Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max - Part 1

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Reg: 02/10/2007
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1. Introduction


For my first NLHE article for Donkr, I have chosen a topic that I know many new players find difficult, namely correct strategies for 3-bet/4-bet/5-bet preflop wars in NLHE 6-max.

1.1 Presenting the problem


Against weak low limit opposition, we can get away with playing an almost completely value-based game. We 3-bet/4-bet/5-bet mainly for value, and it's not a big mistake to assume our opponents are doing the same. If we reraise as a bluff, we usually limit ourselves to the occasional 3-bet bluff. A value-based style with little bluffing works well at small stakes because our opponents use more or less the same strategy, and many of them execute it poorly. Of course, every now and then we run into aggressive players who are capable of reraising as a bluff, but there are plenty of fish that will pay off our straightforward game, even if we bluff much less than is game theoretically optimal.

But let's say our Hero has built a bankroll by patiently grinding the low limits, and now he wants to take a stab at $200NL. He will now experience a lot more 3-betting, especially if he's out of position.

For example:

Example 1.1.1: We get 3-bet out of position


$200NL
6-handed

Hero ($200) raises to $7 with J T from UTG, it's folded to the button ($200) who 3-bets to $24, the blinds fold, and Hero folds.

Straightforward, and although Hero expects to get bluffed some of the time, he really doesn't have any choice but to fold. It's correct that his hand can no longer be played for value, but as we shall see later, it's possible to turn it into a 4-bet bluff.

At any rate, Hero plays on. The players behind him keep 3-betting him frequently when he is out of position, and Hero keeps folding weak hands to 3-bets. After a while, this hand occurs:

Example 1.1.2: We get 3-bet out of position (again)


$200NL
6-handed

Hero ($200) raises to $7 with A J in MP, it's folded to button ($200) who 3-bets to $24, the blinds fold, and Hero folds.

This is getting frustrating. Hero has a decent hand, but it's not strong enough to defend against a 3-bet from out of position, so Hero folds. But he is starting to feel exploited. If only he could get dealt a good hand and punish these bastards!

What an inexperienced player now might do (as his frustration builds up more and more), is to make up his mind to fight back against the loose 3-bettors. But he doesn't quite know what to do, and therefore he will often use poor strategies, and the wrong types of hands!.

Let's look at two common (and sub-optimal) ways to defend against 3-betting, out of position with 100 BB stacks:

Example 1.1.3: We get 3-bet out of position (again) and we call


$200NL
6-handed

Hero ($200) raises til $7 with K Q in MP, button ($200) 3-bets to $24. Hero thinks for a bit, decides that this hand is too good to fold, but too weak to 4-bet, so he calls.

Flop: 944 ($51)
Hero ($176) checks, button ($176) bets $30, Hero folds.

Hero is frustrated, but he doesn't see what else he could have done out of position with a hand of this type. Too strong to fold (at least in Hero's mind) against a loose 3-bettor, but not strong enough to 4-bet. Or? Hmmmmm .... Hero contemplates his next move, and soon another 3-bet pot occurs:

Example 1.1.4: We get 3-bet out of position (again) and we 4-bet for value (or at least that's what we think we are doing)


$200NL
6-handed

Hero ($200) raises to $7 with A J from UTG, MP ($200) 3-bets to $24. Hero decides to fight fire with fire, and he 4-bets pot to $75. Button 5-bets all-in, Hero calls. MP has K K . Hero screams in agony.

Flop: Q T 7 ($403)

Turn: Q T 7 Q ($403)

River: Q T 7 Q 4 ($403)

Hero tears his clothing and sprinkles ashes over his head. Damn!!

What happened throughout this sequence of hands?
OK, I made up this story, but it illustrates several of the problems an ABC low limit player faces when he moves up to tougher games. He will get 3-bet left and right, so he will have to fold a lot out of position (which is correct). He realizes he has to fight back to avoid getting run over (also correct), but he's not quite sure how to do it. So his attempts to counter the aggression are often poorly executed, frustrating and tilt-inducing.

For example, Hero might start calling 3-bets out of position with hands he feels are too good to fold, but not strong enough to 4-bet for value. This leads to many miserable experiences like Example 1.3. Or he might start 4-betting medium/weak hands without a clear understanding of whether he is doing it for value (planning to call a 5-bet), or if he is bluffing (planning to fold to a 5-bet).

What our inexperienced Hero might not realize, is that his opponents' loose 3-betting doesn't necessarily mean they are willing to splash around with lots of weak hands in 4-bet and 5-bet pots. When two good and aggressive NLHE-players engage in 3-bet/4-bet/5-bet warfare preflop, this is what usually happens:


  • Both players operate with wide ranges, and all ranges have a significant percentage of bluffs in them, especially at the early stage (raising and 3-betting)
  • Both players are willing to fold most of their bluffs (but not all of them), when their opponent reraises them back


This results in ranges that start loose, but get more and more (but never completely) weighted towards value. And it's usually plain wrong to assume you can 4-bet a medium hand like AJs for value against a loose 3-bettor, and expect to be a favorite when he 5-bets all-in. Yes, AJs is a decent hand against the range that 3-bet you, but it's crushed by the range that 5-bets you, and it's your opponent who decides when the 5th bet goes in (and that rarely happens unless he has the goods).

Therefore, if you decide on a frustrated whim to "take a stand" against an aggressive and competent 3-bettor with a hand like AJs, you will discover that in some mysterious way he almost always manages to come up with a better hand when you get all-in preflop.

This has lead many an inexperienced NLHE player to lose his stack, since these players:


  • Don't understand the roles different types of hands have in different types of ranges. First and foremost: Do I have a value hand that wants to get all-in, or do I have a bluff hand that I will fold to further aggression?
  • Aren't willing to fold hands that are strong at the early stages, but turn into weak hands when Villain keeps reraising


Let's look at Example 1.4 again. Hero open-raised AJs (correctly), and he got 3-bet. He then decided that his AJs was a good hand against Villain's 3-bet range (debatable, but not a big mistake), so he 4-bet for value (wrong!), planning to call a 5-bet all-in. Playing AJs for value after a 3-bet and going all-in with it was a big mistake. The 4-bet in itself was not a big mistake, since Villain has a lot of bluffs in his 3-betting range, and he will fold most of them to a 4-bet. So it's not a problem to 4-bet AJs as a bluff against a range full of 3-bet bluffs. But when Villain comes over the top with an all-in 5-bet, our AJs crumbles to dust (if Villain knows what he is doing).

But our inexperienced Hero did not realize what had just happened when he got 5-bet, and he stuck with his plan of playing AJs for value against what he perceived to be a wide and weak range. The problem is that the range he faces after a 5-bet from a competent player isn't wide and weak, it's very narrow and very strong.

Note what the real mistake was in this hand. 4-betting AJs against a wide range was not a big mistake in isolation, and neither was calling a 5-bet getting 2: 1. But the combination of 4-betting AJs + planning to always call a 5-bet, now that was a big mistake against a competent opponent. It caused Hero to invest his remaining 96.5bb stack as a huge underdog. The problem was, as mentioned previously, that his opponent controlled when the 5th bet went in, and Villain made sure he had a hand.

Our goal for this article is to give Hero a set of tools he can use to comfortably counter preflop aggression when he is sitting as the raiser out of position. We'll base our work on Hero's opening ranges, and based on these, we can deduce defensive strategies against positional 3-bets. And we will use game theory to design these strategies in such a way that the 3-bettor can not exploit Hero in these scenarios. Our work on Hero's game theory optimal defensive strategies also gives us a set of optimal 3-betting strategies for his opponent, so we kill two birds with one stone.

We have here talked mostly about the ills of getting 3-bet when sitting out of position, and this is what I feel inexperienced players find hardest to deal with. But the mirror image of this scenario, with us being the 3-bettor in position, is also worth discussing. These are easier scenarios to play, but we will benefit a lot from understanding optimal 3-bet/4-bet/5-bet dynamics also from this perspective. We'll learn how to construct optimal 3-betting ranges, based on the raiser's opening range, and we'll learn how to play against a 4-bet.

Regardless of whether we're the raiser or the 3-bettor, we want to understand which hands we can (re)raise for value, and which hands we (re)raise as bluffs. And above all else, we want it to be 100% clear which of these two things we are doing before we engage in a 3-bet/4-bet/5-bet war preflop.

1.2 Our model and overall philosophy


In this article we'll design so-called optimal strategy pairs for the raiser and the 3-bettor in the following scenario:

- The raiser opens some range
- A player behind him 3-bets
- The raiser 4-bets or folds
- The 3-bettor 5-bets, or folds to a 4-bet

Note that the raiser is always out of position (e.g. UTG, MP, or CO), and that no other players interfere.

We'll define a model for this scenario with 100bb stacks and standard bet sizing. Then we'll analyze our model, using mathematics and principles from game theory (but we'll keep it as simple as possible). We then construct game theory optimal(ish) strategy pairs for the raiser and the 3-bettor (one strategy for the raiser, and one matching strategy for the 3-bettor) that they can employ in their 3-bet/4-bet/5-bet wars.

Both players are trying to play perfectly against the other, and both are assuming their opponent is trying to play perfectly as well. The two players now both zoom in on a perfect strategy, designed not to lose against their opponent's perfect strategy. And the result is a pair of strategies that are perfect against each other, and we have our optimal strategy pair.

When we have learned these strategies, we have defensive (e.g. unexploitable) strategies we can use both as the raiser out of position, and as the 3-bettor in position. Using these optimal strategies guarantees that better players can't exploit us. They will also win against players who play poorly, although they will not win the maximum (if we want to exploit opponent leaks maximally, we have to deviate from optimal play ourselves, and use strategies that target specific leaks in our opponent's non-optimal strategies).

Knowing optimal strategies also makes it easier to spot our opponents' mistakes (where we can define "mistake" as a deviation from optimal play). If we know what an opponent should have done if he had played optimally, we can conclude that he has a weakness in his game if he chooses to do something different. And we might be able to exploit these weaknesses and turn them into leaks for him.

1.3 Background material for the article


Before we get started, I want to give credit to Cardrunners instructor Matthew Janda. During the spring of 2010 he published a 3-part video series Optimal Preflop Play I-III at Stoxpoker, which contains most of the theory we use in this article. This video series was inspiring and eye-opening, but sadly it became unavailable after Stoxpoker shut down in May 2010.

Matt Janda is now a Cardrunners instructor, and he continues to produce game theory related videos. His old videos from Stoxpoker might get moved over to Cardrunners, and if that happens, I recommend you check them out.

Without further ado, let's get started:

2. The mathematics behind optimal 3-/4-/5-betting with the raiser out of position


I have chosen an approach where we first go through the necessary math and theory quickly, and then we apply it by constructing optimal strategy pairs for two scenarios:

- The raiser in early position (UTG or MP) with a 15% opening range
- The raiser in CO with a 25% opening range

Lumping UTG and MP together under the label "EP" makes sense, since most players use very similar ranges for these two positions. The percentages we have chosen for EP and CO are typical TAG ranges that can be used under all game conditions.

The exact ranges we use to illustrate the procedures aren't important. Our goal is that you learn to construct optimal strategy pair (one strategy for the raiser and one for the 3-bettor) based on your own opening ranges. And you will of course also be able to design optimal strategy pairs to use against specific opponents (not on the fly, but by doing a bit of analysis work between sessions).

2.1 Our model


We use the following scenario:


  • Alice is sitting with a 100bb stack in EP or CO, and she raises pot to 3.5bb with some opening range
  • Bob is sitting in a position behind Alice with 100bb, and it's folded to him. Bob 3-bets pot to 12bb
  • Alice either 4-bets to 27bb (a bit less than pot), or she folds
  • Bob's response to Alice's 4-bets is to 5-bet all-in or fold
  • Alice's response to Bob's all-in 5-bets is to call or fold


Note that Alice doesn't defend against 3-bets by calling out of position. We could conceivably design a defense strategy where we fold weak hands, 4-bet strong hands, and call with medium hands, but this is not a good strategy out of position with 100bb stacks.

You have poor implied odds (due to low stack/pot ratio and being out of position) when you call for postflop value with implied odds hands. And it's difficult to steal and outplay Villain when you are out of position. And what you absolutely cannot do, is to call and then play fit-or-fold postflop. It will be much more fold than fit, and you are simply burning money by letting Villain c-bet his way to riches and early retirement on your expense.

With regard to Alice's choice of 4-bet size, it's standard to use 25-30bb (where full pot would be 37.5bb) with 100bb stacks. The logic behind this is that with 100bb stacks, we are putting Villain in a shove-or-fold scenario, also when we 4-bet a bit less than pot. His 3-bet bluffs will still fold, and his strong hands will still shove. So we win the same when he folds, but lose less on our bluffs when he doesn't fold. In other words: We risk less for the same reward when we're bluffing, and we don't lose anything when we're 4-betting for value. We simply choose 27bb as a representative value for a less-than-pot 4-bet, and the math won't change much if you use any number between 25bb and 30b instead.

Here are a few assumptions/statements we will use:


  • Bob knows Alice's opening range. Not necessarily all the hands in the range, but he knows the percentage of hands Alice opens
  • Both Alice and Bob are trying to play perfectly, under the assumption that their opponent is also trying to play perfectly
  • The worst hands in a bluffing range or calling range should be break even


The last statement needs an explanation: When we're 3-betting/4-betting/5-betting as a bluff, we should not lose money on our bluffing hands, and the worst of them should be no worse than break even. The same goes for when we're calling for pot odds. This makes sense if you think about it. When we're making a play that loses money, we should stop doing it to increase our EV.

Note that we're not concerned about the effects of deception when we work with game theory. We're only concerned with immediate EV. Also, if we're making money on all our bluffs or our calls, we can make even more money by bluffing more and calling more. So we keep adding bluffs and calling hands until our weakest hands are at the break even point, and then we stop. Conversely, if we're losing money on some of our bluffing or calling hands, we remove them from our ranges. Again, this results in our weakest bluffing/calling hands being no worse than break even.

Under these assumptions, we'll find an optimal strategy pair with a raising strategy (including defense against a 3-bet and against a 5-bet) for Alice, and a 3-betting strategy (including defense against a 4-bet) for Bob. We'll find a unique strategy pair for each of Alice's positions (e.g. for each of her opening ranges). We'll soon see how these strategy pairs follow from Alice's opening range, but first, let's talk a bit about optimal strategy pairs:

What is an optimal strategy pair?
When our two players Alice and Bob are playing optimally against each other, Alice's strategy and Bob's strategy make up an optimal strategy pair. When both are playing optimally, neither of them can gain from changing to a different strategy. If one of them can gain from switching to another strategy, then the original strategy wasn't optimal.

It's important to realize that a game theory optimal strategy doesn't try to maximize +EV against a random opponent. It's trying to maximize EV against an opponent who is also playing perfectly. Sometimes, this means the best result for both players is to break even. A game theory optimal strategy is first and foremost a defensive strategy, designed not to lose. However, an optimal strategy will win against players who are using non-optimal strategies. But If we see an opponent making big mistakes, we will win more by switching to an exploitative strategy, designed to exploit this opponent's specific leaks maximally.

But by changing our strategy from optimal to exploitative, we are moving away from optimal play. By doing so, we are creating weaknesses in our strategy, and other players might be able to exploit those weaknesses (although they might not see them). But if the weak player we are trying to exploit has big leaks, this trade off will usually be worth it. The art of playing against fish and regs at the same time is to exploit the fish, while we're defending ourselves against the regs. Against very poor opponents, we use very exploitative strategies. Against players who are as good as us, or better, we can fall back on optimal strategies so that they can't exploit us.

To balance these two goals well, we need to have an understanding of what optimal play is. Playing optimally (or, more likely, close to optimally) defends us against the good players, and understanding optimal play also makes it easier to spot mistakes in weak players (where "mistake" can be defined as deviating from optimal play).

With these concepts at the back of our mind, we move on to the mathematics behind optimal strategies for raising, 3-betting, 4-betting, and 5-betting with 100bb stacks:

2.2 How opening ranges, 3-betting ranges, 4-betting ranges, and 5-betting ranges are connected mathematically


We work our way through the raise/3-bet/4-bet/5-bet war, one step at a time, and construct all the mathematical tools we need. We jump back and forth between Alice and Bob, and we'll see how they influence each others' strategies when they both are trying to play perfectly against each other, assuming the other player is also trying to play perfectly.

What is Alice's optimal 4-bet%
The process starts with Alice raising some opening range known both to her and to Bob. When Bob 3-bets, Alice's most pressing concern is the following:

Alice can't fold so much that she gives Bob an opportunity to make a profit by 3-bet bluffing any two cards

So how often does Alice have to 4-bet? This follows from the pot odds Bob is getting on his 3-bet bluffs. There's 1.5 + 3.5 =5bb in the pot from the blinds and Alice's raise, and Bob 3-bets to 12bb to win this. Bob is then risking 12bb to win 5bb, and he's getting effective pot odds 5 : 12 on a 3-bet bluff.

He then needs to win more than 12/(5 + 12) =70% to have a profitable bluff. So if Alice folds more than 70%, Bob will have an automatic profit by 3-bet bluffing any two. Alice needs to prevent this, so she has to 4-bet enough to make Bob's bluffs break even.

