Optimal Postflop Play in NLHE 6-max – Part 6

1. Introduction
This is Part 6 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.

I Part 1, Part 2, Part 3, and Part 4 we discussed postflop play heads-up in position after flatting preflop. Then in Part 5 we began working on postflop strategies for the preflop raiser out of position in this scenario.

We’ll continue this work in Part 6. Some of the things we’ll discuss 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

We’ll warm up with a discussion of “follow through” when you have chosen to bet a street:

2. On the consequences of choosing to bet a street

The scenario we studied in Part 5 were based on the following set of assumptions:

  • Alice (100 bb) raises out of position and Bob (100 bb) flats in position
  • Alice’s standard bet sizing on the flop/turn/river (those times she chooses to bet) is 0.75 x pot/0.75 x pot/0.60 x pot
  • Alice c-bets 100% of her preflop range on the flop

Now Bob has position on Alice, and he defends postflop using the optimal strategies we built for him in Parts 1-4 of this article series. He will raise some hands, flat some hands, and fold some hands. The most interesting scenarios for us to study are the ones where Bob flats, so that Alice gets a bet/check-raise/check-call/check-fold decision to make on the next street. The reason these scenarios are the most interesting ones for us is that the rest of the hand will often be automatic when Bob raises anywhere (most of the time Alice will fold her weak hands and 3-bet her best hands for value, and there will be no real decisions to make.

So we will here focus on the postflop scenarios where Alice has the betting lead on the turn after c-betting the flop and getting called. As discussed in Part 5, it’s ow important for her to use a turn strategy adapted to her c-betting range. If she gives up too easily on the turn after c-betting the flop and getting called, Bob can exploit her by floating her c-bet with any two cards, planning to auto-bluff turns those times Alice checks.

To illustrate how this can happen, let’s warm up with some simple math. Let’s say that Alice openraises her default ~25% range from CO, and Bob flats on the button. The flop comes dry and without any possible draws:

Alice now decides to c-bet her entire range, since Bob’s preflop flatting range should mostly miss this flop. This is a reasonable assumption, since Bob’s default preflop flatting range in this case is:

IP flat list after ~25% CO openraise

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

140 combos
11%

We have the tools for checking this assumption, and we can use Pokerazor to calculate how often Bob’s flatting range has flopped a pair or better on this dry texture:

We see from the figure above that there’s only a 39.4% chance that Bob has flopped one pair or better (see the list “Cumulative frequency” to the right). Alice’s c-bet is 0.75 x pot, so if she picks up the pot more than 0.75/(1 + 0.75) =43% of the time, she profits from making a c-bet with any two cards. Therefore, if Bob only calls the c-bet with a pair or better (and there are no draws he can have), a c-bet will be automatically profitable for Alice, since Bob then folds 100 – 39.4 =60.6%.

But this does not mean that Alice’s any-two-cards c-bet is profitable against a good, thinking player that understands the situation. Bob knows that Alice knows that his range has missed the flop more than half the time. He also knows that if he folds more than 100 – 43 =57%, he is giving Alice a license to steal with any two cards. Therefore, Bob will also call the flop with some hands without a pair or a good draw, for example the overcard hands AK and AQ.

These calls that Bob makes with overcards and weak draws (when he has any), are floats. Bob bases this on a combination of several factors:

  • The chance of getting a bluffing opportunity on a later street
  • The chance of checking the hand down on the turn and river and winning a showdown unimproved
  • The chance of checking the hand down on the turn and river and winning a showdown after improving marginally (for example after making a low pair on the turn)
  • The chance of making the best hand on a later street and getting paid (in particular, having good implied odds when he floats with a good draw)

Note that may of the thin flop calls/floats Bob makes can’t be justified based on pot-odds alone, if Bob’s plan is to play strictly fit-or-fold on later streets. We’re only getting pot-odds 1.75 : 0.75 =2.33 : 1 on the flop, and we’re calling with a hand like AQ only to spike a pair, we need (47 – 6) : 6 =7 : 1 to call and draw to 6 outs on the turn (and not planning to sometimes steal the pot when we miss).

For a new NLHE player these thin flop calls might seem “incorrect”, since Bob seems to call with hands like AQ only to draw to two overcards, hoping to make a pair on the turn. But this is not the only reason why Bob calls. Keep in mind he is already beating many hands in Alice’s wide and weak c-betting range, and he will sometimes with unimproved against these hands. For example, he can win with ace high when the turn and the river goes check-check. Bob will also be able to steal some pots on the turn or river if he chooses to use AQ as a bluff when Alice checks to him.

