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

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# 1. Introduction

This is Part 6 in the series

*Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max*, and the next to last theoretical part of the series (there will possibly be a practical part later this year, and we'll talk about that in Part 7). In Part 1 to Part 5 we built a foundation for default NLHE preflop play based on mathematical principles from game theory, plus some common poker sense. In this and the next article we'll test these strategies numerically.

The article series started with a simple scenario in Part 1 where we studied 3/4/5-betting heads-up with the raiser out of position. Then we generalized the strategies we found to other heads-up 3/4/5-bet scenarios, and also to a few select multiway scenarios. Along the way we also defined default ranges for open-raising from all positions.

Below is a summary of the content in Part 1 to Part 5:

**Part 1:**Introduction to the mathematics behind game theory optimal 3/4/5-betting heads-up, studying the scenario where the raiser is out of position**Part 2:**We discussed in greater detail how to implement the theory from Part 1, and we defined default openranges for all positions. Then we defined the heads-up 3/4/5-bet theory for a wide range of openranges with the raiser out of position.**Part 3:**We let the raiser and the 3-bettor switch positions, and we studied the scenario where the raiser opens on the button and gets 3-bet by a player in the blinds.**Part 4:**We generalized the theory from Part 4 and looked at 3/4/5-betting heads-up with the raiser opening from any position outside of the blinds, and the 3-bettor 3betting from out of position in the blinds**Part 5:**We discussed 3/4/5-betting for two multiway scenarios (squeezing in a 3-way pot and cold 4-betting in a 3-way pot).

Throughout Part 1 to Part 5 we have gone through most of the possible preflop scenarios and discussed good default strategies for them. In all 3/4/5-bet scenarios we have used the theory from Part 1 as our starting point, and then adjusted it for similar scenarios. We have used a mix of mathematical reasoning and good poker sense.

The plan for Part 6 is to test the strategies for heads-up 3/4/5-betting using the poker analysis software "Pokerazor". The final test for a strategy is of course to try it out at the tables and see how it performs. But we can also study our strategies numerically using analysis software. Today there are two programs available that let us study complete pre- and postflop strategies for any number of players:

- Pokerazor

- StoxEV

Pokerazor is for the time being no longer commercially available, but a new version is expected some time in the future. StoxEV is available and being actively developed. I have elected to use Pokerazor for this article, since this is the program I am most familiar with. But StoxEV will work just as well if you are interested in doing this type of analysis work on your own.

What we'll do first in this article is to study the typical ABC poker new players are advised to use when they get started with NLHE at the lowest limits ("play tight", "bluff little", "fold a lot when you get 3-bet", etc.) Then we'll show how this ABC poker makes us vulnerable for attacks from aggressive opponents (particularly when they have position on us). We will here only look at preflop play, but the same principles apply postflop as well.

Then we'll go one step further and show how we can improve on ABC preflop strategy by adding strategy components that fully or partly neutralize the attacks aggressive players subject us to (for example, we add 4-bet bluffing to our preflop strategy to defend against 3-bet bluffing). Then we go back to our opponents' strategies and discuss how they can adjust to our adjustments, and so on.

In this manner we'll show how the optimal 3/4/5-bet strategies we have designed can be viewed as the final product of an evolutionary process based on our desire to defend against profitable bluffing with any two cards from aggressive opponents. The main point is that we don't want to put ourselves in a situation where our opponent(s) can exploit us by bluffing profitably with any two cards, be it open-raising, bluff 3-betting, bluff 4-betting, or bluff 5-betting. An optimal strategy "plugs" all such openings for our opponents, but of course this defense does not come entirely without cost.

Through this discussion we'll also shed light on the difference between optimal play and exploitative play, and when we should use one or the other. Optimal strategies put a lot of weight on defense, and they are not necessarily the most profitable strategies against players with big leaks. One reason is that optimal strategies include defensive components (for example, 4-bet bluffing as a defense against light 3-betting) that are often unnecessary against weak players (for example, we don't need to 4-bet bluff against an opponent who only 3-bets premium hands like {JJ+,AK}).

Against players with big and easily exploitable leaks, we'd rather deviate from optimal play and play exploitatively to take full advantage of these leaks. But we need to be aware that by doing so we are creating openings in our strategies that can be exploited by observant opponents. So we have to find a balance between optimal and exploitative play, and we should use different strategies against different opponents. We will do our best to exploit weak players' big mistakes, but we can always fall back on optimal play against good opponents without big leaks. We can also return to optimal play if the player we're trying to exploit with exploitative play suddenly changes his strategies to take advantage of the openings created by our exploitative strategies.

For example, let's say we choose to never 4-bet bluff against a passive player who never 3-bet bluffs. He might now notice this, and adjust to our tight play by starting to 3-bet bluff us. Our exploitative adjustment against this particular opponent then runs the risk of getting counter-exploited if he starts 3-bet-bluffing us often with random weak hands. If this happens, we should return to our optimal optimal 3/4/5-bet strategy. Alternatively, we can make another exploitative adjustment to his adjustment by 4-bet bluffing him a lot (since we know he often is weak and have to fold). But the optimal strategy is always an alternative if we aren't sure whether or not we can exploit his aggressive 3-betting.

In my opinion, this mindset is at the core of the thought processes of a strong NLHE player. He doesn't have to use mathematics like we have done, but he will have a good feel for what an optimal (or near optimal) strategy is in the situation he is in. So he has a strong default strategy to fall back on against unknown players or known strong players, so that he can't be easily exploited. But at the same time he knows how to deviate from optimal strategies to exploit his opponents' systematic leaks. So he can adjust his play in a controlled manner against each individual opponent instead of being locked into a static strategy that he uses against everyone.

Rules of thumb such as "never 4-bet bluff against fish" or "don't 3-bet hands that perform poorly when called" are then replaced by a dynamical mindset that gives is strong control over our choice of strategies. Using optimal play as a starting point (and as a strategy we can always fall back on regardless of who we're playing against), we can move around freely in "strategy land" and exploit opponent leaks as we pick up information about how they play.

Optimal play is never bad play, but exploitative play is always better. But we need information about our opponents' strategies before we can exploit them. If we don't have this information, we can always fall back on optimal play as a good default.

# 2. Testing preflop strategies using the analysis software "Pokerazor"

In this part of the article we'll use Pokerazor to study 2 things:

**1.**How a tight openraise strategy without defense against light 3-betting is vulnerable to 3-bet bluffing with any two cards**2.**What the raiser can do to plug this leak, and how this leads to an optimal strategy pair for the raiser and the 3-bettor

## 2.1 ABC preflop strategies and how these can be exploited

Those of you who have played for a while probably remember the good old days (up to around 2007 or thereabouts) when micro and low limit NLHE was easily beatable by sticking close to the following rules of thumb for preflop play:

Those

- Open tight from all positions (say, 10-12% from UTG/MP, ~20% from MP and ~30% from the button
- 3-bet only for value with {QQ+,AK}, and possibly {JJ+,AQ+} against a loose raiser
- When you get 3-bet and you are out of position, fold everything but {QQ+,AK} regardless of your position and where the 3-bet comes from
- Defend the blinds very tightly (typically 10%)

Believe it or not, but this was more or less the standard getting-started preflop strategy recommended to beginning players at the micro and low limits up to $100NL or so. And it worked well, since the games were so loose and passive that it was correct both to openraise tight, and to fold a lot against 3-bets.

Those of you who have been members of Cardrunners for a while might remember Brystmar's beginner video series "Small Stakes NL" in 6 parts (published during the spring of 2007). This series began with tight-aggressive preflop recommendations based on tight opening ranges and 3-betting only for value:

**Brystmar's preflop strategy for micro/low limit NLHE**

Let's take a trip down memory land and study Brystmar's preflop recommendations given 3.5 years ago. Those who want to read discussion about his video series or download his preflop scheme can look at this Cardrunners forum thread.

Below is a summary of the default openraising ranges (note that "KTs" and "KTo" denote suited and offsuit hands, while "KT+" means both suited and offsuit:

**UTG openraise:**

{22+,AJ+,KQs} =9.8%**MP openraise:**

{22+,AJ+,KQ} =11%**CO openraise:**

{22+,A7s+,A9o+,KT+,QTs+,QJo,J9s+,JTo,T9s} =19%**Button openraise:**

{22+,A4s+,A7o+,KT+,QTs+,QJo,J9s+,JTo,T8s+,

T9o,98s,98o,87s,87o,76s} =26%**Raising from the small blind:**

Openraise the button range if it gets folded to you. In a limped pot, raise {JJ+,AK} for value and overlimp all other hands from your button range, plus all Axs and Kxs.**Raising from the big blind:**

If the small blind openlimps, raise the button range and otherwise check. Out of position in a limped pot, raise {JJ+,AK} for value and otherwise check

Tight opening ranges all around. This is of course not a leak our opponents can exploit, but we might perhaps say that we are exploiting ourselves by folding some profitable hands, particularly on the button.

But the strategies become easy to exploit when we get to playing against a raise:

**In MP with position on a raiser:**

Reraise {JJ+,AK} for value and call with {TT-22,AJs+,AQo}**In CO with position on a raiser:**

Reraise {JJ+,AK} for value and call with {TT-22,AJ+,KQ,QJs,JTs}**On the button with position on a raiser:**

Reraise {JJ+,AK} for value (and AQo if the raise came from CO) and call with {TT-22,AJ+,KQ,QJs,JTs}. With callers between you and the raiser, also call with {JTo,T9s,98s,87s}**In SB after a raise:**

Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}**In BB after a raise:**

Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}. With callers between you and the raiser, also call with {QJs,JTs}

We note two systematic errors in these strategies:

- We're 3-betting more or less the same range regardless of the raiser's position

- We're never 3-bet bluffing

We remember from Part 1 and Part 2 that an optimal 3-betting range on the button varied from 3.6% against a ~15% EP openraise to 8.7% against a ~25% CO openraise. And in all scenarios we used an optimal bluffing frequency of 60%. In other words, more than half of our 3-bets were bluffs- In Brystmar's strategies the 3-betting range is a tight and static value range {JJ+,AK} =3.0%, which is sometimes widened to {JJ+,AQ+} =4.2% against a wide openraising range. We also note that Brystmar chooses to include AQ when loosening up. This is a hand we never 3-bet for value in position when we're playing optimally (since AQ works better as a flatting hand in position).