Alice's optimal 4-betting strategy is therefore to 4-bet 30% of her opening range, and she will 4-bet a mix of value hands (planning to call a 5-bet) and bluffs (planing to fold to a 5-bet). We'll compute Alice's optimal value/bluff ratio in a moment, but first we have to find Bob's optimal ranges for 3-betting and 5-betting. These ranges follow from Alice's opening range:

What is Bob's optimal value/bluff ratio in his 3-bet range?
When Alice 4-bets to 27bb, she is risking 23.5bb (27bb minus he 3.5bb raise) more to win a 17bb pot (1.5bb from the blinds + Alice's 3.5bb raise + Bob's 12bb 3-bet). The effective pot odds for Alice's 4-bet bluffs are 17 : 23.5, and she can make a profit by 4-bet bluffing any two (of the hands she open-raised) if Bob folds his 3-betting hands more than 23.5/(23.5 + 17) =58%.

Bob can't allow Alice to 4-bet bluff any two cards profitably, so he defends optimally by folding exactly 58% of the time, and 5-betting all-in (including some 5-bet bluffs as we shall soon see) 42% of the time. Therefore, 42% of Bob's 3-bets need to be value hands that he plans to 5-bet all-in (including some 5-bet bluffs). We now define a 3-bet for value as a 3-bet where we plan to 5-bet all-in after a 4-bet. If this is not our plan, we are making a 3-bet bluff that we will fold to a 4-bet.

To make these percentages easy to remember, we round Bob's optimal 3-bet value/bluff ratio to 40/60. So now we know that 60% of Bob's 3-bets should be bluffs, and 40% should be for value (including some 5-bet bluffs). But we still don't know how many hands Bob should 3-bet overall. To find this number, we first have to find which hands Bob can 5-bet for value.

What should Bob's 5-betting range look like?
Bob first chooses the type of hands to 5-bet bluff with. He wants hands that have decent equity when called, and we can use Axs hands A5s-A2s for this purpose. Axs hands work as blockers against Alice's AA/AK (an ace in Bob's hand makes it less likely Alice has AA/AK), and they always have at least an overcard when Alice has another high pair. They also have straight and flush potential.

Axs has minimum ~30% equity when we 5-bet and get called, even against a strong range, as shown below:



So Bob will 5-bet a mix of true value hands and some Axs bluff hands, and he expects to have about 30% equity when his bluffs get called. So when he 5-bet bluffs and gets called, he will have ~30% equity in a 201.5bb pot where he invested 88bb with the 5-bet. Bob first 3-bet to 12, so the 5-bet is 88bb more. On average, Bob gets 0.30 x 201.5 =60bb back from the pot, so his net loss after 5-betting and getting called is 88 - 60 =28bb.

The pot size before Bob's 5-bet is 40.5bb (1.5 from the blinds, + 27 from Alice's 4-bet + 12 from Bob's 3-bet). So Bob is effectively risking 28bb to win 40.5bb when he is 5-bet bluffing. The effective pot odds are 40.5 : 28, and Bob needs to win at least 28/(28 + 40.5) =40% to profit from 5-bet bluffing any two (or more precisely, any Axs hand, since we base our calculations on having ~30% equity when called).

For Alice, this means she has to call a 5-bet 60% of the time to prevent Bob from making a profit by 5-betting any two. So Alice's 4-betting range has to contain 60% value hands and 40% bluff hands. Now we know everything we need to know about Alice's 4-betting range. She 4-bets 30% of her opening range, and she uses a 60/40 value/bluff ratio. We'll summarize Alice's total optimal strategy below, but first we'll find out how often Bob should 3-bet.

We know which type of hands Bob should 5-bet bluff (Axs), and we know he should use a 40/60 value/bluff ratio (which, coincidentally is the opposite of the ratio for Alice's 4-bet range). The last piece of information we need is Bob's total 3-bet percentage in an optimal 3-betting strategy. We find the answer by observing that Bob should 5-bet bluff enough to make Alice's weakest value hands break even. He he bluffs more, Alice can gain by calling with more hands, and then Bob's strategy can't be optimal. And if he bluffs less, Alice can gain by folding more hands, and Bob's strategy can't be optimal in this case either.

How many Axs hands we need to make Alice's weakest 5-bet calling hands break even varies with Alice's value range (60% of 30% of her opening range), which follows from her opening range. So we have to compute this result on a per-case basis, for every one of Alice's opening ranges. We'll give a quick example in the summary below, and the procedure will be thoroughly discussed later in the article.

2.3 Summary of Alice's optimal raising strategy


We summarize everything we have learned about Alice's optimal strategy for raising, 4-betting and calling 5-bets:

- She needs to 4-bet 30% of her opening range
- Her 4-betting range should have a 60/40 value/bluff ratio

So Alice's optimal strategy is:


  • Alice open-raises some opening range
  • When she gets 3-bet, she 4-bets 30% of her opening range with a 60/40 ratio between value 4-bets and bluff 4-bets
  • Alice therefore 4-bets 0.60 x 30 =18% of her opening range for value and 0.40 x 30 =12% of her opening range as a bluff
  • If Bob 5-bets all-in, Alice calls with all her value hands, and folds all her 4-bet bluffs


So Alice's value hands are the top 18% of her opening range. For example, if she opens 15% from UTG, this corresponds to a value range of 0.18 x 0.15 =2.7% of all hands. This makes up 0.027 x 1326 =36 combos, e.g approximately the range {QQ+, AK} =34 combos. We'll use this value range example when we summarize Bob's optimal strategy below. And then we'll illustrate each strategy step thoroughly when we apply the theory to Alice's EP and CO openraises.

2.4 Summary of Bob's optimal 3-betting strategy


We summarize everything we have learned about Bob's optimal strategy for 3-betting and 5-betting:


  • Bob starts by finding which hands he can 3-bet for value, planning to 5-bet all-in against Alice's 4-bet value range. For this purpose, he needs hands that have at least 50% equity against Alice's value range
  • Bob then adds enough Axs hands as 5-bet bluffs to make Alice's weakest value hands break even when calling Bob's total 5-bet range
  • Bob's value hands and 5-bet bluffs are joined to a total value range (where value range =the range he 3-bets and 5-bets all-in)
  • Finally, Bob chooses a 3-bet bluff range so that the ratio of his value hands (including 5-bet bluffs) to his bluff hands is 40/60
  • When Alice raises, Bob 3-bets his value range and his bluff range
  • If Alice 4-bets, Bob 5-bets his value range all-in and folds his bluff range


For example, if Alice raises 15% from the UTG, her optimal value range is {QQ+, AK} as shown previously. Bob chooses value hands that are at least 50% against this range, and his pure value range becomes {KK+}. Then he adds Axs hands as 5-bet bluffs until Alice's weakest value hands (QQ and AK) are break even against his total 5-bet range.

Alice then calls her remaining 73 BB to win a 189.5 bb pot (1.5 from the blinds, 100 from Bob, 27 from Alice's 4-bet), so her pot odds are 128.5 : 73. She needs minimum 73(/128.5 + 73) =36% equity to profit from calling, so Bob makes sure her weakest value hands have against his 5-bet-range. Later in the article we'll show that Bob ends up with a total 5-bet range of {KK+, A5s, A4s} when Alice's value range is {QQ+, AK}

This gives Bob {KK+, A5s, A4s} =20 value combos that he 3-bets, planning to 5-bet all-in. Then he picks hands to 3-bet bluff until he has a 40/60 ratio between value combos and bluff combos. Bob needs 60/40 =1.5 bluff combos for every value combo, so he will choose 1.5 x 20 =30 bluff combos against Alice's {QQ+, AK} value range.

You should memorize both Alice's strategy and Bob's strategy until you know them cold. It's not really complicated at all. Just remember that Bob uses a 40/60 value/bluff ratio for his 3-bets, and Alice uses a 60/40 ratio for her 4-bets, and then you know the most of it. Value hands are per definition hands we plan to raise and reraise until we are all-in. Bluff hands are hands we plan to fold if our opponent reraises us back.

We now begin the job of constructing optimal strategy pairs for Alice and Bob. First when Alice raises a 15% range from EP, and then when she raises a 25% range from CO. We'll do this thoroughly and methodically, so that you can learn the procedures inside out. I hope you'll see that these strategies aren't really complicated to construct and then apply at the table.

3. Optimal strategy pairs for raiser/3-bettor with an EP raiser out of position


We'll now find the optimal strategy pair for Alice and Bob when Alice open-raises from early position (EP =UTG or MP), and it's folded to Bob in position.

It's of course possible to vary EP opening ranges a lot, according to opponent tendencies and general game conditions. But the core strategy for a typical TAG is to open somewhere around 15% of his hands (plus/minus a couple of percentage points in both directions) from both EP positions, and slightly tighter from UTG than from MP.

We'll construct all strategies/ranges with great detail for this scenario, so that there won't be any doubt about how to apply the theory. Then we'll move on to the scenario with Alice in CO, and do this quickly, with brief comments along the way.

3.1 Alice's optimal raising strategy in EP (UTG and MP)


We assume Alice is opening with a ~15% EP range. Note that any 15'ish% EP-range will do, since our work is based on the numbers of hands in the range, and not the specific hands it contains. Obvious value hands like high pairs and AK have to be included, since these hands have a job to do in the ranges for 4-betting and calling 5-bets. But the exact mixture of medium and weak hands in Alice's range is irrelevant.

We give Alice the following range:

Alice's EP range
22+
ATs+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s

186 combos
14%

We now place Bob somewhere with position on Alice. Alice open-raises and it's folded to Bob, who 3-bets. Both players want to play perfectly against the other, and both assume the other is also trying to play perfectly.

Alice starts by defining her value range. This is per definition the hands she plans to 4-bet for value and then call a 5-bet with. She counts the total number of combos in her opening range (186), and she knows that she on average has to defend 30% of her total range against a 3-bet. She also knows that the optimal value/bluff ratio of her 4-betting range is 60/40. So she 4-bets 0.60 x 0.30 =18% of her opening range for value, and 0.40 x 0.30 =12% as a bluff.

Alice then 4-bets 18% of the 186 combos for value, e.g. 0.18 x 186 =33 value combos. This corresponds almost exactly to the value range {QQ+, AK} =34 combos (a couple of combos too many or too few doesn't matter much). This is a standard value range from EP, also for players who haven't studied optimal raise/3-bet/4-bet/5-bet strategies.

Now the 4-bet bluff range. These are the hands Alice 4-bets and then folds to a 5-bet. There are two ways to define the bluff 4-bet range: We can choose some specific bluff combos and always 4-bet them, or we can 4-bet all the non-value hands a certain % of the time.

Let's illustrate both methods:

Defining a 4-bet bluff range using the combo method
If we choose specific bluff combos, we need 12% of 186 combos, e.g. 0.12 x 186 =22 bluff combos. For example, we might choose AQ (16) + JJ (6) which gives us exactly 22 combos. Or we can choose something different, since it doesn't matter what we use for bluffs when Bob either folds or 5-bets all-in. When Bob doesn't fold to our bluffs, he 5-bets, and we have to fold, so our 4-bet bluff hands never get to see a flop. And when they never get to see a flop, their postflop value is irrelevant.

But note that a hand like AQ works as a blocker against Bob's premium hands (AA, AK, QQ). So when Alice uses AQ as a bluff, it will be less likely that Bob has a hand he can 5-bet for value. Keep this in mind if you are choosing specific hands to always use for 4-bet bluffing.

Defining a 4-bet bluff range using the percentage method
My preferred method, and also the easiest method to remember. We only need to remember one number, namely the static percentage Alice 4-bet bluffs her non-value hands. Let's find this percentage once and for all:

Alice 4-bets 18% of her opening range for value, and she's left with 82% non-value hands she can use for 4-bet bluffing. We now choose to use all these hands a fixed percentage of the time, so that the effective total value/bluff ration is 60/40. We now want:


value/bluff =60/40
18/82x =60/40
18/82x =1.5
18/82 =1.5x
0.22 =1.5x
x =0.22/1.5 =0.15 =15%


So we 4-bet bluff all non-value hands 15% of the time and fold them the remaining 85% of the time. Note that this percentage is universal for Alice. No matter what her opening range is, she can always use this percentage to obtain a 60/40 value bluff ratio for her 4-bets.

Let's double-check to see that this works the way it should:

When Alice has raised some opening range and gotten 3-bet, we have deduced that her optimal value 4-bet range is 18% of her total range. If she 4-bets the remaining 82% of her range as a bluff 15% of the time, her overall bluff percentage will be 0.15 x 0.82 =0.12 =12%. So her total 4-bet range is he optimal 18 + 12 =30%, with a 18/12 =60/40 value/bluff ration. So the percentages add up perfectly.

Alice's optimal raise/4-bet/call 5-bet strategy in EP
We now have everything we need to specify Alice's total strategy after a 15% open-raise. We can write Alice's total EP range as a sum of value hands (raise, 4-bet for value, call a 5-bet) and bluff hands (raise, 4-bet bluff, fold to a 5-bet):

{Alice's total EP range}
={22+,ATs+,KTs+,QTs+,J9s+,T9s,98s,87s,76s,AJo+,KQo}
={value hands} + {4-bet bluff hands}
={QQ+, AK}
+ (15% 4-bet and 85% fold) x {the rest of the range}


Alice raises {22+, ATs+, KTs+, QTs+, J9s+, T9s, 98s, 87s, 76s, AJo+, KQo} =186 combos from EP. If she gets 3-bet, she 4-bets {QQ+, AK} for value and calls a 5-bet with them. Those times she doesn't have a value hand, (e.g. she has JJ, AJo, 76s, etc.), she 4-bets 15% of the time as a bluff, and otherwise she folds.

The percentage of value hands is then 34/186 =18%, while the effective percentage of bluff hands is 0.15 x (186 - 34)/186 =12%. The value/bluff ratio for her 4-bet range is 18/12 =60/40, which is optimal.

To randomize her 4-bet bluffs and get the correct 15% 4-bet frequency for her non-value hands, Alice uses a random number generator from random.org. She has this on her screen in a small browser window:



Let's illustrate randomized 4-bet bluffing in practice with an example:

Example 3.1.1: Randomized 4-bet bluffing in EP


$100NL
6-handed

Alice ($100) raises pot to $3.50 with 6 6 from UTG, it's folded to Bob ($100) on the button, who 3-bets pot to $12. The blinds fold, and Alice has to make a decision. 4-bet or fold?

Alice does not have one of her value hands {QQ+, AK}, so she knows that this is a 4-bet-bluff-or-fold scenario. She also knows how often she should 4-bet bluff with these hands (15%). Alice clicks the random number generator, planning to 4-bet to 27bb ($27) if it returns a number between 0 and 15, and otherwise she folds.:



The randomize returns 39, so Alice folds this time.

We have now specified Alice's optimal EP strategy for for raising/4-betting/calling a 5-bet when she gets 3-bet by a player in position. Our next step is to turn to Bob. What is Bob's optimal strategy for 3-betting/5-betting in position after a ~15% EP raise from Alice?

3.2 Bob's optimal 3-bet-strategy versus Alice's optimal raising strategy in EP


We're assuming Bob knows Alice's opening range (he only needs to know the % of hands, not the specific hands), either from observation, or by using a HUD. Alice's raise percentage dictates her value 4-bet range, which dictates Bob's strategies for 3-betting and 5-betting.

Bob starts by finding the hands that he 3-bets and 5-bets all-in, purely for value. His value range also includes some 5-bet bluffs, and the next step is to find these. Then we pick a range of 3-bet bluffs that Bob plans to fold to a 4-bet. We'll also talk about Bob's flatting range. These are medium strong hands that are playable, but they are not strong enough to 4-bet for value,and they are too strong to use as bluffs, so Bob flat-calls with them.

Bob's flatting range can be viewed as a completely separate part of Bob's overall strategy, and we don't have to be concerned with it when constructing optimal ranges for 3-betting/4-betting/5-betting. But we will discuss the flatting range briefly, since it helps us understand the big picture. When Alice has raised, Bob can respond in 3-ways: He can 3-bet (for value or as a bluff), he can flat, or he can fold. Different hands have different jobs to do within these ranges. And depending on Alice's opening range, hands can move between Bob's 3-betting/flatting/folding ranges.

For example, we'll see that AK isn't strong enough to be a value hand for Bob against Alice's EP range, so AK goes into the flatting range in this scenario. But when Alice opens a much wider ~25% range in CO, AK is promoted to a value hand that is 3-bet and 5-bet for value. More about that later in the article.

So let's begin defining Bob's optimal 3-bet/5-bet strategy in position against Alice's optimal raise/4-bet/call 5-bet strategy with a ~15% EP range:

Bob's pure value range
Bob knows that Alice EP range is ~15% (14% to be exact), and therefore he can draw the same conclusion Alice just did, namely that her optimal value 4-betting range is {QQ+, AK}. To profitably 3-bet and 5-bet all-in for value against this range, Bob needs a hand with at least 50% equity.

AA is obviously such a hand, and we can easily compute some equities to see that KK is the only other possible hand. So Bob ends up with the super tight pure value range {KK+}.





So Bob will 3-bet {KK+} and 5-bet them all-in if Alice 4-bets. He will also 3-bet/5-bet some 5-bet bluff hands (type Axs), and he will have a wide range of 3-bet bluffs that he folds to a 4-bet. We'll now find Bob's 5-bet bluffing hands, then his 3-bet bluffing range, and then we are done.

