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C-beting in NLHE 6-max- Part 1

1. Introduction
This is the first part of an article series about flop c-betting in NLHE 6-max. In the previous series “Optimal Postflop Play in NLHE 6-max” we looked at postflop play in the scenario where one player has raised preflop and gotten called by another player in position. We discussed how the player in position can defend optimally against c-bets on the flop, and against 2- and 3-barrels on the turn and river. Then we discussed how the preflop raiser can play the turn and river optimally after c-betting the flop and getting called, to prevent his opponent from exploiting him by floating.

For both the preflop raiser out of position and the flatter in position we built postflop strategies that prevents their opponent from exploiting them by betting or floating with any two cards on any street. The flatter in position has to defend enough against c-bets to prevent the preflop raiser from c-bet bluffing any two cards on the flop. And those times the preflop raiser has c-bet the flop and gotten called, she has to play the turn in such a way that she prevents the flatter from floating on the flop with any two cards (planning to steal the pot on the turn).

In our discussion of turn and river play for this scenario, we simply assumed that the preflop raiser had started postflop play by c-betting her entire range on the flop. When we looked at turn and river barreling we limited our study to dry flop textures, so this assumption was reasonable.

In this article we’ll look more closely at c-betting with “air” on the flop, heads-up as the preflop raiser. We’ll use a model where one player (Alice) openraises out of position and gets flatted by another player (Bob) in position. The flop comes, and Alice has a c-bet decision to make. We want to train our ability to recognize flop textures where Bob’s preflop flatting range has connected poorly, so that Alice can c-bet any two cards profitably on the flop.

We are then assuming that Bob is not willing to defend optimally. Because if he does, we can’t profit from c-betting any two cards per definition of optimal play. So we are assuming that Bob will fold more than the optimal amount on flop textures that are bad for his preflop flatting range (for example, a dry flop like J 6 2). In “Optimal Postflop Play in NLHE 6-max” we built optimal postflop strategies for Bob to use in this scenario, but now we’ll assume he behaves more like the players we meet in practice. And they will typically fold too much on flop textures that mostly misses their range.

In Part 1 of this series we’ll study how well different flop textures hit a typical preflop flatting range (we’ll use our standard 10% “IP flat list” range). Based on this, we can estimate the EV of a c-bet bluff with a worthless hand on different textures. In Part 2 we’ll vary the preflop flatting range and see how the EV of the c-bet changes when we’re up against a tight (~5%) and a loose (~15%) flatting range. This analysis will train our ability to identify profitable c-bet bluffing spots based on the flop texture and our knowledge about the preflop flatter’s range.

The modeling we do in these articles is inspired by the video Alans Common C-betting Spots by Bluefirepoker coach Alan Jackson.

Our approach in this article series about c-betting is exploitive. We make assumptions about various opponent mistakes, and then we move away from optimal play in order to exploit these mistakes. Our previous work on optimal play gives us a starting point, and tells us in which direction we should move our strategy. The main mistake we focus on in this article series is the mistake of folding too much to a c-bet. We want to find spots where our opponent is making this mistake, thus giving us an opening for c-bet bluffing any two cards profitably.

2. Our model
Here are the assumptions we’ll use in this article:

2.1 Assumptions about preflop ranges

 

  • Alice (100 bb) openraises in CO
  • Bob (100 bb) is on the button and follows the previously defined optimal strategies for 3/4/5-betting. Other than that he flats his standard range in position (“IP flat list”).
  • Alice knows Bob’s flatting range based on observations and HUD stats

Alice uses our standard 25% opening range from CO:

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

326 combos
25%

We assume Bob uses the optimal 3/4/5bet strategy against a 25% opening range, given in the table of optimal strategy pairs built in “Optimal 3/4/5-betting in NLHE 6-max – Part 2”:

Download link (right-click and choose “Save as”): IP_3-bet_summary.doc

So Bob will use the following preflop strategy against Alice’s 25% CO raise:

  • 3-bet {QQ+,AK, 12 air} for value, planning to 5-bet all-in against a 4-bet
  • 3-bet bluff 70% of “IP 3-bet air list”, planning to fold against a 4-bet
  • Flats the entire “IP flat list”: {JJ-22,AQs-ATs,AQo-AJo,KQs-KTs,KQo,QJs-QTs,JTs,T9s,98s} =140 combos when {QQ+,AK} is 3-bet for value

Bob’s standard preflop flatting range then has 140 combos, which is 140/1326 =10.6% of all hands. This range is representative for what many players will flat on the button in this scenario, and it’s a reasonable assumption to use against unknowns.

When the flop comes, Alice tries to determine whether or not she has a profitable c-bet for 0.75 x pot with any two cards. She bases her analysis on her knowledge about Bob’s preflop flatting range, the flop texture, and assumptions about which hands Bob is willing to defend with.

Since Alice c-bets 0.75 x pot, she is giving herself pot odds 1 : 0.75 on a bluff with any two cards. A c-bet bluff will be automatically profitable if Bob folds more than 0.75/(1 + 0.75) =57%. If Alice’s analysis concludes that Bob in practice will fold more than 57%, she can c-bet her entire range profitably on the flop. If not, she has to check and give up with some of her weakest hands. How much hand strength we need to c-bet proftiably in this case will be discussed in future articles.

The purpose of the work we do in Part 1 is to train our ability to come up with a qualitative yes/no answer to the question about whether or not we can c-bet any two cards profitably. We look at the flop, we think about our opponents preflop flatting range, and we analyze how the flop and the range interact. We then introduce some assumptions about which hands opponent will defend in practice against our c-bet, and we have our answer.

2.2 Assumptions about Bob’s flop strategy
We’ll look at two example flop textures in this article:

– Very coordinated
– Very dry

For both textures we’ll first build Bob’s optimal defense against Alice’s c-bet. The optimal defense strategy is designed to prevent her from c-betting any two cards profitably. If Bob uses this strategy, there is nothing Alice can do to exploit him by bluffing a lot on the flop.

Then we’ll make some assumptions about the strategy Bob will use in practice. We’ll assume that Bob will fold some weak hands (for example overcards and weak pairs) that he should not fold on flops where his range is weak and difficult to defend correctly. Then we’ll analyze whether or not Bob’s deviation from optimal play will make it possible for Alice to exploit him by c-bet bluffing any two cards.

Exactly how Bob deviates from optimal play will be a function of the flop texture. Here are three general assumptions we’ll use for the non-optimal version of Bob:

  • 1. He is not willing to bluffraise against Alice’s c-bet
  • 2. He is not willing to call the c-bet with pairs lower than two of the cards on the board (for example, he will fold 77 and all lover pairs on a A 8 2 flop)
  • 3. He is not willing to call the c-bet with naked overcards and gutshots, with no additional draws

In addition we can make specific assumptions about how Bob will play on specific flop textures. If we do make extra assumptions, we’ll use good poker sense and let Bob play the way a typical opponent in our games will play.

In general, we’ll assume that the non-optimal Bob plays like a typical decent-but-not-great low limit player. He plays mostly straightforward, he bluffs little when others have the initiative, and he has limited knowledge about the interaction between flop texture and hand ranges. Also, he does a poor job changing his postflop strategies and ranges based on the pot odds he’s getting.

The non-optimal version of Bob mostly sees each hand as an isolated case, and he does not think about the hand as a part of an overall range. This is typical for how the majority of poker players think. They think things like “I have top pair, which is a good hand” or “I have bottom pair, which is a very weak hand“, and they don’t think about all the other possible hands they could hold in this particular scenario.

3. C-betting on a coordinated flop
In “Optimal Postflop Strategies in NLHE 6-max” we concluded that our standard positional flatting range “IP flat list” is easy for Bob to defend on coordinated flops like J 9 3, since these flops hit his preflop range hard.

Now we’ll show through analysis why c-bet bluffing any two cards on these flops is a bad idea heads-up and out of position against a preflop flatter, even if our opponent is tight and straightforward. This is something most players intuitively understand, but not we’ll “prove” it using theory, and we’ll get a much clearer picture of exactly why this is so. Then we’ll repeat the process on a dry flop, and we’ll see that dry textures give us opportunities for profitable any-two-cards c-bet bluffing if our opponent is somewhat tight.

3.1 Optimal defense against c-betting on a coordinated flop
Alice (100 bb) raises her standard 25% range from CO, and Bob flats his standard 10.6% flatting range “IP flat list” ={JJ-22,AQs-ATs,AQo-AJo,KQs-KTs,KQo,QJs-QTs,JTs,T9s,98s} =140 combos.

Our coordinated flop is:

When Alice c-bets, she bets 0.75 x pot, and Bob needs to defend at least 1/(1 + 0.75) =57% to prevent her from c-betting any two cards with automatic profit.

Bob has 120 remaining combos in his range after adjusting for card removal effects, as shown below:

Bob’s optimal defends is then to defend 0.57 x 120 =68 combos. We remember from “Optimal Postflop Play in NLHE 6-max” that Bob’s defense on coordinated flops has three components:

– Raise the best hands for value
– Flat the next best hands
– Bluff raise some weak hands using a 1 : 1 value/bluff ratio

Below is a suggestion for a near-optimal flop strategy for Bob. At this point in the analysis our only concern is to defend with 68 combos (or thereabouts) overall. If this leads us to folding or bluffing with hands that could have been played more profitably by calling, this is not a problem for us.

  • Value raise:
    {TT,55,QTs,AQ,AJ,KJ} =23 combos
  • Flat:
    {KQ,QJs,JJ,ATs} =24 combos
  • Bluffraise:
    {KTs,JTs,T9s,KJ,KJ,KJ,98,AJ,AJ,AJ,AJ,AJ,AJ,98,98,98} =22 combos
  • Total: 69 combos (optimal: 68)

As we have seen in previous articles, the optimal flop defense ranges are strong on very coordinated flops after we have flatted our default “IP flat list” preflop. We have so many strong hands to use that we can get away with only flatting top pair + best underpair (JJ) + 2nd pair/top kicker (ATs). All lower pairs can be folded or used as bluff raises.

3.2 Non-optimal defense against c-betting on coordinated flop
Now we’ll limit the strategies Bob is willing to use when he defends against Alice’s c-bet:

  • 1. He is not willing to bluffraise against Alice’s c-bet
  • 2. He is not willing to call the c-bet with pairs lower than two of the cards on the board (for example, he will fold 77 and all lover pairs on a A 8 2 flop)
  • 3. He is not willing to call the c-bet with naked overcards and gutshots, with no additional draws

We remember that Bob has to defend less than 57% to give Alice a profitable any-two-cards bluffing opportunity when she c-bets 0.75 x pot. So the question we want to answer is this:

Will the restrictions above make it impossible for Bob to defend at least 57% on the flop?

If this is the case, Alice can c-bet her entire range profitably. We now try to build a defense strategy for Bob where he defends 57% (68 combos) given the limitations above:

  • Value raise:
    {TT,55,QTs,AQ,AJ,KJ} =23 combos
  • Flat:
    {KQ,QJs,JJ,ATs,KTs,JTs,T9s,98,KJs,AJ,AJ,AJ,AJ,AJ,AJ} =43 combos
  • Bluff raise:
    None
  • Total: 66 combos (optimal: 68)

We can easily get to around the optimal defense, even if we’re not willing to bluffraise, call with 3rd pair or lower, or float with naked overcards and gutshots. The weakest draw Bob has to call with is AdJx/AxJd (overcard + gutshot + backdoor flush draw).

3.3 Conclusion for defense against c-betting on coordinated flop
Both the optimal and the non-optimal versions of Bob could easily defend the optimal 57% on this flop texture. These flops hit Bob’s preflop flatting range so hard that the can get away with folding lots of marginal hands, and still defend enough.

A range analysis with Pokerazor illustrates this with numbers:

On this flop we have 2nd pair or better 45% of the time (see the list “Cumulative frequency” to the right). So we can cover most of the optimal 57% defense with good one pair hands. And the rest is easily covered by our draws. Even if we never bluff raise, flat pairs below 2nd pair, or flat naked overcards/gutshots, we can get to 57% defense.