Brystmar's strategies don't mention defense against 3-betting, but we can assume that default defense is to 4-bet a tight range {QQ+,AK} from all positions. Another significant leak is the squeaky tight blind defense. For example, of button openraises, the preflop scheme tells us to 3-bet {JJ+,AQ} =4.2% from the small blind and flat {TT-22, AJ,KQ} =7.4%. This results in a total defense of 11.6%, which is way lower than the optimal defense threshold of 16% that we estimated in Part 3.

So there are huge openings in Brystmar's preflop recommendations, and these openings can be easily exploited by an aggressive and observant opponent. We're also leaving money at the table because we're openraising to tight, and the main reason for this is that Brystmar does not take full advantage of position. We can openraise a ton of hands on the button when it's folded to us, and we can make life hell for a raiser by 3-bet bluffing him in position, but Brystmar chooses not to do so.

NB! Before we move on I want to point out that I am not trying to put Brystmar's low limit preflop defaults from 2007 in a negative light. His preflop recommendations for beginning NLHE players were very useful back in the day, and gave many new players an easy start. His strategies are best viewed as "training wheels" for staying out of trouble (and he no doubt saw them as such himself) and they were tailored towards the micro/low limit conditions that existed at the time. They will probably still work okay at the lowest micro limits, but I would not recommend anyone to play $25NL and higher with such tight and easily exploitable preflop strategies.

It's clear for everyone who plays $25NL and higher these days that common NLHE strategy has developed in leaps and bounds since Brystmar's 2007 recommendations. Light 3-betting was rare in the "old days", even at $100NL and $200NL. Today it's common, even if you begin as low as $5NL.

The next step of the development of the average low limit NLGE regular back in the day was to add some light 3-bets in position (and Green Plastic's 2006/2007 NLHE videos at Cardrunners inspired many to do so), call more raises and 3-bets in position, and in general get better at using positional advantage. A common mistake many aggressive players did was to 3-bet bluff with hands that were too strong to use as bluffs (for example, JTs). However, this did not cause man problems since most players defended poorly against 3-bets, particularly from out of position.

So the standard recipe in the good old days for an advanced low limit player who wanted to ramp up the aggression was to LAG it up in position. But not necessarily with balance in mind, and not necessarily with a good understanding of how to chose his value range, bluffing range and flatting range in a consistent manner. But this was not a big deal. He played tight out of position, opened a very wide range in position, and 3-bet something fierce in position against weak opponents. The 3-betting was very effective, since the raisers often did one of the following two mistakes:

- Folded a lot out of position and never 3-bet bluffed

- Called a lot with non-premium hands out of position

The first mistake lets the 3-bettor print money by giving him an opening to (in principle) 3-bet bluff any two cards. The second mistake occurs when the raiser tries to correct the first mistake, but he goes about it the wrong way. Defending against 3-bets by flatting weak hands out of position is ineffective, since the raiser now has to play postflop out of position in a scenario where it's difficult for him to win without hitting the flop well. Playing weak starting hands well out of position against a good LAG player is hard, and often results in you losing more money postflop than if you had just folded to the 3-bet preflop.

The cure against light, positional 3-betting is of course to respond by 4-betting a correct value range (which follows from the size of our opening range), balanced with a correct amount of 4-bet bluffing. We have studied this in previous articles, and we have defined optimal strategies for the raiser from all positions. The 3-bettor uses similar thinking to design his 3-bet strategy so that the raiser can not 4bet bluff any two cards profitably. This way an equilibrium gets established.

This equilibrium is given by the optimal strategy pairs for the raiser and the 3-bettor defined in Part 1 and Part 2. We used mathematics to define these strategy pairs, but we can also think about them as a product of an evolutionary process.

The 3-better starts out by exploitative ant-two-cards 3-bet bluffing against a raiser that defends way too tight and folds too much. Then the raiser adjust by choosing a correct value range and introducing 4-bet bluffing. The 3-bettor responds by adjusting his value range and introducing 5-bet bluffing. To prevent the opponent from bluffing with any two cards anywhere, both players fine-tune their ranges until both are using an optimal set of ranges for 3/4/5-betting. "Optimal" here means that neither player can improve his EV by adjusting further. If one of them tries to do so, he is giving the other player an opportunity to increase

*his*EV by making and exploitative adjustment.

We will now illustrate such an evolutionary process using Pokerazor simulations:

## 2.2 Numerical testing of optimal heads-up 3/4/5-bet strategies

We start with the following model:

- Alice (100bb) openraises to 3.5bb with her standard 25% range from CO
- Bob (100bb) is on the button and 3-bets to 12 bb or folds
- Alice defends against 3-betting by 4-betting to 25 bb or folding
- Bob defends against 4-betting by 5-betting all-in or folding
- Alice defends against 5-betting by calling all-in or folding
- The blinds always fold, no matter what Bob does

So we are studying a scenario where Alice openraises, Bob 3-bets or folds, and the blinds never get involved. Alice then makes 1.5 bb (the blinds) per raise when Bob folds, which is a win rate of 150 bb/100. This is her baseline EV for the simulation.

Before we begin the simulations, let's repeat the ranges and optimal strategy pairs we defined in Part 2:

**Alice's 25% open-range from CO**

22+

A2s+ A9o+

K9s+ KTo+

Q9s+ QTo+

J8s+ JTo

T8s+

97s+

87s

76s

65s

326 combos

25%

The corresponding optimal strategy pair that's being used when Bob 3-bets in position can be found from the summary of optimal strategy pairs in Part 2:

Here is a download link for this document (right-click and choose "Save as"):

IP_3-bet_summary.doc

The optimal strategy pair is then:

**Bob:**

3-bets {QQ+,AK,12 air} ={QQ+,AK,A5s-A3s} for value (including 5-bet-bluffing with Axs hands) and 70% av "IP 3-bet air list" as a bluff using a randomizer**Alice:**

4-bets {TT+,AQ+} for value and {AJ,AT,A9s-A7s} as a bluff

**Pokerazor simulation 1 (Bob folds)**

The baseline simulation is to let Bob fold 100%. Alice then picks up the blinds, and makes 1.5 bb each time (=150 bb/10):

- Alice openraises 25% from CO

- Bob folds

EV (baseline for Alice) =150 bb/100

Bob now begins 3-betting so that Alice ends up with EV < 150 bb/100. It's obvious that Alice's EV now becomes lower than the 150 bb/100 baseline, since she can not prevent Bob from making money by 3-betting only his best hands, for example his {QQ+,AK} default value range against her 25% CO openrange. Furthermore, it's correct for Alice to never 4-bet bluff when Bob never 3-bet bluffs, so she has to fold the hands not strong enough to 4-bet for value against Bob's tight value range and let Bob pick up the blinds.

We start by assuming that Bob never 3-bet bluffs and that Alice defends against the 3-bet by 4-betting the value range {KK+} after observing that Bob's 3-betting range is {QQ+,AK}. This is correct since Alice does not want to get all-in preflop with QQ or AK against Bob's {QQ+,AK} range (both are ~40% underdog) with only 3.5 bb invested.

**Pokerazor simulation 2 (Bob 3-bets only for value)**

Bob 3-bets his value hands. We let him use the pure value hands {QQ+,AK} that he would have used in an optimal strategy against Alice's 25% CO openrange, and the drops his 5-bet bluffs for now:

- Alice openraises 25% from CO

- Bob 3-bets {QQ+,AK} for value and 5-bets all-in against a 4-bet

- Alice 4-better {KK+} for value

And the rest follows automatically. We get:

EV (Alice; KK+) =141 bb/100

If Alice also had 4-bet QQ/AK for value, the EV would have become:

EV (Alice; QQ+,AK) =139 bb/100

So Alice's choice of value range against Bob's extremely tight 3-betting range is correct. Note that

*Alice is now exploiting Bob by only 4-betting an extremely tight {KK+} value range!*What happens here is that Bob mostly leaves Alice alone so that she can pick up the blinds almost every time. Bob only pops up with a value 3-bet the 2.56% of the time he has {QQ+,AK}, and Alice then responds by folding everything but {KK+}.

So Alice exploits Bob's squeaky tight 3-betting by not paying off his value hands with weaker hands. And since Bob never 3-bet bluffs, there is never any doubt about his range. Alice can then play perfectly against a 3-bet and drop all her (now unnecessary) 4-bet bluffs as well as her weakest value hands (and we remember that Alice's optimal value range in CO is {TT+,AQ+}). Exploiting someone by folding a lot to his aggression is not the first thing that comes to mind when poker players think about exploitative play. But avoiding paying off a strong range unnecessarily is just as profitable as playing aggressively against players that fold too much.

If Alice had responded with her optimal 4-bet strategy which is {TT+,AQ} for value and {AJ-AT,A9s-A7s} as bluffs, we get:

EV (Alice; optimal strategy) =126 bb/100

And we see that

*Alice's optimal 3/4/5-bet strategy out of position loses relative to the best exploitative strategy*(which gave her 141 bb/100). This is an example of a general rule: If we have an opportunity to exploit someone, we will make more money from this than from continuing to use an optimal strategy. The reason is that Alice now sacrifices EV by defending against a non-existing threat. She has 4-bet bluffs in her 4-betting range to defend against Bob's 3-bet bluffs, but Bob never 3-bet bluffs.

So even if Alice's optimal strategy can not be exploited by any-two-cards 3-bet bluffing from Bob,

*this defense is costing her EV relative to a strategy that exploits Bob's very tight never-bluff 3-betting strategy*. A good analogy would be a nations defense budget in peace time. There has been no warfare on Norwegian soil since 1945, but Norway has still had a defense budget in all the years since then. We are simply paying for a military defense so that other nations don't get an opportunity to invade us without risk.