But first, let's talk about something that I know you're thinking about right now:

Wait a minute, are you saying that we shouldn't 3-bet the mighty strong QQ and AK for value against an EP open-raise?
Correct. Against Alice's tight and optimally played ~15% EP opening range, QQ and AK aren't strong enough to use as value hands, even if they have strong equity against Alice's total opening range. The reason is that they can not profitably get the whole stack in Against the range Alice is willing to get all-in with, namely {QQ+, AK}. Therefore we don't want to 3-bet them for value. Keep in mind that the process of getting all-in starts with a 3-bet, and we know the moment Alice open-raises with a ~15% EP range that her all-in range is a favorite over our QQ and AK hands.

Therefore, since we know this before we have put our first chip into the pot, we don't want to choose a path with QQ/AK that is the first step towards getting all-in with them preflop. This is also true for other hands that are good enough to play for value, but not strong enough to get all-in preflop against Alice's value range {QQ+, AK}. Examples of such hands are QQ-22, AK-AJ, KQ, QJs-T9s, etc. (and you can probably list some more if you think about it).

So should we 3-bet these medium strong hands as bluffs then?. No, because they are too strong to turn into bluffs and waste their postflop value. The alternative, which is a good one, is to flat-call with them and play a raised pot with position against a range we have god equity against (namely Alice's total opening range). Of course, we could always pretend they are trash and use them as 3-bet bluffs, but why should we do that when it's profitable to flat and play for postflop value? It's true that we want to 3-bet bluff a lot, but we have plenty of bad hands to choose from for that purpose, and we don't want to waste the postflop value we gain from flatting with our medium strong hands.

Here is a soccer analogy in these World Cup times:

Moving QQ/AK from the flatting range up to the value range against a ~15% opening range is a bit like moving a defender forward and using him as a striker. Sure, he might score a goal or two, but he isn't quite good enough for the job. But he is too good to sit on the bench, so he should play. Therefore, since there is another job for which he is well qualified (defending), we let him play there. The right man for the right job.

Bob will therefore flat QQ, AK and various other medium strong hands/implied odds hands after a ~15% open-raise from Alice. The optimal flatting range depends on how Bob thinks Alice plays postflop, what he thinks the players in the blinds will do, how they play postflop, their stack sizes, etc. So we leave the construction of an optimal flatting range to Bob.

Note that 3-betting QQ and AK for value against a ~15% EP raiser is equivalent to assuming the raiser isn't playing optimally. If you feel these two hands can always be 3-bet and 5-bet all-in for value against this EP range, you can assume it's because the players you meet don't defend well against 3-bets out of position.

Thinking about these things is useful, because when we know what's game theoretically correct, we know that we can exploit someone if it seems correct to do something else. So feel free to deviate from optimal play in Bob's place, if you have position on a weak player. For example, you might be up against a player who 4-bet bluffs spazzy and way too much, or he raises a lot and calls 3-bets out of position with medium strong hands, and then he plays fit-or-fold on the flop. Against such players, QQ and AK might be used as value 3-bet/5-bet hands, since our opponents play far from optimally against our 3-bets.

But don't 3-bet QQ/AK for value against a ~15% opening range in the hands of a player like Alice. She plays optimally against our 3-bets, so 3-betting QQ/AK won't do anything for us. Against Alice we use QQ/AK as flatting hands, thus setting ourselves up for playing a raised pot in position against a range we have good equity against (Alice's total opening range, and not just her value hands). This will give Alice (and the blinds, should they get involved) opportunities to make postflop mistakes that we can exploit.

But later in the article we'll let Alice open with a ~25% range from CO, and we'll see that QQ/AK now moves up to Bob's value range. Alice's value range is wider and weaker with a 25% opening range, and Bob's optimal 3-bet strategy changes accordingly.

OK, enough about flatting. Let's move on and find Bob's 5-bet bluffs, and then his 3-bet bluffing range:

We include 5-bet bluffs in Bob's value range
Remember the definition of "value range" as the hands we 3-bet, planning to 5-bet all-in after a 4-bet. Some of these hands will be 5-bet bluffs, but for simplicity we'll refer to all the 5-betting hands as Bob's value range.

From the previous theory section, we remember that Bob wants to have enough Axs 5-bet bluffs in his value range to make Alice's weakest value hands break even. This accomplishes two things for Bob:


  • He forces Alice to fold more of her 4-bet bluffs
  • He makes it impossible for Alice to "cheat" by not paying off Bob's value 5-bets with {KK+}. If she tries to be "smart" and fold QQ/AK, Bob will just collect his profit with his 5-bet bluffs instead.


So Bob's 5-bet bluffs with some Axs hands attack Alice's 4-bet bluffs, and they also make it impossible for her to profitably tighten up her value range, even if she knows Bob's value range is the squeaky tight {KK+}. Keep in mind that Alice knows Bob's strategy, since this follows from her own strategy, which follows from her opening range, which both players know.

So she knows Bob only 3-bets/5-bets {KK+} for pure value, and if Bob's doesn't 5-bet bluff a bit, Alice can improve her 5-bet-calling strategy by folding the big underdog's QQ/AK from her value range {QQ+, AK}. And when one of the players can improve his/her EV by a strategy change, the original strategy pair can't be optimal (per definition). So Bob has to 5-bet bluff.

The next step for Bob is to add enough Axs to make Alice's weakest value hands break even when they call a 5-bet. Alice then calls off her last 73bb to win the blinds + Alice's 4-bet + Bob's stack =1.5 + 27 + 100 =128.5 bb. The pot odds are 128.5 : 73 =1.76 : 1, so Alice needs minimum 1/(1 + 1.76) =36% equity against Bob's 5-betting range to call profitably.

We add A5s to Bob's value range, and check Alice's equity with QQ/AK:



AK is above the threshold, but QQ is way below 36%. We add A4s and try again:



QQ is now slightly better than break even, and Bob can use A5s/A4s as his optimal 5-bet bluffing hands. However, if we want Alice's equity to be exactly break even, we have to remove a 5-bet bluff or two. Let's remove A 4 and see what we get:



Bingo, and Bob's optimal 5-bet bluffing hands are {A5s, A 4 , A 4 , A 4 }. But here I'll say that we don't have to be this strict. A combo or two too much or too little doesn't change things much, and we can use A5s/A4s in practice. Also, as we'll discuss further in the summary at the end of the article, it's debatable whether we need to 5-bet bluff at all in most games, unless we are playing against people like Durrrr.

People generally don't 4-bet bluff enough, and they are also reluctant to tighten up their 4-bet value ranges when they get exploited by very tight 5-betting (e.g. 5-bets that are 100% for value). For example. a typical low limit TAG with a ~15% EP range might have decided to never 4-bet bluff, and always 4-bet QQ and AK for value and call a 5-bet with them. And he is unlikely to change that plan, even if Bob's exploitative response is to drop all 5-bet bluffs from his value range, and only 5-bet-shove {KK+}, purely for value.

These things happen because a) people are blinded by seemingly strong hands, even after they get trapped in situations where their hands suddenly aren't strong anymore, and b) because people are reluctant to change their initial plan, even after if becomes clear it's a bad plan.

Against an opponent who makes the dual mistake of not 4-betting bluffing enough, and also paying off our value 5-bets too much, Bob can gain a lot from not having to think about 5-bet bluffing. Bob simply 3-bets {KK+} for value, plus a wide range of 3-bet bluffs, and after a 4-bet he 5-bets {KK+} for value and folds everything else.

This way Bob exploits Villains lack of 4-bet bluffing, since his 3-bet bluffs forces Villain to fold most of his non-value hands (since Villain is unwilling to 4-bet bluff with these hands). And Bob also doesn't need to attack Villain's 4-bet bluffs with 5-bet bluffs of his own, since Villain isn't 4-bet bluffing. Finally, Bob exploits Villain's static 5-bet-calling range by only 5-betting for value (and getting called as a big favorite), and not having to include 5-bet bluffs for deception. Easy game.

At any rate, Bob's final value 3-bet range (including his 5-bet bluffs) against Alice's optimally played ~15% EP range is {KK+, A5s, A4s}. Bob's last job is to construct the 3-bet bluff range. These are the hands we 3-bet, and always fold to a 4-bet.

Bob's 3-bet bluff range
We remember the strength principle for poker hands:

- Bet/raise your strongest hands for value
- Check/call with your medium hands
- Fold/bluff with your weakest hands

We have already defined Bob's value range (including 5-bet bluffs) as {KK+, A5s, A4s}, and we have mentioned that he also flats some range of good-but-not-great medium strong hands. Against Alice's ~15% EP range this means flatting with hands like QQ, JJ, TT, AK, AQ, AJ, KQ, etc.

So when we pick hands for Bob's 3-bluffing range, we drop down to the "cellar" and pick hands that aren't god enough to 3-bet for value preflop, and not good enough to flat for postflop value. Against Alice, who either 4-bets or folds, it doesn't matter which hands we choose to 3-bet bluff with, since these hands will never see a flop. Alice either 4-bets or folds, and when she 4-bets, we 5-bet our value range all-in, and fold our 3-bet bluff range.

But in practice the choice of 3-bet bluff range matters a bit, since the raiser will sometimes call our 3-bet with his medium strong hands out of position and force us to play postflop. Therefore, since we can choose freely from our worst hands, we might as well choose the best of our worst hands.

In other words, we'd rather 3-bet a hand like K8s as a bluff than a hand like 72o. K8s has some postflop value those times the raiser calls and forces us to see a flop, while 72o doesn't. So 3-bet bluffing with hands like K8s dominates (e.g. is sometimes better than, and never worse than) 3-bet bluffing with hands like 72o.

So let's list some 3-bet bluff candidates à la K8s that are too weak to flat, but have some postflop value when we get called. We make a list of ace high, king high and queen high candidate hands:

Candidate list for 3-bet bluffing:
- Ace high: A9s-A6s ATo-A8o (52 combos)
- King high: K9s-K6s, KJo-K9o (52 combos)
- Queen high: Q9s-Q6s, QJo-Q9o (52 combos)

If you don't approve of this list, feel free to make your own. The specific hands are irrelevant, what matters is that we use hands with the right properties, namely hands that aren't quite strong enough to flat. NB! A5s-A2s are reserved for 5-bet bluffing, so we can't include them in this list.

This gives us a list of 156 "pretty" combos for 3-bet bluffing, and the next question is which hands to choose and when. We remember that the optimal value/bluff ratio for Bob's 3-betting range is 40/60, so he can use 60/40 =1.5 bluff combos for each of the combos in his value range (including his 5-bet bluffs). His total value range is {KK+, A5s, A4s} =20 combos, so Bob can pick 1.5 x 20 =30 3-bet bluff combos.

As mentioned previously, there are two techniques Bob can use:

- Pick 30 specific combos and always 3-bet them
- 3-bet all hands from the candidate list a certain percentage of the time

I prefer the percentage method. To use it, we only need to memorize the candidate range once and for all, plus one number (the % we 3-bet bluff the candidate hands). Let's compute the number to use against Alice's EP range:

To effectively have 30 bluff combos from the candidate list in our 3-betting range, we need to use each of them 30/152 =20% of the time. Note that this percentage isn't universal, like Alice's fixed 4-bet bluff percentage (15%) is for all her opening ranges. To see this, note that Bob's value range varies with Alice's opening range, but the candidate list of 3-bet bluff hands is static (we have simply chosen some hands to use).

So Bob will have to calculate a new bluff% to use for his candidate list against each of Alice's opening ranges. However, this isn't a big job, we simply do the math once and for all against each of Alice's ranges and memorize the numbers we need (and we'll look at Alice's CO range in a minute).

So, finally:

Bob's optimal 3-bet strategy against Alice's optimal raising strategy in EP
{Bob's total 3-bet range}
={value hands and 5-bet-bluff hands} + {3-bet bluff hands}
={KK+, A5s, A4s}
+ 20% x {A9s-A6s,ATo-A8o,K9s-K6s,KJo-K9o,Q9s-Q6s,QJo-Q9o}


Bob always 3-bets {KK+, A5s, A4s} and 5-bets all-in after a 4-bet. If he has one of the 152 combos from his candidate list for 3-bet bluffing, he uses a randomizer and 3-bet bluffs 20% of the time, and he folds to a 4-bet. We had to do a bit of work to construct all these ranges, but it was worth it, and we have learned a lot in the process.

Let's see what Bob's optimal total 3-bet% is in this case:

- Value part: 20 combos (1.5% of all hands)
- Bluff part: Effectively 20% of 152 =30 combos (2.3% of all hands)

This results in a total 3-bet% of 1.5 + 2.3 =3.8% against Alice's ~15% EP raises. His value/bluff ratio is the desired optimal 20/30 =40/60. Later, when we construct an optimal strategy against Alice's 25% CO range, we'll see that Bob's 3-bet% skyrockets as a consequence of Alice raising a much wider opening range.

Note that the combination of a candidate list of 3-bet bluff hands and a fixed (but adjustable) bluff% to use with these hands, gives us a lot of flexibility to adjust our 3-bet bluffing as we please. Against an unknown opponent, we can start with the optimal 20% frequency, and 3-bet {KK+, A5s, A4s} always, and the candidate list 20% of the time. But if we note that the raiser doesn't defend optimally, we might want to adjust this bluff percentage.

For example, of the raiser never 4-bet bluffs and only 4-bets a tight value range like {QQ+, AK}, we can go bananas with our 3-bet bluffs. We might decide to double the bluff frequency from 20% to 40% for our list of 152 bluff candidate combos. Now we have 20 value combos, and effectively 0.40 x 152 =61 bluff combos. This means 20/(20 + 61) =25% of our 3-bets are for value, and 75% are bluffs. Our first adjustment to exploit this particular opponent is therefore to lower the optimal value/bluff ratio from the optimal 40/ to the more exploitative 25/75.

Then we can also drop 5-bet-bluffing against this tight player, as discussed previously. The simplest adjustment is to keep 3-betting our 5-bet bluffing hands A5s/A4s, but we move them from the value range down to the 3-bet bluff range, and fold them to a 4-bet. The only hands we 5-bet against this player and his {QQ+, AK} 4-bet range is {KK+}, purely for value.

Here is an example of randomized 3-bet-bluffing, using the randomizer from random.org:

Example 3.2.1: Randomized 3-bet bluffing against a ~15% EP raise


$100NL
6-handed

Alice ($100) raises to $3.5 from UTG, and it's folded to Bob ($100) who has Q 9 on the button. This hand is on the candidate list of 3-bet bluff hands, and we remember that the optimal bluff frequency to use against a ~15% opening range is 20%. Bob clicks the randomizer, planning to 3-bet if it returns a number between 0 and 20, and otherwise fold:



The randomizer returns 18, so Bob 3-bets to $12. Alice quickly 4-bets to $27, and Bob folds.

Everything according to plan, and with total control, so there is no reason to feel frustrated after this clash. Our Q 9 did it's job (attacking the weakest part of Alice's opening range) perfectly, regardless of the outcome, and it's irrelevant that Alice had a 4-betting hand this time.

Remember that we know Alice's strategy just as well as she knows our strategy, and we know that she will 3-bet us 30% of the time and fold 70%. When the 4-bet comes, we quietly fold our 3-bet bluffs and 5-bet-shove our value/5-bet bluff range of {KK+, A5s, A4s}. And we do these things calmly, without emotion.

4. Optimal strategy pairs for raiser/3-bettor with a CO raiser out of position


After the thorough work with Alice raising ~15% in EP, we can now reap the rewards and quickly run through the same procedure with Alice raising a ~25% range in CO. She now opens a wider range, as a consequence, all other ranges get wider as well.

4.1 Alice's optimal raising strategy for CO


Raising from CO is a bit more situational than raising from EP. It's now easier to isolate the blinds, and with a tight player on the button, it might be correct to play very loosely to get heads-up with position on the blinds. Still, everybody has a core range of hands that they always play, regardless of whether they have written this range down or not.

We'll assume Alice is using a TAG core range of ~25% in CO. More specifically, this range:

Alice's CO range
22+
A2s+ A9o+
K9s+ KTo+
Q9s+ QTo+
J8s+ JTo
T8s+
97s+
87s
76s
65s

326 combos
25%

Alice's value range
Alice defends against 3-bets 30% of the time, and she does it by 4-betting 18% of her opening range for value and 12% as a bluff. So she needs 0.18 x 326 =59 value combos that she can 4-bet and call a 5-bet with. In EP she used [QQ+, AK} =34 combos, and in CO we simply add the next tier of hands and use {JJ+, AQ+} =56 combos (precise enough).

Then she needs 0.12 x 326 =39 bluff combos. She can pick ~39 specific combos and always 4-bet them (e.g. AJ, AT, TT =38 combos), or she can 4-bet all her non-value hands 15% of the time, as explained previously. We choose the latter approach, and write Alice's complete raise strategy for CO as:

Alice's optimal raise/4-bet/call 5-bet-strategy in CO:
{Alice's total CO range}
={22+,A2s+,K9s+,Q9s+,J8s+,T8s+,97s+,87s,76s,65s,
A9o+,KTo+,QTo+,JTo}
={value hands} + {4-bet bluff hands}
={JJ+, AQ+}
+ (15% 4-bet and 85% fold) x {the rest of the range}


Alice raises {22+,A2s+,K9s+,Q9s+,J8s+,T8s+,97s+,87s,76s,65s,
A9o+,KTo+,QTo+,JTo} =326 combos from CO. If she gets 3-bet, she 4-bets {JJ+, AQ+} for value, planning to call a 5-bet. Those times she doesn't have a value hand (e.g. 88, A9o, T9s, etc.), she 4-bets 15% of the time, and the rest of the time she folds. Using a random number generator fromrandom.org to randomize 4-bet bluffs has been illustrated in a previous example.

