Optimal Postflop Play in NLHE 6-max – Part 5

1. Introduction
This is Part 5 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, Part 2, Part 3 and Part 4 we discussed postflop play heads-up in position after flatting preflop. This is an important postflop scenario for us, since our preflop strategies include lots of flatting in position.

When we have position on the raiser it’s important that we defend enough postflop to prevent her from c-betting any two cards profitably on the flop. When we flat on the flop, we have to defend enough against her turn bets to prevent her from 2-barreling any two cards as a bluff, and the same goes for river play after we flat the turn. How often we defend on each street depends on the raiser’s bet sizing. The smaller she bets, the more hands we defend. This makes sens intuitively, since smaller bets means the raiser is getting a better prize on her bluffs (we should defend more), while we’re getting better pot-odds to continue (so more of our weak hands are getting the right prize to see the next street).

We have used the following standard bet sizes in the postflop articles:

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

If Alice raises preflop and Bob flats in position, Alice is getting pot-odds 1 : 0.75 on her flop and turn bets. She then automatically makes a profit if Bob folds more than 0.75/(1 + 0.75) =43% , so Bob has to defend at least 100 – 43 =57% against Alice’s c-bets and turn bets. On the river Alice’s pot-odds on a 0.6 x pot river bet are 1 : 0.60. She automatically makes a profit if Bob folds more than 0.6/(1 + 0.6) =38%, so Bob should defend the river at least 100 – 38 =62% to prevent this.

Bob’s total postflop strategy in position after flatting preflop is made up of of value raising, bluff raising and flatting on each street. But as we discussed in previous articles, it will be better for him to only defend by flatting on the driest flops (like 2 6 6 ) to prevent his flatting range from being weak and easy for Alice to read and play against on later streets.

Bob did not have this problem when flatting on coordinated flops (like J 9 3 ), since these flops hit his preflop flatting range much harder and gives him many strong hands/strong draws that he can raise for value. Furthermore, his flatting hands on this type of flop will often improve to strong hands on the turn. So Alice can’t assume Bob’s turn range is weak on a coordinated board, just because he flatted the flop. Therefore, it is on the dry flops that we often have scenarios where the raiser c-bets the flop, 2-barrels the turn, and 3-barrels the river, while the raiser is calling down in position with a weak range.

In these scenarios both players rarely have anything better than one pair. Forcing the other player to fold his weak one pair hands and good overcards is therefor an important value component in both players’ postflop strategies. For example, if the raiser c-bets A A on a Q 8 4 flop and the flatter folds 2 2 , the raiser has gained a lot.

The raiser out of position tries to achieve this by c-betting a lot as a bluff, and then sometimes bluffing again on the turn when called, and again on the river when called on the turn. And the player in position tries to win pots by calling down a lot with his one pair hands, but also sometimes floating with very weak hands, planning to bluff with these hands if the raiser checks and gives up on a later street.

We define a float as a call done either with a weak hand that can’t win a showdown unimproved (so we plan to often bluff on later streets if we get the chance) or a hand with mediocre showdown value that we are hoping to take cheaply to showdown (but we are too weak to call down if the raiser bets all 3 streets). Using this definition, calling with both T 9 and 2:heart: 2:spade: on a Q 8 4 flop would be floats.

In previous articles we have studies Bob’s strategies in position. In this article we’ll turn the tables and study Alice’s strategies out of position. We’ll start with the following model:

– Both players begin with 100 bb stacks
– Alice openraises preflop and Bob flats in position
– Alice c-bets her entire preflop range on the flop

This creates a turn/river dynamic between the two players those times Bob calls the flop. In this article we’ll only look at dry flops, since this lets us use two simplifying assumptions:

1. Alice begins by c-betting her entire preflop range (reasonable, since Bob’s preflop flatting range will be weak on dry flops)
2. Bob never raises the flop (reasonable, since it makes sense for him to slowplay his best hands on dry flops for reasons previously discussed)

Whether or non Alice should c-bet her entire range on dry flops is not something we’ll discuss here, but it is reasonable on dry flops. We’ll use this as an assumption in our model, since it can never be a big mistakes when we are heads-up against a preflop flatter that will often have missed a dry flop. Furthermore, we’ll limit our discussion to scenarios where Bob never has a hand strong enough to raise for value on any street. This puts him in a situation where he is either calling or folding on each street. This creates a postflop dynamic where:

– Bob needs to defend enough against Alice’s barreling on all 3 streets
– Alice needs to defend enough against Bob’s floats on the flop and turn

Bob’s task is to prevent Alice from having an automatically profitable bet/bet/bet strategy (3-barreling) with any two cards. Alice’s task is to prevent Bob from having an automatically profitable float with weak hands on the flop and turn.