Therefore, since Bob will often (and correctly) float the flop without a pair or a good draw on dry flops, Alice can’t check and give up on the turn every time her flop c-bet gets called. She knows that her 25% CO range is weak on the flop, and she knows that Bob knows this as well:

Alice’s Default 25% CO-range

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

326 combos
25%

Not only is Alice’s c-betting range (which is her entire preflop range) weak on this flop, but it’s weaker than Bob’s range!. We see from the figure above that the chance of Alice having flopped one pair or better is a measly 25.4%, compared to 39.4% for Bob’s range. The observant and optimally playing Bob can therefore easily make many light flop floats, hoping Alice will screw up on the turn or river and give him opportunities to steal profitably with any two cards.

If Alice also plays optimally postflop, Bob can’t expect to profit from any-two-cards floating (as we’ll see in a minute), but at the very least he can float enough to prevent her from c-betting any two cards profitably (and we showed that this was possible for him in Part 5).

The gist of it is that Alice knows that Bob’s range is weak, but her range is weak as well. She knows this, Bob knows this, and Alice knows that Bob knows this. Therefore Alice can expect Bob to call his optimal 57% on this flop (and as discussed in previous articles, Bob chooses to slowplay his few monster hands on thus type of flop). So Alice can’t c-bet her entire preflop range on this dry flop without having a plan for how to 2-barrel/check-call/check-raise on later streets to prevent Bob from floating her profitably with any two cards. A player that thinks he is exploiting the player in position by c-betting a lot on dry flop, expecting lots of folds, runs the risk of getting counter-exploited by the player in position if this player understands what is going on.

To illustrate what can happen without a good turn/river plan, assume that Alice gives up on 50% of turns (and she will have no pair/no draw far more often than this after c-betting her entire 25% CO range on a dry flop). Now Bob can call her 0.75 x pot c-bet on the flop and then auto-bet turns when Alice checks. If Alice never check-calls or check-raises, Bob knows that he wins when she checks. Bob’s risk when floating the flop was then only 0.75 x flop-pot, since his turn bluff has zero risk (Always check-folds).

So Bob risked 0.75 x flop pot (P) to win 1.75P (the flop pot + Alice’s c-bet). 50% of the time he loses 0.75P and 50% of the time he wins 1.75P. The EV for his any-two-cards flop float is then:

EV (float) =0.50(+1.75P) + 0.50(-0.75P) =0.5P

Bob’s EV for floating the flop with a random hand against a weakly playing Alice was 1/2 of the flop pot. Not bad! So what should Alice do? In Part 5 we designed the following defense equation for Alice’s turn play after c-betting the flop and getting called:

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

where check-continue =check-raise or check-call. And the same equation also applied to river play after 2-barreling the turn and getting called:

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

Note that the mathematics does not tell us whether or not it’s correct for Alice to c-bet her entire range on the flop. What it does tell us is that when she has chosen to do so she will be vulnerable to any-two-cards floating if she is not willing to play the next street according to these defense equations.

There’s a subtle point buried here:

If you bet a street, and the thought of barreling 70% (or an equivalent combination of barreling, check-calling and check-raising) on the next street will make you feel sick, almost regardless of which card falls, you are probably betting too much on the current street

An obvious example would be if Alice elected to c-bet her entire 25% CO range on a coordinated flop like this one:

This range hits Bob’s solid preflop flatting range hard, as shown below:

Bob’s flatting rage is full of pocket pairs and suited/coordinated medium/high cards, and this is a very good flop for him. There a whopping 63.9% chance he has one pair or better, and he also has lots of gutshots, open-enders and flush draws in his range. Alice’s range has also connected often with this flop, but rarely hard (wide ranges hits lots of flops in various ways, but often in weak ways), and she is out of position to boot.

Therefore, c-betting this coordinated flop, planning to 2-barrel/check-call/check-raise optimally on the flop seems like a very bad and unprofitable idea for Alice. She ha to respect the fact that Bob’s preflop range has hit this flop harder than her range, and that he also has advantage of position. So Alice should check some of her weak hands (hands like 22, A2s, etc) instead of c-betting her entire range.