That's it for Alice's CO strategy. Over to Bob:

4.2 Bob's optimal 3-bet-strategy against Alice's optimal raising strategy in CO


Bob needs a value range, including an optimal number of 5-bet bluffs, and he needs a range of hands to 3-bet bluff.

Bob's pure value range
Bob knows that Alice now uses {JJ+, AQ+} as her value range, so he builds a range of pure value hands that have at least 50% equity against {JJ+, AQ+}. AA and KK obviously belong in this range. To see what else is included, we run equity calculations for the next tier of hands (QQ and AK):



QQ and AK are barely above the threshold, while all weaker hands will be big underdogs. Thus, Bob's pure value range is {QQ+, AK}, and he happily 3-bets these hands, and then 5-bets them all-in, purely for value.

We add 5-bet bluffs to Bob's value range
We now want to add enough Axs hands so that Alice weakest value hands (JJ and AQ) are break even when they call our 5-bet range (and the threshold is 36% equity, as shown previously). We start with A5s/A4s and see what we get:



Alice's weakest value hand is AQ, and it's a small loser with 34% equity against Bob's total value range {QQ+, AK, A5s, A4s}. Close enough for us, so the 5-bet bluffs in this case are the same as we used against Alice's EP range. However, if we want it to be exact, we need to add a couple more bluffs (for example, A 3 and A 3 ) to lift AQ up to 36%:



Bob's 3-bet bluffing
Bob's value range, including 5-bet bluffs, is {QQ+, AK, A5s, A4s} =42 combos. He wants an optimal 40/60 value/bluff-ratio, so he needs 60/40 =1.5 times as many bluff combos. This amounts to 1.5 x 42 =63 bluff combos.

We use the previously defined candidate list for 3-bet bluff hands:

Candidate list for 3-bet bluffing:
- Ace high: A9s-A6s ATo-A8o (52 combos)
- King high: K9s-K6s, KJo-K9o (52 combos)
- Queen high: Q9s-Q6s, QJo-Q9o (52 combos)

We bluff these hands some fixed percentage x, and for this to effectively correspond to 63 bluff combos, we need x =63/152 =41%. We can round this to x =40% to make it easy to remember.

We remember that we used a 20% bluff candidate frequency against Alice's ~15% EP range. So when Alice moves from a ~15% EP range to a ~25% CO range, our 3-bet bluff candidate frequency doubles. We only need to memorize the candidate list, and these two numbers (20% vs EP and 40% vs CO), and then we know all we need to know about 3-bet bluffing optimally against Alice's EP/CO ranges.

At any rate, against Alice's optimal CO raising strategy, Bob gets:

Bob's optimal 3-bet strategy against Alice's optimal raising strategy in CO
{Bob's total 3-bet range}
={value hands and 5-bet bluff hands} + {3-bet bluff hands}
={QQ+, AK, A5s, A4s}
+ 40% x {A9s-A6s,ATo-A8o,K9s-K6s,KJo-K9o,Q9s-Q6s,QJo-Q9o}


Using a randomizer from random.org to randomize 3-bet bluffing has been illustrated in a previous example.

Bob's total 3-bet% for this scenario is:

- Value part: 42 combos (3.2% of all hands)
- Bluff part: Effectively 40% of 152 =61 combos (4.6% of all hands)

This results in a total 3-bet range of 3.2 + 4.6 =7.8% against Alice's ~25% CO range. The value/bluff ratio, using our numerical rounding, is 42/61, which is very close to the optimal 40/60.

When Alice moves from EP to CO and her opening range changes from ~15% to ~25%, Bob responds by loosening up his 3-betting range dramatically. This is an interesting observation. Those of you who use a more or less static 3-betting range (for example, the generic {JJ+, AQ} without any 3-bet bluffing that is recommended on many low limit forums) now have game theoretical "proof" that we can get away with a lot of 3-bet bluffing on the button against a CO raiser.

Even against a TAG CO raiser with a solid ~25% opening range, you can 3-bet almost 8% on the button and there isn't anything he can do to exploit your loose 3-bets. And if he defends poorly, for example by not 4-bet-bluffing enough, or not being willing to use JJ/AQ as value hands, you can deviate from optimal play and attack him even harder. The first adjustment against a weak/passive CO raiser who folds a lot to 3-bets is to increase your fixed 3-bet bluff percentage for the candidate list. You might increase the bluff candidate 3-betting frequency from 40% to 60%. If Villain (and the blinds) doesn't adjust to your exploitative, loose 3-betting, you'll be printing money.

5. Summary


We have gone through the theory for game theory optimal(ish) raising/3-betting/4-betting/5-betting with the raiser out of position, and then we demonstrated how the theory can be implemented and used at the table.

We constructed optimal strategy pairs (one strategy for the raiser, and one for the 3-bettor) for two scenarios. First with the raiser in EP (UTG or MP) with a ~15% range, and then with the raiser in CO with a ~25% range. In both scenarios we gave the raiser a standard TAG opening range. We then deduced optimal strategies for both players as a function of the raisers opening range. We observed that the strategies for the CO scenario involved considerably looser ranges than the strategies for the EP scenario.

Our optimal strategy pairs confirmed that it's correct to 3-bet a wide range on the button against a CO raiser, even if he starts with a solid opening range, and defends optimally against a 3-bet. And if he doesn't defend optimally, we can loosen up even more. When you see a good and aggressive NL player dominate the table by 3-betting loosely in position, this is what happens. Loose, positional 3-betting is game theoretically correct, even against strong players. And against weak players, it's even more correct.

As a result of our work, we ended up with specific and concrete implementations of the theory, both as the raiser and as the 3-bettor. You can implement these strategies immediately in your own game by following the procedures outlined in this article. The strategy pairs depend on the raiser's opening range, but the ~15% and ~25% EP and CO ranges are relatively standard, and you will meet many opponents who play close to these ranges. If you need to apply the theory to other ranges, just plug them into the method, and construct the strategy pairs yourself.

We didn't look at small blind vs big blind in this article, even if it falls under the same category with the raiser out of position. I elected to leave this situation out, since blind vs blind dynamics is very dependent on the players involved, and the history between them. So it's difficult (and probably not very useful) to try and generalize and assign SB a standard opening range. But if you want to do this, you can use the method and construct the optimal strategy pair yourself.

Those of you who enjoy experimenting with ranges and numbers can now start to apply the optimal strategies in your own game, using your own ranges. Plug your own opening ranges for EP and CO into the theoretical "machinery" outlined in this article, and produce optimal strategy pairs, based on the ranges you use at the table. Remember that everything follows from the opening ranges, and remember that you will get both an optimal strategy for the raiser (you), and the positional 3-bettors optimal strategy against you.

Learn both parts of every optimal strategy pair. When you are the raiser OOP against an unknown 3-bettor, you can simply play optimally and assume that he is playing optimally too. You now have 100% knowledge about the raiser's range (since this is your range), and you know the optimal strategy pair for this situation exactly. Since the 3-bettor doesn't know these things precisely, he will make mistakes, and you won't.

When you have position on the raiser, things are slightly less straightforward, since he is the one who chooses the opening range. But against an unknown raiser, you can start by assuming he uses opening ranges that are close to your default ranges. Then you simply respond with the corresponding optimal 3-betting strategy. If he uses ranges that are only slightly different from yours, the optimal strategy pairs will be similar.

And if you should need optimal strategy pairs for opening ranges that are very different from your own (for example, if you meet a CO raiser who opens 45% of his hands), you can quickly construct the corresponding optimal strategy pair for him and yourself. Remember that you don't need to know his opening range in detail, you only need to know the number of hands that he opens. This number is relatively easy to estimate from a HUD, even if the sample isn't big.

To be prepared for any opening range you might encounter as a 3-bettor, you can sit down and do the work for 10%, 35% and 45% opening ranges on your own. Then you'll have have a set of optimal strategy pairs that cover almost all cases of EP and CO open-raising you are likely to encounter in practice.

Again, when you are the raiser, everything follows from your ranges, and you can do this work once and for all (assuming you have a well-defined set of default core opening ranges) and memorize it. Then you can play optimally from out of position, and sniff around for opponent leaks. If you don't find any, keep playing optimally. If you find some exploitable leaks, think about how you can adjust to increase your EV. But you don't have to adjust until you are sure. Remember, if you are playing optimally and your opponent isn't, you gain from his mistakes (although you might gain more by switching to an exploitative strategy).

A classic opponent mistake at the low limits is not 3-bet bluffing enough (or at all) in position. Love these guys, because it's easy to exploit them. For starters, they are "exploiting themselves" by allowing you to run over them by not 3-betting you nearly as often as they should. And when they do 3-bet, you know that they are strong. So you simply drop all your 4-bet bluffs from your range and continue with a 4-betting range of only value hands, planning to call a 5-bet. Easy decisions and easy game.

When someone has raised in front of you, you ideally want to use an optimal strategy for each opponent, and for each of his positions (since optimal 3-bet strategy is a function of the raiser's range). This might sound like a lot of work, but in practice it all follows from estimates about the ranges you meet. And small deviations don't change things dramatically. For example, when you know the strategy pair corresponding to a 15% opening range, you can apply the same strategies against a 12% raiser and an 18% raiser without losing much accuracy. You won't play optimally in these cases, but near-optimally is close enough. Besides, pin-pointing opponent opening ranges to within +/-1% or less is difficult, so using near-optimal strategies is the best we can hope for in practice.

The nest step of the process is the most interesting one. When you have trained optimal play, you will discover that it's now much easier to spot opponent mistakes. For example, when you come across an opponent who doesn't 4-bet bluff (and these are common at the low limits), you immediately know that this is a leak, and you know how to exploit it. Tight and straightforward players who refuse to 4-bet bluff can be exploited by 3-betting a lot, and not 5-bet bluffing at all. You can 3-bet a metric fuckton of bluff hands, and when they finally pick up a hand good enough to 4-bet, you fold all your bluffs and ship a tight value range (sometimes as tight as {KK+}). Just keep an eye on the other players to see if they are trying to exploit your loose 3-betting (tighten up a bit if they do), and you'll do very well in this spot.

Another leak you'll see is spazzy 4-betting from players with insufficient understanding of the theory behind optimal 3-bet/4-bet/5-bet wars. This might happen when you have driven someone crazy with your loose 3-betting, and he starts to tilt. Or when someone tries to fight back in a controlled manner, but he doesn't quite know how to do it (so he starts 4-bet bluffing way too much).

The first thing you have to realize when you are playing optimally, and then spotting a leak, is this: It's not necessary to deviate from optimal play to benefit from his mistakes. If you keep playing optimally, and your opponent doesn't, you will win from him in the long run, period. The question is now whether you should deviate from optimal play yourself, in order to win more. If you have a clear idea about how to exploit your opponent maximally, by all means go ahead and make the adjustment.

But be cautious when you adjust to spazzy and unpredictable opponents. Remember that your optimal 3bet/4-bet/5-bet strategies are designed to protect you, and there is nothing a maniac can do to exploit you in these scenarios, even if he raises and reraises at every opportunity. If you see concrete adjustments you can make to win more, go for it, but be careful if you tilt easily (preflop raising wars have a tendency to trigger tilt). Then you might be better off sticking to optimal play against hyper-aggressive opponents, let the ranges do the work for you. You can use your focus to terrorize the passive and easily exploitable players instead.

Finally, if you meet tough regs who don't give up preflop edge in these scenarios (at least any edge you can see), these optimal strategies will protect you from getting exploited. They can't take advantage of you in preflop 3-bet/4-bet/5-bet wars, so don't worry about it if they try. Follow the optimal strategies, and the mathematics of the situation will protect you. But don't forget to sniff for leaks against regs. Everybody has leaks, and your knowledge about optimal 3-/4-/5-betting will make it easier for you to find them. And pay close attention if you see a reg starting to tilt! Now he might blow up completely in preflop raising wars, and you can adjust accordingly.

I hope this article will be useful for those of you who find it difficult to play well in preflop 3-/4-/5-bet wars, and that you have learned to implement the optimal strategies in your own game. And for those who already knew these things, I hope that this systematic discussion of the topic has given you things to think about.

I chose to name this article "Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max - Part 1", even if a Part 2 hasn't been planned yet. But I do have some more ideas about the topic, and I might write more. For example, we could do one article about optimal strategy pairs with the raiser in position (e.g. after a 3-bet from the blinds). Then we could dedicate one article to discussion about optimal versus exploitative play, and talk about how to apply one or the other against different opponent types.

Good luck!
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Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max - Part 1
[h1]1. Introduction[/h1]
For my first NLHE article for Donkr, I have chosen a topic that I know many new players find difficult, namely correct strategies for 3-bet/4-bet/5-bet preflop wars in NLHE 6-max.

[h2]1.1 Presenting the problem[/h2]
Against weak low limit opposition, we can get away with playing an almost completely value-based game. We 3-bet/4-bet/5-bet mainly for value, and it[apostrophe]s not a big mistake to assume our opponents are doing the same. If we reraise as a bluff, we usually limit ourselves to the occasional 3-bet bluff. A value-based style with little bluffing works well at small stakes because our opponents use more or less the same strategy, and many of them execute it poorly. Of course, every now and then we run into aggressive players who are capable of reraising as a bluff, but there are plenty of fish that will pay off our straightforward game, even if we bluff much less than is game theoretically optimal.

But let[apostrophe]s say our Hero has built a bankroll by patiently grinding the low limits, and now he wants to take a stab at $200NL. He will now experience a lot more 3-betting, especially if he[apostrophe]s out of position.

For example:

[h2]Example 1.1.1: We get 3-bet out of position[/h2]
$200NL
6-handed

Hero ($200) raises to $7 with :JS :TS from UTG, it[apostrophe]s folded to the button ($200) who 3-bets to $24, the blinds fold, and Hero folds.

Straightforward, and although Hero expects to get bluffed some of the time, he really doesn[apostrophe]t have any choice but to fold. It[apostrophe]s correct that his hand can no longer be played for value, but as we shall see later, it[apostrophe]s possible to turn it into a 4-bet bluff.

At any rate, Hero plays on. The players behind him keep 3-betting him frequently when he is out of position, and Hero keeps folding weak hands to 3-bets. After a while, this hand occurs:

[h2]Example 1.1.2: We get 3-bet out of position (again)[/h2]
$200NL
6-handed

Hero ($200) raises to $7 with :AH :JC in MP, it[apostrophe]s folded to button ($200) who 3-bets to $24, the blinds fold, and Hero folds.

This is getting frustrating. Hero has a decent hand, but it[apostrophe]s not strong enough to defend against a 3-bet from out of position, so Hero folds. But he is starting to feel exploited. If only he could get dealt a good hand and punish these bastards!

What an inexperienced player now might do (as his frustration builds up more and more), is to make up his mind to fight back against the loose 3-bettors. But he doesn[apostrophe]t quite know what to do, [i]and therefore he will often use poor strategies, and the wrong types of hands![/i].

Let[apostrophe]s look at two common (and sub-optimal) ways to defend against 3-betting, out of position with 100 BB stacks:

[h2]Example 1.1.3: We get 3-bet out of position (again) and we call[/h2]
$200NL
6-handed

Hero ($200) raises til $7 with :KH :QH in MP, button ($200) 3-bets to $24. Hero thinks for a bit, decides that this hand is too good to fold, but too weak to 4-bet, so he calls.

[b]Flop:[/b] :9S:4S:4C ($51)
Hero ($176) checks, button ($176) bets $30, Hero folds.

Hero is frustrated, but he doesn[apostrophe]t see what else he could have done out of position with a hand of this type. Too strong to fold (at least in Hero[apostrophe]s mind) against a loose 3-bettor, but not strong enough to 4-bet. Or? Hmmmmm .... Hero contemplates his next move, and soon another 3-bet pot occurs:

[h2]Example 1.1.4: We get 3-bet out of position (again) and we 4-bet for value (or at least that[apostrophe]s what we think we are doing)[/h2]
$200NL
6-handed

Hero ($200) raises to $7 with :AC :JC from UTG, MP ($200) 3-bets to $24. Hero decides to fight fire with fire, and he 4-bets pot to $75. Button 5-bets all-in, Hero calls. MP has :KH :KC . Hero screams in agony.

[b]Flop:[/b] :QH :TS :7C ($403)

[b]Turn:[/b] :QH :TS :7C :QS ($403)

[b]River:[/b] :QH :TS :7C :QS :4H ($403)

Hero tears his clothing and sprinkles ashes over his head. Damn!!

[b]What happened throughout this sequence of hands?[/b]
OK, I made up this story, but it illustrates several of the problems an ABC low limit player faces when he moves up to tougher games. He will get 3-bet left and right, so he will have to fold a lot out of position (which is correct). He realizes he has to fight back to avoid getting run over (also correct), but he[apostrophe]s not quite sure how to do it. So his attempts to counter the aggression are often poorly executed, frustrating and tilt-inducing.

For example, Hero might start calling 3-bets out of position with hands he feels are too good to fold, but not strong enough to 4-bet for value. This leads to many miserable experiences like Example 1.3. Or he might start 4-betting medium/weak hands without a clear understanding of whether he is doing it for value (planning to call a 5-bet), or if he is bluffing (planning to fold to a 5-bet).