Now, let's assume that Bob has been sitting there and 3-betting only {QQ+,AK} for value and folding everything else, while Alice has responded by 4-betting only {KK+} for value and folding everything else. Bob has observed that Alice almost always folds. It's now easy for him to reach the conclusion that he could make a lot more money by throwing in some 3-bet bluffs, planning to fold then when Alice 4-bets. Since Bob does not have to worry about the blind players waking up with a hand, let's allow him to 3-bet any two cards to maximally exploit Alice's super tight defense against 3-bets:

**Pokerazor simulation (Bob 3-bets any two cards)**

Bob uses his read on Alice, and changes his strategy to exploit her. He keeps 3-betting {QQ+,AK} for value, and then he 3-bets all other hands as a 3-bet bluff. Alice, not knowing that Bob suddenly has changed his strategy, keeps 4-betting {KK+} for value and folding everything else:

- Alice openraises 25% from CO

- Bob 3-bets {QQ+,AK} for value + any two cards as a 3-bet bluff

- Alice 4-bets {KK+} for value

EV (Alice) =-286 bb/100

Alice now gets slaughtered by Bob's any-two-cards 3-bet-bluffing

*and she actually loses money from trying to steal the blinds from CO*. Her first adjustment is to return to the optimal value range {TT+,AQ+} associated with her 25% CO openrange. This gives us:

EV (Alice; optimal 4-bet value range) =-39 bb/100

It helps, but not enough. She will still lose money for every hand she openraises, unless she also starts 4-bet bluffing. Alice now returns to her complete optimal strategy in CO, designed to defend against any-two-cards 3-bet bluffing:

EV (Alice; optimal total 4-bet-range) =+179 bb/100

Bingo! Alice's optimal 4-bet-strategy not only prevents Bob from exploiting her by 3-bet bluffing with any two cards, it also punishes him for it. Alice now makes 179 - 150 =+29 bb/100 more than if Bob had simply folded every time and let her pick up the blinds.

But is Alice's optimal strategy the most profitable strategy against Bob's any-two-cards 3-bet bluff strategy? No, and to find Alice's most profitable strategy we use common sense. When Bob is 3-betting any two cards, but only continues with {QQ+,AK} after a 4-bet, he is extremely vulnerable for 4-bet bluffing. He only continues with 2.56% of his hands and folds the remaining 97.44% against Alice's 4-bets

Alice can then maximally exploit Bob's attempt to exploit her

*by 4-bet bluffing him with any two cards*(or rather, any two cards in her original opening 25% opening range). We let Alice continue with her optimal {TT+,AQ+} value range. Then she deviates from the optimal strategy by widening her 4-bet bluffing range to the rest of her 25% CO opening range. Bob, unaware that Alice has suddenly changed her strategy to exploit his strategy, keeps 3-bet bluffing any two cards:

EV (Alice; 4-bet bluff any two cards) =+1261 bb/100

A huge increase in EV for Alice,

*and Alice now makes about 8 times more than if Bob had folded every hand*. This is an extremely clear illustration of the difference between optimal and exploitative play. Alice's optimal defense strategy against Bob's 3-bets guarantees that he can not exploit her by 3-bet bluffing any two cards, but he optimal defense did nothing to counter-exploit Bob extreme strategy. Alice made a little bit more than if Bob had folded every hand, but not a lot ((179 bb/100 vs 150 bb/100) .

But when Alice responds by choosing the strategy that exploits Bob's any-two-cards-bluffing strategy maximally, her EV explodes. Still, this does not come without risk for her, since Bob can take his exploitative strategy to the next level and begin 5-bet bluffing any two cards to exploit Alice's any-two-cards 4-bet bluffing. Alice must then make a new exploitative adjustment (for example, widening her value range dramatically and calling Bob's 5-bets with a very wide range of value hands) to stay one step ahead of Bob.

We can view this exploit/counter-exploit process as "strategic ping-pong" where both players zig and zag, using extreme strategy changes in different directions to maximally exploit their opponent. When one of them has made a big strategy change to exploit her opponent, she also creates an opening that the opponent can exploit by making an adjustment of his own. Then the first player has to make another big strategy adjustment, and the process repeats itself ad infinitum.

When we are playing optimally, we are using a different mindset. Instead of trying to stay one step ahead of our opponents by sudden "gear changes", we can fall back on a strategy that performs more or less equally well no matter what our opponent does. If he uses an extreme strategy, our optimal strategy will win a bit from him, but maximizing our profit from his mistake(s) is not our main goal. Instead, we simply want to prevent him from exploiting us. In the simulations above we saw that this worked well for Alice, but not as well as the maximally exploitative strategy she can use against Bob's any-two-cards 3-bet bluffing.

Okay Bob, now what? Bob can of course respond to Alice's any-two-cards 4-bet bluffing by going to the next level and start 5-bet bluffing with any two cards. Alice then gets exploited for a while, until she realizes this and adjusts. But we will not take any-two-cards bluffing beyond what we did in the previous simulations, and we assume that Bob now adjusts by falling back on his optimal 3-betting strategy. Alice in turn falls back on her optimal 4-betting strategy:

**Pokerazor simulation 4 (both players use optimal strategies)**

Both Alice and Bob now uses the optimal strategy. As we have seen in previous articles, this optimal strategy pair follows from Alice's openrange. Both players are now protecting themselves against any-two-cards bluffing from their opponent in all phases of the 3/4/5-bet war.

Alice's EV now becomes:

EV (Alice; both players 3/4/5-bet optimally) =+129 bb/100

In the last round of simulations we will let one player stick to the optimal strategy while the other player is let "off the leash", free to try anything to increase her or his EV against the other player's optimal strategy. Then we compare the resulting EV with EV when both players play optimally (129 bb/100 for Alice). We start out with Bob playing optimally, while Alice is testing out some deviations from her optimal strategy:

**Pokerazor simulation 5 (Bob plays optimally and Alice can do what she wants)**

We remember that Alice makes 150 bb/100 when Bob doesn't interfere, and Bob's optimal 3-betting strategy reduces this to 129 bb/100 when Alice plays optimally too. We shall now see that Alice can't do anything to increase her EV significantly when Bob sticks with his optimal strategy.

For example, let's assume that Alice drops the weakest hands TT/AQ from her value range when she sees that Bob uses the strong value range {QQ+,AK} (and remember that he also 3-bets/5-bets with his 5-bet bluffs A5s-A3s). Alice then chooses to 4-bet TT/AQ as before, but she folds them when Bob 5-bets all-in:

EV (Alice; folds TT/AQ to 5-bet) =+128 bb/100

Alice's EV drops a little, and she can't increase her EV by playing a tighter value range against Bob's strong value range. The reason is that Bob has put exactly so many 5-bet bluffs in his 5-betting range that Alice becomes indifferent to folding or calling with her weakest value hands. When Alice tries to avoid paying off Bob's better value hands, his 5-bet bluffs makes more money.

Can Alice increase her EV by 4-bet-bluffing more than optimally? We test this by letting her 4-bet bluff any two cards against Bob's optimal strategy, keeping everything else optimal:

EV (Alice; 4-bet-bluffs any two cards) =+139 bb/100

A small EV increase, but nothing comparable to the results of the extreme exploitative any-two-cards adjustments in previous simulations. Note that a perfectly optimal strategy for Bob should make it impossible for Alice to increase her EV, but our optimal strategy implementations probably contain some "numerical noise", since we have made some approximations and rounding along the way.

At any rate, Alice's attempt to exploit Bob's optimal strategy with any-two-cards 4-bet bluffing only results in a small EV change. Bob's optimal strategy therefore protects him well against exploitative 4-bet bluffing from Alice. And if Bob wanted to, he could probably fine-tune his strategy (for example, remove a couple of 3-bet bluff combos) to eliminate Alice's small EV increase completely.

So we have seen that Bob's side of the optimal 3/4/5-bet strategy pair seems robust against Alice's extreme adjustments. Let's now turn to Alice, and let her play the optimal strategy while Bob is allowed to do whatever he wants:

**Pokerazor simulation 6 (Alice plays optimally, while Bob can do what he wants)**

We saw previously that Bob's attempts to exploit Alice's optimal strategy by 3-bet bluffing any two cards didn't work for him. He lost against her optimal strategy (her EV increased from the baseline 150 bb/100 to 179 bb/100), and he lost a lot when Alice counter-exploited him by 4-bet bluffing any two cards (her EV increased to 1261 bb/100). We shall now repeat this simulation by using Bob's optimal strategy as a starting point, and then we make the adjustment that he 3-bet bluffs any two cards on top of that. All other ranges for value and 5-bet bluffing are as in the optimal strategy:

EV (Alice; Bob 3-bet-bluffs any two cards) =+179 bb/100

Alice's EV increases with +50 bb/100 (from 129 bb/100) relative to Bob's optimal strategy, and +29 bb/100 relative to the baseline EV when Bob always folds (150 bb/100). And we remember that if Alice wants to, she can exploit Bob hard by 4-bet bluffing any two cards and pocket more than 1200 bb/100 until Bob adjusts back. So Bob can't increase his EV by aggressive 3-bet bluffing, which is what we expected.

Then we let Bob 3-bet bluff any two cards and also 5-bet bluff any two cards. In other words, we let him play like a complete maniac, where he 3-bets any two cards and then 5-bets any two cards if he gets 4-bet.

EV (Alice; Bob 3-bet-bluffs/5-bet-bluffs any two cards)

=+300 bb/100

This causes Alice's EV to more than double relative to the 129 bb/100 she has against Bob's optimal strategy. We conclude that Alice's optimal strategy is waterproof against any-two-cards 3-bet bluffing, and that Bob only hurts himself if he tries.

The results from the last two simulations are worth noticing, since they can be uncomfortable scenarios to play when you don't know whether or not you are defending correctly. But as we have seen, even against a total maniac who 3-bets you from position at every opportunity, you don't have to do anything else than respond with your memorized optimal 4-bet strategy. If you do this, he will loose money relative to playing an optimal strategy himself, and he will probably end up losing money overall (so that 3-betting is worse for him than folding).

Note that an aggressive 3-bettor can still reduce our EV relative to the baseline EV (i.e. we will make less when he sometimes 3-bets than when he always folds), which is of course intuitively obvious since he has a lot of strong hands in his 3-betting range as well. For example, he could choose to 3-bet only AA, and we could not do anything to prevent him from making money in this situation and reduce our EV. ). So if we always respond with our optimal strategy, we can't deny the 3-bettor some +EV.

For example, if Bob deviates from his optimal strategy by increasing his 3-bet bluffing from 70% of "IP 3-bet air list" to 100% of the list, we get:

EV (Alice; Bob 3-bet-bluffs all of "IP 3-bet air list")

=+130 bb/100

We still make a little bit more (129 bb/100 --> 130 bb/100) when Bob increases his 3-bet bluff percentage beyond the optimal percentage, but he still reduces our EV relative to our baseline EV when he always folds (150 bb/100 --> 130 bb/100). We can't prevent Bob from making some money in this situation, and we just have to accept that a player in position has the right to make money by 3-betting us. Of course, our openraise will still be nicely profitable overall, just less profitable than if he had always folded behind us.