We therefore conclude:

Alice can’t c-bet any two cards profitably on our very coordinated flop texture, even if Bob plays tightly and isn’t necessarily willing to defend an optimal amount. He can easily build defense strategies that defend the optimal amount, even with strong limitations on the hands he is willing to defend.

In future articles we’ll talk more about how much hand strength we need to have a profitable c-bet on these flops. We obviously have to be willing to semibluff a bit, and c-bet some weak draws. But we should check-fold our pure air, like 76, 22 and A4 on this flop. Bob simply doesn’t fold often enough to make it profitable, even if he plays somewhat tight postflop.

4. C-betting on a dry flop
Next we’ll show why c-bet bluffing with any two cards on very dry flops generally is a good idea Even players who defend loosely on the flop will find it difficult to defend the optimal amount, since this requires them to float with lots of air.

4.1 Optimal defense against c-betting on dry flops
Alice (100 bb) raises her standard 25% range from CO, and Bob (100 bb) flats his standard 10.6% flatting range “IP flat list” ={JJ-22,AQs-ATs,AQo-AJo,KQs-KTs,KQo,QJs-QTs,JTs,T9s,98s} =140 combos.

Our dry flop is:

This is the classic super-dry flop with one Broadway card, two medium/low cards, and no flush or open-ended straight draws possible. Again, Alice c-bets 0.75 x pot, and Bob needs to defend 57% to prevent her from bluffing with any two cards, as in the previous case.

After adjusting for card removal effects, Bob has 126 remaining combos in his range:

Bob’s optimal defense requires him to defend 0.57 x 126 =72 combos. We remember from “Optimal Postflop Play in NLHE 6-max” that Bob defends only by flatting on very dry flops. So he slowplays all his monsters (only sets are monsters on this flop), together with hos good hands, and some weak hands. He is often forced to flat with all his one pair hands, and perhaps also float some naked overcards and gutshots in order to defend optimally.

Below is a suggestion for an optimal flop defense strategy for Bob:

  • Value raise:
    None
  • Flat:
    {99,22,KQ,KJs,KTs,JJ-TT,T9s,98s,88-44} =72 combos
  • Bluff raise:
    None
  • Total: 72 combos (optimal: 72)

Bob has to flat all his pocket pairs, except 33. As an alternative, he can fold some low pocket pairs and float his best overcards instead (AQ):

  • Value raise:
    None
  • Flat:
    {99,22,KQ,KJs,KTs,JJ-TT,T9s,98s,88-66,AQ} =76 combos
  • Bluff raise:
    None
  • Total: 76 combos (optimal: 72)

But regardless of how he chooses to do it, Bob has to flat lots of weak hands on this flop texture in order to defend the optimal 57%.

4.2 Non-optimal defense against c-betting on a dry flop:
Again, we introduce limitations for Bob:

  • 1. He is not willing to bluffraise against Alice’s c-bet
  • 2. He is not willing to call the c-bet with pairs lower than two of the cards on the board (for example, he will fold 77 and all lover pairs on a A 8 2 flop)
  • 3. He is not willing to call the c-bet with naked overcards and gutshots, with no additional draws

Then we see how far he can go:

  • Value raise:
    None
  • Flat:
    {99,22,KQ,KJs,KTs,JJ-TT,T9s,98s} =42 combos
  • Bluff raise:
    None
  • Total: 42 combos (optimal: 72)

It turns our that if Bob is unwilling to flat with naked gutshots, naked overcards, and pairs lower than two of the cards on the board, it is impossible for him to defend the optimal amount. He gets to 42/126 =33% defense, and folds 100 – 33 =67%.

Let’s say Alice c-bets with a worthless hand that will never win the pot when Bob doesn’t fold on the flop. Her EV for the bet is:

EV (c-bet)
=0.67 (P) + 0.33 (-0.75P)
=+0.42P

Where P is the pot size on the flop. If the preflop raise was 3.5 bb, the pot is P =2(3.5) + 0.5 + 1 =8.5 bb on the flop. Alice’s c-bet bluff is then worth 0.42 x 8.5 bb =3.6 bb. This is a very nice profit for an any-two-cards bluff with a hand that can only win when Bob folds.

4.3 Conclusion for defense against c-betting on dry flops
Bob could defend our dry flop texture optimally without floating with extremely weak hands, but he had to drop down to the “cellar” and use his lowest one pair hands. Alternatively, he could fold some low pairs and float with some of his best overcard hands instead.

When Bob’s strategies were limited, it was impossible for him to defend enough. If he is unwilling to call with his lowest one pair hand, good ace high hands, or gutshots, he can’t defend our dry example flop optimally. This opens him up for getting exploited by Alice’s any-two-cards c-bet bluffs.

A range analysis with Pokerazor illustrates this with numbers:

On the coordinated example flop we had 2nd pair or better 45% of the time, in addition to many draws. On the dry example flop we have 2nd pair or better only 33% of the time, and we have no strong draws, only naked overcards and gutshots.

Most NLHE players know (or intuitively see) that our dry example flop is an excellent flop to bluff at. So you can expect the preflop raiser to c-bet a lot when you are the preflop flatter on such a flop. Therefore, if you believe the preflop raiser will try to exploit you by c-bet bluffing with any two cards, don’t be afraid to float!.

Remember that you will also call with some good hands like sets, top pair, and good 2nd pair/underpair hands. So if he 2-barrels a lot with air, he will get punished by your strong flatting hands. Think about what his range looks like on this type of flop. If he has raised from CO, his range is full of garbage like A8o, 76s, etc. Force him to play turns with these hands if he is aggressive enough to c-bet any two cards on the flop.

If he keeps betting on the turn, you have to fold low pairs like 55 and floats like AQ, but you will still plenty of hands to continue with (remember, you have slowplayed sets and top pair hands in your range). So your turn range will be decently strong, even if you floated the flop with a weak range.

5. Summary
In this article we have begun studying c-betting on the flop in heads-up pots as the preflop raiser.

We saw that coordinated flops are easy to defend optimally for the preflop flatter, even if he isn’t necessarily willing to defend optimally. When we did the same model study on a dry flop, we saw that it was impossible for the flatter to defend optimally if he was unwilling to float his weakest one pair hands, and/or some floats (overcards/gutshots type hands. When you have identified such players at the table (and they are common at the low limits), you can c-bet dry flops with your entire range against them, and “print money”.

The gist of it is that all flops can be defended optimally, in principle (it’s only a matter of including more and more weak hands, as the flop texture gets drier), but many players are unwilling to do so if it requires them to defend with very weak hands. These players can be exploited by c-bet bluffing a lot on dry flop textures. On the driest of flops, you can c-bet your entire range profitably,.

In Part 2 we’ll continue our modeling. Now we’ll let Bob use two other preflop flatting ranges (tight =5% and loose =15%) in addition to his standard 10% “IP flat list”. This gives us an opportunity to learn about how various preflop flatting ranges hit various types of flop textures, and the consequences this has for our c-betting strategy.

Being able to classify flop textures as coordinated or dry gives us possibilities to c-bet profitable with any two cards, and this was what we learned in this article. If we also train our ability to distinguish between different opponent ranges, we add one extra dimension to our analysis. This will enable us to find even more profitable c-bet bluffing spots. A particular flop texture can give us a profitable c-bet bluff against one opponent range, but not against another. This is the topic for the next article.

Note that the work done in this article defines a method for training our ability to recognize profitable c-bet bluffing opportunities. You can generate random flops using Flopgenerator.com and perform this type of analysis, using assumptions about your opponent’s flatting range and postflop tendencies. This will tell you whether or not you have a profitable any-two-cards c-betting opportunity on the given flop.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/c-beting-in-nlhe-6-max–part-1-263#sthash.rntipxOO.dpuf

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Optimal Postflop Play in NLHE 6-max – Part 7

1. Introduction
This is Part 7 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

In this article we’ll continue the work started in Part 5 and Part 6, where we studied postflop strategy for a preflop raiser out of position in a heads-up scenario. In Part 5 we designed an optimal barreling strategy for the raiser that protected her against random floating done by her opponent in position. In Part 6 we verified mathematically that this strategy made her opponents any-two-cards-floats break even, which means he can not float random weak hands profitably against her flop c-bet. We also studied the effect of changing the raiser’s preflop opening range. We found that a looser preflop range forced her to play looser ranges postflop, if she began postflop play by c-betting her entire range on the flop.

The topic for this article is to look more closely at:

– The effect of the preflop raiser’s postflop bet sizing
– The effect of her opponent slowplaying his strong hands postflop

In previous postflop articles we have assumed our players are using a standard postflop bet sizing scheme of 0.75 x pot on the flop, 0.75 x pot on the turn, and 0.60 x pot on the river. But if we always stick with standard bet sizing, we risk giving up +EV in some spots. What makes NLHE one of the most profitable games for a strong player is the freedom she has to vary her bet sizing. This enables her to exploit weaker players’ mistakes maximally.

Here we’ll look at a specific example where the raiser c-bets the flop, 2-barrels the turn, and 3-barrels the river with an overpair on a dry flop texture. Conventional wisdom is we can/should use small bet sizing on dry flop textures, since our opponent will have fewer draws on such flops. So there’s less risk of getting drawn out on, and we can bet smaller to protect our hand against draws. But this does not necessarily mean we maximize our EV for the hand by betting small on these board textures.

If we find ourselves heads-up against a player who we know has a range full of medium/weak hands (so that it’s easy for us to know when we’re ahead and when we’re behind(, we’ll see that we maximize our EV by using big value bets on all streets. But of course with a balanced mix of value hands and bluffs, since we’re trying to play close to optimally.

The scenario we’ll study in this article is valuebetting/barreling an overpair on a dry flop texture heads-up and out of position against a weak opponent range. We’ll study the effect of varying bet sizing for the raiser out of position, and the effect of slowplaying for the player in position.

We begin by defining the model scenario we’ll work with throughout the article. Then we define the two bet sizing schemes (“standard” and “alternate”) that the preflop raiser (Alice) will be using postflop. Next, we define the postflop strategies for the player in position (Bob), and we use Pokerazor to compute the EV for Alice’s c-bet/2-barrel/3-barrel postflop line with overpairs against Bob’s range/strategy.

We end up with EV calculations for Alice’s postflop play under 4 combinations of circumstances:

  • Alice’s standard bet sizing against Bob who doesn’t slowplay
  • Alice’s standard bet sizing against Bob who slowplays
  • Alice’s alternate bet sizing against Bob who doesn’t slowplay
  • Alice’s alternate bet sizing against Bob who slowplays

Based on this we can draw conclusions about how Bob should defend in position on dry flops. We’ll verify that slowplaying on dry flops is a good strategy for him, which is something we have simply assumed in previous articles. We’ll also draw conclusions about how Alice can vary her bet sizing to increase her EV against an opponent that she knows has a weak postflop range.

What Alice wants is to use the information she has about Bob’s postflop range after he flats preflop (with a medium strong preflop range) and the flop comes dry and uncoordinated (which means it mostly misses Bob’s preflop range). On these flops Alice’s good one pair hands (e.g. her overpairs) can extract lots of value from Bob’s weaker pairs. One way to achieve this is to use big turn and river bets so that her final bet is all-in on the river (as opposed to the standard bet sizing scheme where ~1/2 the stack has been put into the pot after the river bet).

We’ll test this alternate bet sizing scheme for Alice by computing the EV for her barreling the three overpairs AA-QQ on a dry flop against Bob who defends according to the strategies designed for him in Part 1, Part 2, Part 3 and Part 4 of this article series. Alice will of course also barrel other hands on the turn and river, including an optimal number of bluffs, but here we simply want to find the EV for her best overpair hands in a vacuum. They are a part of an overall optimal barreling strategy for her, but we don’t have to know her total strategy in order to find the EV for these hands in isolation. However, we will need Bob’s complete defense strategy in position, in order to find Alice’s EV with AA-QQ against his strategy.