Bob starts by calling Alice’s c-bet with many medium/weak hands that are not strong enough to call down. Alice’s job on the turn and river is then to play these streets in such a way that Bob can’t call the flop or turn with any two cards and make a profit. For example, if Alice c-bets 100% of her range on the flop, but then check-folds 2/3 of her range on the turn without ever check-calling or check-raising, Bob can call her c-bet with any two cards, planning to auto-bet the turn as a pure bluff those times Alice checks and gives up.

If Bob can call a flop c-bet with automatic profit with a hand as weak as 2 2 on a J T 4 flop, Alice is probably doing something wrong on the turn and river. Note that when Alice checks the turn and gives up after getting floated on the flop, she has in reality lost the hand. If Bob has floated with a worthless hand, he will now bet and Alice will fold. If he has a hand with weak showdown value, as in the 2 2 hand above, he can choose between betting it as a bluff or checking it to showdown (we’re assuming Alice isn’t planning to bluff the river when the turn goes check-check). If the hand get checked down, Bob will usually win, since Alice on average will have few outs those times she checks and gives up on the turn.

At any rate, Alice can not allow Bob to sit behind her and call c-bet and turn bets profitably with any two cards, so she has to make sure she defends her betting range on the current street by not giving up too easily on the next street after getting called. In this article we’ll show how Alice can build turn and river strategies, based on pot-odds and simple theory, that prevents a player in position from floating her with any two cards on the turn or river.

Alice does this by betting, check-calling and check-raising enough on the next street after betting the current street and getting called. This prevents the player in position from getting enough profitable bluffing opportunities, or opportunities to get cheaply to showdown with weak hands that have some showdown value. Precisely how often Alice needs to continue on the next street after betting the current street and getting called is something we can estimate using mathematics and simple assumptions.

We’ll use theory borrowed from Matthew Janda’s excellent game theory videos at /Cardrunners.com. Then we adapt this theory to the “model game” we have designed throughout the NLHE preflop article series and this NLHE postflop article series. We’ll use our default preflop “core ranges” as a starting point for out postflop ranges.

Before we begin building Alice’s postflop strategy, we’ll warm up by verifying that Bob’s calling strategy in position (discussed in Parts 1-4 in this article series) does what it was designed to do, namely prevent Alice from c-betting/2-barreling/3-barreling profitably with any two cards those times Bob doesn’t have a hand strong enough to raise for value on any street.

2. How Bob’s calldown strategies makes Alice’s any-two-cards bluffs break even
Let’s quickly repeat an example from Part 4 where Alice c-bets the flop, 2-barrels the turn, and 3-barrels the river. We’re only looking at the region of possible outcomes where Bob only has a calling hand on each street.

Alice (100 bb) raises her default ~25% range from CO, Bob (100 bb) flats on the button with his standard flatting range in position (“IP flat list”):

IP flat list after a ~25% CO openraise

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

140 combos

The flop comes:

Bob’s preflop flatting range of 140 combos was reduced to 130 combos on this flop (card removal effects):

Bob then had to defend 57% against Alice’s c-bets on the flop, which is 0.57 x 130 =74 combos. We estimated Bob’s optimal flop strategy as:

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

Bob slowplayed all his strong hands on this very dry flop, and the reasons for this choice were discussed previously. Then the turn came:

The flop flatting range of 77 combos was reduced to 73 combos, given this turn card:

Again, Bob has to defend 57% of his range, which is 0.57 x 73 =42 combos. On the turn he will use a raising range of strong hands (some slowplayed monsters from the flop) and he balances this with bluffs in a 1 : 1 value/bluff ratio. The rest of the defense is done by flatting. We estimated his optimal turn strategy to be:

  • Raise for value
    {88,55,33} =9 combos
  • Flat
    {AQ,JJ,TT} =24 combos
  • Bluffraise
    {AJs,9 9 ,9 9 ,9 9 ,9 9 ,9 9 } =9 combos

we then moved on to playing the river after Bob had flatted the turn:

The river card had no effect on Bob’s range, and his 24 turn flatting combos were intact on the river:

Bob then had 24 combos in his river range, and he had to defend them optimally against Alice’s 0.60 x pot river bet. As calculated previously, Bob then has to defend 62% of his range to prevent Alice from bluffing profitably with any two cards. He has no hands strong enough to raise for value (he only has one pair hands to use as bluffcatchers), so he needs to defend 0.62 x 24 =15 combos by flatting them. We estimated Bob’s optimal river strategy to be:

  • Raise for value
    None
  • Flat
    {AQ,J J , J J , J J } =15 combos
  • Bluffraise
    None

What generally happens from street to street those times Bob finds himself inn a call-down process (those times he has medium/weak hands) on a dry flop texture is that he begins by flatting the flop with a wide range of almost any pair plus his best overcard hands. The overcard hands are floats that he doesn’t plan to call down with, but he has to call the flop with them in order to defend enough. Then he typically drops his overcards and lowest pairs to a turn bet when Alice bets again. And finally, he calls a 3rd bet with his best pairs on the river and folds his lowest pairs.

This makes sense intuitively, since Bob needs to balance two factors:

– He has to prevent Alice from often bluffing him out of the pot with any two cards
– But he has to avoid paying off her better hands too often

The optimal call-down strategy outlined above makes sure Bob isn’t giving Alice a big opening for bluffing profitably with any two cards on any street. He calls down enough to prevent this, but he also folds enough to prevent Alice’s strong hands from extracting a lot of value from his bluffcatchers.

We’ll now use mathematics to show that Bob’s optimal call-down strategy prevents Alice from running a profitable any-two-cards bluff against him. We’ll assume that:

– Bob has a bluffcatcher that always beats Alice’s bluffs
– Alice has a pure bluff that never draws out on Bob’s hand
– Alice decides to run a 3-barrel bluff with her worthless hand
– Bob calls down optimally

Bob’s defense on the flop
Let the pot size on the flop be P. Alice now c-bets 0.75P with her worthless hand. Bob calls 57% of the time with his bluffcatcher (he can use a randomizer to determine when he calls and when he folds) and folds 43% of the time. Those times he calls, the pot grows from P to P + 0.75P + 0.75P =2.5P. Both players have now put 0.75P into the pot postflop.

– % Bob folds the flop: 43%
– Alice’s profit when Bob folds the flop: P

Alice wins the flop pot when Bob folds.

Bob’s defense on the turn
The pot is 2.5P on the turn. Alice now 2-barrels 0.75x pot with her worthless hand. Bob calls 57% and folds 43% to this turn bet. When he calls, the pot again grows with a factor 2.5 and becomes 2.5 x 2.5 x P =6.25P. Both players have now put (6.25P – P)/2 =2.625P into the pot postflop.

– % Bob calls the flop and folds the turn: 0.57×0.43 =25%
– Alice’s profit when Bob folds the turn: P + 0.75P =1.75P

Alice wins the flop pot + Bob’s flop call when Bob calls the flop and folds the turn.

Bob’s defense on the river
The pot is 6.25P on the river. Alice now 3-barrels 0.60 x pot with her worthless hand. Bob calls (and wins against Alice’s bluff) 62% and folds 38%. When he calls, the pot grows from 6.25P to 6.25P + 2 x 0.6 x 6.25P =13.75P. Both players have now put (13.75P – P)/2 =6.375P into the pot postflop.

– % Bob calls flop and turn, and the folds river: 0.57×0.57×0.38 =12%
– Alice’s profit when Bob folds the river: P + 2.625P =3.625P

Alice wins the flop pot + Bob’s flop call + Bob’s turn call when Bob calls the flop + turn, and then folds the river.