As we saw previously, we can exploit a complete maniac by deviating from optimal play to take advantage of the gaping holes in his strategy, particularly if he folds too much to 4-bets. If he lets us exploit him, we can make

*more*money from an exploitative strategy than from our optimal strategy. But then we have to play guessing games with him, and we also run the risk of offering big openings to the other players at the table (they can deviate from optimal play to exploit our non-optimal play). Since an optimal strategy will protect us (and then some) from getting exploited by a wild 3-bettor, this trade-off might not be worth it

A couple of obvious adjustments we can use to exploit a very aggressive with position on us 3-bettor are:

- 4-bet bluff more, if he folds easily to 4-bets (in other words, he defends his loose 3-betting range to tightly)
- Drop 4-bet bluffing, but 4-bet more hands like AJ, AT, 99, 88, etc. for value, if he folds too little to 4-bets and calls and 5-bets a lot with weak hands

But we don't have to make these adjustments to defend out of position against overly aggressive 3-betting. Our optimal strategy is more than enough. It might

*feel*like we're getting exploited, and some of the reason for that is that a strategy where we fold a lot (70% in the optimal strategy, as explained in Part 1) feels "weak". But the reality is that a maniac 3-bettor in position ends up costing himself if he starts 3-betting any two cards against our optimal strategy. Keep this in mind every time you feel exploited by a 3-bettor in position.

# 3. Summary

We have tested optimal strategy pairs for heads-up 3/4/5-betting using the analysis software Pokerazor. We started with a discussion of ABC preflop strategies without 3-bet bluffing or 4-bet bluffing. We then used simulations to show how ineffective and vulnerable these strategies are against players who are capable of reraising as a bluff with any two cards. As a part of this simulation we looked at exploitative adjustments we can make against players with big leaks in their 3/4/5-bet strategies.

Then we tested the robustness of the optimal 3/4/5-bet strategies we defined in previous articles, with the raiser out of position. We concluded that both the raiser's and the 3-bettor's optimal strategies were robust, and that they did not give the opponent openings he could exploit by bluffing with any two cards.

In Part 7 we'll do numerical simulations for flatting heads-up in position. Among other things we'll compare EV for flatting versus 3-betting for value with hands that are in between clear value hands and clear flatting hands (for example QQ against a tight UTG raiser). In the last half of Part 7 we'll adjust our heads-up 3/4/5-bet strategies for blind vs blind scenarios.

Good luck!

Bugs

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Edited 7 years ago

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

[h1]1. Introduction[/h1]

This is Part 6 in the series [i]Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max[/i], and the next to last theoretical part of the series (there will possibly be a practical part later this year, and we[apostrophe]ll talk about that in Part 7). In Part 1 to Part 5 we built a foundation for default NLHE preflop play based on mathematical principles from game theory, plus some common poker sense. In this and the next article we[apostrophe]ll test these strategies numerically.

The article series started with a simple scenario in Part 1 where we studied 3/4/5-betting heads-up with the raiser out of position. Then we generalized the strategies we found to other heads-up 3/4/5-bet scenarios, and also to a few select multiway scenarios. Along the way we also defined default ranges for open-raising from all positions.

Below is a summary of the content in Part 1 to Part 5:

[LIST]

[*][b]Part 1:[/b] Introduction to the mathematics behind game theory optimal 3/4/5-betting heads-up, studying the scenario where the raiser is out of position

[*][b]Part 2:[/b] We discussed in greater detail how to implement the theory from Part 1, and we defined default openranges for all positions. Then we defined the heads-up 3/4/5-bet theory for a wide range of openranges with the raiser out of position.

[*][b]Part 3:[/b] We let the raiser and the 3-bettor switch positions, and we studied the scenario where the raiser opens on the button and gets 3-bet by a player in the blinds.

[*][b]Part 4:[/b] We generalized the theory from Part 4 and looked at 3/4/5-betting heads-up with the raiser opening from any position outside of the blinds, and the 3-bettor 3betting from out of position in the blinds

[*][b]Part 5:[/b] We discussed 3/4/5-betting for two multiway scenarios (squeezing in a 3-way pot and cold 4-betting in a 3-way pot).

[/LIST]

Throughout Part 1 to Part 5 we have gone through most of the possible preflop scenarios and discussed good default strategies for them. In all 3/4/5-bet scenarios we have used the theory from Part 1 as our starting point, and then adjusted it for similar scenarios. We have used a mix of mathematical reasoning and good poker sense.

The plan for Part 6 is to test the strategies for heads-up 3/4/5-betting using the poker analysis software "[url=http://www.pokerazor.com/]Pokerazor[/url]". The final test for a strategy is of course to try it out at the tables and see how it performs. But we can also study our strategies numerically using analysis software. Today there are two programs available that let us study complete pre- and postflop strategies for any number of players:

- [url=http://www.pokerazor.com]Pokerazor[/url]

- [url=http://www.stoxev.com]StoxEV[/url]

Pokerazor is for the time being no longer commercially available, but a new version is expected some time in the future. StoxEV is available and being actively developed. I have elected to use Pokerazor for this article, since this is the program I am most familiar with. But StoxEV will work just as well if you are interested in doing this type of analysis work on your own.

What we[apostrophe]ll do first in this article is to study the typical ABC poker new players are advised to use when they get started with NLHE at the lowest limits ("play tight", "bluff little", "fold a lot when you get 3-bet", etc.) Then we[apostrophe]ll show how this ABC poker makes us vulnerable for attacks from aggressive opponents (particularly when they have position on us). We will here only look at preflop play, but the same principles apply postflop as well.

Then we[apostrophe]ll go one step further and show how we can improve on ABC preflop strategy by adding strategy components that fully or partly neutralize the attacks aggressive players subject us to (for example, we add 4-bet bluffing to our preflop strategy to defend against 3-bet bluffing). Then we go back to our opponents[apostrophe] strategies and discuss how they can adjust to our adjustments, and so on.

In this manner we[apostrophe]ll show how the optimal 3/4/5-bet strategies we have designed can be viewed as the final product of an evolutionary process based on our desire to defend against profitable bluffing with any two cards from aggressive opponents. The main point is that we don[apostrophe]t want to put ourselves in a situation where our opponent(s) can exploit us by bluffing profitably with any two cards, be it open-raising, bluff 3-betting, bluff 4-betting, or bluff 5-betting. An optimal strategy "plugs" all such openings for our opponents, but of course this defense does not come entirely without cost.

Through this discussion we[apostrophe]ll also shed light on the difference between optimal play and exploitative play, and when we should use one or the other. Optimal strategies put a lot of weight on defense, and they are not necessarily the most profitable strategies against players with big leaks. One reason is that optimal strategies include defensive components (for example, 4-bet bluffing as a defense against light 3-betting) that are often unnecessary against weak players (for example, we don[apostrophe]t need to 4-bet bluff against an opponent who only 3-bets premium hands like {JJ+,AK}).

Against players with big and easily exploitable leaks, we[apostrophe]d rather deviate from optimal play and play exploitatively to take full advantage of these leaks. But we need to be aware that by doing so we are creating openings in our strategies that can be exploited by observant opponents. So we have to find a balance between optimal and exploitative play, and we should use different strategies against different opponents. We will do our best to exploit weak players[apostrophe] big mistakes, but we can always fall back on optimal play against good opponents without big leaks. We can also return to optimal play if the player we[apostrophe]re trying to exploit with exploitative play suddenly changes his strategies to take advantage of the openings created by our exploitative strategies.

For example, let[apostrophe]s say we choose to never 4-bet bluff against a passive player who never 3-bet bluffs. He might now notice this, and adjust to our tight play by starting to 3-bet bluff us. Our exploitative adjustment against this particular opponent then runs the risk of getting counter-exploited if he starts 3-bet-bluffing us often with random weak hands. If this happens, we should return to our optimal optimal 3/4/5-bet strategy. Alternatively, we can make another exploitative adjustment to his adjustment by 4-bet bluffing him a lot (since we know he often is weak and have to fold). But the optimal strategy is always an alternative if we aren[apostrophe]t sure whether or not we can exploit his aggressive 3-betting.

In my opinion, this mindset is at the core of the thought processes of a strong NLHE player. He doesn[apostrophe]t have to use mathematics like we have done, but he will have a good feel for what an optimal (or near optimal) strategy is in the situation he is in. So he has a strong default strategy to fall back on against unknown players or known strong players, so that he can[apostrophe]t be easily exploited. But at the same time he knows how to deviate from optimal strategies to exploit his opponents[apostrophe] systematic leaks. So he can adjust his play in a controlled manner against each individual opponent instead of being locked into a static strategy that he uses against everyone.

Rules of thumb such as "never 4-bet bluff against fish" or "don[apostrophe]t 3-bet hands that perform poorly when called" are then replaced by a dynamical mindset that gives is strong control over our choice of strategies. Using optimal play as a starting point (and as a strategy we can always fall back on regardless of who we[apostrophe]re playing against), we can move around freely in "strategy land" and exploit opponent leaks as we pick up information about how they play.

Optimal play is never bad play, but exploitative play is always better. But we need information about our opponents[apostrophe] strategies before we can exploit them. If we don[apostrophe]t have this information, we can always fall back on optimal play as a good default.

[h1]2. Testing preflop strategies using the analysis software "Pokerazor"[/h1]

In this part of the article we[apostrophe]ll use [url=http://pokerazor.com]Pokerazor[/url] to study 2 things:

[LIST]

[*][b]1.[/b] How a tight openraise strategy without defense against light 3-betting is vulnerable to 3-bet bluffing with any two cards

[*][b]2.[/b] What the raiser can do to plug this leak, and how this leads to an optimal strategy pair for the raiser and the 3-bettor

[/LIST]

[h2]2.1 ABC preflop strategies and how these can be exploited[/h2]

Those of you who have played for a while probably remember the good old days (up to around 2007 or thereabouts) when micro and low limit NLHE was easily beatable by sticking close to the following rules of thumb for preflop play:

Those

[LIST]

[*]Open tight from all positions (say, 10-12% from UTG/MP, ~20% from MP and ~30% from the button

[*]3-bet only for value with {QQ+,AK}, and possibly {JJ+,AQ+} against a loose raiser

[*]When you get 3-bet and you are out of position, fold everything but {QQ+,AK} regardless of your position and where the 3-bet comes from

[*]Defend the blinds very tightly (typically 10%)

[/LIST]

Believe it or not, but this was more or less the standard getting-started preflop strategy recommended to beginning players at the micro and low limits up to $100NL or so. And it worked well, since the games were so loose and passive that it was correct both to openraise tight, and to fold a lot against 3-bets.