2. Definition of our model scenario
Alice (100 bb) raises her ~15% UTG-range:

Alice’s Default 15% UTG-range

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

194 combos
15%

Bob (100 bb) flats his standard “IP flat list” on the button:

IP flat list after ~15% UTG openraise

QQ-22
AKs-ATs AKo-AJo
KTs+ KQo
QTs+
JTs
T9s
98s

162 combos
12%

The flop comes:

Alice then begins postflop play by c-betting 0.75 x pot with her entire preflop range on the flop. Bob now calls. We give Bob the option of choosing between always slowplaying and never slowplaying his strong hands on the flop:

Bob’s postflop strategy 1: Bob never slowplays
In this case we can assume that Bob’s flat on the flop eliminates the few possible monster hands (66 or 22) from his range, since he would have raised them for value. We will also assume that Bob would have raised for value with his 3 best overpairs (QQ-TT) as well. Beyond his choice of slowplaying/nor slowplaying his monster hands, Bob’s strategies follow the principles outlined in previous articles. So when he flats a dry flop in a situation where he would have raised all his strong hands, he must have a range of mostly weak one pair hands and overcards. His plan for the turn and river (barring improvement) is to call down optimally, in order to prevent Alice to profitably barrel any two cards as a bluff.

If Alice has a read on Bob as a player who never slowplays the flop, she now knows that his postflop range is weak after the flop call. He can never have anything better than a medium one pair hand, and Alice can use this knowledge to make big turn and river value bets with her good one pair hands, mixed with an optimal number of bluffs. Note that this is something she can do because the flop is dry and because she knows Bob’s range is weak (and likely to stay weak all the way to the river). On a coordinated flop, where Bob’s flatting range would have been stronger (and more likely to improve on many turn and river cards) value betting hard with her good one pair hands would have been much more dangerous for Alice.

Bob’s postflop strategy 2: Bob slowplays until the river
If Bob slowplays the flop, he will also slowplay the turn with his few monster hands to give Alice a chance to lose more money by bluffing the river with her weakest barreling hands. This is a reasonable strategy for Bob, and by slowplaying his strongest hands he also protects the weakest hands in his call-down range. His range is weak overall, and Alice can put pressure on him, but she can’t automatically fire big turn and river bets with her optimal value/bluff range without sometimes getting punished.

If Alice knows that Bob slowplays, there isn’t really a lot she can do with this information, since Bob’s range is still pretty weak. So she should still bet for value with her good one pair hands. But we expect that Bob’s slowplaying will counter the positive effect of Alice alternate bet sizing, where she bets big on the turn and river to get all-in for value with her good (and probably winning) one pair hands (as well as her monster hands, and some bluffs for balance. Whether or not Alice should revert to standard bet sizing against a slowplaying Bob remains to be seen.

Regardless of her turn/river betting scheme, Alice starts out with a 0.75 x pot c-bet on the flop. We’ll then estimate the EV for Alice’s turn/river barreling with her 3 best overpairs AA-QQ, using the Pokerazor analysis software.

The turn is:

Alice will now 2-barrel the turn with AA-QQ after Bob’s flop flat, and Bob calls again, regardless of whether he’s using a slowplay strategy or not (since he will always slowplay to the river, when he slowplays). Bob then uses the theory for optimal postflop play in position, defined in Parts 1-4 in this article series, and he calls with a range designed to make Alice’s weakest 2-barreling hands (i.e. her bluffs) break even). Note that we have chosen a turn card that doesn’t improve Bob, so that we won’t have to think about how the few cases where one of his medium/weak flop flatting hands improves to a value hand on the turn.

Here Alice can use two different bet sizing schemes, and we’ll study her EV for both using Pokerazor.

The river is:

Alice will now 3-barrel the river for value with AA-QQ after Bob’s turn flat. We have let the river card be a card that could have improved Bob. But if Bob doesn’t slowplay, it can’t have improved him to anything better than one pair, since he would have raised TT for value on the flop. So if Bob isn’t slowplaying postflop, he will now have a range of bluffcatchers on the river after flatting the flop and turn. He will defend against Alice’s riverbet by calling down an optimal amount that prevents her from profitably 3-barreling any two cards against him.

In the case that Bob slowplays, he will now raise all-in with all his slowplayed monster hands. If Alice has used the alternate bet sizing scheme, her 3-barrel will put bob all-in, and he will of course call with his monsters. And he will also call with enough bluffcatchers to prevent a profitable any-two-cards bluff from Alice. Bob’s monsters are 66 (1 combo), 22 (1 combo) and TT (3 combos). In the case where he raises all-in for value, he also raises some bluffs for balance.

3. Defining Alice’s two bet sizing schemes

Standard bet sizing

– 0.75 x pot on the flop
– 0.75 x pot on the turn
– 0.60 x pot on the river

Alice and Bob then get to the river with 74 bb left in their stacks, and the pot is 53.5 bb (100 – 74 =26 bb from each of them, plus 1.5 bb from the blinds). Alice then bets 0.60 x 53.5 =32 bb on the river, and Bob calls or shoves all-in to 74 bb. When Bob shoves, Alice gets pot-odds 159.5 : 42 =3.8 : 1 on a call. Since Bob is shoving a balanced range, she is indifferent towards calling or folding with her overpairs (they are now bluffcatchers). Since her EV is the same (0) for calling or folding against Bob’s optimal river raising strategy, we simply choose to let her bet-call the river.

Alternate bet sizing
Alice and Bob have 96.5 bb in their stacks after preflop play, and the pot is 8.5 bb before postflop betting begins. Alice c-bets 0.75 x pot (6.5 bb rounded to the nearest half big blind), and the pot grows to 21.5 bb with 90 bb behind.

Alice now chooses her turn and river bet sizing so that she bets the same fraction of the pot on both streets, and her river bet is all-in. To accomplish this, she bets the same fraction (r) of the pot on both the turn and river so that the final pot becomes 201.5 bb when Bob calls the river.

She begins by betting r times the pot on the turn, and the pot grows to:

flop-pot + 2r x flop-pot =flop-pot x (1 + 2r)

Then she bets r times the pot on the river, and the pot grows to:

turn-pot + 2r x turn-pot
=turn-pot x (1 + 2r)
=flop-pot x (1 + 2r) x (1 + 2r)
=flop-pot x (1 + 2r)^2

The flop pot is 21.5 bb, and we know that the final river pot should be 201.5 bb, so we can write:

21.5(1 + 2r)^2 =201.5
(1 + 2r)^2 =201.5/21.5
(1 + 2r)^2 =9.37

We take the square root on both sides and get:

1 + 2r =3.06
r =(3.06 - 1)/2
r =1.03

We find that Alice should bet 1.03 x pot on both the turn and the river. This puts her all-in on the river, using two bets slightly bigger than pot. Let’s check that this is correct:

Alice bets 1.03 x 21.5 =22 bb on the turn, the pot grows to 21.5 + 2 x 22 =65.5 bb, and both players have 90 – 22 =68 bb behind. Then she bets the remaining 68 bb on the river into the 65.5 bb pot (ratio: 68 : 65.5 =1.04) and gets all-in. So we get very close to the desired bet sizing of 1.03 x pot on both streets.

Before we move on, lets ask: Why does Alice want to use a bet sizing scheme where she bets the same fraction of the pot on the turn and the river, planning to get all-in?

We will not delve into the theory here, but simply accept that this is a reasonable thing to do. Matthew Janda has discussed this in his game theory video series at Cardrunners, and you can also find a more in-depth discussion in the book The Mathematics of Poker (Chen/Ankenman)). If Alice has a range of nuts/air hands (i.e. hands that either always win or always lose at a showdown), and Bob has a range of bluffcatchers (i.e. hands that lose to all of Alice’s value hand and beat all her bluffs), Alice maximizes her EV by betting in such a way that she:

– Gets all-in on the river
– Bets the same fraction of the pot on each street

Alice then bets a balanced ratio of nuts air, so that Bob becomes indifferent towards calling down or folding with his bluffcatchers. If Bob folds too much, Alice’s bluffs become more profitable, and if he calls too much, her value hands become more profitable. When Alice’s value/bluff ratio is optimally balanced, she is guaranteed a minimum profit regardless of what Bob does.

We choose this alternate bet sizing scheme for Alice, since the situation after Bob calls the flop is similar to the nuts/air scenario described above. For example, Alice knows that when Bob calls the flop, and he never slowplays, her overpairs AA-TT have to be ahead on our example flop:

This is because Bob would have:

– Raised AA-KK preflop
– Raised QQ-TT and house/quads on the flop (we assumed this earlier in the article)

Therefore Alice can bomb away with big turn and river bets against Bob’s very weak range, after he has revealed is as such by calling the flop (assuming Alice knows that Bob doesn’t slowplay). It’s easy for her to know which of her hands are value hands (all monsters and her highest overpairs), which hands are bluffcatchers (medium one pair hands), and which hands are air (everything else). She balances her value/bluff ratio according to the postflop strategies we designed for her in Part 5 and Part 6, and we’ll use Pokerazor to show that this alternate bet sizing scheme (0.75x/1.03x/1.03x) yields a higher EV than the standard scheme (0.75x/0.75x/0.60x) when Bob never slowplays

The next step is to build Bob’s postflop strategies on the flop, turn and river. Then we’ll use these strategies as Pokerazor input, and estimate the EV for Alice’s turn/river betting with AA-QQ. If you need to brush up on these strategies, read Parts 1-4.

4. Bob’s postflop strategies as a function of Alice’s bet sizing
Alice’s choice of bet sizing scheme (“standard” or “alternate”) determines the pot-odds Bob is getting on the flop and turn, so his defense strategies will vary with the bet sizing. This means we have to build two sets of postflop strategies for him, one for standard bet sizing and one for alternate bet sizing.

We remember that regardless of Alice’s bet sizing scheme, and regardless of whether or not Bob slowplays, the postflop play goes like this:

– Alice c-bets the flop, Bob calls
– Alice 2-barrels the turn, Bob calls
– Alice 3-barrels the river, Bob calls or shoves

And this is because:

When Bob slowplays, he always slowplays to the river, so he will always call the flop and the turn when he defends. Those times he doesn’t slowplay, the turn and river cards will not improve him to a monster hand, so he will be stuck with a calling range on all streets.