– % Bob calls the flop + turn, and then folds river: 0.57×0.57×0.62 =20%
– Alice’s loss when Bob calls down: -6.375P

Alice loses her flop c-bet + turn bet + river bet when Bob calls down.

Total EV for Alice’s 3-barrel bluff
Below is a summary of all the possible outcomes, with Alice’s profit/loss for each of then in parentheses:

  • Bob folds flop: 43% (P)
  • Bob calls flop/folds turn: 25% (1.75P)
  • Bob calls flop/calls turn/folds river: 12% (3.625P)
  • Bob calls flop/calls turn/calls river: 20% (-6.375P)
  • Total: 100%

 

EV (3-barrel bluff)
=0.43(P) + 0.25(1.75P) + 0.12(3.625P) + 0.20(-6.375P)
=0

Bingo! Alice’s 3-barrel bluff project is exactly break even when Bob sits behind her with a bluffcatcher and calls down optimally. His call/fold percentages on each street are functions of Alice’s bet sizes on each street. If Alice had changed her bet sizes, Bob would have adjusted his call/fold percentages correspondingly (smaller bets =Bob calls more, bigger bets =Bob folds more). For example if Alice had bet the pot on each street, Bob would have called 50% and folded 50% on each street (since Alice’s pot-odds on a bluff are now 1 : 1 on each street). You can easily verify that Alice’s 3-barrel bluff EV would have been zero with this bet sizing scheme as well.

This verifies that when Bob is inn a call/fold scenario that stretches over multiple streets, his optimal postflop strategies will prevent Alice from running a profitable any-two-cards 3-barrel bluff against him. So Alice can’t exploit Bob by bluffing aggressively, but note that Bob isn’t doing anything to exploit Alice’s bluffing either.

To exploit Alice’s any-two-cards bluffing strategy (if she is in fact using such a strategy) Bob needs to call down more than optimally to exploit the opening Alice is offering him. For example, he can choose to call down 100% with his bluffcatcher if he believes that Alice is betting 100% of her range on every street in an attempt to bluff him off his weak hands.

This should be profitable for him, since there should be many more bluffs than value hands in Alice’s range on a dry flop. However, by doing so he is offering Alice an opening for exploiting him back by stopping to bluff and only betting her value hands. But Bob can always return to the optimal call-down strategy if he isn’t sure whether or not Alice is bluffing way too much, or if he suspects she will quickly adjust to his attempts to exploit her bluffing.

Now we have warmed up, and we move on to the main topic for this article:

3. Optimal 2- and 3-barreling heads-up and out of position
We’ll now look at the scenario where:

– Both players start with 100 bb stacks
– Alice raises preflop and Bob flats in position
– Alice c-bets her entire preflop range on a dry flop, and Bob flats
– Alice then uses a turn/river barreling strategy designed to prevent Bob from floating profitably with any two cards on the turn or river

We’ll do this in to steps:

1. Study a simple mathematical model
2. Implement the theory working through an example

3.1 Modeling barreling out of position
First, let’s define barreling. This is simply to keep betting on the next street after you have bet the current street and gotten called (and it doesn’t matter whether you’re weak or strong). So if Alice raises preflop, c-bets the flop, and then bets the turn, she has done a 2-barrel. If she also bets the river after getting called on the turn, she has done a 3-barrel.

When Alice is out of position versus Bob, c-bets the flop and gets called, it’s important for her to have a balanced strategy for turn play in order to prevent Bob from exploiting her by floating with any two cards on the flop (planning to steal the pot on later streets). If Alice checks and gives up on too many turns, it will be profitable for Bob to call her c-bet regardless of what he has, planning to auto-bluff the turn when checked to (for example if he floated the flop with a gutshot straight or overcards), or planning to check down a hand with marginal showdown value (for example, if he floated the flop with a low pair).

Alice can counter Bob’s floating strategy with random weak hands by 2-barreling enough on the turn and we’ll see how often she needs to do that in a minute). But Alice can’t only defend her flop betting range by 2-barreling, since this makes her turn checking range transparent and easy to exploit (since Bob then knows that Alice is always weak when she checks). So Alice needs to mix in some check-calling and check-raising on the turn as well.