By removing weak hands from her flop c-betting range, Alice is setting herself up for reaching the turn with a stronger range those times she chooses to c-bet and she gets called. When her turn range is stronger, it will be easier and much more comfortable for her to play the turn optimally, according to the defense equations, since a larger fraction of her turn range now will be strong enough to 2-barrel, check-calling, or check-raising without feeling sick about having to do so.

The main point is that if you often find yourself on the turn, out of position after having c-bet the flop and gotten called, and without a hand you feel comfortable 2-barreling, check-calling or check-raising, you have a problem. You might try to fix this problem by check-folding a lot of turns so that you don’t spew more chips, but you will probably (and correctly) feel that the player in position is bluffing you a lot.

And then you might conclude “Playing the turn out of position is hard, I need to get better at it” without realizing that the root of your problem is located in your flop c-betting strategy. You should fix the problem by starting with your flop c-bet decisions on textures that are bad for you and good for your opponent. Check and give up with more weak hands on these flops, and I can guarantee that your turn decisions will become easier and more pleasurable those times you do c-bet and get called.

The next step for us is to verify that the raiser’s optimal turn/river barreling/check-calling/check-raising strategies that we designed in Part 5 in fact do defend her sufficiently against any-two-cards floating.

3. Verifying mathematically that the preflop raiser’s turn/river strategies defend her against any-two-cards floating
We’ll now show that Alice’s turn/river strategies according to the defense equation protects her from getting exploited by a player who floats her with random weak hands in position.

In Part 5 we verified that Bob’s optimal calling with a bluffcatcher in position defended him correctly against any-two-cards barreling from Alice. She could not make money by c-betting a random worthless hand on the flop and then continuing to bet the turn or river when called.

Here we’ll use the same method to show that Alice’s optimal turn/river strategies defends her against Bob’s floating with random worthless hands. We calculate the probabilities associated with all possible outcomes, find Bob’s EV for each of them, and then write out the total EV equation for his float.

We’re assuming that Bob is floating with a worthless hand on a dry flop (where Bob is calling with all hands he defends with). To keep the math simple, we’ll assume that Bob’s only chance to win is when Alice checks and gives up on a later street (he has 0% pot equity, and will never win a showdown). Bob’s plan is to call the c-bet on the flop, and then auto-bluff the turn when Alice checks. Those times Alice 2-barrels the turn, Bob always folds.

Alice’s strategy is to play the turn and river in such a way that random floating is not automatically profitable for Bob. She does this by building barreling/check-calling/check-raising ranges that satisfy the defense equations defined previously.

Alice’s strategy on the flop
Let the pot size be P on the flop. We begin by assuming Alice c-bets 0.75P with 100% of her opening range on a dry flop. Bob calls with his worthless float, planning to bluff the turn if checked to. The pot grows to P + 2 x 0.75P =2.5 P, and both players have put 0.75P into the pot postflop.

Alice’s strategy on the turn
We’ll show that Alice can make Bob’s flop floats break even by playing the turn according to the defense equation:

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

First, assume that Alice defends by only 2-barreling, so that check-continue% =0 and 2-barrel% =70.

– Alice 2-barrels: 70%:
– Alice check-raises/check-calls: 0%
– Alice check-folds: 30%

Bob then folds his float to Alice’s 2-barrel 70% of the time and loses his 0.75P flop call. 30% of the time he gets the opportunity to bluff the turn. Alice always check-folds, and Bob makes +1.75P (the flop pot + Alice’s c-bet).

The EV equation for Bob’s flop float is:

EV (float)
=0.70(-0.75P) + 0.30(+1.75P)
=-0.525P + 0.525P
=0

So Alice’s 2-barrel strategy makes it impossible for Bob to profit from floating the flop with any two cards. Now we look at the more general form of the equation where Alice also check-calls and check-raises. For example, assume Alice 2-barrels 35% (Bob folds), check-raises 10% (Bob bets and folds to the check-raise), and check-calls 10% (Bob bets and gives up when called). Bob then folds to Alice’s 2-barrels, auto-bets the turn when checked to, and gives up with his worthless hand when check-called or check-raised.