What our inexperienced Hero might not realize, is that his opponents[apostrophe] loose 3-betting doesn[apostrophe]t necessarily mean they are willing to splash around with lots of weak hands in 4-bet and 5-bet pots. When two good and aggressive NLHE-players engage in 3-bet/4-bet/5-bet warfare preflop, this is what usually happens:

[LIST]
[*]Both players operate with wide ranges, and all ranges have a significant percentage of bluffs in them, especially at the early stage (raising and 3-betting)
[*]Both players are willing to fold most of their bluffs (but not all of them), when their opponent reraises them back
[/LIST]

This results in ranges that start loose, but get more and more (but never completely) weighted towards value. And it[apostrophe]s usually plain wrong to assume you can 4-bet a medium hand like AJs for value against a loose 3-bettor, and expect to be a favorite when he 5-bets all-in. Yes, AJs is a decent hand against the range that 3-bet you, but it[apostrophe]s crushed by the range that 5-bets you, and it[apostrophe]s your opponent who decides when the 5th bet goes in (and that rarely happens unless he has the goods).

Therefore, if you decide on a frustrated whim to "take a stand" against an aggressive and competent 3-bettor with a hand like AJs, you will discover that in some mysterious way he almost always manages to come up with a better hand when you get all-in preflop.

This has lead many an inexperienced NLHE player to lose his stack, since these players:

[LIST]
[*]Don[apostrophe]t understand the roles different types of hands have in different types of ranges. First and foremost: Do I have a value hand that wants to get all-in, or do I have a bluff hand that I will fold to further aggression?
[*]Aren[apostrophe]t willing to fold hands that are strong at the early stages, but turn into weak hands when Villain keeps reraising
[/LIST]

Let[apostrophe]s look at Example 1.4 again. Hero open-raised AJs (correctly), and he got 3-bet. He then decided that his AJs was a good hand against Villain[apostrophe]s 3-bet range (debatable, but not a big mistake), so he 4-bet for value (wrong!), planning to call a 5-bet all-in. Playing AJs for value after a 3-bet and going all-in with it was a big mistake. The 4-bet in itself was not a big mistake, since Villain has a lot of bluffs in his 3-betting range, and he will fold most of them to a 4-bet. So it[apostrophe]s not a problem to 4-bet AJs as a bluff against a range full of 3-bet bluffs. But when Villain comes over the top with an all-in 5-bet, our AJs crumbles to dust (if Villain knows what he is doing).

But our inexperienced Hero did not realize what had just happened when he got 5-bet, and he stuck with his plan of playing AJs for value against what he perceived to be a wide and weak range. The problem is that the range he faces [i]after a 5-bet from a competent player[/i] isn[apostrophe]t wide and weak, it[apostrophe]s very narrow and very strong.

Note what the real mistake was in this hand. 4-betting AJs against a wide range was not a big mistake in isolation, and neither was calling a 5-bet getting 2: 1. But the combination of 4-betting AJs + [i]planning to always call a 5-bet[/i], now that was a big mistake against a competent opponent. It caused Hero to invest his remaining 96.5bb stack as a huge underdog. The problem was, as mentioned previously, that his opponent controlled when the 5th bet went in, and Villain made sure he had a hand.

Our goal for this article is to give Hero a set of tools he can use to comfortably counter preflop aggression when he is sitting as the raiser out of position. We[apostrophe]ll base our work on Hero[apostrophe]s opening ranges, and based on these, we can deduce defensive strategies against positional 3-bets. And we will use game theory to design these strategies in such a way that [i]the 3-bettor can not exploit Hero[/i] in these scenarios. Our work on Hero[apostrophe]s game theory optimal defensive strategies also gives us a set of optimal 3-betting strategies for his opponent, so we kill two birds with one stone.

We have here talked mostly about the ills of getting 3-bet when sitting out of position, and this is what I feel inexperienced players find hardest to deal with. But the mirror image of this scenario, with us being the 3-bettor in position, is also worth discussing. These are easier scenarios to play, but we will benefit a lot from understanding optimal 3-bet/4-bet/5-bet dynamics also from this perspective. We[apostrophe]ll learn how to construct optimal 3-betting ranges, based on the raiser[apostrophe]s opening range, and we[apostrophe]ll learn how to play against a 4-bet.

Regardless of whether we[apostrophe]re the raiser or the 3-bettor, we want to understand which hands we can (re)raise for value, and which hands we (re)raise as bluffs. And above all else, we want it to be 100% clear which of these two things we are doing [i]before we engage in a 3-bet/4-bet/5-bet war preflop[/i].

[h2]1.2 Our model and overall philosophy[/h2]
In this article we[apostrophe]ll design so-called [i]optimal strategy pairs[/i] for the raiser and the 3-bettor in the following scenario:

- The raiser opens some range
- A player behind him 3-bets
- The raiser 4-bets or folds
- The 3-bettor 5-bets, or folds to a 4-bet

Note that the raiser is always out of position (e.g. UTG, MP, or CO), and that no other players interfere.

We[apostrophe]ll define a [i]model[/i] for this scenario with 100bb stacks and standard bet sizing. Then we[apostrophe]ll analyze our model, using mathematics and principles from game theory (but we[apostrophe]ll keep it as simple as possible). We then construct game theory optimal(ish) strategy pairs for the raiser and the 3-bettor (one strategy for the raiser, and one matching strategy for the 3-bettor) that they can employ in their 3-bet/4-bet/5-bet wars.

Both players are trying to play perfectly against the other, and both are assuming their opponent is trying to play perfectly as well. The two players now both zoom in on a perfect strategy, designed not to lose against their opponent[apostrophe]s perfect strategy. And the result is a pair of strategies that are perfect against each other, and we have our optimal strategy pair.

When we have learned these strategies, we have defensive (e.g. unexploitable) strategies we can use both as the raiser out of position, and as the 3-bettor in position. Using these optimal strategies guarantees that better players can[apostrophe]t exploit us. They will also win against players who play poorly, although they will not win the maximum (if we want to exploit opponent leaks maximally, we have to deviate from optimal play ourselves, and use strategies that target specific leaks in our opponent[apostrophe]s non-optimal strategies).

Knowing optimal strategies also makes it easier to spot our opponents[apostrophe] mistakes (where we can define "mistake" as a deviation from optimal play). If we know what an opponent [i]should have done[/i] if he had played optimally, we can conclude that he has a weakness in his game if he chooses to do something different. And we might be able to exploit these weaknesses and turn them into leaks for him.

[h2]1.3 Background material for the article[/h2]
Before we get started, I want to give credit to Cardrunners instructor Matthew Janda. During the spring of 2010 he published a 3-part video series [i]Optimal Preflop Play I-III[/i] at Stoxpoker, which contains most of the theory we use in this article. This video series was inspiring and eye-opening, but sadly it became unavailable after Stoxpoker shut down in May 2010.

Matt Janda is now a Cardrunners instructor, and he continues to produce game theory related videos. His old videos from Stoxpoker might get moved over to Cardrunners, and if that happens, I recommend you check them out.

Without further ado, let[apostrophe]s get started:

[h1]2. The mathematics behind optimal 3-/4-/5-betting with the raiser out of position[/h1]
I have chosen an approach where we first go through the necessary math and theory quickly, and then we apply it by constructing optimal strategy pairs for two scenarios:

- The raiser in early position (UTG or MP) with a 15% opening range
- The raiser in CO with a 25% opening range

Lumping UTG and MP together under the label "EP" makes sense, since most players use very similar ranges for these two positions. The percentages we have chosen for EP and CO are typical TAG ranges that can be used under all game conditions.

The exact ranges we use to illustrate the procedures aren[apostrophe]t important. Our goal is that you learn to construct optimal strategy pair (one strategy for the raiser and one for the 3-bettor) based on [i]your own[/i] opening ranges. And you will of course also be able to design optimal strategy pairs to use against specific opponents (not on the fly, but by doing a bit of analysis work between sessions).

[h2]2.1 Our model[/h2]
We use the following scenario:

[LIST]
[*]Alice is sitting with a 100bb stack in EP or CO, and she raises pot to 3.5bb with some opening range
[*]Bob is sitting in a position behind Alice with 100bb, and it[apostrophe]s folded to him. Bob 3-bets pot to 12bb
[*]Alice either 4-bets to 27bb (a bit less than pot), or she folds
[*]Bob[apostrophe]s response to Alice[apostrophe]s 4-bets is to 5-bet all-in or fold
[*]Alice[apostrophe]s response to Bob[apostrophe]s all-in 5-bets is to call or fold
[/LIST]

Note that Alice doesn[apostrophe]t defend against 3-bets by calling out of position. We [i]could[/i] conceivably design a defense strategy where we fold weak hands, 4-bet strong hands, and call with medium hands, but this is not a good strategy out of position with 100bb stacks.

You have poor implied odds (due to low stack/pot ratio and being out of position) when you call for postflop value with implied odds hands. And it[apostrophe]s difficult to steal and outplay Villain when you are out of position. And what you absolutely cannot do, is to call and then play fit-or-fold postflop. It will be much more fold than fit, and you are simply burning money by letting Villain c-bet his way to riches and early retirement on your expense.

With regard to Alice[apostrophe]s choice of 4-bet size, it[apostrophe]s standard to use 25-30bb (where full pot would be 37.5bb) with 100bb stacks. The logic behind this is that with 100bb stacks, we are putting Villain in a shove-or-fold scenario, also when we 4-bet a bit less than pot. His 3-bet bluffs will still fold, and his strong hands will still shove. So we win the same when he folds, but lose less on our bluffs when he doesn[apostrophe]t fold. In other words: We risk less for the same reward when we[apostrophe]re bluffing, and we don[apostrophe]t lose anything when we[apostrophe]re 4-betting for value. We simply choose 27bb as a representative value for a less-than-pot 4-bet, and the math won[apostrophe]t change much if you use any number between 25bb and 30b instead.

Here are a few assumptions/statements we will use:

[LIST]
[*]Bob knows Alice[apostrophe]s opening range. Not necessarily all the hands in the range, but he knows the percentage of hands Alice opens
[*]Both Alice and Bob are trying to play perfectly, under the assumption that their opponent is also trying to play perfectly
[*]The worst hands in a bluffing range or calling range should be break even
[/LIST]

The last statement needs an explanation: When we[apostrophe]re 3-betting/4-betting/5-betting as a bluff, we should not lose money on our bluffing hands, and the worst of them should be no worse than break even. The same goes for when we[apostrophe]re calling for pot odds. This makes sense if you think about it. When we[apostrophe]re making a play that loses money, we should stop doing it to increase our EV.

Note that we[apostrophe]re not concerned about the effects of deception when we work with game theory. We[apostrophe]re only concerned with immediate EV. Also, if we[apostrophe]re making money on all our bluffs or our calls, we can make even more money by bluffing more and calling more. So we keep adding bluffs and calling hands until our weakest hands are at the break even point, and then we stop. Conversely, if we[apostrophe]re losing money on some of our bluffing or calling hands, we remove them from our ranges. Again, this results in our weakest bluffing/calling hands being no worse than break even.

Under these assumptions, we[apostrophe]ll find an [i]optimal strategy pair[/i] with a raising strategy (including defense against a 3-bet and against a 5-bet) for Alice, and a 3-betting strategy (including defense against a 4-bet) for Bob. We[apostrophe]ll find a unique strategy pair for each of Alice[apostrophe]s positions (e.g. for each of her opening ranges). We[apostrophe]ll soon see how these strategy pairs follow from Alice[apostrophe]s opening range, but first, let[apostrophe]s talk a bit about optimal strategy pairs:

[b]What is an optimal strategy pair?[/b]
When our two players Alice and Bob are playing optimally against each other, Alice[apostrophe]s strategy and Bob[apostrophe]s strategy make up an optimal strategy pair. When both are playing optimally, neither of them can gain from changing to a different strategy. If one of them can gain from switching to another strategy, then the original strategy wasn[apostrophe]t optimal.

It[apostrophe]s important to realize that a game theory optimal strategy doesn[apostrophe]t try to maximize +EV against a random opponent. It[apostrophe]s trying to maximize EV against an opponent who is also [i]playing perfectly[/i]. Sometimes, this means the best result for both players is to break even. A game theory optimal strategy is first and foremost a [i]defensive strategy[/i], designed not to lose. However, an optimal strategy will win against players who are using non-optimal strategies. But If we see an opponent making big mistakes, we will win [i]more[/i] by switching to an [i]exploitative strategy[/i], designed to exploit this opponent[apostrophe]s specific leaks maximally.

But by changing our strategy from optimal to exploitative, we are moving away from optimal play. By doing so, we are creating weaknesses in our strategy, and other players might be able to exploit those weaknesses (although they might not see them). But if the weak player we are trying to exploit has big leaks, this trade off will usually be worth it. The art of playing against fish and regs at the same time is to exploit the fish, while we[apostrophe]re defending ourselves against the regs. Against very poor opponents, we use very exploitative strategies. Against players who are as good as us, or better, we can fall back on optimal strategies so that they can[apostrophe]t exploit us.

To balance these two goals well, we need to have an understanding of what optimal play is. Playing optimally (or, more likely, close to optimally) defends us against the good players, and understanding optimal play also makes it easier to spot mistakes in weak players (where "mistake" can be defined as deviating from optimal play).

With these concepts at the back of our mind, we move on to the mathematics behind optimal strategies for raising, 3-betting, 4-betting, and 5-betting with 100bb stacks:

[h2]2.2 How opening ranges, 3-betting ranges, 4-betting ranges, and 5-betting ranges are connected mathematically[/h2]
We work our way through the raise/3-bet/4-bet/5-bet war, one step at a time, and construct all the mathematical tools we need. We jump back and forth between Alice and Bob, and we[apostrophe]ll see how they influence each others[apostrophe] strategies when they both are trying to play perfectly against each other, assuming the other player is also trying to play perfectly.

[b]What is Alice[apostrophe]s optimal 4-bet%[/b]
The process starts with Alice raising some opening range known both to her and to Bob. When Bob 3-bets, Alice[apostrophe]s most pressing concern is the following:

[i]Alice can[apostrophe]t fold so much that she gives Bob an opportunity to make a profit by 3-bet bluffing any two cards[/i]

So how often does Alice have to 4-bet? This follows from the pot odds Bob is getting on his 3-bet bluffs. There[apostrophe]s 1.5 + 3.5 =5bb in the pot from the blinds and Alice[apostrophe]s raise, and Bob 3-bets to 12bb to win this. Bob is then risking 12bb to win 5bb, and he[apostrophe]s getting effective pot odds 5 : 12 on a 3-bet bluff.

He then needs to win more than 12/(5 + 12) =70% to have a profitable bluff. So if Alice folds more than 70%, Bob will have an automatic profit by 3-bet bluffing any two. Alice needs to prevent this, so she has to 4-bet enough to make Bob[apostrophe]s bluffs break even.

Alice[apostrophe]s optimal 4-betting strategy is therefore to 4-bet 30% of her opening range, and she will 4-bet a mix of value hands (planning to call a 5-bet) and bluffs (planing to fold to a 5-bet). We[apostrophe]ll compute Alice[apostrophe]s optimal value/bluff ratio in a moment, but first we have to find Bob[apostrophe]s optimal ranges for 3-betting and 5-betting. These ranges follow from Alice[apostrophe]s opening range:

[b]What is Bob[apostrophe]s optimal value/bluff ratio in his 3-bet range?[/b]
When Alice 4-bets to 27bb, she is risking 23.5bb (27bb minus he 3.5bb raise) more to win a 17bb pot (1.5bb from the blinds + Alice[apostrophe]s 3.5bb raise + Bob[apostrophe]s 12bb 3-bet). The effective pot odds for Alice[apostrophe]s 4-bet bluffs are 17 : 23.5, and she can make a profit by 4-bet bluffing any two (of the hands she open-raised) if Bob folds his 3-betting hands more than 23.5/(23.5 + 17) =58%.

Bob can[apostrophe]t allow Alice to 4-bet bluff any two cards profitably, so he defends optimally by folding exactly 58% of the time, and 5-betting all-in (including some 5-bet bluffs as we shall soon see) 42% of the time. Therefore, 42% of Bob[apostrophe]s 3-bets need to be value hands that he plans to 5-bet all-in (including some 5-bet bluffs). We now define a [i]3-bet for value[/i] as a 3-bet where we plan to 5-bet all-in after a 4-bet. If this is not our plan, we are making a [i]3-bet bluff[/i] that we will fold to a 4-bet.

To make these percentages easy to remember, we round Bob[apostrophe]s optimal 3-bet value/bluff ratio to 40/60. So now we know that 60% of Bob[apostrophe]s 3-bets should be bluffs, and 40% should be for value (including some 5-bet bluffs). But we still don[apostrophe]t know how many hands Bob should 3-bet overall. To find this number, we first have to find which hands Bob can 5-bet for value.

[b]What should Bob[apostrophe]s 5-betting range look like?[/b]
Bob first chooses the type of hands to 5-bet bluff with. He wants hands that have decent equity when called, and we can use Axs hands A5s-A2s for this purpose. Axs hands work as blockers against Alice[apostrophe]s AA/AK (an ace in Bob[apostrophe]s hand makes it less likely Alice has AA/AK), and they always have at least an overcard when Alice has another high pair. They also have straight and flush potential.