The same logic applies to river play after Bob flats Alice’s turn bet. She has to 3-barrel/check-call/check-raise enough to prevent Bob from floating the turn with any two cards, planning to steal the pot on the river, or win a showdown with a weak hand that has showdown value (but not strong enough to call both the turn and the river.

We’ll use a simple model and a bit of math to estimate how often Alice needs to defend on the next street after betting the current street and getting called. We use our standard postflop bet sizing scheme:

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

When Alice c-bets 0.75 x pot on the flop, Bob is getting pot-odds (1 + 0.75) : 0.75 =1.75 : 0.75 on a call. If Alice never check-raises or check-calls the turn, Bob can float a random weak hand with automatic profit if Alice checks and gives up more than 0.75/(1.75 + 0.75) =30% on the turn. Therefore, if Alice defends against Bob’s flop floats by only 2-barreling, she needs to 2-barrel 100 – 30 =70% of her flop betting range on the turn. We can express this as:

2-barrel%=70%

This is a mathematically acceptable defense strategy against flop floats, but Alice can make things easier for herself by also check-calling and check-raising some on the turn. This makes it more expensive on average for Bob to steal the pot (which means Alice can get away with less 2-barreling). It also makes Alice’s turn checking range much harder to read, since she isn’t always ready to give up the pot when she checks.

Those times Alice 2-barrels the turn and Bob folds his random flop float, his loss is limited to his flop call of 0.75 x flop-pot. Now, assume Bob always bets his floats as a turn bluff when Alice checks to him. His plan is to fold to a turn checkraise, and give up his steal attempt if Alice check-calls Bob is then prepared to check down the hand and lose a showdown). Bob’s turn bet is 0.75 x turn-pot, and the turn-pot is 1 + 0.75 + 0.75 =2.5 x flop-pot. Bob then invests 0.75 x 2.5 =1.875 x flop-pot with his turn bluff.

Then his total risk for trying to steal the pot with a flop float + turn bluff is (0.75 + 1.875) =2.625 x flop-pot. When Alice check-calls or check-raises the turn, Bob’s expense is then 2.625/0.75 =3.5 x higher than when Alice 3-bets (so that Bob only loses his flop call of 0.75 x flop-pot).

To make Bob’s steal attempt break even, the following equation needs to be satisfied:

2-barrel%(-0.75P) + check-continue%(-2.625P)
+ (100 - 2-barrel% - check-continue%)(+1.75P) =0

In words:

The amount Bob loses by floating the flop and getting 2-barreled (-0.75P each time), plus the amount he loses by floating the flop and getting his turn bluff check-called or check-raised, plus the amount he makes when his turn bluff succeeds, should sum to zero. That makes his float flop + bluff turn strategy break even, which is what Alice’s wants her turn strategy to do for her.

We simplify this equation to get:

2-barrel%(-0.75P) + check-continue%(-2.625P)
+ 175P - 2-barrel%(1.75P) - check-continue(1.75P) =0
2-barrel%(-0.75P - 1.75P)
+ check-continue%(-2.625P - 1.75P) + 175P =0
-2.5P x 2-barrel% - 4.375P x check-continue% + 175P =0
2.5P x 2-barrel% + 4.375P x check-continue% =175P
P x 2-barrel% + 1.75P x check-continue% =70P

And the above equation for Alice’s turn defense strategy against flop floats can be generalized to:

2-barrel% + 1.75 x check-continue% =70%

The term check-continue is the label we use for all of Alice’s check-calling and check-raising. We have here assumed that Bob always loses the hand when he bets the turn and Alice doesn’t fold. Note that we are ignoring the equity of Bob’s hand, and we assume that he never wins a showdown after Alice check-calls the turn. Bob is always behind when this happens, he never improves to the best hand on the river, and he never bluffs the river. These are simplifying assumptions, but this is fine when we’re modeling a situation. Also, keep in mind that sometimes Alice bets or check-calls the worst hand, and then she draws out on the river. So as a first approximation we can assume that these two effects cancel out.

We’ll now put the above equation to work by studying an example scenario heads-up with the raiser out of position on a dry flop. On these flops we’ll often get a call-down scenario where the raiser c-bets any two cards on the flop, and then the preflop flatter sits in position with a medium/weak range of mostly one pair hands and overcards. usually the caller is not strong enough to raise anywhere along the way, so he will often be faced with a call/fold decision on every street those times the raiser fires multiple barrels.

