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

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