– Alice 2-barrels: 35%:
– Alice check-raises: 10%
– Alice check-calls: 10%
– Alice check-folds: 45%

Note that this strategy satisfies the defense equation since:

35% + 1.75(10% + 10%) =70%

35% of the time Bob folds his float to Alice’s 2-barrel and loses his 0.75P flop-call. 10% + 10% =20% of the time he bluffs the turn with a 0.75 x turn-pot bet, gets check-raised or check-called and gives up. Hen then bets 0.75 x turn-pot =0.75 x 2.5P =1.875P, and loses this amount in addition to his 0.75P flop-call for a total loss of -0.75P – 1.875P =-2.625P. The remaining 100 – 35 – 20 =45% of the time he bluffs the turn successfully and picks up the 2.5P pot, where 1.75P is profit (the flop pot + Alice’s 0.75P flop c-bet).

The EV equation for Bob’s float now becomes:

EV (float)
=0.35(-0.75P) + 0.20(-2.625P) + 0.45(+1.75P)
=-0.2625P -0.525P + 0.7875P
=0

And we see that Alice can also defend optimally and make Bob’s random floats break even by going from a 2-barrel/check-fold strategy to a 2-barrel/check-raise/check-call/check-fold strategy. She builds her 2-barreling, check-calling, and check-raising ranges so that they satisfy the defense equation, and Bob’s random flop floats can not make money.

4. The effect of the raiser’s preflop range on her postflop strategies
We end this article with a new set of flop/turn/river strategies for the flop example we worked through in Part 5. In that example, Alice started out with her 15% EP range:

~15% UTG range

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

194 combos
15%

Bob called with his standard “IP flat list” against an UTG raiser:

IP flat list after ~15% EP openraise

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

162 combos

Flop/turn/river came

4.1 Alice’s postflop strategy after 15% UTG-raise
Alice began by c-betting 100% of her preflop range on the flop, and Bob called (we know that he will defend on this type of dry flop by only calling). Then Alice used turn/river strategies designed to prevent Bob from floating her profitably with any two cards on the flop or turn. Alice’s flop7turn/river bet sizing was 0.75 x pot/0.75 x pot/0.60 x pot, and Bob called flop and turn. We found the following turn/river strategies for Alice, based on this bet sizing and the defense equation we derived previously:

  • 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

 

  • 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

We’ll now estimate Alice’s turn/river strategies after starting out with a 25% openraise in CO. Bob flats the same preflop range as before, except for 3-betting QQ/AK for value instead of flatting them.

4.2 Alice’s postflop strategy after a 25% CO-raise
Alice openraises:

Alice’s default 25% CO-range

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

326 combos
25%

And the flop comes as before:

As before, Alice begins postflop play by c-betting 100% of her preflop range, and Bob calls. We have to estimate her new turn/river strategies, based on her opening range, card removal effects and the requirement that the defense equation should be satisfied.

On the turn Alice’s range is reduced from 326 to 282 combos:

She must now play the turn so that:

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

If she only 2-barrels, this corresponds to barreling 70% of 282 combos, which is 0.70 x 282 =197 combos. We can rewrite the defense equation as:

2-barrel-combos + 1.75 x check-continue-combos =197

Below is a suggestion for a turn strategy for Alice that satisfies the defense equation. The corresponding turn strategy after a 15% UTG openraise is listed for comparison:

Turn strategy after 25% CO-raise

  • Check-raise:
    {JJ} =3 combos
    Value bet:
    {99,66,33,J9s,AA-QQ,AJ,KJ,QJ,JT} =77 combos
  • Check-call
    {J8s,TT,A9s,T9s,98s,97s} =21 combos
    Bluff:
    {QTs,T8s,AK,AQ,KQ,KT,87s} =76 combos

Test of defense equation:

(77 + 76) + 1.75(3 + 21) =195 (optimal =197)

Turn strategy after 15% UTG-raise

  • 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

Test of defense equation:

(41 + 40) + 1.75(3 + 18) =118 (optimal =118)

Compared to play after an UTG raise Alice is now forced to barrel and check-call much thinner in order to protect herself against Bob’s floats. We will not discuss whether or not these ranges are too loose, but keep in mind what we discussed previously about setting ourselves up for weak turn ranges by c-betting too wide a range on the flop. The defense equation does not mention the quality of our turn ranges, only that they should defend against random floating. The looser we c-bet the flop, the looser we have to barrel/check-call/check-raise the turn in order to avoid getting exploited by floating. If we’re not careful, we might take this too far.

The solution to this problem (if in fact it becomes a real problem for us) is to check more weak hands on the flop so that we get to the turn with a stronger range after c-betting and getting called. As discussed previously, this is very important on draw-heavy flops that hit the preflop flatters range hard.