What typically happens when two good, thinking players clash in this type of scenario is that both will be playing wide ranges on the flop (the raiser c-bets a lot and the player in position flats a lot). Then both players drop many (but not all) of their bluffs, floats and weak one pair hands on the turn, and then again on the river. And both players are trying to prevent the other player from bluff-barreling/floating profitably with any two cards on any street.

3.2 Example of optimal c-betting/2-barreling/3-barreling heads-up and out of position on a dry flop
Alice raises her default ~15% opening range from UTG:

~15% UTG-range

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

194 combos
15%

Bob flats on the button. At this moment we’re not particularly concerned with Bob’s flatting range or postflop strategy, but we can assume he uses his standard flatting range outside of the blinds (“IP flat list”):

IP flat list after ~15% EP openraise

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

162 combos

The flop comes:

We’ll now focus on Alice’s postflop strategy from street to street. She begins by c-betting her entire preflop range on this dry, low flop, since it mostly misses Bob’s preflop flatting range, and she expects him to fold a lot. We don’t know what Bob has, but we can assume his range is weak. Alice must now have a strategy ready for the turn, so that Bob can’t exploit her by floating the c-bet with any two cards.

We saw previously that Alice can achieve this by 2-barreling, check-calling and check-raising the turn so that the following equation is satisfied:

2-barrel% + 1.75 x check-continue% =70%

The turn comes

Before Alice builds a turn strategy, we take card removal effects into consideration and count the number of combos in her turn range, given the cards on the board. Since she c-bet her entire preflop range on the flop, her turn range equals her preflop range minus the combos that are eliminated due to card removal effects:

There are 168 combos remaining in Alice’s range. If she only 2-barrels and never check-calls or check-folds, she needs to bet 70% of these combos, which is 0.70 x 168 =118 combos. If she also check-calls and check-raises, we can rewrite the defense equation as:

2-barrel-combos + 1.75 x check-continue-combos =118

Alice now uses a turn strategy where she:

– Check-raises a few of her best hands
– Bets the rest of her best hands for value
– Check-calls with some medium strong hands
– Balances her value bets with some bluffs in a 1 : 1 ratio
– Check-folds the rest of her hands

Here we’ll not go into detail about which hands are good enough to check-raise, value bet or check-call, and we’ll use good poker sense when putting hands into different categories. Furthermore, we haven’t shown mathematically that 1 : 1 is the best value/bluff ratio to use for Alice’s 2-barrels, but we’ll assume this is reasonable (and it’s easy to remember).

Let’s estimate a reasonable total turn strategy for Alice and check whether or not this gives her enough protection against floats according to the defense equation:

  • Check-raise:
    {JJ} =3 combos
    Value bet:
    {99,66,33,J9s,AA-QQ,AJ} =41 combos
  • Check-call
    {KJs,QJs,JTs,TT,A9s} =18 combos
    Bluff:
    {QTs,AK,AQ,KQs} =40 combos

So Alice 2-barrels 41 + 40 =81 combos using an approximate 1 : 1 value/bluff ration, and she check-calls/check-raises 3 + 18 =21 combos. She makes things simple and choose top pair/top kicker or better as her value hands, check-calls with the remaining top pair hands + the best of the lower pairs, and bluffs with an open-ended straight draw and the best overcard hands.

The defense equation gives:

2-barrel-combos + 1.75 x check-continue-combos
=81 + 1.75 x 21
=118 (optimal =118)

Our estimate of Alice’s turn strategy satisfies the defense equation exactly. Now we can go back to our ranges and do some polish if we want to, particularly for the hands in between the obvious check-calling hands and the “air hands” (our 2-barrel bluffs). For example, we chose to check-fold T9s since we had enough better one pair hands to use for check-calling, and we preferred to bluff with draws and overcards, since these on average have more outs than marginal one pair hand.