4.1 Bob’s postflop play against standard bet sizing
We begin with Bob’s defense on the flop:

Standard 0.75 x pot c-bet sizing means that Alice is getting pot-odds 1 : 0.75, and she will automatically profit if Bob folds more than 1/(1 + 0.75) =43%. Bob prevents this by defending 100 – 43 =57% of his range on the flop. His preflop flatting range is reduced from 162 to 154 combos on this particular flop:

So Bob needs to defend 0.57 x 154 =88 combos on the flop. In the case where he doesn’t slowplay, we’ll assume he raises 66, 22, QQ-TT =22 combos for value. He balances this with 2 bluff combos, and raises a total of 22 + 22 0 44 combos. Then he needs to flat 88 – 44 =44 combos in order to defend 88 combos in total:

Flop defense against standard bet sizing, without slowplay:

  • Value raise:
    {66,22,QQ-TT} =22 combos
  • Flat:
    {99-77,55-44,AK} =46 combos
  • Bluff raise:
    {KQ,KJs,K JK J} =22 combos
  • Total: 90 combos (Optimal: 88)

If Bob slowplays, he will not use a raising range, and he flats with his ~88 best combos:

Flop defense against standard bet sizing, with slowplay:

  • Value raise:
    None
  • Flat:
    {66,22,QQ-77,55-33,AK,AQ} =90 combos
  • Bluff raise:
    None
  • Total: 90 combos (Optimal: 88)

So when Bob flats the flop, he has a range of marginal one pair hands and overcards ({99,88,77,55,44,AK} =46 combos) when he doesn’t slowplay, and a somewhat stronger range of monsters, marginal one pair hands and overcards ({66,22,QQ-77,55-33,AK,AQ}) =90 combos) when he slowplays. He brings these two ranges with him to the turn:

Alice now bets 0.75 x pot on the turn, and Bob defends 57% like he did on the flop. When he doesn’t slowplay, he has the flop range {99,88,77,55,44,AK} =46 combos, which doesn’t change with this turn card (no card removal effects). When he slowplays, he has the range {66,22,QQ-77,55-33,AK,AQ} =90 combos, which is reduced to 88 combos given this turn card:

When Bob doesn’t slowplay, he has no value raising hands on the turn, and he defends the optimal 57% by flatting 0.57 x 46 =26 combos:

Turn defense against standard bet sizing, without slowplay:

  • Value raise:
    None
    None
  • Flat:
    {99-77,55,4 4,4 4,} =26 combos
  • Bluff raise:
    None
  • Total: 26 combos (Optimal: 26)

When he slowplays, he has some value hands on the turn, but he keeps slowplaying them to the river and he defends the optimal 57% by flatting 0.57 x 88 =50 combos:

Turn defense against standard bet sizing, with slowplay:

  • Value raise:
    None
  • Flat:
    {66,22,QQ-77,55-44} =50 combos
  • Bluff raise:
    None
  • Total: 50 combos (Optimal: 50)

Bob brings the ranges {99-77,55,4 4,4 4,} =26 combos and {66,22,QQ-77,55-44} =50 combos to the river

In the case where Bob isn’t slowplaying, he gets to the river with the range {99-77,55,4 4,4 4} =26 combos which doesn’t change with this river card. In the case where he’s slowplaying, he gets to the river with the range {66,22,QQ-77,55-44} =50 combos, which is reduced to 47 combos:

Alice now bets 0.6 x pot, which gives her pot-odds 1 : 0.6 on a bluff. She has an automatic profit with any two cards if Bob folds more than 0.6/(1 + 0.6) =38%. Bob prevents this by defending 100 – 38 =62% of his river ranges. So Bob defends 0.62 x 26 =16 combos when he hasn’t slowplayed, and 0.62 x 47 =29 combos when he has slowplayed.

In the case where he has slowplayed, Bob gets to the river with the range {99-77,55,4 4,4 4} =26 combos, all of them bluffcatchers. He calls Alice’s river bet with the 16 best combos:

River defense against standard bet sizing, without slowplay:

  • Value raise:
    None
  • Flat:
    {99-88, 7 77 7,7 7,7 7} =16 combos
  • Bluff raise:
    None
  • Total: 16 combos (Optimal: 16)

Bob’s slowplayed range now has value hands he can raise, namely {66,22,TT} =5 combos. The stacks are 74 bb on the river after Alice’s standard 0.75x/0.75x/0.60x betting scheme, and her river bet is 32 bb into a 53 bb pot. Bob then raises his value hands all-in, and the pot grows to 159 bb with 42 bb for Alice to call. Her pot-odds are 159 : 42 =3.8 : 1, and Bob bluffs just enough to make her indifferent towards calling or folding with her bluffcatchers (and all her overpairs are now bluffcatchers).

Bob accomplishes this by raising 1 bluff combo for every 3.8 value combos, which is 1/3.8 =0.26 bluff combos per value combo. So he needs 5 x 0.26 =1.3 bluff combos, which we round to 1. Since it’s the same for Alice whether she calls or folds against an optimally balanced raising range, we’ll simply assume she is bet-calling with all her value hands on the river. When Bob has built his raising range, he does the rest of the defense by adding calls with bluffcatchers until he’s defending 29 combos in total:

River defense against standard bet sizing, with slowplay:

  • Value raise:
    {66,22,TT} =5
  • Flat:
    {QQ-JJ,99,all 88 except 8 8} =23 combos
  • Bluff raise:
    {8 8} =1 combo
  • Total: 29 combos (Optimal: 29)

The next step is to find Bob’s flop, turn and river strategies for the alternate 0.75x/1.03x/1.03x betting scheme.

4.2 Bob’s postflop play against alternate bet sizing
Since the flop c-bet is the same in both the 0.75x/1.03x/1.03x scheme and the 0.75x/0.75x/0.60x scheme, Bob’s flop play is the same in both. So we begin by finding is new turn strategies:

When Bob isn’t slowplaying, he has the range {99,88,77,55,44,AK} =46 combos, which doesn’t change with this turn card. When he is slowplaying, his range is {66,22,QQ-77,55-33,AK,AQ} =90 combos, which is reduced to 88:

Alice now bets 1.03 x pot, and gives herself pot-odds 1 : 1.03. She will have an automatic profit if Bob folds more than 1.03/(1 + 1.03) =51%. Bob prevents this by defending 100 – 51 =49% of his range. So he defends 0.49 x 46 =23 combos when he isn’t slowplaying, and 0.49 x 88 =43 combos when he is slowplaying.

In both cases he defends the turn entirely by flatting, and we get the turn strategies.

Turn defense against alternate bet sizing, without slowplay

  • Value raise:
    None
  • Flat:
    {99-77,5 5,5 5,5 5,5 5,5 5} =23 combos
  • Bluff-raise:
    None
  • Total: 23 combos (Optimal: 23)

Turn defense against alternate bet sizing, with slowplay:

  • Value raise:
    None
  • Flat:
    {66,22,QQ-77,5 5,5 5,5 5,5 5,5 5} =43 combos
  • Bluff raise:
    None
  • Total: 43 combos (Optimal: 43)

Bob brings the ranges {99-77,5 5,5 5,5 5,5 5,5 5} =23 combos and {66,22,QQ-77,5 5,5 5,5 5,5 5,5 5} =43 combos with him to the river:

His two ranges are reduced to 23 and 40 combos, given the river card:

Alice now bets the rest of her stack all-in with a 1.03 x pot river bet, and the pot-odds are identical to the situation on the flop. Bob defends 49% of his ranges, and he has to do this by calling all-in. He calls 0.49 x 23 =12 combos when he isn’t slowplaying, and 0.49 x 40 =20 combos when he is slowplaying:

River defense against alternate bet sizing, without slowplay

  • Value raise:
    None
  • Flat:
    {99-88} =12 combos
  • Bluff raise:
    None
  • Total: 12 combos (Optimal: 12)

River defense against alternate bet sizing, with slowplay:

  • Value raise:
    None
  • Flat:
    {66,22,TT,QQ-JJ,9 9,9 9,9 9} =20 combos
  • Bluff raise:
    None
  • Total: 20 combos (Optimal: 20)

Now we have built Bob’s postflop strategies against Alice’s barreling, and we can plug them into Pokerazor and estimate the EV for Alice’s barreling with the overpairs AA-QQ:

5. EV simulations for Alice’s 3-barreling with the overpairs AA-QQ
In the standard 0.75x/0.75x/0.60x betting scheme, Alice bets AA-QQ for value on the flop, turn and river on this dry board, and then she calls those times raises the river (but since Bob’s river raising range is optimally balanced, it doesn’t matter whether she calls or folds). Bob follows the strategies outlined above. In the alternate 0.75x/1.03x/1.03x betting scheme, Alice bets for value on the flop, turn and river, and gets all-in with the river bet. So Bob’s river defense is done by calling all-in.

Note that we have built Bob’s postflop strategies without taking our knowledge about Alice’s hands into consideration (since Bob can’t know that we’re only looking at AA-QQ in isolation in our model study). For example, we haven’t reduced the number of AK combos in Bob’s ranges to adjust for the fact that many of the aces and kings are in Alice’s range (card removal effects). We accept this as a simplifying approximation.

We now compute the EV for Alice’s turn/river bet-bet line with her overpairs AA-QQ:

5.1 Results from the Pokerazor simulations:
Standard betting scheme, without slowplay
EV (AA) =+44.9 bb
EV (KK) =+44.9 bb
EV (QQ) =+40.7 bb

In the case where Bob raises all his strong hands on the flop, he defends the turn and river with a weak calling range of one pair hands and overcards. Alice’s overpair are basically “the nuts” against Bob’s weak range, and we extract a lot of value by betting the turn and river. Checking the turn or river for pot control is NOT recommended in this scenario, and we’ll see in a minute that we profit even more from “bombing” the turn and river with big value bets, putting ourselves all-in with the final bet.

Note that AA and KK are basically the same hand against Bob’s weak range. The same goes for QQ, but the for QQ differs from the EV for AA/KK because of the card removal effects discussed previously. For example, AA/KK makes it less likely that Bob has AK. We’ll ignore these effects for simplicity.

Bob can reduce Alice’s EV significantly by slowplaying his monsters, as shown by the next set of simulations:

Standard betting scheme, with slowplay
EV (AA) =+34.7 bb
EV (KK) =+33.2 bb
EV (QQ) =+30.0 bb

Bob’s slowplay strategy reduces the EV for Alice’s overpair by 23-26%. This confirms that slowplaying is a much better strategy on this type of dry flop than raising our few monsters on the flop and being stuck with a very weak calling range on later streets. As we’ll see in a minute, Alice’s alternate betting scheme can really punish Bob when he only flats the flop with weak hands. If Alice knows this, she can punish him by overbetting the turn and the river:

Alternate betting scheme, without slowplay
EV (AA) =+55.2 bb (+44.9 bb)
EV (KK) =+55.2 bb (+44.9 bb)
EV (QQ) =+49.1 bb (+40.7 bb)

The EVs for the standard betting scheme is given in parentheses for comparison. The effect is what we expected. When Bob never slowplays, Alice can increase her EV for the turn/river betting by 21-23% relative to the standard betting scheme. She does this by making sure she gets her entire stack in with her overpairs against Bob’s weak range of bluffcatchers. Interestingly, this increase is of the same order of magnitude as the effect of Bob slowplaying in the standard betting scheme (23-26%).

Alternate betting scheme, with slowplay:
EV (AA) =+38.2 bb (+34.7 bb)
EV (KK) =+36.4 bb (+33.2 bb)
EV (QQ) =+30.0 bb (+30.0 bb)

The EVs for the standard betting scheme is given in parentheses for comparison. When Alice uses pot-sized betting on the turn and river, the effect of Bob’s slowplaying is increased. He can now reduce Alice’s EV by 29-39%, relative to not slowplaying. Note that even if Bob slowplays against Alice alternate scheme of big turn and river bets, she still makes more money than from the standard betting scheme. Bob’s slowplaying keeps her profit down, but Bob can’t stop Alice from overbetting profitably.

5.1 Conclusions from our Pokerazor simulations
The best strategy for Bob is to always slowplay dry flops. Below are Alice’s EVs for the standard betting scheme, with and without slowplay:

Standard betting scheme with/without slowplay
EV (AA) =+34.7 bb / +44.9 bb
EV (KK) =+33.2 bb / +44.9 bb
EV (QQ) =+30.0 bb / +40.7 bb

The difference between slowplaying/not slowplaying is 11-12 bb in favor of Bob, when Alice uses the standard bet sizing.

When Alice uses big turn and river bets, it’s even more important for Bob to slowplay:

Alternate betting scheme with/without slowplay
EV (AA) =+38.2 bb / +55.2 bb
EV (KK) =+36.4 bb / +55.2 bb
EV (QQ) =+30.0 bb / +49.1 bb

The difference between slowplaying/not slowplaying is now 16-18 bb in favor of Bob, when Alice maximizes her EV with big turn and river bets.

We conclude:

Bob should always slowplay his monster hands on dry flops, regardless of Alice’s betting scheme. If he chooses to not slowplay, he can get lucky and lose less than maximum, if Alice chooses to use small turn and river bets. But if Alice bets big on the turn and river, Bob will loose significantly by not slowplaying. Since Bob can use slowplaying to keep Alice’s EV down, regardless of her bet sizing, he should always do so. 

Note that our conclusion isn’t necessarily valid on coordinated flops where both players have many draws in their ranges. But on dry and uncoordinated flops, Bob should slowplay.

6. Summary:
We have studied the scenario where the preflop raiser 3-barrels overpairs in a dry flop against a flatter in position. We studied the effects of bet sizing for the preflop raiser, and slowplaying for the flatter.