Here we’ll simply assume that Alice has chosen to c-bet her entire range on the flop, and that she is willing to take the consequences of her flop actions on the turn. She 2-barrels the turn with the value/bluff ranges above, and Bob calls again. Alice now has the following range on the river:

99,66,33,J9s,AA-QQ,AJ,KJ,QJ,JT} + {QTs,T8s,AK,AQ,KQ,KT,87s} =77 + 76 =153 combos

The river card doesn’t touch these ranges, and Alice still has 153 combos in her range after accounting for card removal effects:

If Alice defends her turn betting range only by 3-barreling, she needs to defend 70% of 153 combos which is 0.70 x 153 =107 combos. Using the defense equation we get:

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

 

2-barrel-combos + 1.75 x check-continue-combos =107

With the bet sizing 0.60 x pot on the river, Bob is getting 1.60 : 0.60 on a call, so Alice uses 0.60/(1.60 + 0.60) =27% bluffs in her 3-barreling range to make it break even for Bob to call with a bluffcatcher. So she uses 27/73 =0.37 bluff combos per value combo.

Below is a suggestion for a river strategy for Alice that satisfies the defense equation. The strategy corresponding to a 15% UTG openraise is listed for comparison:

River strategy after 25% CO-raise:

  • Check-raise:
    {99} =3 combos
    Value bet:
    {66,33,J9s,AA-QQ,AJ} =38 combos
  • Check-call:
    {KJ,QJ,JTs} =28 combos
    Bluff:
    {AK} =16 combos

Test of defense equation:

(38 + 16) + 1.75(3 + 28) =108 (optimal =107)

River strategy after 15% UTG-raise:

  • 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

Test of defense equation:

(26 + 10) + 1.75(3 + 12) =62 (optimal =57)

The widening of our postflop ranges that we observed on the turn is carried over to the river, and Alice is forced to value bet and check-call thinner on the river in order to prevent Bob from floating her turn bet profitably with random weak hands. But note that there should also be an adjustment for Bob.

We have let Bob flat the same preflop range in both cases (except that he 3-bets QQ/AK against the CO raise but flats them against the UTG raise). But an observant and optimally playing Bob should adjust his preflop flatting range to Alice’s position. When Alice moves from UTG to CO her opening range widens and more difficult to play out of position. This means more preflop flatting hands should become profitable for Bob.

And since Bob also needs to defend his preflop range enough against Alice’s postflop barreling, he will be forced to widen his postflop ranges as well, if he starts by widening his preflop range when Alice widens hers. So a certain symmetry should develop in this scenario where both players loosen up preflop, and as a result are forced to loosen up postflop as well. When both players are forced to play wider and weaker ranges postflop, Alice can value bet and check-call thinner.

So even if Alice’s two postflop strategies for the 15% UTG range and the 25% CO range seem very different, it’s not necessarily a big problem for Alice in practice. If Bob has started out with a wider preflop range as well, he will have to call and value bet weaker hands himself.

5. Summary
We have gone one step further with our study of optimal postflop strategies as the preflop raiser out of position. We started with a discussion of what it means to follow up a bet made on the current street. Simply put, we’re committing ourselves to a certain amount of betting, check-calling and check-raising on the next street. If we’re not willing to do this, we’re opening ourselves up for getting exploited by loose floating by a player with position on us.

Then we used mathematics to show that the turn/river strategies we designed for the raiser defended her optimally against random floating (by making them break even).

Finally, we studied the effect of the raiser’s opening range by building a new set of turn/river strategies for Alice, corresponding to her opening a 25% CO range instead of the 15% UTG range used in the previous example from Part 5. This resulted in significantly looser postflop strategies. We noted that starting postflop play by c-betting 100% of our preflop range on the flop leads to looser turn/river ranges, and that c-betting 100% of a wide preflop opening range perhaps isn’t optimal, even if the flop is dry and without draws.

In Part 7 we’ll talk about:

  • Optimal bet sizing for the raiser out of position on a dry flop, when he knows that the flatter in position has a weak range (he can use bigger bets to maximize value)
  • The effect of the player in position slowplaying his monster hands on dry flops (the raiser now must be a bit cautious when value betting big on the turn and river)
  • Some simulations of EV where we let the raiser’s and the flatter’s postflop strategies meet, and use Pokerazor to calculate EV for the raiser’s barreling line

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-6-823#sthash.cVW6OWN8.dpuf

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