Here it’s important to note that T9s will win some showdowns, since Bob will sometimes check the turn and give up. So check-folding these marginal one pair hands does not automatically mean they lose, since the player in position will sometimes be willing to check down weaker hands. And of the turn and river goes check-check we’d rather have T9s than two overcards. So it makes more sense to check turns with our weakest one pair hands, instead of turning them into bluffs. And then we pick our bluffs from hands that can’t win showdowns unimproved.

At any rate, when we’re building a mathematically sound total turn strategy this type of marginal decision making is not very important. For the moment we’re only concerned with building a reasonable turn strategy for Alice, and then we can polish it later.

Now we let Alice bet her turn 2-barrel range, which is a 1 : 1 mix of value hands and bluffs (note that top set JJ is not a part of our range, since we put it in our turn check-raising range):

  • Value bet:
    {99,66,33,J9s,AA-QQ,AJ} =41 combos
  • Bluff:
    {QTs,AK,AQ,KQs} =40 combos

Bob calls turn turn bet, and the river comes:

Alice’s 2-barrel-range of 40 + 41 =81 combos is unaffected by this river card, and she still has 81 combos on the river:

We remember that Alice’s bet sizing is 0.60 x pot on the river. Now there aren’t any more cards to come that can change hand strength, and only one round of betting remains. we can now calculate the exact optimal value/bluff ratio for Alice’s 3-barrels. When she bets 0.6 x pot, Bob is getting pot-odds (1 + 0.60) : 0.60 =1.60 : 0.6. Alice now wants to bluff exactly so often than Bob becomes indifferent to calling or folding with his bluffcatchers (those of his hands that can only win if Alice is bluffing, for example a marginal one pair hand).

The logic behind this is that if Alice bluffs less, Bob can exploit her by always folding his bluffcatcher and save money (since he isn’t getting the right pot-odds to call). But if she bluffs more than optimally, relative to her bet sizing, Bob can exploit her by calling with even more bluff catchers (since he is getting better pot odds than he needs), and Alice now loses money.

Therefore, Alice wants to bluff just enough to make Bob’s EV zero when he calls with a bluffcatcher. Then she has a guaranteed minimum profit from betting the river. If Bob tries to save chips by not paying off with his bluffcatchers, Alice will steal some pots with her bluffs. If he tries to snap off a possible bluff by calling with all his bluffcatchers, he will mostly be paying off Alice’s value hands.

Alice now makes her value/bluff ratio for the 3-barrel equal to the pot-odds Bob is getting, namely 1.60 : 0.60. Alice then bluffs 0.60/(1.60 + 0.60) =27% of the time and value bets 100 – 27 =73% of the time. She then needs 27/73 =0.37 bluff combos per value combo.

In addition to the requirement of optimal value betting/bluffing on the river, Alice needs to 3-barrel/check-raise/check-call the river so that:

3-barrel% + 1.75 x check-continue% =70%

This follows from her 0.75 x pot turn bet, which gives us the same mathematics as her 0.75 x pot flop bet, and the same defense equation (she has to play the river in such a way that Bob can’t flop her turn bet with any two cards). Alice’s 2-barrel-range on the turn had 81 combos and she still has 81 combos on the river. 70% of this is 0.70 x 81 =57 combos. The defense equation can be written as:

3-barrel-combos + 1.75 x check-continue-combos =57

The range Alice brought with her from the turn to the river after c-betting and 2-barreling is {99,66,33,J9s,AA-QQ,AJ} + {QTs,AK,AQ,KQs} =40 + 41 =81 combos. Below is a suggestion for a total river strategy that satisfies the defense equation and also has the optimal value/bluff ratio for her river 3-barreling range:

  • Check-raise:
    {99} =3 combos
    Value bet:
    {66,33,J9s,AA-QQ} =26 combos
  • Check-call:
    {AJ} =12 combos
    Bluff:
    {10 AK-combos} =10 combos

Alice check-raises one of her sets and value bets all other sets, two pair and overpairs. She check-calls top pair/top kicker, and bluffs with 10 of the 16 AK combos (for example all AKs and the 6 remaining A Kx and A Kx). She 3-barrels a total of 26 + 10 =36 combos with a bluff% of 10/(26 + 10) =28% (close to the optimal 27%), and she check-calls/check-raises 3 + 12 =15 combos.