Those of you who have been members of Cardrunners for a while might remember Brystmar[apostrophe]s beginner video series "Small Stakes NL" in 6 parts (published during the spring of 2007). This series began with tight-aggressive preflop recommendations based on tight opening ranges and 3-betting only for value:

[b]Brystmar[apostrophe]s preflop strategy for micro/low limit NLHE[/b]

Let[apostrophe]s take a trip down memory land and study Brystmar[apostrophe]s preflop recommendations given 3.5 years ago. Those who want to read discussion about his video series or download his [url=http://rapidshare.com/files/155660787/crchartimg.jpg]preflop scheme[/url] can look at [url=http://www.cardrunners.com/cr_forums/showthread.php?t=25318#Post168442]this Cardrunners forum thread[/url].

Below is a summary of the default openraising ranges (note that "KTs" and "KTo" denote suited and offsuit hands, while "KT+" means both suited and offsuit:

[LIST]

[*][b]UTG openraise:[/b]

{22+,AJ+,KQs} =9.8%

[*][b]MP openraise:[/b]

{22+,AJ+,KQ} =11%

[*][b]CO openraise:[/b]

{22+,A7s+,A9o+,KT+,QTs+,QJo,J9s+,JTo,T9s} =19%

[*][b]Button openraise:[/b]

{22+,A4s+,A7o+,KT+,QTs+,QJo,J9s+,JTo,T8s+,

T9o,98s,98o,87s,87o,76s} =26%

[*][b]Raising from the small blind:[/b]

Openraise the button range if it gets folded to you. In a limped pot, raise {JJ+,AK} for value and overlimp all other hands from your button range, plus all Axs and Kxs.

[*][b]Raising from the big blind:[/b]

If the small blind openlimps, raise the button range and otherwise check. Out of position in a limped pot, raise {JJ+,AK} for value and otherwise check

[/LIST]

Tight opening ranges all around. This is of course not a leak our opponents can exploit, but we might perhaps say that we are exploiting ourselves by folding some profitable hands, particularly on the button.

But the strategies become easy to exploit when we get to playing against a raise:

[LIST]

[*][b]In MP with position on a raiser:[/b]

Reraise {JJ+,AK} for value and call with {TT-22,AJs+,AQo}

[*][b]In CO with position on a raiser:[/b]

Reraise {JJ+,AK} for value and call with {TT-22,AJ+,KQ,QJs,JTs}

[*][b]On the button with position on a raiser:[/b]

Reraise {JJ+,AK} for value (and AQo if the raise came from CO) and call with {TT-22,AJ+,KQ,QJs,JTs}. With callers between you and the raiser, also call with {JTo,T9s,98s,87s}

[*][b]In SB after a raise:[/b]

Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}

[*][b]In BB after a raise:[/b]

Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}. With callers between you and the raiser, also call with {QJs,JTs}

[/LIST]

We note two systematic errors in these strategies:

- We[apostrophe]re 3-betting more or less the same range regardless of the raiser[apostrophe]s position

- We[apostrophe]re never 3-bet bluffing

We remember from Part 1 and Part 2 that an optimal 3-betting range on the button varied from 3.6% against a ~15% EP openraise to 8.7% against a ~25% CO openraise. And in all scenarios we used an optimal bluffing frequency of 60%. In other words, more than half of our 3-bets were bluffs- In Brystmar[apostrophe]s strategies the 3-betting range is a tight and static value range {JJ+,AK} =3.0%, which is sometimes widened to {JJ+,AQ+} =4.2% against a wide openraising range. We also note that Brystmar chooses to include AQ when loosening up. This is a hand we never 3-bet for value in position when we[apostrophe]re playing optimally (since AQ works better as a flatting hand in position).

Brystmar[apostrophe]s strategies don[apostrophe]t mention defense against 3-betting, but we can assume that default defense is to 4-bet a tight range {QQ+,AK} from all positions. Another significant leak is the squeaky tight blind defense. For example, of button openraises, the preflop scheme tells us to 3-bet {JJ+,AQ} =4.2% from the small blind and flat {TT-22, AJ,KQ} =7.4%. This results in a total defense of 11.6%, which is way lower than the optimal defense threshold of 16% that we estimated in Part 3.

So there are huge openings in Brystmar[apostrophe]s preflop recommendations, and these openings can be easily exploited by an aggressive and observant opponent. We[apostrophe]re also leaving money at the table because we[apostrophe]re openraising to tight, and the main reason for this is that Brystmar does not take full advantage of position. We can openraise a ton of hands on the button when it[apostrophe]s folded to us, and we can make life hell for a raiser by 3-bet bluffing him in position, but Brystmar chooses not to do so.

NB! Before we move on I want to point out that I am not trying to put Brystmar[apostrophe]s low limit preflop defaults from 2007 in a negative light. His preflop recommendations for beginning NLHE players were very useful back in the day, and gave many new players an easy start. His strategies are best viewed as "training wheels" for staying out of trouble (and he no doubt saw them as such himself) and they were tailored towards the micro/low limit conditions that existed at the time. They will probably still work okay at the lowest micro limits, but I would not recommend anyone to play $25NL and higher with such tight and easily exploitable preflop strategies.

It[apostrophe]s clear for everyone who plays $25NL and higher these days that common NLHE strategy has developed in leaps and bounds since Brystmar[apostrophe]s 2007 recommendations. Light 3-betting was rare in the "old days", even at $100NL and $200NL. Today it[apostrophe]s common, even if you begin as low as $5NL.

The next step of the development of the average low limit NLGE regular back in the day was to add some light 3-bets in position (and Green Plastic[apostrophe]s 2006/2007 NLHE videos at Cardrunners inspired many to do so), call more raises and 3-bets in position, and in general get better at using positional advantage. A common mistake many aggressive players did was to 3-bet bluff with hands that were too strong to use as bluffs (for example, JTs). However, this did not cause man problems since most players defended poorly against 3-bets, particularly from out of position.

So the standard recipe in the good old days for an advanced low limit player who wanted to ramp up the aggression was to LAG it up in position. But not necessarily with balance in mind, and not necessarily with a good understanding of how to chose his value range, bluffing range and flatting range in a consistent manner. But this was not a big deal. He played tight out of position, opened a very wide range in position, and 3-bet something fierce in position against weak opponents. The 3-betting was very effective, since the raisers often did one of the following two mistakes:

- Folded a lot out of position and never 3-bet bluffed

- Called a lot with non-premium hands out of position

The first mistake lets the 3-bettor print money by giving him an opening to (in principle) 3-bet bluff any two cards. The second mistake occurs when the raiser tries to correct the first mistake, but he goes about it the wrong way. Defending against 3-bets by flatting weak hands out of position is ineffective, since the raiser now has to play postflop out of position in a scenario where it[apostrophe]s difficult for him to win without hitting the flop well. Playing weak starting hands well out of position against a good LAG player is hard, and often results in you losing more money postflop than if you had just folded to the 3-bet preflop.

The cure against light, positional 3-betting is of course to respond by 4-betting a correct value range (which follows from the size of our opening range), balanced with a correct amount of 4-bet bluffing. We have studied this in previous articles, and we have defined optimal strategies for the raiser from all positions. The 3-bettor uses similar thinking to design his 3-bet strategy so that the raiser can not 4bet bluff any two cards profitably. This way an equilibrium gets established.

This equilibrium is given by the optimal strategy pairs for the raiser and the 3-bettor defined in Part 1 and Part 2. We used mathematics to define these strategy pairs, but we can also think about them as a product of an evolutionary process.

The 3-better starts out by exploitative ant-two-cards 3-bet bluffing against a raiser that defends way too tight and folds too much. Then the raiser adjust by choosing a correct value range and introducing 4-bet bluffing. The 3-bettor responds by adjusting his value range and introducing 5-bet bluffing. To prevent the opponent from bluffing with any two cards anywhere, both players fine-tune their ranges until both are using an optimal set of ranges for 3/4/5-betting. "Optimal" here means that neither player can improve his EV by adjusting further. If one of them tries to do so, he is giving the other player an opportunity to increase [i]his[/i] EV by making and exploitative adjustment.

We will now illustrate such an evolutionary process using Pokerazor simulations:

[h2]2.2 Numerical testing of optimal heads-up 3/4/5-bet strategies[/h2]

We start with the following model:

[LIST]

[*]Alice (100bb) openraises to 3.5bb with her standard 25% range from CO

[*]Bob (100bb) is on the button and 3-bets to 12 bb or folds

[*]Alice defends against 3-betting by 4-betting to 25 bb or folding

[*]Bob defends against 4-betting by 5-betting all-in or folding

[*]Alice defends against 5-betting by calling all-in or folding

[*]The blinds always fold, no matter what Bob does

[/LIST]

So we are studying a scenario where Alice openraises, Bob 3-bets or folds, and the blinds never get involved. Alice then makes 1.5 bb (the blinds) per raise when Bob folds, which is a win rate of 150 bb/100. This is her baseline EV for the simulation.

Before we begin the simulations, let[apostrophe]s repeat the ranges and optimal strategy pairs we defined in Part 2:

[b]Alice[apostrophe]s 25% open-range from CO[/b]

[pre]22+

A2s+ A9o+

K9s+ KTo+

Q9s+ QTo+

J8s+ JTo

T8s+

97s+

87s

76s

65s

326 combos

25%[/pre]

The corresponding optimal strategy pair that[apostrophe]s being used when Bob 3-bets in position can be found from the summary of optimal strategy pairs in Part 2:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-6/IP_3-bet_summary.png[/img]

Here is a download link for this document (right-click and choose "Save as"):

[url=http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-6/IP_3-bet_summary.doc]IP_3-bet_summary.doc[/url]

The optimal strategy pair is then:

[LIST]

[*][b]Bob:[/b]

3-bets {QQ+,AK,12 air} ={QQ+,AK,A5s-A3s} for value (including 5-bet-bluffing with Axs hands) and 70% av "IP 3-bet air list" as a bluff using a [url=http://www.random.org/]randomizer[/url]

[*][b]Alice:[/b]

4-bets {TT+,AQ+} for value and {AJ,AT,A9s-A7s} as a bluff

[/LIST]

[b]Pokerazor simulation 1 (Bob folds)[/b]

The baseline simulation is to let Bob fold 100%. Alice then picks up the blinds, and makes 1.5 bb each time (=150 bb/10):

- Alice openraises 25% from CO

- Bob folds

[pre]EV (baseline for Alice) =150 bb/100[/pre]

Bob now begins 3-betting so that Alice ends up with EV [lt] 150 bb/100. It[apostrophe]s obvious that Alice[apostrophe]s EV now becomes lower than the 150 bb/100 baseline, since she can not prevent Bob from making money by 3-betting only his best hands, for example his {QQ+,AK} default value range against her 25% CO openrange. Furthermore, it[apostrophe]s correct for Alice to never 4-bet bluff when Bob never 3-bet bluffs, so she has to fold the hands not strong enough to 4-bet for value against Bob[apostrophe]s tight value range and let Bob pick up the blinds.