We concluded that:

  • On dry flop textures where the flatters preflop range has flopped mostly marginal one pair hands and overcards, the raiser can maximize her EV by using big turn and river bets that puts her all-in on the river
  • The flatter should always slowplay in these flops to keep the raiser’s EV down
  • But even if the flatter slowplays, the raiser can profitably overbet the pot on the turn and river, so she should do so

These very dry flop textures give the preflop raiser an opportunity to extract additional EV by putting pressure on the flatter’s weak postflop range with big bets. The flatter can limit the damage by slowplaying, but he can’t eliminate all of the raiser’s advantage from using big bet sizing.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-7-834#sthash.GgJlIV1g.dpuf

Optimal Postflop Play in NLHE 6-max – Part 3

1. Introduction
This is Part 3 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

In Part 1 and Part 2 we introduced fundamental theory for heads-up flop play in position after flatting preflop. Alice raises from some position, Bob flats in position, and all other players fold. Alice c-bets most flops, and Bob has to defend enough to prevent Alice from c-betting any two cards profitably.

Bob’s response to Alice’s c-bet is to choose:

– A range for value-raising
– A range for flatting
– A range for bluff-raising

And the he folds the rest of his hands. We found that Bob had to defend minimum 57% against a 0.75 x pot c-bet. We also estimated that Bob should use a 1 : 1 ratio of value hands to bluffs when he raises. Our method for estimating Bob’s flop ranges are:

  • 1. Choose a value range (for example, top pair/top kicker or better, plus monster draws). Then we also know how many bluff combos we need (number of bluffs =number of value hands)
  • 2. When the number of value hands/bluffs is counted, we pick enough flatting hands to give us a total defense of 57%. Our flatting hands are chosen from the best hands not strong enough to raise for value (for example, top pair hands weaker than top pair/top kicker, some lower pairs, and non-monster draws).
  • 3. Lastly, we choose our bluff combos from the best hands not strong enough to raise for value or flat (typically the weakest one pair hands, the best overcard hands, and gutshot draws)

In this article we’ll put these principles to work on two different flops:

– A coordinated flop with many draws
– A dry flop without draws

And we’ll place Bob in two different preflop flat scenarios:

– On the button after a CO openraise from Alice
– In the big blind after an SB openraise from Alice

So Bob will defend against Alice’s c-bet on two different flop types, and with two different preflop flatting ranges. This gives us 4 scenarios:

– Bob on the button with a coordinated flop
– Bob on the button with a dry flop
– Bob in the big blind with a coordinated flop
– Bob in the big blind with a dry flop

We’ll work through these scenarios systematically for practice. After reading this article you should be able to do the same type of analysis on your own, so that you can practice optimal heads-up flop play away from the table, using your own standard preflop flatting ranges.

2. Our two practice flops
We go to FlopgGenerator.Com and generate a coordinated (wet) flop and an uncoordinated (dry) flop:

2.1. Coordinated flop

A coordinated flop with two possible straights, and also high cards that will connect with many hands in Bob’s preflop flatting ranges. So we expect this flop to be an easy one to defend.

2.2 Uncoordinated flop flop

A low, rainbow flop that mostly misses Bob’s preflop flatting ranges. There are some possible straight draws, but few of Bob’flatting hands connects with these draws (none, when he has flatted on the button). So we expect this flop to be a tough one to defend enough.

This is also a flop where we have to consider slowplaying the few monster hands in our flop range (basically, our sets) in order to make it harder for Alice to play the turn and river after we flat the flop (since our flop flatting range will be weak on this type of low, dry flop texture). More about this later.

2. Alice’s and Bob’s preflop ranges
We’ll work with two scenarios:

Scenario 1: Alice in CO and Bob on the button
Alice opens her default 25% CO range:

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

326 combos
25%

Bob flats with his default flatting range outside of the blinds (“IP flat list”) given in the overview below:

Here is a download link for this document (right click and choose “Save as”):
IP_3-bet_summary.doc

With Alice in CO, Bob 3-bets {QQ+,AK} for value, so his flatting range contains 140 combos:

“IP flat list” after a 25% CO openraise:

JJ-22
AQs-ATs AQo-AJo
KQs-KTs KQo
QTs+
JTs
T9s
98s

140 combos

This flatting range is weighted towards high/medium suited and coordinated hands. So it will connect well with high/medium coordinated flop textures and be easy to defend on these flops.

But on low, uncoordinated flop textures it might be difficult for us to defend enough, since we simply don’t have enough strong combos in our range. So we might have to accept that we won’t be able to defend the required minimum 57% on very dry flops. But this is not necessarily a problem for us, since we should be able to defend a bit more than minimum on the coordinated flops. So in the long run, these two factors should even out.

Scenario 2: Alice in the small blind and Bob in the big blind
Alice now opens her 35% button range as default from the small blind:

35% button openrange:

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

458 combos
35%

We discussed this flatting scenario in detail in Part 7 of the preflop series. Since Bob is the only player left to defend the blinds, he has all of the responsibility of defending the blinds enough to prevent Alice from stealing with any two cards. We found that Bob needs to defend with 37.5% of his hands preflop, and he will use a combination of optimal 3/4/5-bet strategies and flatting.

We assumed Bob would 3-bet {JJ+,AK} for value, together with an optimal amount of 3-bet bluffs, and the rest of the defense was done by flatting. We ended up with the following suggestion for a default flatting range (“Blind vs blind flat list”) for Bob to use in the big blind after an openraise from the small blind:

Blind vs Blind flat list

TT-22
ATs-A6s AJo-A7o
K8s+ K9o+
Q8s+ Q9o+
J7s+ J9o+
T7s+ T8o+
96s+
86s+
75s+
65s

362 combos

This flatting range contains many more low combos that the flatting range we use on the button (“IP flat list”), so it will hit more of the low/dry flops. Since we can hit any flop hard, we have the possibility to credibly represent strength on any flop, and thereby create postflop difficulties for the small blind.

But on the other hand we will now have lots of low hand combos in our range that can’t be used to defend high/medium coordinated flops. Whether or not this will create problems for us on the coordinated flops remains to be seen.

In all scenarios we’ll use the strength principle when designing ranges:

– Raise the best hands for value
– Flat with the next best hands
– Bluff with the best of the weakest hands, and fold the rest

4. Bob’s flop strategies after flatting on the button
We now go through Bob’s flop play systematically. First for the coordinated flop, then for the dry flop:

4.1 Play on the coordinated flop after flatting on the button

“IP flat list” after 25% CO openraise:

JJ-22
AQs-ATs AQo-AJo
KQs-KTs KQo
QTs+
JTs
T9s
98s

140 combos

First we count all remaining combos in Bob’s preflop flatting range, given the cards on the board. ProPokerTool’s count function gives us:

So Bob has 117 combos in his range on the flop. In order to defend a total of 57%, he needs to defend 0.57 x 117 =67 combos in total. We choose his value combos first.

Assume Bob will value-raise all his made hands two pair and better on this coordinated flop (so we let all top pair hand go in the flatting range). Bob with then raise a range made up of two pair (QTs, T9s), sets (TT, 99) and straights (KJs). In addition we let him raise the monster draw combo QJs (top pair + open-ended straight draw). This gives us 17 value combos as shown below:

Bob balances this with 17 bluff-raise combos, but before we choose these we pick his flatting hands. Bob needs 67 – 2 x 17 =33 flatting combos to get to 57% total defense. We pick his flatting hands from the next tier of hands on the equity ladder:

– One pair hands
– Open-ended straight draws

It seems obvious to choose from top pair/top kicker (AQs, AQo), top pair/2nd kicker (KQs, KQo), underpair + open-ender (JJ), middle pair + open-ender (JTs). This gives us the 33 combos we need:

Note how strong the ranges for value-raising and flatting are on this flop. We only raise two pair or better + monster draws for value, and our weakest flatting hand is middle pair + open-ended straight draw.

So we have somewhat of a “luxury problem” on these flops after flatting our tight and solid “IP flat” list on the button. We can pick and choose from some very good hands, and we can easily defend the required 57% by only continuing past the flop with quality hands that have good equity.

The last step of the process is to choose Bob’s 17 bluff combos. We step down to the last rung on the equity ladder and choose hands from the low pairs and weak draws (weak one pair hands, overcard hands, gutshots). Note that some open-ended straight draws are counted as weak draws on this flop, since we have so many better made hand and draws to use.

For example, we can pick KTs (2nd pair + gutshot + overcard), ATs (middle pair + overcard), 98s (3rd pair + gutshot) and AJ (open-ender + overcard). This gives us a few too many bluff combos, so we can drop some of the AJ combos. We end up with the following bluffraising range:

Summary
Bob’s total flop strategy on the coordinated flop Q T 9 after flatting on the button is:

  • Raise for value
    {QTs,T9s,TT,99,KJs,QJs} =17 combos
  • Flat
    {AQs,AQo,KQs,KQo,JJ,JTs} =33 combos
  • Bluffraise
    {KTs,ATs,AJs,A J , A J , A J , A J , A J , A J , A J } =17 combos

Bob then defends 17 + 33 + 17 =67 combos in total, which is exactly 67/117 =57% of his total range on the flop. This is the optimal defense percentage we found in Part 1, and Bob’s flop strategy now makes Alice’s random c-bet bluffs break even. We could have designed Bob’s flop strategy in slightly different ways, but our strategy is very reasonable.

But note that we haven’t bothered to defend more than the optimal 57% here, even if we could have. For example, we let Bob fold some draw combos like A J , and the weak pair + draw combos 987s. We have also used potential flatting hands (the weakest middle pair hands) as bluffs, since we had so many better hands to use for value raising and flatting.

We won’t be able to defend the very dry flops as easily, and we should consider overdefending a bit on the coordinated flops to make up for this. For example, we could have moved ATs up to the flatting range and moved the AJ/98s combos we folded up from the folding range to the bluffing range. We have lots of flexibility on this type of coordinated flop, and if we can easily defend more than 57%, we should consider doing so.

We now move on to Bob’s defense with “IP flat list” on the button when the flop comes low and uncoordinated. We’ll see that this flop texture is much harder to defend sufficiently:

4.2 Play on dry flop after flatting on the button

“IP flat list” after 25% CO openraise:

JJ-22
AQs-ATs AQo-AJo
KQs-KTs KQo
QTs+
JTs
T9s
98s

140 combos

As before we begin by counting the remaining combos in Bob’s preflop flatting range:

The poor match-up between this flop texture and Bob’s preflop flatting range is reflected in the number of remaining combos (130 of the original 140). On the coordinated flop we lost a much bigger chunk of our preflop range (117 of the original 140 remained), since our range connected much harder with that flop.

Our standard procedure is to begin by choosing Bob’s value range, but before we do this we should ask: Should Bob have a value range at all on this extremely dry flop?

There are no draws on this flop, and our only monster hands are 9 set combos (3 of each of 88, 55 and 33). If we decide to raise these for value, together with our best overpairs (e.g. JJ and TT), we’ll have an extremely strong value range, but also an extremely weak and easily readable flop flatting range. The reason is that our flatting range will then be made up of two types of hands: Mediocre one pair hands, and some strong overcards (e.g. AQ).

This makes it easy for Alice to play the turn with her value hands. For example, when she has QQ she can bet confidently for value on basically all turn cards, knowing that the best hand we could have on the flop was a pair lower than her. Remember that we would have raised AA/KK preflop, we would have raised all sets for value on the flop, and there are no two pair hands in our range on this flop.

To avoid this problem we can drop all value/bluff raising on the flop and defend entirely by flatting. Then we put all hands worth playing (sets, one pair, good overcards) into our flatting range. Our flop defense range will still be a bit weak, but now Alice can’t bet safely for value with all of her good one pair hands without risking running into a concealed monster hand. If she does, she will every so often get punished by a slowplayed set.