Axs has minimum ~30% equity when we 5-bet and get called, even against a strong range, as shown below:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/Axs_vs_5bet_callerange.png[/img]

So Bob will 5-bet a mix of true value hands and some Axs bluff hands, and he expects to have about 30% equity when his bluffs get called. So when he 5-bet bluffs and gets called, he will have ~30% equity in a 201.5bb pot where he invested 88bb with the 5-bet. Bob first 3-bet to 12, so the 5-bet is 88bb more. On average, Bob gets 0.30 x 201.5 =60bb back from the pot, so his net loss after 5-betting and getting called is 88 - 60 =28bb.

The pot size before Bob[apostrophe]s 5-bet is 40.5bb (1.5 from the blinds, + 27 from Alice[apostrophe]s 4-bet + 12 from Bob[apostrophe]s 3-bet). So Bob is effectively risking 28bb to win 40.5bb when he is 5-bet bluffing. The effective pot odds are 40.5 : 28, and Bob needs to win at least 28/(28 + 40.5) =40% to profit from 5-bet bluffing any two (or more precisely, any Axs hand, since we base our calculations on having ~30% equity when called).

For Alice, this means she has to call a 5-bet 60% of the time to prevent Bob from making a profit by 5-betting any two. So Alice[apostrophe]s 4-betting range has to contain 60% value hands and 40% bluff hands. Now we know everything we need to know about Alice[apostrophe]s 4-betting range. She 4-bets 30% of her opening range, and she uses a 60/40 value/bluff ratio. We[apostrophe]ll summarize Alice[apostrophe]s total optimal strategy below, but first we[apostrophe]ll find out how often Bob should 3-bet.

We know which type of hands Bob should 5-bet bluff (Axs), and we know he should use a 40/60 value/bluff ratio (which, coincidentally is the opposite of the ratio for Alice[apostrophe]s 4-bet range). The last piece of information we need is Bob[apostrophe]s [i]total[/i] 3-bet percentage in an optimal 3-betting strategy. We find the answer by observing that Bob should 5-bet bluff enough to make Alice[apostrophe]s weakest value hands break even. He he bluffs more, Alice can gain by calling with more hands, and then Bob[apostrophe]s strategy can[apostrophe]t be optimal. And if he bluffs less, Alice can gain by folding more hands, and Bob[apostrophe]s strategy can[apostrophe]t be optimal in this case either.

How many Axs hands we need to make Alice[apostrophe]s weakest 5-bet calling hands break even varies with Alice[apostrophe]s value range (60% of 30% of her opening range), which follows from her opening range. So we have to compute this result on a per-case basis, for every one of Alice[apostrophe]s opening ranges. We[apostrophe]ll give a quick example in the summary below, and the procedure will be thoroughly discussed later in the article.

[h2]2.3 Summary of Alice[apostrophe]s optimal raising strategy[/h2]
We summarize everything we have learned about Alice[apostrophe]s optimal strategy for raising, 4-betting and calling 5-bets:

- She needs to 4-bet 30% of her opening range
- Her 4-betting range should have a 60/40 value/bluff ratio

So Alice[apostrophe]s optimal strategy is:

[LIST]
[*]Alice open-raises some opening range
[*]When she gets 3-bet, she 4-bets 30% of her opening range with a 60/40 ratio between value 4-bets and bluff 4-bets
[*]Alice therefore 4-bets 0.60 x 30 =18% of her opening range for value and 0.40 x 30 =12% of her opening range as a bluff
[*]If Bob 5-bets all-in, Alice calls with all her value hands, and folds all her 4-bet bluffs
[/LIST]

So Alice[apostrophe]s value hands are the top 18% of her opening range. For example, if she opens 15% from UTG, this corresponds to a value range of 0.18 x 0.15 =2.7% of all hands. This makes up 0.027 x 1326 =36 combos, e.g approximately the range {QQ+, AK} =34 combos. We[apostrophe]ll use this value range example when we summarize Bob[apostrophe]s optimal strategy below. And then we[apostrophe]ll illustrate each strategy step thoroughly when we apply the theory to Alice[apostrophe]s EP and CO openraises.

[h2]2.4 Summary of Bob[apostrophe]s optimal 3-betting strategy[/h2]
We summarize everything we have learned about Bob[apostrophe]s optimal strategy for 3-betting and 5-betting:

[LIST]
[*]Bob starts by finding which hands he can 3-bet for value, planning to 5-bet all-in against Alice[apostrophe]s 4-bet value range. For this purpose, he needs hands that have at least 50% equity against Alice[apostrophe]s value range
[*]Bob then adds enough Axs hands as 5-bet bluffs to make Alice[apostrophe]s weakest value hands break even when calling Bob[apostrophe]s total 5-bet range
[*]Bob[apostrophe]s value hands and 5-bet bluffs are joined to a total value range (where value range =the range he 3-bets and 5-bets all-in)
[*]Finally, Bob chooses a 3-bet bluff range so that the ratio of his value hands (including 5-bet bluffs) to his bluff hands is 40/60
[*]When Alice raises, Bob 3-bets his value range and his bluff range
[*]If Alice 4-bets, Bob 5-bets his value range all-in and folds his bluff range
[/LIST]

For example, if Alice raises 15% from the UTG, her optimal value range is {QQ+, AK} as shown previously. Bob chooses value hands that are at least 50% against this range, and his pure value range becomes {KK+}. Then he adds Axs hands as 5-bet bluffs until Alice[apostrophe]s weakest value hands (QQ and AK) are break even against his total 5-bet range.

Alice then calls her remaining 73 BB to win a 189.5 bb pot (1.5 from the blinds, 100 from Bob, 27 from Alice[apostrophe]s 4-bet), so her pot odds are 128.5 : 73. She needs minimum 73(/128.5 + 73) =36% equity to profit from calling, so Bob makes sure her weakest value hands have against his 5-bet-range. Later in the article we[apostrophe]ll show that Bob ends up with a total 5-bet range of {KK+, A5s, A4s} when Alice[apostrophe]s value range is {QQ+, AK}

This gives Bob {KK+, A5s, A4s} =20 value combos that he 3-bets, planning to 5-bet all-in. Then he picks hands to 3-bet bluff until he has a 40/60 ratio between value combos and bluff combos. Bob needs 60/40 =1.5 bluff combos for every value combo, so he will choose 1.5 x 20 =30 bluff combos against Alice[apostrophe]s {QQ+, AK} value range.

You should memorize both Alice[apostrophe]s strategy and Bob[apostrophe]s strategy until you know them cold. It[apostrophe]s not really complicated at all. Just remember that Bob uses a 40/60 value/bluff ratio for his 3-bets, and Alice uses a 60/40 ratio for her 4-bets, and then you know the most of it. Value hands are per definition hands we plan to raise and reraise until we are all-in. Bluff hands are hands we plan to fold if our opponent reraises us back.

We now begin the job of constructing optimal strategy pairs for Alice and Bob. First when Alice raises a 15% range from EP, and then when she raises a 25% range from CO. We[apostrophe]ll do this thoroughly and methodically, so that you can learn the procedures inside out. I hope you[apostrophe]ll see that these strategies aren[apostrophe]t really complicated to construct and then apply at the table.

[h1]3. Optimal strategy pairs for raiser/3-bettor with an EP raiser out of position[/h1]
We[apostrophe]ll now find the optimal strategy pair for Alice and Bob when Alice open-raises from early position (EP =UTG or MP), and it[apostrophe]s folded to Bob in position.

It[apostrophe]s of course possible to vary EP opening ranges a lot, according to opponent tendencies and general game conditions. But the core strategy for a typical TAG is to open somewhere around 15% of his hands (plus/minus a couple of percentage points in both directions) from both EP positions, and slightly tighter from UTG than from MP.

We[apostrophe]ll construct all strategies/ranges with great detail for this scenario, so that there won[apostrophe]t be any doubt about how to apply the theory. Then we[apostrophe]ll move on to the scenario with Alice in CO, and do this quickly, with brief comments along the way.

[h2]3.1 Alice[apostrophe]s optimal raising strategy in EP (UTG and MP)[/h2]
We assume Alice is opening with a ~15% EP range. Note that any 15[apostrophe]ish% EP-range will do, since our work is based on the numbers of hands in the range, and not the specific hands it contains. Obvious value hands like high pairs and AK have to be included, since these hands have a job to do in the ranges for 4-betting and calling 5-bets. But the exact mixture of medium and weak hands in Alice[apostrophe]s range is irrelevant.

We give Alice the following range:

[b]Alice[apostrophe]s EP range[/b]
22+
ATs+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s

186 combos
14%

We now place Bob somewhere with position on Alice. Alice open-raises and it[apostrophe]s folded to Bob, who 3-bets. Both players want to play perfectly against the other, and both assume the other is also trying to play perfectly.

Alice starts by defining her value range. This is per definition the hands she plans to 4-bet for value and then call a 5-bet with. She counts the total number of combos in her opening range (186), and she knows that she on average has to defend 30% of her total range against a 3-bet. She also knows that the optimal value/bluff ratio of her 4-betting range is 60/40. So she 4-bets 0.60 x 0.30 =18% of her opening range for value, and 0.40 x 0.30 =12% as a bluff.

Alice then 4-bets 18% of the 186 combos for value, e.g. 0.18 x 186 =33 value combos. This corresponds almost exactly to the value range {QQ+, AK} =34 combos (a couple of combos too many or too few doesn[apostrophe]t matter much). This is a standard value range from EP, also for players who haven[apostrophe]t studied optimal raise/3-bet/4-bet/5-bet strategies.

Now the 4-bet bluff range. These are the hands Alice 4-bets and then folds to a 5-bet. There are two ways to define the bluff 4-bet range: We can choose some specific bluff combos and always 4-bet them, or we can 4-bet all the non-value hands a certain % of the time.

Let[apostrophe]s illustrate both methods:

[b]Defining a 4-bet bluff range using the combo method[/b]
If we choose specific bluff combos, we need 12% of 186 combos, e.g. 0.12 x 186 =22 bluff combos. For example, we might choose AQ (16) + JJ (6) which gives us exactly 22 combos. Or we can choose something different, since it doesn[apostrophe]t matter what we use for bluffs when Bob either folds or 5-bets all-in. When Bob doesn[apostrophe]t fold to our bluffs, he 5-bets, and we have to fold, so our 4-bet bluff hands never get to see a flop. And when they never get to see a flop, their postflop value is irrelevant.

But note that a hand like AQ works as a blocker against Bob[apostrophe]s premium hands (AA, AK, QQ). So when Alice uses AQ as a bluff, it will be less likely that Bob has a hand he can 5-bet for value. Keep this in mind if you are choosing specific hands to always use for 4-bet bluffing.

[b]Defining a 4-bet bluff range using the percentage method[/b]
My preferred method, and also the easiest method to remember. We only need to remember one number, namely the static percentage Alice 4-bet bluffs her non-value hands. Let[apostrophe]s find this percentage once and for all:

Alice 4-bets 18% of her opening range for value, and she[apostrophe]s left with 82% non-value hands she can use for 4-bet bluffing. We now choose to use all these hands a fixed percentage of the time, so that the effective total value/bluff ration is 60/40. We now want:

[pre]
value/bluff =60/40
18/82x =60/40
18/82x =1.5
18/82 =1.5x
0.22 =1.5x
x =0.22/1.5 =0.15 =15%[/pre]

So we 4-bet bluff all non-value hands 15% of the time and fold them the remaining 85% of the time. Note that this percentage is universal for Alice. No matter what her opening range is, she can always use this percentage to obtain a 60/40 value bluff ratio for her 4-bets.

Let[apostrophe]s double-check to see that this works the way it should:

When Alice has raised some opening range and gotten 3-bet, we have deduced that her optimal value 4-bet range is 18% of her total range. If she 4-bets the remaining 82% of her range as a bluff 15% of the time, her overall bluff percentage will be 0.15 x 0.82 =0.12 =12%. So her total 4-bet range is he optimal 18 + 12 =30%, with a 18/12 =60/40 value/bluff ration. So the percentages add up perfectly.

[b]Alice[apostrophe]s optimal raise/4-bet/call 5-bet strategy in EP[/b]
We now have everything we need to specify Alice[apostrophe]s total strategy after a 15% open-raise. We can write Alice[apostrophe]s total EP range as a sum of value hands (raise, 4-bet for value, call a 5-bet) and bluff hands (raise, 4-bet bluff, fold to a 5-bet):

[pre]{Alice[apostrophe]s total EP range}
={22+,ATs+,KTs+,QTs+,J9s+,T9s,98s,87s,76s,AJo+,KQo}
={value hands} + {4-bet bluff hands}
={QQ+, AK}
+ (15% 4-bet and 85% fold) x {the rest of the range}[/pre]

Alice raises {22+, ATs+, KTs+, QTs+, J9s+, T9s, 98s, 87s, 76s, AJo+, KQo} =186 combos from EP. If she gets 3-bet, she 4-bets {QQ+, AK} for value and calls a 5-bet with them. Those times she doesn[apostrophe]t have a value hand, (e.g. she has JJ, AJo, 76s, etc.), she 4-bets 15% of the time as a bluff, and otherwise she folds.

The percentage of value hands is then 34/186 =18%, while the effective percentage of bluff hands is 0.15 x (186 - 34)/186 =12%. The value/bluff ratio for her 4-bet range is 18/12 =60/40, which is optimal.

To randomize her 4-bet bluffs and get the correct 15% 4-bet frequency for her non-value hands, Alice uses a random number generator from [url=http://random.org]random.org[/url]. She has this on her screen in a small browser window:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/random1.png[/img]

Let[apostrophe]s illustrate randomized 4-bet bluffing in practice with an example:

[h2]Example 3.1.1: Randomized 4-bet bluffing in EP[/h2]
$100NL
6-handed

Alice ($100) raises pot to $3.50 with :6H :6C from UTG, it[apostrophe]s folded to Bob ($100) on the button, who 3-bets pot to $12. The blinds fold, and Alice has to make a decision. 4-bet or fold?

Alice does not have one of her value hands {QQ+, AK}, so she knows that this is a 4-bet-bluff-or-fold scenario. She also knows how often she should 4-bet bluff with these hands (15%). Alice clicks the random number generator, planning to 4-bet to 27bb ($27) if it returns a number between 0 and 15, and otherwise she folds.:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/random2.png[/img]

The randomize returns 39, so Alice folds this time.

We have now specified Alice[apostrophe]s optimal EP strategy for for raising/4-betting/calling a 5-bet when she gets 3-bet by a player in position. Our next step is to turn to Bob. What is Bob[apostrophe]s optimal strategy for 3-betting/5-betting in position after a ~15% EP raise from Alice?

[h2]3.2 Bob[apostrophe]s optimal 3-bet-strategy versus Alice[apostrophe]s optimal raising strategy in EP[/h2]
We[apostrophe]re assuming Bob knows Alice[apostrophe]s opening range (he only needs to know the % of hands, not the specific hands), either from observation, or by using a HUD. Alice[apostrophe]s raise percentage dictates her value 4-bet range, which dictates Bob[apostrophe]s strategies for 3-betting and 5-betting.

Bob starts by finding the hands that he 3-bets and 5-bets all-in, purely for value. His value range also includes some 5-bet bluffs, and the next step is to find these. Then we pick a range of 3-bet bluffs that Bob plans to fold to a 4-bet. We[apostrophe]ll also talk about Bob[apostrophe]s [i]flatting range[/i]. These are medium strong hands that are playable, but they are not strong enough to 4-bet for value,and they are too strong to use as bluffs, so Bob flat-calls with them.

Bob[apostrophe]s flatting range can be viewed as a completely separate part of Bob[apostrophe]s overall strategy, and we don[apostrophe]t have to be concerned with it when constructing optimal ranges for 3-betting/4-betting/5-betting. But we will discuss the flatting range briefly, since it helps us understand the big picture. When Alice has raised, Bob can respond in 3-ways: He can 3-bet (for value or as a bluff), he can flat, or he can fold. Different hands have different jobs to do within these ranges. And depending on Alice[apostrophe]s opening range, hands can move between Bob[apostrophe]s 3-betting/flatting/folding ranges.

For example, we[apostrophe]ll see that AK isn[apostrophe]t strong enough to be a value hand for Bob against Alice[apostrophe]s EP range, so AK goes into the flatting range in this scenario. But when Alice opens a much wider ~25% range in CO, AK is promoted to a value hand that is 3-bet and 5-bet for value. More about that later in the article.

So let[apostrophe]s begin defining Bob[apostrophe]s optimal 3-bet/5-bet strategy in position against Alice[apostrophe]s optimal raise/4-bet/call 5-bet strategy with a ~15% EP range:

[b]Bob[apostrophe]s pure value range[/b]
Bob knows that Alice EP range is ~15% (14% to be exact), and therefore he can draw the same conclusion Alice just did, namely that her optimal value 4-betting range is {QQ+, AK}. To profitably 3-bet and 5-bet all-in for value against this range, Bob needs a hand with at least 50% equity.

AA is obviously such a hand, and we can easily compute some equities to see that KK is the only other possible hand. So Bob ends up with the super tight pure value range {KK+}.