We start by assuming that Bob never 3-bet bluffs and that Alice defends against the 3-bet by 4-betting the value range {KK+} after observing that Bob[apostrophe]s 3-betting range is {QQ+,AK}. This is correct since Alice does not want to get all-in preflop with QQ or AK against Bob[apostrophe]s {QQ+,AK} range (both are ~40% underdog) with only 3.5 bb invested.

[b]Pokerazor simulation 2 (Bob 3-bets only for value)[/b]

Bob 3-bets his value hands. We let him use the pure value hands {QQ+,AK} that he would have used in an optimal strategy against Alice[apostrophe]s 25% CO openrange, and the drops his 5-bet bluffs for now:

- Alice openraises 25% from CO

- Bob 3-bets {QQ+,AK} for value and 5-bets all-in against a 4-bet

- Alice 4-better {KK+} for value

And the rest follows automatically. We get:

[pre]EV (Alice; KK+) =141 bb/100[/pre]

If Alice also had 4-bet QQ/AK for value, the EV would have become:

[pre]EV (Alice; QQ+,AK) =139 bb/100[/pre]

So Alice[apostrophe]s choice of value range against Bob[apostrophe]s extremely tight 3-betting range is correct. Note that [i]Alice is now exploiting Bob by only 4-betting an extremely tight {KK+} value range![/i] What happens here is that Bob mostly leaves Alice alone so that she can pick up the blinds almost every time. Bob only pops up with a value 3-bet the 2.56% of the time he has {QQ+,AK}, and Alice then responds by folding everything but {KK+}.

So Alice exploits Bob[apostrophe]s squeaky tight 3-betting by not paying off his value hands with weaker hands. And since Bob never 3-bet bluffs, there is never any doubt about his range. Alice can then play perfectly against a 3-bet and drop all her (now unnecessary) 4-bet bluffs as well as her weakest value hands (and we remember that Alice[apostrophe]s optimal value range in CO is {TT+,AQ+}). Exploiting someone by folding a lot to his aggression is not the first thing that comes to mind when poker players think about exploitative play. But avoiding paying off a strong range unnecessarily is just as profitable as playing aggressively against players that fold too much.

If Alice had responded with her optimal 4-bet strategy which is {TT+,AQ} for value and {AJ-AT,A9s-A7s} as bluffs, we get:

[pre]EV (Alice; optimal strategy) =126 bb/100[/pre]

And we see that [i]Alice[apostrophe]s optimal 3/4/5-bet strategy out of position loses relative to the best exploitative strategy[/i] (which gave her 141 bb/100). This is an example of a general rule: If we have an opportunity to exploit someone, we will make more money from this than from continuing to use an optimal strategy. The reason is that Alice now sacrifices EV by defending against a non-existing threat. She has 4-bet bluffs in her 4-betting range to defend against Bob[apostrophe]s 3-bet bluffs, but Bob never 3-bet bluffs.

So even if Alice[apostrophe]s optimal strategy can not be exploited by any-two-cards 3-bet bluffing from Bob, [i]this defense is costing her EV relative to a strategy that exploits Bob[apostrophe]s very tight never-bluff 3-betting strategy[/i]. A good analogy would be a nations defense budget in peace time. There has been no warfare on Norwegian soil since 1945, but Norway has still had a defense budget in all the years since then. We are simply paying for a military defense so that other nations don[apostrophe]t get an opportunity to invade us without risk.

Now, let[apostrophe]s assume that Bob has been sitting there and 3-betting only {QQ+,AK} for value and folding everything else, while Alice has responded by 4-betting only {KK+} for value and folding everything else. Bob has observed that Alice almost always folds. It[apostrophe]s now easy for him to reach the conclusion that he could make a lot more money by throwing in some 3-bet bluffs, planning to fold then when Alice 4-bets. Since Bob does not have to worry about the blind players waking up with a hand, let[apostrophe]s allow him to 3-bet any two cards to maximally exploit Alice[apostrophe]s super tight defense against 3-bets:

[b]Pokerazor simulation (Bob 3-bets any two cards)[/b]

Bob uses his read on Alice, and changes his strategy to exploit her. He keeps 3-betting {QQ+,AK} for value, and then he 3-bets all other hands as a 3-bet bluff. Alice, not knowing that Bob suddenly has changed his strategy, keeps 4-betting {KK+} for value and folding everything else:

- Alice openraises 25% from CO

- Bob 3-bets {QQ+,AK} for value + any two cards as a 3-bet bluff

- Alice 4-bets {KK+} for value

[pre]EV (Alice) =-286 bb/100[/pre]

Alice now gets slaughtered by Bob[apostrophe]s any-two-cards 3-bet-bluffing [i]and she actually loses money from trying to steal the blinds from CO[/i]. Her first adjustment is to return to the optimal value range {TT+,AQ+} associated with her 25% CO openrange. This gives us:

[pre]EV (Alice; optimal 4-bet value range) =-39 bb/100[/pre]

It helps, but not enough. She will still lose money for every hand she openraises, unless she also starts 4-bet bluffing. Alice now returns to her complete optimal strategy in CO, designed to defend against any-two-cards 3-bet bluffing:

[pre]EV (Alice; optimal total 4-bet-range) =+179 bb/100[/pre]

Bingo! Alice[apostrophe]s optimal 4-bet-strategy not only prevents Bob from exploiting her by 3-bet bluffing with any two cards, it also punishes him for it. Alice now makes 179 - 150 =+29 bb/100 more than if Bob had simply folded every time and let her pick up the blinds.

But is Alice[apostrophe]s optimal strategy the most profitable strategy against Bob[apostrophe]s any-two-cards 3-bet bluff strategy? No, and to find Alice[apostrophe]s most profitable strategy we use common sense. When Bob is 3-betting any two cards, but only continues with {QQ+,AK} after a 4-bet, he is extremely vulnerable for 4-bet bluffing. He only continues with 2.56% of his hands and folds the remaining 97.44% against Alice[apostrophe]s 4-bets

Alice can then maximally exploit Bob[apostrophe]s attempt to exploit her [i]by 4-bet bluffing him with any two cards[/i] (or rather, any two cards in her original opening 25% opening range). We let Alice continue with her optimal {TT+,AQ+} value range. Then she deviates from the optimal strategy by widening her 4-bet bluffing range to the rest of her 25% CO opening range. Bob, unaware that Alice has suddenly changed her strategy to exploit his strategy, keeps 3-bet bluffing any two cards:

[pre]EV (Alice; 4-bet bluff any two cards) =+1261 bb/100[/pre]

A huge increase in EV for Alice, [i]and Alice now makes about 8 times more than if Bob had folded every hand[/i]. This is an extremely clear illustration of the difference between optimal and exploitative play. Alice[apostrophe]s optimal defense strategy against Bob[apostrophe]s 3-bets guarantees that he can not exploit her by 3-bet bluffing any two cards, but he optimal defense did nothing to counter-exploit Bob extreme strategy. Alice made a little bit more than if Bob had folded every hand, but not a lot ((179 bb/100 vs 150 bb/100) .

But when Alice responds by choosing the strategy that exploits Bob[apostrophe]s any-two-cards-bluffing strategy maximally, her EV explodes. Still, this does not come without risk for her, since Bob can take his exploitative strategy to the next level and begin 5-bet bluffing any two cards to exploit Alice[apostrophe]s any-two-cards 4-bet bluffing. Alice must then make a new exploitative adjustment (for example, widening her value range dramatically and calling Bob[apostrophe]s 5-bets with a very wide range of value hands) to stay one step ahead of Bob.

We can view this exploit/counter-exploit process as "strategic ping-pong" where both players zig and zag, using extreme strategy changes in different directions to maximally exploit their opponent. When one of them has made a big strategy change to exploit her opponent, she also creates an opening that the opponent can exploit by making an adjustment of his own. Then the first player has to make another big strategy adjustment, and the process repeats itself ad infinitum.

When we are playing optimally, we are using a different mindset. Instead of trying to stay one step ahead of our opponents by sudden "gear changes", we can fall back on a strategy that performs more or less equally well no matter what our opponent does. If he uses an extreme strategy, our optimal strategy will win a bit from him, but maximizing our profit from his mistake(s) is not our main goal. Instead, we simply want to prevent him from exploiting us. In the simulations above we saw that this worked well for Alice, but not as well as the maximally exploitative strategy she can use against Bob[apostrophe]s any-two-cards 3-bet bluffing.

Okay Bob, now what? Bob can of course respond to Alice[apostrophe]s any-two-cards 4-bet bluffing by going to the next level and start 5-bet bluffing with any two cards. Alice then gets exploited for a while, until she realizes this and adjusts. But we will not take any-two-cards bluffing beyond what we did in the previous simulations, and we assume that Bob now adjusts by falling back on his optimal 3-betting strategy. Alice in turn falls back on her optimal 4-betting strategy:

[b]Pokerazor simulation 4 (both players use optimal strategies)[/b]

Both Alice and Bob now uses the optimal strategy. As we have seen in previous articles, this optimal strategy pair follows from Alice[apostrophe]s openrange. Both players are now protecting themselves against any-two-cards bluffing from their opponent in all phases of the 3/4/5-bet war.

Alice[apostrophe]s EV now becomes:

[pre]EV (Alice; both players 3/4/5-bet optimally) =+129 bb/100[/pre]

In the last round of simulations we will let one player stick to the optimal strategy while the other player is let "off the leash", free to try anything to increase her or his EV against the other player[apostrophe]s optimal strategy. Then we compare the resulting EV with EV when both players play optimally (129 bb/100 for Alice). We start out with Bob playing optimally, while Alice is testing out some deviations from her optimal strategy:

[b]Pokerazor simulation 5 (Bob plays optimally and Alice can do what she wants)[/b]

We remember that Alice makes 150 bb/100 when Bob doesn[apostrophe]t interfere, and Bob[apostrophe]s optimal 3-betting strategy reduces this to 129 bb/100 when Alice plays optimally too. We shall now see that Alice can[apostrophe]t do anything to increase her EV significantly when Bob sticks with his optimal strategy.