So let’s design a flop flatting for Bob. We want to defend 57% of our range, so we need to find 0.57 x 130 =74 playable combos. It might be impossible to do so without having to flat some unreasonably weak hands, but we’ll see.

We begin with all sets and one pair hands: {88,55,33,JJ,TT,99,77,66,44,22}. This gives us 51 combos, so sets and pairs do most of the work for us. Then we add the best overcard hands: {AQ,AJ} =32 combos.

This gives us 83 combos, and a bit more than we need. We can now use a bit of good poker sense and drop the 6 22 combos. Note that if we are behind a better pair on the flop, it’s better to have AQ/AJ than 22, since the overcard hands have more outs. So we land on the following defense strategy for Bob on the 8 5 3 flop after flatting on the button preflop:

We defend 3 combos more than we need, but that’s fine.

Summary:
Bob’s total flop strategy on the dry flop 8 5 3 after flatting on the button:

  • Raise for value
    None
  • Flat
    {88,55,33,JJ,TT,99,77,66,44,AQ,AJ} =77 combos
  • Bluffraise
    None

So we managed to defend the minimum 57%, but we had to use overcard hands to get there. Of course we technically don’t have the pot-odds to draw to overcard outs, but keep in mind that our overcards are sometimes ahead of Alice on the flop (she has lots of low card hands in her c-betting range). We should also have a bit of implied odds, since Alice might barrel a lot of turn cards that hit our overcards, assuming they are scare cards for us. So she might choose to bluff the turn if a Q falls to barrel us off our weakest one pair hands. Then she donates implied odds to our top pair with AQ, and sometimes she will bet into our slowplayed sets.

We’ll now go through the two example flops one more time, but now with Bob in the big blind after flatting a preflop steal raise from Alice in the small blind. Bob’s preflop flatting range is now wider, and therefore more difficult to defend.

5. Bob’s flop strategies after flatting in the big blind
We’ll now go through Bob’s flop strategies on the coordinated flop and then on the dry flop after flatting in the big blind after a steal raise from the small blind.

5.1 Play on coordinated flop after flatting in the big blind

Blind vs Blind flat list

TT-22
ATs-A6s AJo-A7o
K8s+ K9o+
Q8s+ Q9o+
J7s+ J9o+
T7s+ T8o+
96s+
86s+
75s+
65s

362 combos

Bob has 294 remaining combos in his range, given this flop:

To defend this preflop flatting range optimally, Bob needs to defend 57% of 294 combos on the flop, which is 0.57 x 294 =168 combos. So compared to playing the button preflop range, we will now have to climb further down on the equity ladder and “promote” some button folding hands to flatting and bluffraising hands in the big blind. Note that this is consistent with the fact that we’re up against a weaker raising range (Alice opens her 35% button range in the small blind, but her 25% CO range in CO). So it makes sense that we can value raise and flat with weaker hands than we could on the button.

We now have much more worthless trash in our range, but on the other hands we also have more two pair combos (wider ranges make more “raggedy” two pair combos postflop), and this helps our defense. Which of these two effects is more significant remains to be seen.

We do as we did on the button and put top pair in the flop flatting range. So we value raise two pair(T9s, T9o, Q9s, Q9o, QTs, QTo), sets (TT,99), and straight straighter (J8s, KJs,KJo). This gives us 53 value combos of strong made hands. Then we can add the best pair + draw combos QJs/QJo (top pair + open-ender), and we end up with a value range of 65 combos:

Now we need 65 bluff combos and and 168 – 2 x 65 =38 flatting combos. We pick the flatting hands first from the next rung on the equity ladder (one pair hands and non-monster draws):

For example:

– The remaining top pair hands: AQs,AQo,KQs,KQo,Q8s
– The best middle pair + gutshot hands: KTs,KTo

This gives us 39 combos as shown below:

So we end up with a situation similar to the one we had on the button. We use a tight value range of only two pair and better plus monster draws, and we have plenty of good hands to use as flatting hands. We also have a wide range of mediocre hands to use as bluffs (weak one pair hands and weak draws).

Again, note that we’re not particularly concerned with how to best play a hand like AT on this flop. We simply use the strength principle together with the requirement of 57% total defense, and then we let the hands fall into reasonable categories. In this example AT ended up in the bluffraising range, but this is not very important for us. What counts the most is that we end up with a solid total defense strategy, and that we have a reasonable system for labeling hands as value hands, flatting hands, bluffraising hands and folding hands.

At any rate, what remains is to choose the 65 bluff combos. We pick hands from the remaining one pair hands and draws. For example.:

– The remaining middle pair hands: ATs,ATo,T8s,T8o,T7s
– Bottom pair + open-ender/gutshot: J9s,J9o,98s
– Underpair + gutshot: 88
– Remaining open-enders: AJ,J7s

This gives us 64 combos (close enough) as shown below:

Summary:
Bob’s total flop strategy on coordinated flop Q T 9 after flatting in the big blind is:

  • Raise for value
    {T9s,T9o,Q9s,Q9o,QTs,QTo,TT,99,J8s,KJs,KJo,QJs,QJo} =65 combos
  • Flat
    {AQs,AQo,KQs,KQo,Q8s,KTs,KTo} =39 combos
  • Bluffraise
    {ATs,ATo,T8s,T8o,T7s,J9s,J9o,98s,88,AJ,J7s} =64 combos

Bob then defends 65 + 39 + 64 =168 combos in total, which is 168/294 =57% of his total flop range. Again we see that it’s easy to design a strategy that defends the minimum requirement 57% when the flop comes medium/high and coordinated. We have more weak hands in our preflop range after flatting in the big blind, but we also flop more value hands (more two pair combos).

Like we did in the button scenario we ended up putting some potential flatting hands in the bluffraising range. We used the strength principle as our starting point, chose a solid value range, and let the rest more or less follow from mathematics.

Our last scenario is the most difficult one, namely defending on a dry flop with a wide and weak preflop flatting range:

5.2 Play on dry flop after flatting in the big blind

Blind vs Blind flat list

TT-22
ATs-A6s AJo-A7o
K8s+ K9o+
Q8s+ Q9o+
J7s+ J9o+
T7s+ T8o+
96s+
86s+
75s+
65s

362 combos

Bob has 337 remaining combos in his range, given this flop:

Again we see that most of Bob’s preflop range is intact on a low and dry flop, since the flop connects poorly with our range. We have a flop range of 337 combos and we have to defend with 57%, which corresponds to 0.57 x 337 =192 combos. We use the same philosophy as before, and choose to defend this low and dry flop with only a flatting range.

As we’ll see in a minute, it’s impossible to get to 57% defense without flatting a very wide range of overcard hands. But we start by counting all our combos of one pair or better, and see what we get:

– Sets: 88,55,33
– One pair: TT,99,77,66,44,22,A8,K8s,Q8s,J8s,T8,98s,87s,86s,75s,65s

We have 93 combos of one pair or better:

So with a theoretical 57% total defense, we have to flat 192 – 93 =99 no pair combos. This means we have to reach far down the overcard hierarchy, and we conclude that:

Defending an extremely low/dry flop optimally with a very wide preflop flatting range might me impossible in practice

So we have to accept lots of folding in this scenario, unless we want to defend with lots of ace high and king high hands. We remember that with a tight/solid “IP flat list” on the button (with only 140 preflop combos) we managed to defend this flop 57% by only flatting sets, one pair, and the best overcard hands AQ/AJ. But with a big blind flatting range we have to play many more overcard hands.

Let’s build an optimal 57% defense range, so that we can see what it looks like. We begin by adding our only decent draw (an open-ender with 76s) and then we add overcard hands. If we flat all ace high/king high combos with minimum a T kicker, we get 193 combos (1 more then the 12 we need):

Here we could also have chosen the gutshot + overcard combos 97s/96s, but this will not make a big difference. The gist of it is that we have to defend a very wide and weak range on the flop, and that more than half our flats are no-pair hands.

Summary
Bob’s total flop strategy on the dry flop 8 5 3 after flatting in the big blind is:

  • Raise for value
    None
  • Flat
    {88,55,33,TT,99,77,66,44,22,A8,K8s,Q8s,J8s,T8,98s,87s,86s,75s,65s,AQ-AT,KQ-KT} =193 combos
  • Bluff raise
    None

6. Summary
We have worked our way through 4 flop scenarios where we tried to defend optimally against a c-bet after flatting preflop. We looked at the following scenarios:

– Coordinated flop with a tight preflop flatting range
– Dry flop with a tight preflop flatting range
– Coordinated flop with a loose preflop flatting range
– Dry flop with a loose preflop flatting range

We saw that defending a coordinated flop is an easy task with both preflop flatting ranges. On dry flops we run into the problem of not having enough one-pair-or-better hand or good draws, so we have to resort to overcard hands to reach 57% total defense. On the driest flops we might have to give up more than optimally, but we might be able to make up for this by defending a bit more than optimally on the coordinated flops.

On the extremely dry flops we chose to defend with only a flatting range to avoid polarizing our flop defense ranges into a very strong raising range and a very weak flatting range. If we choose to defend this way, we slowplay all strong hands by flatting them on the flop, planning to raise for value on later streets.

In the next article in this series we’ll go one step further and discuss play on the turn and river after executing our defense strategies in position on the flop.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-3-808#sthash.m4RmAhOM.dpuf

Optimal Postflop Play in NLHE 6-max – Part 2

In this series we apply game theory principles to heads-up postflop play in singly raised pots. Our goal is to design strategies that are hard to exploit. We focus mainly on flop play, but turn and river play will also be mentioned.

1. Introduction
This is Part 2 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

In Part 1 we introduced a model and some basic theory for playing the flop heads-up in position after flatting a raise preflop. Alice raises from some position, Bob flats her raise in position, and all other players fold. Alice now c-bets the flop, and Bob has to defend enough to prevent Alice from profitably c-betting any two cards as a bluff.

Bob defends against the c-bet by raising his best hands for value, flatting the c-bet with his next best hands, bluff raising some of the best hands too weak to raise for value or flat, and folding the rest. In Part 1 we found that Bob needs to defend 57% against a 0.75 x pot c-bet to prevent a profitable any-two-card c-bet. We then estimated the optimal value/bluff ratio for Bob’s flop raising range to be 1 : 1. So Bob should raise one bluff combo for every value combo.

Bob’s method for defending against Alice’s c-bets is then:

  • Bob first chooses his value hands. These are the hands Bob raises on the flop with the intent of getting all-in if Alice 3-bets (and keep betting on most turn cards if Alice calls the raise and checks the turn). He now automatically knows how many bluff combos he needs, and the total number of combos he will be raising
  • Then he picks enough flatting hands to bring his total defense percentage (the sum of raising hands and flatting hands) to 57% of his total range on the flop
  • Then he picks his bluffing hands from the best of the remaining hands (the hands not good enough to raise for value or flat). These are the hands Bob raises on the flop, planning to fold to a 3-bet. If Alice calls the raise and checks the turn, he will sometimes keep bluffing on the turn, sometimes bet for value (when he improves), and sometimes check and give up

In Part 1 we worked through a detailed example, where we let Alice raise with our default ~15% UTG range:

Alice’s default ~15% UTG range:

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

194 combos
15%

Then Bob flatted on the button with the default “IP flat list” defined in Part 2 of the NLHE preflop article series:

Download link for this table (right click and choose “save as”):
IP_3-bet_summary.doc

The flop came:

We estimated the following optimal value, flat and bluff ranges for Bob:

Value raise:

  • Sets: {JJ,99,44} =9 combos
  • Overpairs: {QQ} =6 combos
  • Top pair/top kicker: {AJ} =12 combos
  • Monster draws: {A K , A Q , K Q , Q T } =4 combos
  • Total: 31 combos

Flat:

  • Top pair without top kicker {K J , K J , K J , Q J , Q J , Q J , J T ,J T ,J T } =9 combos
  • 2nd pair: {T 9 , T 9 , T 9 } =3 combos
  • Underpair higher than 2nd pair: {TT} =6 combos
  • Draws: {A T , K T } =2 combos
  • Total: 20 combos

Bluff raise:

{9 8 , 9 8 , 9 8 , Q TQ T , Q T , A K , A K , A K , A K , A K , A K , A K , A Q , A Q , A Q , A Q , A Q , A Q , A Q , A T , K Q , K Q , K Q , K Q , K Q , K Q , K Q , K T } =29 combos

By going through this process many times on many different flop textures, we can train our ability to estimate these ranges quickly at the table (we won’t have the time to find the ranges as precisely as in the example). In Part 1 we focused on the method for estimating Bob’s flop strategy, and how to train this method by repeating it over and over on many flops.

