Part 7 and beyond in detail, so I will put on my thinking cap and come up with ideas for how to proceed.

Good luck!