For example, let[apostrophe]s assume that Alice drops the weakest hands TT/AQ from her value range when she sees that Bob uses the strong value range {QQ+,AK} (and remember that he also 3-bets/5-bets with his 5-bet bluffs A5s-A3s). Alice then chooses to 4-bet TT/AQ as before, but she folds them when Bob 5-bets all-in:

[pre]EV (Alice; folds TT/AQ to 5-bet) =+128 bb/100[/pre]

Alice[apostrophe]s EV drops a little, and she can[apostrophe]t increase her EV by playing a tighter value range against Bob[apostrophe]s strong value range. The reason is that Bob has put exactly so many 5-bet bluffs in his 5-betting range that Alice becomes indifferent to folding or calling with her weakest value hands. When Alice tries to avoid paying off Bob[apostrophe]s better value hands, his 5-bet bluffs makes more money.

Can Alice increase her EV by 4-bet-bluffing more than optimally? We test this by letting her 4-bet bluff any two cards against Bob[apostrophe]s optimal strategy, keeping everything else optimal:

[pre]EV (Alice; 4-bet-bluffs any two cards) =+139 bb/100[/pre]

A small EV increase, but nothing comparable to the results of the extreme exploitative any-two-cards adjustments in previous simulations. Note that a perfectly optimal strategy for Bob should make it impossible for Alice to increase her EV, but our optimal strategy implementations probably contain some "numerical noise", since we have made some approximations and rounding along the way.

At any rate, Alice[apostrophe]s attempt to exploit Bob[apostrophe]s optimal strategy with any-two-cards 4-bet bluffing only results in a small EV change. Bob[apostrophe]s optimal strategy therefore protects him well against exploitative 4-bet bluffing from Alice. And if Bob wanted to, he could probably fine-tune his strategy (for example, remove a couple of 3-bet bluff combos) to eliminate Alice[apostrophe]s small EV increase completely.

So we have seen that Bob[apostrophe]s side of the optimal 3/4/5-bet strategy pair seems robust against Alice[apostrophe]s extreme adjustments. Let[apostrophe]s now turn to Alice, and let her play the optimal strategy while Bob is allowed to do whatever he wants:

[b]Pokerazor simulation 6 (Alice plays optimally, while Bob can do what he wants)[/b]

We saw previously that Bob[apostrophe]s attempts to exploit Alice[apostrophe]s optimal strategy by 3-bet bluffing any two cards didn[apostrophe]t work for him. He lost against her optimal strategy (her EV increased from the baseline 150 bb/100 to 179 bb/100), and he lost a lot when Alice counter-exploited him by 4-bet bluffing any two cards (her EV increased to 1261 bb/100). We shall now repeat this simulation by using Bob[apostrophe]s optimal strategy as a starting point, and then we make the adjustment that he 3-bet bluffs any two cards on top of that. All other ranges for value and 5-bet bluffing are as in the optimal strategy:

[pre]EV (Alice; Bob 3-bet-bluffs any two cards) =+179 bb/100[/pre]

Alice[apostrophe]s EV increases with +50 bb/100 (from 129 bb/100) relative to Bob[apostrophe]s optimal strategy, and +29 bb/100 relative to the baseline EV when Bob always folds (150 bb/100). And we remember that if Alice wants to, she can exploit Bob hard by 4-bet bluffing any two cards and pocket more than 1200 bb/100 until Bob adjusts back. So Bob can[apostrophe]t increase his EV by aggressive 3-bet bluffing, which is what we expected.

Then we let Bob 3-bet bluff any two cards and also 5-bet bluff any two cards. In other words, we let him play like a complete maniac, where he 3-bets any two cards and then 5-bets any two cards if he gets 4-bet.

[pre]EV (Alice; Bob 3-bet-bluffs/5-bet-bluffs any two cards)

=+300 bb/100[/pre]

This causes Alice[apostrophe]s EV to more than double relative to the 129 bb/100 she has against Bob[apostrophe]s optimal strategy. We conclude that Alice[apostrophe]s optimal strategy is waterproof against any-two-cards 3-bet bluffing, and that Bob only hurts himself if he tries.

The results from the last two simulations are worth noticing, since they can be uncomfortable scenarios to play when you don[apostrophe]t know whether or not you are defending correctly. But as we have seen, even against a total maniac who 3-bets you from position at every opportunity, you don[apostrophe]t have to do anything else than respond with your memorized optimal 4-bet strategy. If you do this, he will loose money relative to playing an optimal strategy himself, and he will probably end up losing money overall (so that 3-betting is worse for him than folding).

Note that an aggressive 3-bettor can still reduce our EV relative to the baseline EV (i.e. we will make less when he sometimes 3-bets than when he always folds), which is of course intuitively obvious since he has a lot of strong hands in his 3-betting range as well. For example, he could choose to 3-bet only AA, and we could not do anything to prevent him from making money in this situation and reduce our EV. ). So if we always respond with our optimal strategy, we can[apostrophe]t deny the 3-bettor some +EV.

For example, if Bob deviates from his optimal strategy by increasing his 3-bet bluffing from 70% of "IP 3-bet air list" to 100% of the list, we get:

[pre]EV (Alice; Bob 3-bet-bluffs all of "IP 3-bet air list")

=+130 bb/100[/pre]

We still make a little bit more (129 bb/100 --[gt] 130 bb/100) when Bob increases his 3-bet bluff percentage beyond the optimal percentage, but he still reduces our EV relative to our baseline EV when he always folds (150 bb/100 --[gt] 130 bb/100). We can[apostrophe]t prevent Bob from making some money in this situation, and we just have to accept that a player in position has the right to make money by 3-betting us. Of course, our openraise will still be nicely profitable overall, just less profitable than if he had always folded behind us.

As we saw previously, we can exploit a complete maniac by deviating from optimal play to take advantage of the gaping holes in his strategy, particularly if he folds too much to 4-bets. If he lets us exploit him, we can make [i]more[/i] money from an exploitative strategy than from our optimal strategy. But then we have to play guessing games with him, and we also run the risk of offering big openings to the other players at the table (they can deviate from optimal play to exploit our non-optimal play). Since an optimal strategy will protect us (and then some) from getting exploited by a wild 3-bettor, this trade-off might not be worth it

A couple of obvious adjustments we can use to exploit a very aggressive with position on us 3-bettor are:

[LIST]

[*]4-bet bluff more, if he folds easily to 4-bets (in other words, he defends his loose 3-betting range to tightly)

[*]Drop 4-bet bluffing, but 4-bet more hands like AJ, AT, 99, 88, etc. for value, if he folds too little to 4-bets and calls and 5-bets a lot with weak hands

[/LIST]

But we don[apostrophe]t have to make these adjustments to defend out of position against overly aggressive 3-betting. Our optimal strategy is more than enough. It might [i]feel[/i] like we[apostrophe]re getting exploited, and some of the reason for that is that a strategy where we fold a lot (70% in the optimal strategy, as explained in Part 1) feels "weak". But the reality is that a maniac 3-bettor in position ends up costing himself if he starts 3-betting any two cards against our optimal strategy. Keep this in mind every time you feel exploited by a 3-bettor in position.

[h1]3. Summary[/h1]

We have tested optimal strategy pairs for heads-up 3/4/5-betting using the analysis software Pokerazor. We started with a discussion of ABC preflop strategies without 3-bet bluffing or 4-bet bluffing. We then used simulations to show how ineffective and vulnerable these strategies are against players who are capable of reraising as a bluff with any two cards. As a part of this simulation we looked at exploitative adjustments we can make against players with big leaks in their 3/4/5-bet strategies.

Then we tested the robustness of the optimal 3/4/5-bet strategies we defined in previous articles, with the raiser out of position. We concluded that both the raiser[apostrophe]s and the 3-bettor[apostrophe]s optimal strategies were robust, and that they did not give the opponent openings he could exploit by bluffing with any two cards.

In Part 7 we[apostrophe]ll do numerical simulations for flatting heads-up in position. Among other things we[apostrophe]ll compare EV for flatting versus 3-betting for value with hands that are in between clear value hands and clear flatting hands (for example QQ against a tight UTG raiser). In the last half of Part 7 we[apostrophe]ll adjust our heads-up 3/4/5-bet strategies for blind vs blind scenarios.

Good luck!

Bugs

This is Part 6 in the series [i]Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max[/i], and the next to last theoretical part of the series (there will possibly be a practical part later this year, and we[apostrophe]ll talk about that in Part 7). In Part 1 to Part 5 we built a foundation for default NLHE preflop play based on mathematical principles from game theory, plus some common poker sense. In this and the next article we[apostrophe]ll test these strategies numerically.

The article series started with a simple scenario in Part 1 where we studied 3/4/5-betting heads-up with the raiser out of position. Then we generalized the strategies we found to other heads-up 3/4/5-bet scenarios, and also to a few select multiway scenarios. Along the way we also defined default ranges for open-raising from all positions.

Below is a summary of the content in Part 1 to Part 5:

[LIST]

[*][b]Part 1:[/b] Introduction to the mathematics behind game theory optimal 3/4/5-betting heads-up, studying the scenario where the raiser is out of position

[*][b]Part 2:[/b] We discussed in greater detail how to implement the theory from Part 1, and we defined default openranges for all positions. Then we defined the heads-up 3/4/5-bet theory for a wide range of openranges with the raiser out of position.

[*][b]Part 3:[/b] We let the raiser and the 3-bettor switch positions, and we studied the scenario where the raiser opens on the button and gets 3-bet by a player in the blinds.

[*][b]Part 4:[/b] We generalized the theory from Part 4 and looked at 3/4/5-betting heads-up with the raiser opening from any position outside of the blinds, and the 3-bettor 3betting from out of position in the blinds

[*][b]Part 5:[/b] We discussed 3/4/5-betting for two multiway scenarios (squeezing in a 3-way pot and cold 4-betting in a 3-way pot).

[/LIST]

Throughout Part 1 to Part 5 we have gone through most of the possible preflop scenarios and discussed good default strategies for them. In all 3/4/5-bet scenarios we have used the theory from Part 1 as our starting point, and then adjusted it for similar scenarios. We have used a mix of mathematical reasoning and good poker sense.