In Part 2 we’ll study this method in more detail, and we’ll discuss some points we only touched on in Part 1. Among other things, we’ll talk about:

  • How Bob’s value range changes as a function of the raiser’s preflop range and the flop texture
  • Play on coordinated flops versus play on dry flops (when should we slowplay?)
  • What should Bob do when Alice checks instead of c-betting?

We’ll illustrate the principles with some simulations done with the analysis softwarePokerazor (which lets us estimate EV for postflop play). So Part 2 will be mostly about theory and modeling. When this work is done, we’ll work through some more examples in Part 2, where we’ll estimate Bob’s flop strategies on two different flop textures (one very coordinated flop, and one very dry flop). We’ll then let Bob use his default “IP flat list” (defined in Part 2 of the preflop series) on the button after a raise from early position, and the “Blind vs blind flat list” (defined in Part 7 of the preflop series) in the big blind after a small blind openraise.

2. Principles for choosing a value range on the flop
When we build a value range for Bob to raise on the flop, we want this to be hands that profit from raising and getting all-in when Alice 3-bets us. For simplicity, assume Alice either 3-bets our raise or folds, and that she never 3-bet bluffs. In this model, our equity has two components:

– What we make when Alice c-bets and folds to our raise
– The all-in equity we have when she c-bets, we raise, she 3-bets, we shove and she calls

It’s obvious that our value hands and our bluff hands make the same when Alice bet-folds (ignoring card removal effects), and the difference between them is the equity from getting all-in against Alice’s value hands.

Choosing value hands that are the favorite against Alice’s presumed all-in range is a good starting point (then both value components are positive), but it’s not an absolute must if Alice bet-folds a lot. It could be that we have a +EV raise with a hands that is a small underdog when getting all-in, if Alice bet-folds so much that the chips we win when she folds outweigh the chips we lose when she 3-bets us and we get all-in. However, for this type of hand we might be better off flatting the c-bet and playing the turn against her total c-betting range. This is analogous to flatting QQ preflop in position against a tight openraiser instead of 3-betting for value, even if 3-betting is +EV in isolation (but flatting is more +EV).

Let’s begin the process of building a value range by making some assumptions:

  • All hands two pair or better are automatic value hands on all flops (unless we elect to slowplay some monster hands)
  • Overpairs and top pair/top kicker can be value hands, but not necessarily. We shall see that this depends on the flop texture (coordinated or dry), the raisers range (tight or loose), and how high our pair is
  • Monster draws can be played for value. The most common monster draws are strong flush draws with extra (flush draws with a straight draw, a pair, or overcards)
  • Made hands top pair without top kicker, and all lower pairs, are candidates for the flatting range

We’ll now do some simple modeling on 4 scenarios to illustrate the effect of Alice’s open range (tight or loose) on two different flop types (coordinated or dry). We will then generalize and draw some conclusions about how to select our value range under different circumstances

2.1 Modeling of a value range
We’ll let Alice openraise two different ranges:

Alice’s default ~15% UTG range:

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

194 combos
15%

Alice’s default ~25% UCO range:

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

326 combos
25%

And we’ll use two different flops:

Coordinated flop:

This is the flop we worked with in Part 1. There we used top pair/top kicker as a value hand, and now we’ll check whether or not this was a good choice:

Dry flop:

This is a very dry rainbow flop without strong draws. This limits Bob’s possibilities for defending against the c-bet, since he now doesn’t have any draws to use. On the driest flops it might be impossible for him to defend sufficiently (57%) without flatting very weak hands. We’ll also see that we run into trouble on these flops if we automatically raise the few monster hands we have (mostly sets) on these flops, since this makes our flatting range weak and transparent. The turn can then be difficult to play when Alice knows we flatted the flop with a weak range of mostly one pair hands and overcards. .

Having a medium/weak flatting range is not a problem on a coordinated flop, since we’re flatting many draws with decent potential for improvement. So Alice can’t simply barrel her whole range again on every turn card, just because she suspects our flop flatting range was weak. If she does, she will often bet into our improved hands. But when we flat on a dry flop, the weak hands in our flatting range are unlikely to improve on the turn (since one pair hands have few outs).

If we never slowplay on dry flops, a flat tells Alice the following:

  • Our flatting range on a dry flop consists mostly of marginal one pair hands and overcards
  • These hands rarely improve on the turn

So Alice can safely bet for value with both her monster hands and her good one pair hands, and she can balance her value bets with bluffs and put a lot of pressure on our marginal hands. We’ll talk more about this problem later, but it’s obvious that we can fix this problem (at least partly) by slowplaying on dry flops, thus making our flatting range stronger.

We now define a model for Alice’s and Bob’s postflop play. We define top pair/top kicker as a hand in between the obvious value hands (where two pair or better are always value hands) and the obvious flatting hands (where one pair hands lower than top pair/top kicker are always flatting hands). Note that we have made these choices to get a simple model that is easy to work with. The assumptions will not always be the best for all situations, but they are reasonable.

Now we’ll model how Bob’s top pair/top kicker perform against:

– Alice’s UTG range on the coordinated flop J 9 4
– Alice’s CO range on the coordinated flop J 9 4
– Alice’s UTG range on the dry flop J 3 2
– Alice’s CO range on the dry flop J 3 2

In all scenarios Bob uses his default “IP flat list” for flatting on the button preflop. We’ll use Pokerazor to compute the EV for Bob’s value raise with top pair/top kicker in these 4 scenarios. Before we can do this, we also need some assumptions about how Alice plays on the flop.

  • Alice raises pot (3.5 bb) preflop, Bob flats his “IP flat list” on the button, and all other players fold. Both players start with 100 bb stacks, so the pot is 8.5 bb on the flop with 96.5 bb left in the stacks.
  • Alice c-bets 6.5 bb with all her hands, Bob raises to 17 bb (~1/2 pot) with top pair/top kicker, Alice folds, calls, or 3-bets to 34 bb ((~1/2 pot) and calls a shove
  • Bob shoves all-in if Alice 3-bets his raise. If she calls his raise, both players check the hand down

Alice’s postflop strategy after Bob’s raise is:

  • 3-bet (and call a shove) with top pair/top kicker or better, together with flushdraw + pair, flushdraw + straight draw, and flushdraw + 2 overcards
  • Call the raise with top pair without top kicker, and let the hand be checked to showdown
  • Fold everything else

Note that we’re not trying to design a perfect flop strategy for Alice. We’re simply making some reasonable assumptions that we can model over. What we are interested in is finding out how the EV for Bob’s value raise with top pair/top kicker changes with the flop texture and with Alice’s openrange, given the assumptions we have chosen for the model.

So it’s the EV differences between the various scenarios that are of most interest to us, not the absolute EVs for each separate scenario. The trends we find for Bob’s EV when he raises to pair/top kicker for value will tell us something about how he should select his value range, given Alice’s openrange and the flop texture. Top pair/top kicker is a hand that can be used both as a value hand and as a flatting hand, so a model study for this hand will tell us a lot about where to draw the line between value range and flatting range.

2.2 Model scenario 1: 15% UTG openrange on coordinated flop J 9 4
Bob’s “IP flat-list” after Alice’s 15% UTG raise is {QQ-22,AKs-ATs,AK-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, where QQ and AK are in the flatting range preflop. The top pair/top kicker hands on this flop are the 12 AJ combos, and we use all of them to get an average EV (they are not 100% equivalent, since some of them have backdoor flush draws).

Alice’s presumed value and flatting hands from her UTG range on this flop are:

  • Top pair/top kicker or better:
    {JJ,99,44,J9s,AA,KK,QQ,AJ}
  • Monster draws:
    {A K , A Q , K Q , K T , Q T , 8 7 }
  • Flatting hands (top pair without top kicker):
    {KJs,QJs,JTs,J9s}

The EV for Bob’s flop raise with his AJ hands on the J 9 4 flop against Alice’s 15% UTG range is:

EV (raise TPTK) =-3.55 bb

Raising top pair/top kicker in this scenario does not quite work for Bob against Alice’s tight openrange, even if the loss is not catastrophic. Bob makes money when Alice folds to his raise, and when she calls with some worse top pair hands, but he loses too much when he gets all-in against her tight value range.

Let’s see what happens against Alice’s CO range:

2.3 Model scenario 2: 25% CO openrange on coordinated flop J 9 4
Bob’s “IP flat-list” after Alice’s 25% CO raise is {JJ-22,AQs-ATs,AQ-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, where QQ and AK now gets 3-bet for value preflop. Everything else is as in Scenario 1.

Alice’s presumed value and flatting hands for her CO range on this flop are:

  • Top pair/top kicker or better:
    {JJ,99,44,J9s,AA,KK,QQ,AJ}
  • Monster draws:
    {A 4 , A K , A Q , K Q , K T , Q T , T 8 , 8 7 }
  • Flatting hands:
    {KJs,QJs,JTs,J9s,J8s}

Alice has two monster draws more than in Scenario 1 (since she now also opens A4s and T8s), and she now also flats the raise with the top pair hand J8s. These rest of her postflop strategy is identical to Scenario 1.

The EV for Bob’s flop raise with AJ on the J 9 4 flop against Alice’s 25% CO range is:

EV (raise TPTK) =+1.33 bb

Bob’s EV goes from negative to positive. Since Alice uses almost the same 3-betting and flatting ranges as in Scenario 1, most of the difference must come from Alice’s folding. She c-bets a much wider range now (25% vs 15%), but the hands she defends with against the range has not changed much. Bob’s value raise then picks up a lot of pots uncontested, and his EV increases.

Note that this problem for Alice is something we have worked with in the preflop series as well. It’s much easier to defend a tight range correctly against aggression than a wide and weak range. This is true both preflop and postflop.

Now we do the same simulations on the dry flop:

2.4 Model scenario 3: 15% UTG openrange on dry flop J 3 2
Bob’s “IP flat-list” after Alice’s 15% UTG raise is {QQ-22,AKs-ATs,AK-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, where QQ and AK are in the flatting range. His top pair/top kicker hands on this flop are the 12 AJ combos, and we use all of them to get an average, as before.

Alice’s presumed value and flatting hands from her UTG range on this flop are:

  • Top pair/top kicker or better:
    {AA,KK,QQ,JJ,33,22,AJ}
  • Monster draws:
    None
  • Flatting hands:
    {KJs,QJs,JTs,J9s}

The EV for Bob’s value raise with AJ on the J 3 2 flop against Alice’s 15% UTG range is:

EV (raise TPTK) =-0.96 bb

The EV for value raising top pair/top kicker against Alice’s 15% UTG range is a bit better on the dry flop texture than on the coordinated one, but still negative. Alice must fold a lot, but a tight c-betting range is still easy to defend.

Then we let Alice openraise from CO:

2.4 Model scenario 4: 25% CO openrange on dry flop J 3 2
Bob’s “IP flat-list” after Alice’s 25% CO raise is {JJ-22,AQs-ATs,AQ-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, where QQ and AK now bet 3-bet for value preflop. Everything else is as before.