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/KK_vs_EP_value.png[/img]
[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/QQ_vs_EP_value.png[/img]
[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/AK_vs_EP_value.png[/img]

So Bob will 3-bet {KK+} and 5-bet them all-in if Alice 4-bets. He will also 3-bet/5-bet some 5-bet bluff hands (type Axs), and he will have a wide range of 3-bet bluffs that he folds to a 4-bet. We[apostrophe]ll now find Bob[apostrophe]s 5-bet bluffing hands, then his 3-bet bluffing range, and then we are done.

But first, let[apostrophe]s talk about something that I know you[apostrophe]re thinking about right now:

[b]Wait a minute, are you saying that we shouldn[apostrophe]t 3-bet the mighty strong QQ and AK for value against an EP open-raise?[/b]
Correct. Against Alice[apostrophe]s tight and optimally played ~15% EP opening range, QQ and AK aren[apostrophe]t strong enough to use as value hands, [i]even if they have strong equity against Alice[apostrophe]s total opening range[/i]. The reason is that they can not profitably get the whole stack in Against the range Alice is willing to get all-in with, namely {QQ+, AK}. Therefore we don[apostrophe]t want to 3-bet them for value. Keep in mind that the process of getting all-in starts with a 3-bet, and we know the moment Alice open-raises with a ~15% EP range that her all-in range is a favorite over our QQ and AK hands.

Therefore, since we know this before we have put our first chip into the pot, we don[apostrophe]t want to choose a path with QQ/AK that is the first step towards getting all-in with them preflop. This is also true for other hands that are good enough to play for value, but not strong enough to get all-in preflop against Alice[apostrophe]s value range {QQ+, AK}. Examples of such hands are QQ-22, AK-AJ, KQ, QJs-T9s, etc. (and you can probably list some more if you think about it).

So should we 3-bet these medium strong hands as bluffs then?. No, [i]because they are too strong to turn into bluffs and waste their postflop value[/i]. The alternative, which is a good one, is to flat-call with them and play a raised pot with position against a range we have god equity against (namely Alice[apostrophe]s total opening range). Of course, we could always pretend they are trash and use them as 3-bet bluffs, but why should we do that when it[apostrophe]s profitable to flat and play for postflop value? It[apostrophe]s true that we want to 3-bet bluff a lot, but we have plenty of bad hands to choose from for that purpose, and we don[apostrophe]t want to waste the postflop value we gain from flatting with our medium strong hands.

Here is a soccer analogy in these World Cup times:

Moving QQ/AK from the flatting range up to the value range against a ~15% opening range is a bit like moving a defender forward and using him as a striker. Sure, he might score a goal or two, but he isn[apostrophe]t quite good enough for the job. But he is too good to sit on the bench, so he should play. Therefore, since there is another job for which he is well qualified (defending), we let him play there. The right man for the right job.

Bob will therefore flat QQ, AK and various other medium strong hands/implied odds hands after a ~15% open-raise from Alice. The optimal flatting range depends on how Bob thinks Alice plays postflop, what he thinks the players in the blinds will do, how they play postflop, their stack sizes, etc. So we leave the construction of an optimal flatting range to Bob.

Note that 3-betting QQ and AK for value against a ~15% EP raiser [i]is equivalent to assuming the raiser isn[apostrophe]t playing optimally[/i]. If you feel these two hands can always be 3-bet and 5-bet all-in for value against this EP range, you can assume it[apostrophe]s because the players you meet don[apostrophe]t defend well against 3-bets out of position.

Thinking about these things is useful, because when we know what[apostrophe]s game theoretically correct, [i]we know that we can exploit someone if it seems correct to do something else[/i]. So feel free to deviate from optimal play in Bob[apostrophe]s place, if you have position on a weak player. For example, you might be up against a player who 4-bet bluffs spazzy and way too much, or he raises a lot and calls 3-bets out of position with medium strong hands, and then he plays fit-or-fold on the flop. Against such players, QQ and AK might be used as value 3-bet/5-bet hands, since our opponents play far from optimally against our 3-bets.

But don[apostrophe]t 3-bet QQ/AK for value against a ~15% opening range in the hands of a player like Alice. She plays optimally against our 3-bets, so 3-betting QQ/AK won[apostrophe]t do anything for us. Against Alice we use QQ/AK as flatting hands, thus setting ourselves up for playing a raised pot in position against a range we have good equity against (Alice[apostrophe]s total opening range, and not just her value hands). This will give Alice (and the blinds, should they get involved) opportunities to make postflop mistakes that we can exploit.

But later in the article we[apostrophe]ll let Alice open with a ~25% range from CO, and we[apostrophe]ll see that QQ/AK now moves up to Bob[apostrophe]s value range. Alice[apostrophe]s value range is wider and weaker with a 25% opening range, and Bob[apostrophe]s optimal 3-bet strategy changes accordingly.

OK, enough about flatting. Let[apostrophe]s move on and find Bob[apostrophe]s 5-bet bluffs, and then his 3-bet bluffing range:

[b]We include 5-bet bluffs in Bob[apostrophe]s value range[/b]
Remember the definition of "value range" as the hands we 3-bet, planning to 5-bet all-in after a 4-bet. Some of these hands will be 5-bet bluffs, but for simplicity we[apostrophe]ll refer to all the 5-betting hands as Bob[apostrophe]s value range.

From the previous theory section, we remember that Bob wants to have enough Axs 5-bet bluffs in his value range to make Alice[apostrophe]s weakest value hands break even. This accomplishes two things for Bob:

[LIST]
[*]He forces Alice to fold more of her 4-bet bluffs
[*]He makes it impossible for Alice to "cheat" by not paying off Bob[apostrophe]s value 5-bets with {KK+}. If she tries to be "smart" and fold QQ/AK, Bob will just collect his profit with his 5-bet bluffs instead.
[/LIST]

So Bob[apostrophe]s 5-bet bluffs with some Axs hands attack Alice[apostrophe]s 4-bet bluffs, and they also make it impossible for her to profitably tighten up her value range, even if she knows Bob[apostrophe]s value range is the squeaky tight {KK+}. Keep in mind that Alice knows Bob[apostrophe]s strategy, since this follows from her own strategy, which follows from her opening range, which both players know.

So she knows Bob only 3-bets/5-bets {KK+} for pure value, and if Bob[apostrophe]s doesn[apostrophe]t 5-bet bluff a bit, Alice can improve her 5-bet-calling strategy by folding the big underdog[apostrophe]s QQ/AK from her value range {QQ+, AK}. And when one of the players can improve his/her EV by a strategy change, the original strategy pair can[apostrophe]t be optimal (per definition). So Bob has to 5-bet bluff.

The next step for Bob is to add enough Axs to make Alice[apostrophe]s weakest value hands break even when they call a 5-bet. Alice then calls off her last 73bb to win the blinds + Alice[apostrophe]s 4-bet + Bob[apostrophe]s stack =1.5 + 27 + 100 =128.5 bb. The pot odds are 128.5 : 73 =1.76 : 1, so Alice needs minimum 1/(1 + 1.76) =36% equity against Bob[apostrophe]s 5-betting range to call profitably.

We add A5s to Bob[apostrophe]s value range, and check Alice[apostrophe]s equity with QQ/AK:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/bob_valuerange+A5s.png[/img]

AK is above the threshold, but QQ is way below 36%. We add A4s and try again:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/bob_valuerange+A5s+A4s.png[/img]

QQ is now slightly better than break even, and Bob can use A5s/A4s as his optimal 5-bet bluffing hands. However, if we want Alice[apostrophe]s equity to be exactly break even, we have to remove a 5-bet bluff or two. Let[apostrophe]s remove :AC :4C and see what we get:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/bob_valuerange+A5s+A4s-Ac4c.png[/img]

Bingo, and Bob[apostrophe]s optimal 5-bet bluffing hands are {A5s, :AS :4S , :AH :4H , :AD :4D }. But here I[apostrophe]ll say that we don[apostrophe]t have to be this strict. A combo or two too much or too little doesn[apostrophe]t change things much, and we can use A5s/A4s in practice. Also, as we[apostrophe]ll discuss further in the summary at the end of the article, it[apostrophe]s debatable whether we need to 5-bet bluff at all in most games, unless we are playing against people like Durrrr.

People generally don[apostrophe]t 4-bet bluff enough, and they are also reluctant to tighten up their 4-bet value ranges when they get exploited by very tight 5-betting (e.g. 5-bets that are 100% for value). For example. a typical low limit TAG with a ~15% EP range might have decided to never 4-bet bluff, and always 4-bet QQ and AK for value and call a 5-bet with them. And he is unlikely to change that plan, even if Bob[apostrophe]s exploitative response is to drop all 5-bet bluffs from his value range, and only 5-bet-shove {KK+}, purely for value.

These things happen because a) people are blinded by seemingly strong hands, even after they get trapped in situations where their hands suddenly aren[apostrophe]t strong anymore, and b) because people are reluctant to change their initial plan, even after if becomes clear it[apostrophe]s a bad plan.

Against an opponent who makes the dual mistake of not 4-betting bluffing enough, and also paying off our value 5-bets too much, Bob can gain a lot from not having to think about 5-bet bluffing. Bob simply 3-bets {KK+} for value, plus a wide range of 3-bet bluffs, and after a 4-bet he 5-bets {KK+} for value and folds everything else.

This way Bob exploits Villains lack of 4-bet bluffing, since his 3-bet bluffs forces Villain to fold most of his non-value hands (since Villain is unwilling to 4-bet bluff with these hands). And Bob also doesn[apostrophe]t need to attack Villain[apostrophe]s 4-bet bluffs with 5-bet bluffs of his own, since Villain isn[apostrophe]t 4-bet bluffing. Finally, Bob exploits Villain[apostrophe]s static 5-bet-calling range by only 5-betting for value (and getting called as a big favorite), and not having to include 5-bet bluffs for deception. Easy game.

At any rate, Bob[apostrophe]s final value 3-bet range (including his 5-bet bluffs) against Alice[apostrophe]s optimally played ~15% EP range is {KK+, A5s, A4s}. Bob[apostrophe]s last job is to construct the 3-bet bluff range. These are the hands we 3-bet, and always fold to a 4-bet.

[b]Bob[apostrophe]s 3-bet bluff range[/b]
We remember the [i]strength principle[/i] for poker hands:

- Bet/raise your strongest hands for value
- Check/call with your medium hands
- Fold/bluff with your weakest hands

We have already defined Bob[apostrophe]s value range (including 5-bet bluffs) as {KK+, A5s, A4s}, and we have mentioned that he also flats some range of good-but-not-great medium strong hands. Against Alice[apostrophe]s ~15% EP range this means flatting with hands like QQ, JJ, TT, AK, AQ, AJ, KQ, etc.

So when we pick hands for Bob[apostrophe]s 3-bluffing range, we drop down to the "cellar" and pick hands that aren[apostrophe]t god enough to 3-bet for value preflop, and not good enough to flat for postflop value. Against Alice, who either 4-bets or folds, it doesn[apostrophe]t matter which hands we choose to 3-bet bluff with, since these hands will never see a flop. Alice either 4-bets or folds, and when she 4-bets, we 5-bet our value range all-in, and fold our 3-bet bluff range.

But [i]in practice[/i] the choice of 3-bet bluff range matters a bit, since the raiser will sometimes call our 3-bet with his medium strong hands out of position and force us to play postflop. Therefore, since we can choose freely from our worst hands, we might as well choose [i]the best[/i] of our worst hands.

In other words, we[apostrophe]d rather 3-bet a hand like K8s as a bluff than a hand like 72o. K8s has some postflop value those times the raiser calls and forces us to see a flop, while 72o doesn[apostrophe]t. So 3-bet bluffing with hands like K8s [i]dominates[/i] (e.g. is sometimes better than, and never worse than) 3-bet bluffing with hands like 72o.

So let[apostrophe]s list some 3-bet bluff candidates à la K8s that are too weak to flat, but have some postflop value when we get called. We make a list of ace high, king high and queen high candidate hands:

[b]Candidate list for 3-bet bluffing:[/b]
- Ace high: A9s-A6s ATo-A8o (52 combos)
- King high: K9s-K6s, KJo-K9o (52 combos)
- Queen high: Q9s-Q6s, QJo-Q9o (52 combos)

If you don[apostrophe]t approve of this list, feel free to make your own. The specific hands are irrelevant, what matters is that we use hands [i]with the right properties[/i], namely hands that aren[apostrophe]t quite strong enough to flat. NB! A5s-A2s are reserved for 5-bet bluffing, so we can[apostrophe]t include them in this list.

This gives us a list of 156 "pretty" combos for 3-bet bluffing, and the next question is which hands to choose and when. We remember that the optimal value/bluff ratio for Bob[apostrophe]s 3-betting range is 40/60, so he can use 60/40 =1.5 bluff combos for each of the combos in his value range (including his 5-bet bluffs). His total value range is {KK+, A5s, A4s} =20 combos, so Bob can pick 1.5 x 20 =30 3-bet bluff combos.

As mentioned previously, there are two techniques Bob can use:

- Pick 30 specific combos and always 3-bet them
- 3-bet all hands from the candidate list a certain percentage of the time

I prefer the percentage method. To use it, we only need to memorize the candidate range once and for all, plus one number (the % we 3-bet bluff the candidate hands). Let[apostrophe]s compute the number to use against Alice[apostrophe]s EP range:

To effectively have 30 bluff combos from the candidate list in our 3-betting range, we need to use each of them 30/152 =20% of the time. Note that this percentage isn[apostrophe]t universal, like Alice[apostrophe]s fixed 4-bet bluff percentage (15%) is for all her opening ranges. To see this, note that Bob[apostrophe]s value range varies with Alice[apostrophe]s opening range, but the candidate list of 3-bet bluff hands is static (we have simply chosen some hands to use).

So Bob will have to calculate a new bluff% to use for his candidate list against each of Alice[apostrophe]s opening ranges. However, this isn[apostrophe]t a big job, we simply do the math once and for all against each of Alice[apostrophe]s ranges and memorize the numbers we need (and we[apostrophe]ll look at Alice[apostrophe]s CO range in a minute).

So, finally:

[b]Bob[apostrophe]s optimal 3-bet strategy against Alice[apostrophe]s optimal raising strategy in EP[/b]
[pre]{Bob[apostrophe]s total 3-bet range}
={value hands and 5-bet-bluff hands} + {3-bet bluff hands}
={KK+, A5s, A4s}
+ 20% x {A9s-A6s,ATo-A8o,K9s-K6s,KJo-K9o,Q9s-Q6s,QJo-Q9o}[/pre]

Bob always 3-bets {KK+, A5s, A4s} and 5-bets all-in after a 4-bet. If he has one of the 152 combos from his candidate list for 3-bet bluffing, he uses a randomizer and 3-bet bluffs 20% of the time, and he folds to a 4-bet. We had to do a bit of work to construct all these ranges, but it was worth it, and we have learned a lot in the process.

Let[apostrophe]s see what Bob[apostrophe]s optimal total 3-bet% is in this case:

- Value part: 20 combos (1.5% of all hands)
- Bluff part: Effectively 20% of 152 =30 combos (2.3% of all hands)

This results in a total 3-bet% of 1.5 + 2.3 =3.8% against Alice[apostrophe]s ~15% EP raises. His value/bluff ratio is the desired optimal 20/30 =40/60. Later, when we construct an optimal strategy against Alice[apostrophe]s 25% CO range, we[apostrophe]ll see that Bob[apostrophe]s 3-bet% skyrockets as a consequence of Alice raising a much wider opening range.

Note that the combination of a candidate list of 3-bet bluff hands and a fixed (but adjustable) bluff% to use with these hands, gives us a lot of flexibility to adjust our 3-bet bluffing as we please. Against an unknown opponent, we can start with the optimal 20% frequency, and 3-bet {KK+, A5s, A4s} always, and the candidate list 20% of the time. But if we note that the raiser doesn[apostrophe]t defend optimally, we might want to adjust this bluff percentage.

For example, of the raiser never 4-bet bluffs and only 4-bets a tight value range like {QQ+, AK}, we can go bananas with our 3-bet bluffs. We might decide to double the bluff frequency from 20% to 40% for our list of 152 bluff candidate combos. Now we have 20 value combos, and effectively 0.40 x 152 =61 bluff combos. This means 20/(20 + 61) =25% of our 3-bets are for value, and 75% are bluffs. Our first adjustment to exploit this particular opponent is therefore to lower the optimal value/bluff ratio from the optimal 40/ to the more exploitative 25/75.

Then we can also drop 5-bet-bluffing against this tight player, as discussed previously. The simplest adjustment is to keep 3-betting our 5-bet bluffing hands A5s/A4s, but we move them from the value range down to the 3-bet bluff range, and fold them to a 4-bet. The only hands we 5-bet against this player and his {QQ+, AK} 4-bet range is {KK+}, purely for value.

Here is an example of randomized 3-bet-bluffing, using the randomizer from [url=http://random.org]random.org[/url]:

[h2]Example 3.2.1: Randomized 3-bet bluffing against a ~15% EP raise[/h2]
$100NL
6-handed

Alice ($100) raises to $3.5 from UTG, and it[apostrophe]s folded to Bob ($100) who has :QS :9H on the button. This hand is on the candidate list of 3-bet bluff hands, and we remember that the optimal bluff frequency to use against a ~15% opening range is 20%. Bob clicks the randomizer, planning to 3-bet if it returns a number between 0 and 20, and otherwise fold:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/random3.png[/img]

The randomizer returns 18, so Bob 3-bets to $12. Alice quickly 4-bets to $27, and Bob folds.