We plug these numbers into the defense equation and get:

3-barrel-combos + 1.75 x check-continue-combos
=36 + 1.75 x 15
=62 (optimal =57)

We see that it’s easy for Alice to defend enough on this river when she starts out with a strong UTG range preflop and then gets called on the flop and turn on a dry board. She has enough sets and overpairs in her barreling range to comfortably get to showdown with only top pair or better, without giving Bob any opportunities to float her profitably with any two cards anywhere along the way. Note that Alice does not need to make crying check-calls on the river to defend optimally. But as we shall see in the next article, Alice’s opening range is an important factor. The tighter her opening range, the more of our range will be made up of top pair or better postflop, and the easier it becomes to defend optimally out of position.

For example, had we opened our default 25% range from CO and gotten the same flop, we would have had a much larger percentage of worthless hands in our postflop range after c-betting our entire range on the flop. Compared to a 15% UTG open range we would now be forced to defend with a much weaker range on the turn to satisfy the defense requirement of 70% 2-barreling (or the equivalent amount of 2-barreling, check-calling and check-raising, according to the defense equation). We would have carried this problem with us to the river, and we would have to defend a weaker range there as well. We’ll talk more about this in Part 6.

So what can we learn from the work done in this article? For example, we see that out one pair hands drop steadily in value from flop –> turn –> river. At the river top pair/top kicker became a check-calling hand in this example. Further more, all worse one pair hands (if we had had any) would have been put in the check-folding range, since we don’t need to check-call these hands to satisfy the defense equation.

Does all this make sense intuitively? Yes, since we can’t expect to win many pots by betting or check-calling a mediocre one pair hand after we have bet for value on the flop and turn and gotten called twice on a dry board. Villain will often have a better hand, and we will pay off a lot if we insist on taking all our mediocre one pair hands to showdown.

So the ranges we build based on pot-odds, mathematics and principles from game theory correlate well with our intuitive understanding of the situation. But of course, if you’re at the river in such a scenario and you expect Villain to bluff enough to make check-calling profitable with a mediocre one pair hand, by all means go ahead and check-call. The main point of the optimal strategy is that it gives us a good starting point for playing correctly.

If we follow the optimal strategy, Villain can’t exploit us by loose floating, that’s the big picture idea here. If we have additional information that tells us he is likely to bluff way too much if we check the river, we can exploit him by check-calling more than optimally.

Therefore, if the strategies above seem to loose or too tight to use as a default at the limits you are playing, you can view this as a sign that you usually have additional information that allows you to build exploitative strategies that are better than the optimal default strategy. But you will still benefit from training a good understanding of what the optimal strategies look like, so that you know where to start when you adjust to individual opponents’ mistakes. You will also have a solid default strategy to use against unknown players.

4. Summary
In this article we moved from postflop play heads-up in position after flatting preflop to postflop play out of position as the preflop raiser. We used simple mathematics and modeling to estimate an optimal c-bet/2-barrel/3-barrel strategy for the raiser.

We assumed that the raiser began postflop play by c-betting her entire preflop raising range on a dry flop texture, and the player in position called. On the turn and river the raiser used strategies that prevented the player in position from floating the flop or the turn profitably with any two cards- The raiser did this by barreling/check-calling/check-raising enough to make it mathematically impossible for the player in position to make a profit from floating a street with a weak hand, planning to bluff or sneak cheaply to showdown when the raiser checks the next street. We worked thoroughly through an example to illustrate how the theory can be implemented at the table.

In the next article we’ll continue with this topic. Some of the things we’ll talk about are:

  • More about the consequences of choosing to bet a street
  • Show mathematically that the raiser’s optimal turn/river strategies defends her against any-two-cards floating
  • Study the effect of the raise’s opening range on her postflop strategies

The plan for the rest of the article series is to move on from heads-up play in singly raised pots to heads-up play in 3-bet pots (which new players tend to find difficult). But before we move on to 3-bet pots we will gain a lot of insight from studying play in singly raised pots both as the raiser out of position and the flatter in position. The mathematics and models we use will come in handy when we learn about play in 3-bet pots later.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-5-821#sthash.0OkTQEBC.dpuf

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