Alice’s presumed value/calling hands from her CO range on this flop are identical to the hands she played on the coordinated flop, except one additional flatting hand (since she now also openraises J8s):

  • Top pair/top kicker or better:
    {AA,KK,QQ,JJ,33,22,AJ}
  • Monster draws:
    None
  • Flatting hands:
    {KJs,QJs,JTs,J9s,J8s}

The EV for Bob’s flop raise with his AJ hands on the J 3 2 flop against Alice’s 25% CO range becomes

EV (raise TPTK) =+3.02 bb

Raising TPTK on this dry flop becomes even better when Alice starts with a wide range. This is obvious, since Alice now has to bet-fold a lot in our model (she has no draws to defend with, and only a few value hands). We are now in poor shape when we get 3-bet, but all the fold equity we have makes this a nicely +EV raise.

2.5 Summary of the modeling of EV for value raising flops with top pair/top kicker

– Against Alice’s UTG range on coordinated flop: -3.55 bb
– Against Alice’s CO- range on coordinated flop: +1.33 bb
– Against Alice’s UTG range on dry flop: -0.96 bb
– Against Alice’s CO range on dry flop: +3.02 bb

We can draw some conclusions from this:

1. Raising top pair/top kicker for value can be a slightly losing play
In all model scenarios we were either a small loser or a small winner. This is not totally unexpected, since top pair/top kicker is a good-but-not-great hand somewhere in the region between obvious value hands and obvious flatting hands. We have good equity against all other one pair hands and against draws, but we struggle against better one pair hands, and all hands two pair or better. But overall we did not lose much when raising was -EV.

2. Raising top pair/top kicker goes down in value against a tight openraising range
When Alice starts out with a tight UTG range, she has an easy job defending it against Bob’s flop raises, even if she bet-folds a lot. On both the dry and the coordinated flops her UTG range contained enough value and flatting hands to make Bob’s value raise with top pair/top kicker a slightly losing play, given the assumptions in our model.

But Bob’s raise was +EV on both flops when Alice started with the much looser CO range. Now she had to bet-fold much more, so Bob cashed in on fold equity.

3. Raising top pair/top kicker for value is more profitable on dry flops than on coordinated flops against a raiser that c-bets her entire preflop range on the flop
Note the assumptions used in our model. We have assumed that Alice c-bets her entire preflop raising range on both flops, which isn’t entirely realistic. A good NLHE player will check more of her weak hands on coordinated flops, because she knows that these pots will be hard to win uncontested (she is out of position with a weak hand, and coordinated flops hits her opponent’s preflop flatting range hard). So she will check-fold more weak hands, and her c-betting range will become stronger.

But against a “primitive” player who c-bets too much, and who does not distinguish between flop textures, our model and our conclusions are more valid. We then have much more fold equity on dry flops than on coordinated flops, so the EV for any raising hand increases (and against such a player we can also consider bluff raising a lot more than against a good player).

So we have learned that top pair/top kicker isn’t always a value hand. We have to take the preflop raiser’s c-betting range into consideration, as well as flop texture, before we raise for value, planning to shove all-in after a 3-bet. Furthermore, tight c-betting ranges contain much less “air” than loose ranges, regardless of flop texture. So we have less fold equity when we raise against a tight range, and this reduces the EV for all raising hands.

As a rule of thumb for later postflop modeling, we can assume that it’s fine to draw the line for value raising at top pair/top kicker, regardless of the flop. As a default we will never raise worse hands for value, and we will always raise better hands for value (assuming we don’t want to slowplay). With top pair/top kicker we will sometimes raise for value and sometimes flat. But as we shall see, we won’t necessarily have a value raising range on all flop textures. On the driest flops we might want to slowplay all our strong hands, planning to raise the turn instead.

3. Slowplaying on dry flops
As mentioned previously in this article, we have to take care not to give away too much information about our range when we flat on very dry flops like J 3 2 . For example, let’s say we elect to defend against Alice’s c-bet on this flop by raising all hands top pair/top kicker or better, flatting all lower one pair hands and the best overcard hands (KJ/99/AK/AQ, etc.), and bluffraising some of the better weak hands (KQ, QT, etc).

Some consequences of this flop strategy are:

  • Our value raising range is now easier for Alice to read than when we raise on a coordinated flop (where we can have many draws). This is not a big problem, since we balance our value raises with bluffs, but it will be easy for Alice to see which hands we are representing
  • Our flatting range becomes very transparent. This is our biggest problem when we raise all our best hands on a dry flop and flat with our marginal hands. Now Alice knows that our flatting range contains only marginal one pair hands and some overcard hands, and that we have few outs to improve on the turn.

Let’s study this in more detail on the example flop J 3 2 where we have flatted out default “IP flat list” after a CO raise from Alice. So we begin with the preflop range {JJ-22,AQs-ATs,AQ-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, and we decide to never slowplay on the flop. Alice c-bets the flop, and we call. We would have raised all hands top pair/top kicker or better on the flop, and Alice knows this based on the reads she has picked up on us. She also knows our preflop flatting range.

Alice can now draw some strong conclusions:

  • We don’t have any overpairs in our flop flatting range (we would have raised AA-QQ preflop).
  • We don’t have top pair/top kicker or sets in our flop flatting range (we would have raised these on the flop)
  • So the best hand we can have after flatting the flop is KJs, and most of our flatting hands are weaker than this

Alice now has easy pickings on the turn. She can continue to bet safely for value with all her made hands top pair/top kicker and better, regardless of the turn card. Of course some turn cards will improve some of our flatting hands, but usually they won’t, and the percentage play for Alice is to keep value betting all her best hands. And on a flop without draws, Alice don’t have to worry about clashing with a flush or a straight on the turn.

Furthermore, Alice can easily take advantage of both scare cards and blanks on the turn to force us to fold our weak one pair hands and overcard hands. Let’s say we’re at the turn with a mix of marginal hands like AK, 99, 88, etc after flatting the flop c-bet. It will be difficult for us to call another big bet on the turn, even if it’s a blank. An in addition, scare cards can come (typically overcards) that will make it extremely difficult for us if Alice bets again.

Problem:
Playing straightforwardly with strong hands on dry flops makes it difficult to play our flop flatting range well on later streets.

Solution:
On dry flops we have to consider slowplaying some (or all) of our best hands, and flat them together with our marginal hands. This serves two purposes:

  • We make our flatting range stronger
  • We make more money from Alice’s bluffs. Sometimes she improves and keeps betting (but now for value). Other times she keeps bluffing, assuming our range is weak after we flatted the flop

Do we need to slowplay on coordinated flops like J 9 4 ? We can if we want to, but our flop flatting range will be strong on many turn cards, since we flat with a lot of draws. So Alice can’t simply barrel away on the turn, assuming our flop flatting range is still weak on the turn.

The thing about coordinated flops is that many of our flatting hands can improve to the nuts or near-nuts on the turn. And since our flop flatting range in practice will cover all turn scare cards, Alice can’t simply bluff at any turn scare card and get away with it.

Also, slowplaying on coordinated flops is more risky with regards to giving Alice a cheap shot at drawing out. Slowplaying top set on a J 9 4 flop is much more risky than on a J 3 2 flop, since Alice will have many flush and straight draws on the first flop.

We’ll talk more about slowplaying on dry flops in Part 3. There we will look at Bob’s flop play on two different flop textures (coordinated and dry) with two different preflop flatting ranges (IP flat list” and “Blind vs blind flat list”). On very dry flops we might not want to raise at all. If this is the case, we do all the flop defense with a flatting range, planning to raise our monster hands for value on a later street.

4. What do we do when Alice checks the flop?
It should now be clear from examples and discussion that Bob’s flop strategy of value raising, bluff raising, flatting and sometimes slowplaying will prevent Alice from profitably c-betting blindly on all flop textures, particularly on the coordinated flops that hit Bob’s preflop flatting range hard. So Alice will have to check some hands, planning to give up. What should Bob do when Alice checks?

It’s obvious that we want to bet our value hands and balance this by buffing with some weak hands, particularly if Alice rarely checkraises in these scenarios. There are always lots of hands without a pair, a draw or showdown value in our range. We should bluff a lot of these hands when Alice has checked, since she has told us that she is probably weak, and it will be difficult for us to win the pot unless we bluff. We can also consider turning our weakest one pair hands into bluffs when checked to, if we believe this will be more profitable than trying to sneak cheaply to showdown.

For example, if Bob has 22 on the flop J 9 4 , it’s fine to bet the hand when checked to, mostly to win it right there, even if his bet isn’t for value (he can’t continue after a checkraise). Betting can be a bad idea when no worse hands call and no better hands fold, but there can be merit to bet to collect dead money, even if we never get action from hands we beat. If Alice is usually weak and will rarely checkraise, betting to protect a hand with weak showdown value can be fine.

But if our marginal hand is strong enough to call a turn bet if we check the flop, and/or if it’s difficult for Alice’s worse hands to draw out on us, we have more reason to check behind. For example with QQ on a A 9 3 rainbow flop. Now we can check the flop, planning to call at least one bet, should Alice come out betting later. And giving Alice a free card will rarely cost us the pot, since she her worse hands can never have many outs against us.

If Alice checks the flop, we can also semibluff with the draws we would have flatted a c-bet with, and we can bet for thin value/protection with some of the better one pair hands that we would have flatted with (typically the top pair hands). In general, we don’t want to give Alice a free card when we have a vulnerable hand, but we have to be cautious about betting marginal hands behind players that also slowplay good hands. Against players that checkraise a lot and give us tough decisions, we should check more of our marginal hands on the flop, and then we play the turn (planning to call a lot of turn bets if we get bet into by an aggressive player).

On the other hand, against a straightforward player who rarely check good hands, we can bet more turns. Either for value (our flop value raising range), as a semibluff (draws not in our value range), or as a thin bet for value/protection (some marginal one pair hands that we would have flatted against a c-bet). And of course we will bet lots of pure bluffs (the bluff raising hands, and some more).

Getting these betting opportunities is a bonus effect of defending strongly against Alice’s c-bets, using our optimal value raise/bluff/raise/flat strategy. Alice sees that we defend more than half the time she c-bets (57% to be exact), so she has to give up any-two-cards bluffing, and she is forced to check many weak hands to us. Note that she can in principle balance her weak checks with some strong checks (slowplaying) but this is difficult to do well. Her strong hands also need to balance her c-betting range, and she only has so many strong hands to use. Balancing both a c-betting range and a checking range is very hard, and against most players you can get away with a lot of betting after they check to you on the flop. And many of your bets should be pure bluffs.

5. Summary
In this article we have talked about choosing our value raising range against a c-bet on the flop, and we have modeled value raising with top pair/top kicker in 4 scenarios:

– Against a tight UTG range on a coordinated flop
– Against a loose CO range on a coordinated flop
– Against a tight UTG range on a dry flop
– Against a loose CCO range on a dry flop

We saw that top pair/top kicker is a hand that can sometimes be used as a value hand (particularly against a loose c-betting range), even if it’s an underdog against the range the c-bettor continues with after we raise. But top pair/top kicker was a marginal value hand at best in our model, and sometimes flatting the c-bet will be a better way to play it, for example against an UTG raiser with a tight range.

Then we discussed slowplaying on dry flops, and the consequences of always value raising our best hands for value on these flops. Our flatting range then becomes transparent and easy to play against on later streets, so we have to consider slowplaying some (or all) of our best hands as well. On extremely dry flops we might not want to raise for value at all, and only defend with a flatting range (more about this in Part 3).

Lastly, we talked about what to do when Alice checks to us instead of c-betting. Our defense strategy against her c-bets forces her to check some hands, and her checking range will usually be weak and easy to play against. This gives us the opportunity to bluff a lot when checked to, particularly against a player that rarely checks strong hands or checkraise bluffs in this situation.

In Part 3 we’ll see how these principles are applied in practice. We’ll pick a coordinated flop and a dry flop, and see how Bob designs his defense against Alice’s c-bets on these flops, taking the positive effects of slowplaying into account on the dry flop. To practice using our default preflop ranges postflop, we’ll let Bob use two different preflop flatting ranges: First the standard “IP flat list” on the button, and then the “Blind vs blind flat list” in a blind vs blind scenario where he flats an openraise made by the small blind.

God luck!
Bugs

– See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-2-761#sthash.dzqhKRUQ.dpuf