Everything according to plan, and with total control, so there is no reason to feel frustrated after this clash. Our :QS :9H did it[apostrophe]s job (attacking the weakest part of Alice[apostrophe]s opening range) perfectly, regardless of the outcome, and it[apostrophe]s irrelevant that Alice had a 4-betting hand this time.

Remember that we know Alice[apostrophe]s strategy just as well as she knows our strategy, and we know that she will 3-bet us 30% of the time and fold 70%. When the 4-bet comes, we quietly fold our 3-bet bluffs and 5-bet-shove our value/5-bet bluff range of {KK+, A5s, A4s}. And we do these things calmly, without emotion.

[h1]4. Optimal strategy pairs for raiser/3-bettor with a CO raiser out of position[/h1]
After the thorough work with Alice raising ~15% in EP, we can now reap the rewards and quickly run through the same procedure with Alice raising a ~25% range in CO. She now opens a wider range, as a consequence, all other ranges get wider as well.

[h2]4.1 Alice[apostrophe]s optimal raising strategy for CO[/h2]
Raising from CO is a bit more situational than raising from EP. It[apostrophe]s now easier to isolate the blinds, and with a tight player on the button, it might be correct to play very loosely to get heads-up with position on the blinds. Still, everybody has a core range of hands that they always play, regardless of whether they have written this range down or not.

We[apostrophe]ll assume Alice is using a TAG core range of ~25% in CO. More specifically, this range:

[b]Alice[apostrophe]s CO range[/b]
22+
A2s+ A9o+
K9s+ KTo+
Q9s+ QTo+
J8s+ JTo
T8s+
97s+
87s
76s
65s

326 combos
25%

[b]Alice[apostrophe]s value range[/b]
Alice defends against 3-bets 30% of the time, and she does it by 4-betting 18% of her opening range for value and 12% as a bluff. So she needs 0.18 x 326 =59 value combos that she can 4-bet and call a 5-bet with. In EP she used [QQ+, AK} =34 combos, and in CO we simply add the next tier of hands and use {JJ+, AQ+} =56 combos (precise enough).

Then she needs 0.12 x 326 =39 bluff combos. She can pick ~39 specific combos and always 4-bet them (e.g. AJ, AT, TT =38 combos), or she can 4-bet all her non-value hands 15% of the time, as explained previously. We choose the latter approach, and write Alice[apostrophe]s complete raise strategy for CO as:

[b]Alice[apostrophe]s optimal raise/4-bet/call 5-bet-strategy in CO:[/b]
[pre]{Alice[apostrophe]s total CO range}
={22+,A2s+,K9s+,Q9s+,J8s+,T8s+,97s+,87s,76s,65s,
A9o+,KTo+,QTo+,JTo}
={value hands} + {4-bet bluff hands}
={JJ+, AQ+}
+ (15% 4-bet and 85% fold) x {the rest of the range}[/pre]

Alice raises {22+,A2s+,K9s+,Q9s+,J8s+,T8s+,97s+,87s,76s,65s,
A9o+,KTo+,QTo+,JTo} =326 combos from CO. If she gets 3-bet, she 4-bets {JJ+, AQ+} for value, planning to call a 5-bet. Those times she doesn[apostrophe]t have a value hand (e.g. 88, A9o, T9s, etc.), she 4-bets 15% of the time, and the rest of the time she folds. Using a random number generator from[url=http://random.org]random.org[/url] to randomize 4-bet bluffs has been illustrated in a previous example.

That[apostrophe]s it for Alice[apostrophe]s CO strategy. Over to Bob:

[h2]4.2 Bob[apostrophe]s optimal 3-bet-strategy against Alice[apostrophe]s optimal raising strategy in CO[/h2]
Bob needs a value range, including an optimal number of 5-bet bluffs, and he needs a range of hands to 3-bet bluff.

[b]Bob[apostrophe]s pure value range[/b]
Bob knows that Alice now uses {JJ+, AQ+} as her value range, so he builds a range of pure value hands that have at least 50% equity against {JJ+, AQ+}. AA and KK obviously belong in this range. To see what else is included, we run equity calculations for the next tier of hands (QQ and AK):

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/QQ_and_AK_vs_CO_value.png[/img]

QQ and AK are barely above the threshold, while all weaker hands will be big underdogs. Thus, Bob[apostrophe]s pure value range is {QQ+, AK}, and he happily 3-bets these hands, and then 5-bets them all-in, purely for value.

[b]We add 5-bet bluffs to Bob[apostrophe]s value range[/b]
We now want to add enough Axs hands so that Alice weakest value hands (JJ and AQ) are break even when they call our 5-bet range (and the threshold is 36% equity, as shown previously). We start with A5s/A4s and see what we get:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/bob_valuerange_vs_CO+A5s+A4s.png[/img]

Alice[apostrophe]s weakest value hand is AQ, and it[apostrophe]s a small loser with 34% equity against Bob[apostrophe]s total value range {QQ+, AK, A5s, A4s}. Close enough for us, so the 5-bet bluffs in this case are the same as we used against Alice[apostrophe]s EP range. However, if we want it to be exact, we need to add a couple more bluffs (for example, :AS :3S and :AH :3H ) to lift AQ up to 36%:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-1/bob_valuerange_vs_CO+A5s+A4s+As3s+Ah3h.png[/img]

[b]Bob[apostrophe]s 3-bet bluffing[/b]
Bob[apostrophe]s value range, including 5-bet bluffs, is {QQ+, AK, A5s, A4s} =42 combos. He wants an optimal 40/60 value/bluff-ratio, so he needs 60/40 =1.5 times as many bluff combos. This amounts to 1.5 x 42 =63 bluff combos.

We use the previously defined candidate list for 3-bet bluff hands:

[b]Candidate list for 3-bet bluffing:[/b]
- Ace high: A9s-A6s ATo-A8o (52 combos)
- King high: K9s-K6s, KJo-K9o (52 combos)
- Queen high: Q9s-Q6s, QJo-Q9o (52 combos)

We bluff these hands some fixed percentage x, and for this to effectively correspond to 63 bluff combos, we need x =63/152 =41%. We can round this to x =40% to make it easy to remember.

We remember that we used a 20% bluff candidate frequency against Alice[apostrophe]s ~15% EP range. So when Alice moves from a ~15% EP range to a ~25% CO range, our 3-bet bluff candidate frequency doubles. We only need to memorize the candidate list, and these two numbers (20% vs EP and 40% vs CO), and then we know all we need to know about 3-bet bluffing optimally against Alice[apostrophe]s EP/CO ranges.

At any rate, against Alice[apostrophe]s optimal CO raising strategy, Bob gets:

[b]Bob[apostrophe]s optimal 3-bet strategy against Alice[apostrophe]s optimal raising strategy in CO[/b]
[pre]{Bob[apostrophe]s total 3-bet range}
={value hands and 5-bet bluff hands} + {3-bet bluff hands}
={QQ+, AK, A5s, A4s}
+ 40% x {A9s-A6s,ATo-A8o,K9s-K6s,KJo-K9o,Q9s-Q6s,QJo-Q9o}[/pre]

Using a randomizer from [url=http://random.org]random.org[/url] to randomize 3-bet bluffing has been illustrated in a previous example.

Bob[apostrophe]s total 3-bet% for this scenario is:

- Value part: 42 combos (3.2% of all hands)
- Bluff part: Effectively 40% of 152 =61 combos (4.6% of all hands)

This results in a total 3-bet range of 3.2 + 4.6 =7.8% against Alice[apostrophe]s ~25% CO range. The value/bluff ratio, using our numerical rounding, is 42/61, which is very close to the optimal 40/60.

When Alice moves from EP to CO and her opening range changes from ~15% to ~25%, Bob responds by loosening up his 3-betting range dramatically. This is an interesting observation. Those of you who use a more or less static 3-betting range (for example, the generic {JJ+, AQ} without any 3-bet bluffing that is recommended on many low limit forums) now have game theoretical "proof" that we can get away with [i]a lot[/i] of 3-bet bluffing on the button against a CO raiser.

Even against a TAG CO raiser with a solid ~25% opening range, you can 3-bet almost 8% on the button [i]and there isn[apostrophe]t anything he can do to exploit your loose 3-bets[/i]. And if he defends poorly, for example by not 4-bet-bluffing enough, or not being willing to use JJ/AQ as value hands, you can deviate from optimal play and attack him even harder. The first adjustment against a weak/passive CO raiser who folds a lot to 3-bets is to increase your fixed 3-bet bluff percentage for the candidate list. You might increase the bluff candidate 3-betting frequency from 40% to 60%. If Villain (and the blinds) doesn[apostrophe]t adjust to your exploitative, loose 3-betting, you[apostrophe]ll be printing money.

[h1]5. Summary[/h1]
We have gone through the theory for game theory optimal(ish) raising/3-betting/4-betting/5-betting with the raiser out of position, and then we demonstrated how the theory can be implemented and used at the table.

We constructed optimal strategy pairs (one strategy for the raiser, and one for the 3-bettor) for two scenarios. First with the raiser in EP (UTG or MP) with a ~15% range, and then with the raiser in CO with a ~25% range. In both scenarios we gave the raiser a standard TAG opening range. We then deduced optimal strategies for both players as a function of the raisers opening range. We observed that the strategies for the CO scenario involved considerably looser ranges than the strategies for the EP scenario.

Our optimal strategy pairs confirmed that it[apostrophe]s correct to 3-bet a wide range on the button against a CO raiser, [i]even if he starts with a solid opening range, and defends optimally against a 3-bet[/i]. And if he doesn[apostrophe]t defend optimally, we can loosen up even more. When you see a good and aggressive NL player dominate the table by 3-betting loosely in position, this is what happens. Loose, positional 3-betting is game theoretically correct, even against strong players. And against weak players, it[apostrophe]s even more correct.

As a result of our work, we ended up with specific and concrete implementations of the theory, both as the raiser and as the 3-bettor. You can implement these strategies immediately in your own game by following the procedures outlined in this article. The strategy pairs depend on the raiser[apostrophe]s opening range, but the ~15% and ~25% EP and CO ranges are relatively standard, and you will meet many opponents who play close to these ranges. If you need to apply the theory to other ranges, just plug them into the method, and construct the strategy pairs yourself.

We didn[apostrophe]t look at small blind vs big blind in this article, even if it falls under the same category with the raiser out of position. I elected to leave this situation out, since blind vs blind dynamics is very dependent on the players involved, and the history between them. So it[apostrophe]s difficult (and probably not very useful) to try and generalize and assign SB a standard opening range. But if you want to do this, you can use the method and construct the optimal strategy pair yourself.

Those of you who enjoy experimenting with ranges and numbers can now start to apply the optimal strategies in your own game, using your own ranges. Plug your own opening ranges for EP and CO into the theoretical "machinery" outlined in this article, and produce optimal strategy pairs, based on the ranges you use at the table. Remember that everything follows from the opening ranges, and remember that you will get both an optimal strategy for the raiser (you), and the positional 3-bettors optimal strategy against you.

Learn both parts of every optimal strategy pair. When you are the raiser OOP against an unknown 3-bettor, you can simply play optimally and assume that he is playing optimally too. You now have 100% knowledge about the raiser[apostrophe]s range (since this is your range), and you know the optimal strategy pair for this situation exactly. Since the 3-bettor doesn[apostrophe]t know these things precisely, he will make mistakes, and you won[apostrophe]t.

When you have position on the raiser, things are slightly less straightforward, since he is the one who chooses the opening range. But against an unknown raiser, you can start by assuming he uses opening ranges that are close to your default ranges. Then you simply respond with the corresponding optimal 3-betting strategy. If he uses ranges that are only slightly different from yours, the optimal strategy pairs will be similar.

And if you should need optimal strategy pairs for opening ranges that are very different from your own (for example, if you meet a CO raiser who opens 45% of his hands), you can quickly construct the corresponding optimal strategy pair for him and yourself. Remember that you don[apostrophe]t need to know his opening range in detail, you only need to know the [i]number of hands[/i] that he opens. This number is relatively easy to estimate from a HUD, even if the sample isn[apostrophe]t big.

To be prepared for any opening range you might encounter as a 3-bettor, you can sit down and do the work for 10%, 35% and 45% opening ranges on your own. Then you[apostrophe]ll have have a set of optimal strategy pairs that cover almost all cases of EP and CO open-raising you are likely to encounter in practice.

Again, when you are the raiser, everything follows from [i]your[/i] ranges, and you can do this work once and for all (assuming you have a well-defined set of default core opening ranges) and memorize it. Then you can play optimally from out of position, and sniff around for opponent leaks. If you don[apostrophe]t find any, keep playing optimally. If you find some exploitable leaks, think about how you can adjust to increase your EV. But you don[apostrophe]t have to adjust until you are sure. Remember, if you are playing optimally and your opponent isn[apostrophe]t, [i]you gain from his mistakes[/i] (although you might gain more by switching to an exploitative strategy).

A classic opponent mistake at the low limits is not 3-bet bluffing enough (or at all) in position. Love these guys, because it[apostrophe]s easy to exploit them. For starters, they are "exploiting themselves" by allowing you to run over them by not 3-betting you nearly as often as they should. And when they do 3-bet, you know that they are strong. So you simply drop all your 4-bet bluffs from your range and continue with a 4-betting range of only value hands, planning to call a 5-bet. Easy decisions and easy game.

When someone has raised in front of you, you ideally want to use an optimal strategy for each opponent, and for each of his positions (since optimal 3-bet strategy is a function of the raiser[apostrophe]s range). This might sound like a lot of work, but in practice it all follows from estimates about the ranges you meet. And small deviations don[apostrophe]t change things dramatically. For example, when you know the strategy pair corresponding to a 15% opening range, you can apply the same strategies against a 12% raiser and an 18% raiser without losing much accuracy. You won[apostrophe]t play optimally in these cases, but near-optimally is close enough. Besides, pin-pointing opponent opening ranges to within +/-1% or less is difficult, so using near-optimal strategies is the best we can hope for in practice.

The nest step of the process is the most interesting one. When you have trained optimal play, you will discover that it[apostrophe]s now much easier to spot opponent mistakes. For example, when you come across an opponent who doesn[apostrophe]t 4-bet bluff (and these are common at the low limits), you immediately know that this is a leak, and you know how to exploit it. Tight and straightforward players who refuse to 4-bet bluff can be exploited by 3-betting a lot, and not 5-bet bluffing at all. You can 3-bet a metric fuckton of bluff hands, and when they finally pick up a hand good enough to 4-bet, you fold all your bluffs and ship a tight value range (sometimes as tight as {KK+}). Just keep an eye on the other players to see if they are trying to exploit your loose 3-betting (tighten up a bit if they do), and you[apostrophe]ll do very well in this spot.

Another leak you[apostrophe]ll see is spazzy 4-betting from players with insufficient understanding of the theory behind optimal 3-bet/4-bet/5-bet wars. This might happen when you have driven someone crazy with your loose 3-betting, and he starts to tilt. Or when someone tries to fight back in a controlled manner, but he doesn[apostrophe]t quite know how to do it (so he starts 4-bet bluffing way too much).

The first thing you have to realize when you are playing optimally, and then spotting a leak, is this: [i]It[apostrophe]s not necessary to deviate from optimal play to benefit from his mistakes[/i]. If you keep playing optimally, and your opponent doesn[apostrophe]t, you will win from him in the long run, period. The question is now whether you should deviate from optimal play yourself, in order to [i]win more[/i]. If you have a clear idea about how to exploit your opponent maximally, by all means go ahead and make the adjustment.

But be cautious when you adjust to spazzy and unpredictable opponents. Remember that your optimal 3bet/4-bet/5-bet strategies are designed to [i]protect you[/i], and there is nothing a maniac can do to exploit you in these scenarios, even if he raises and reraises at every opportunity. If you see concrete adjustments you can make to win more, go for it, but be careful if you tilt easily (preflop raising wars have a tendency to trigger tilt). Then you might be better off sticking to optimal play against hyper-aggressive opponents, let the ranges do the work for you. You can use your focus to terrorize the passive and easily exploitable players instead.

Finally, if you meet tough regs who don[apostrophe]t give up preflop edge in these scenarios (at least any edge you can see), these optimal strategies will protect you from getting exploited. They can[apostrophe]t take advantage of you in preflop 3-bet/4-bet/5-bet wars, so don[apostrophe]t worry about it if they try. Follow the optimal strategies, and the mathematics of the situation will protect you. But don[apostrophe]t forget to sniff for leaks against regs. Everybody has leaks, and your knowledge about optimal 3-/4-/5-betting will make it easier for you to find them. And pay close attention if you see a reg starting to tilt! Now he might blow up completely in preflop raising wars, and you can adjust accordingly.

I hope this article will be useful for those of you who find it difficult to play well in preflop 3-/4-/5-bet wars, and that you have learned to implement the optimal strategies in your own game. And for those who already knew these things, I hope that this systematic discussion of the topic has given you things to think about.

I chose to name this article "Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max - Part 1", even if a Part 2 hasn[apostrophe]t been planned yet. But I do have some more ideas about the topic, and I might write more. For example, we could do one article about optimal strategy pairs with the raiser in position (e.g. after a 3-bet from the blinds). Then we could dedicate one article to discussion about optimal versus exploitative play, and talk about how to apply one or the other against different opponent types.

Good luck!
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