The plan for Part 6 is to test the strategies for heads-up 3/4/5-betting using the poker analysis software "[url=http://www.pokerazor.com/]Pokerazor[/url]". The final test for a strategy is of course to try it out at the tables and see how it performs. But we can also study our strategies numerically using analysis software. Today there are two programs available that let us study complete pre- and postflop strategies for any number of players:

- [url=http://www.pokerazor.com]Pokerazor[/url]

- [url=http://www.stoxev.com]StoxEV[/url]

Pokerazor is for the time being no longer commercially available, but a new version is expected some time in the future. StoxEV is available and being actively developed. I have elected to use Pokerazor for this article, since this is the program I am most familiar with. But StoxEV will work just as well if you are interested in doing this type of analysis work on your own.

What we[apostrophe]ll do first in this article is to study the typical ABC poker new players are advised to use when they get started with NLHE at the lowest limits ("play tight", "bluff little", "fold a lot when you get 3-bet", etc.) Then we[apostrophe]ll show how this ABC poker makes us vulnerable for attacks from aggressive opponents (particularly when they have position on us). We will here only look at preflop play, but the same principles apply postflop as well.

Then we[apostrophe]ll go one step further and show how we can improve on ABC preflop strategy by adding strategy components that fully or partly neutralize the attacks aggressive players subject us to (for example, we add 4-bet bluffing to our preflop strategy to defend against 3-bet bluffing). Then we go back to our opponents[apostrophe] strategies and discuss how they can adjust to our adjustments, and so on.

In this manner we[apostrophe]ll show how the optimal 3/4/5-bet strategies we have designed can be viewed as the final product of an evolutionary process based on our desire to defend against profitable bluffing with any two cards from aggressive opponents. The main point is that we don[apostrophe]t want to put ourselves in a situation where our opponent(s) can exploit us by bluffing profitably with any two cards, be it open-raising, bluff 3-betting, bluff 4-betting, or bluff 5-betting. An optimal strategy "plugs" all such openings for our opponents, but of course this defense does not come entirely without cost.

Through this discussion we[apostrophe]ll also shed light on the difference between optimal play and exploitative play, and when we should use one or the other. Optimal strategies put a lot of weight on defense, and they are not necessarily the most profitable strategies against players with big leaks. One reason is that optimal strategies include defensive components (for example, 4-bet bluffing as a defense against light 3-betting) that are often unnecessary against weak players (for example, we don[apostrophe]t need to 4-bet bluff against an opponent who only 3-bets premium hands like {JJ+,AK}).

Against players with big and easily exploitable leaks, we[apostrophe]d rather deviate from optimal play and play exploitatively to take full advantage of these leaks. But we need to be aware that by doing so we are creating openings in our strategies that can be exploited by observant opponents. So we have to find a balance between optimal and exploitative play, and we should use different strategies against different opponents. We will do our best to exploit weak players[apostrophe] big mistakes, but we can always fall back on optimal play against good opponents without big leaks. We can also return to optimal play if the player we[apostrophe]re trying to exploit with exploitative play suddenly changes his strategies to take advantage of the openings created by our exploitative strategies.

For example, let[apostrophe]s say we choose to never 4-bet bluff against a passive player who never 3-bet bluffs. He might now notice this, and adjust to our tight play by starting to 3-bet bluff us. Our exploitative adjustment against this particular opponent then runs the risk of getting counter-exploited if he starts 3-bet-bluffing us often with random weak hands. If this happens, we should return to our optimal optimal 3/4/5-bet strategy. Alternatively, we can make another exploitative adjustment to his adjustment by 4-bet bluffing him a lot (since we know he often is weak and have to fold). But the optimal strategy is always an alternative if we aren[apostrophe]t sure whether or not we can exploit his aggressive 3-betting.

In my opinion, this mindset is at the core of the thought processes of a strong NLHE player. He doesn[apostrophe]t have to use mathematics like we have done, but he will have a good feel for what an optimal (or near optimal) strategy is in the situation he is in. So he has a strong default strategy to fall back on against unknown players or known strong players, so that he can[apostrophe]t be easily exploited. But at the same time he knows how to deviate from optimal strategies to exploit his opponents[apostrophe] systematic leaks. So he can adjust his play in a controlled manner against each individual opponent instead of being locked into a static strategy that he uses against everyone.

Rules of thumb such as "never 4-bet bluff against fish" or "don[apostrophe]t 3-bet hands that perform poorly when called" are then replaced by a dynamical mindset that gives is strong control over our choice of strategies. Using optimal play as a starting point (and as a strategy we can always fall back on regardless of who we[apostrophe]re playing against), we can move around freely in "strategy land" and exploit opponent leaks as we pick up information about how they play.

Optimal play is never bad play, but exploitative play is always better. But we need information about our opponents[apostrophe] strategies before we can exploit them. If we don[apostrophe]t have this information, we can always fall back on optimal play as a good default.

[h1]2. Testing preflop strategies using the analysis software "Pokerazor"[/h1]

In this part of the article we[apostrophe]ll use [url=http://pokerazor.com]Pokerazor[/url] to study 2 things:

[LIST]

[*][b]1.[/b] How a tight openraise strategy without defense against light 3-betting is vulnerable to 3-bet bluffing with any two cards

[*][b]2.[/b] What the raiser can do to plug this leak, and how this leads to an optimal strategy pair for the raiser and the 3-bettor

[/LIST]

[h2]2.1 ABC preflop strategies and how these can be exploited[/h2]

Those of you who have played for a while probably remember the good old days (up to around 2007 or thereabouts) when micro and low limit NLHE was easily beatable by sticking close to the following rules of thumb for preflop play:

Those

[LIST]

[*]Open tight from all positions (say, 10-12% from UTG/MP, ~20% from MP and ~30% from the button

[*]3-bet only for value with {QQ+,AK}, and possibly {JJ+,AQ+} against a loose raiser

[*]When you get 3-bet and you are out of position, fold everything but {QQ+,AK} regardless of your position and where the 3-bet comes from

[*]Defend the blinds very tightly (typically 10%)

[/LIST]

Believe it or not, but this was more or less the standard getting-started preflop strategy recommended to beginning players at the micro and low limits up to $100NL or so. And it worked well, since the games were so loose and passive that it was correct both to openraise tight, and to fold a lot against 3-bets.

Those of you who have been members of Cardrunners for a while might remember Brystmar[apostrophe]s beginner video series "Small Stakes NL" in 6 parts (published during the spring of 2007). This series began with tight-aggressive preflop recommendations based on tight opening ranges and 3-betting only for value:

[b]Brystmar[apostrophe]s preflop strategy for micro/low limit NLHE[/b]

Let[apostrophe]s take a trip down memory land and study Brystmar[apostrophe]s preflop recommendations given 3.5 years ago. Those who want to read discussion about his video series or download his [url=http://rapidshare.com/files/155660787/crchartimg.jpg]preflop scheme[/url] can look at [url=http://www.cardrunners.com/cr_forums/showthread.php?t=25318#Post168442]this Cardrunners forum thread[/url].

Below is a summary of the default openraising ranges (note that "KTs" and "KTo" denote suited and offsuit hands, while "KT+" means both suited and offsuit:

[LIST]

[*][b]UTG openraise:[/b]

{22+,AJ+,KQs} =9.8%

[*][b]MP openraise:[/b]

{22+,AJ+,KQ} =11%

[*][b]CO openraise:[/b]

{22+,A7s+,A9o+,KT+,QTs+,QJo,J9s+,JTo,T9s} =19%

[*][b]Button openraise:[/b]

{22+,A4s+,A7o+,KT+,QTs+,QJo,J9s+,JTo,T8s+,

T9o,98s,98o,87s,87o,76s} =26%

[*][b]Raising from the small blind:[/b]

Openraise the button range if it gets folded to you. In a limped pot, raise {JJ+,AK} for value and overlimp all other hands from your button range, plus all Axs and Kxs.

[*][b]Raising from the big blind:[/b]

If the small blind openlimps, raise the button range and otherwise check. Out of position in a limped pot, raise {JJ+,AK} for value and otherwise check

[/LIST]

Tight opening ranges all around. This is of course not a leak our opponents can exploit, but we might perhaps say that we are exploiting ourselves by folding some profitable hands, particularly on the button.

But the strategies become easy to exploit when we get to playing against a raise:

[LIST]

[*][b]In MP with position on a raiser:[/b]

Reraise {JJ+,AK} for value and call with {TT-22,AJs+,AQo}

[*][b]In CO with position on a raiser:[/b]

Reraise {JJ+,AK} for value and call with {TT-22,AJ+,KQ,QJs,JTs}

[*][b]On the button with position on a raiser:[/b]

Reraise {JJ+,AK} for value (and AQo if the raise came from CO) and call with {TT-22,AJ+,KQ,QJs,JTs}. With callers between you and the raiser, also call with {JTo,T9s,98s,87s}

[*][b]In SB after a raise:[/b]

Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}

[*][b]In BB after a raise:[/b]

Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}. With callers between you and the raiser, also call with {QJs,JTs}

[/LIST]

We note two systematic errors in these strategies:

- We[apostrophe]re 3-betting more or less the same range regardless of the raiser[apostrophe]s position

- We[apostrophe]re never 3-bet bluffing

We remember from Part 1 and Part 2 that an optimal 3-betting range on the button varied from 3.6% against a ~15% EP openraise to 8.7% against a ~25% CO openraise. And in all scenarios we used an optimal bluffing frequency of 60%. In other words, more than half of our 3-bets were bluffs- In Brystmar[apostrophe]s strategies the 3-betting range is a tight and static value range {JJ+,AK} =3.0%, which is sometimes widened to {JJ+,AQ+} =4.2% against a wide openraising range. We also note that Brystmar chooses to include AQ when loosening up. This is a hand we never 3-bet for value in position when we[apostrophe]re playing optimally (since AQ works better as a flatting hand in position).

Brystmar[apostrophe]s strategies don[apostrophe]t mention defense against 3-betting, but we can assume that default defense is to 4-bet a tight range {QQ+,AK} from all positions. Another significant leak is the squeaky tight blind defense. For example, of button openraises, the preflop scheme tells us to 3-bet {JJ+,AQ} =4.2% from the small blind and flat {TT-22, AJ,KQ} =7.4%. This results in a total defense of 11.6%, which is way lower than the optimal defense threshold of 16% that we estimated in Part 3.

So there are huge openings in Brystmar[apostrophe]s preflop recommendations, and these openings can be easily exploited by an aggressive and observant opponent. We[apostrophe]re also leaving money at the table because we[apostrophe]re openraising to tight, and the main reason for this is that Brystmar does not take full advantage of position. We can openraise a ton of hands on the button when it[apostrophe]s folded to us, and we can make life hell for a raiser by 3-bet bluffing him in position, but Brystmar chooses not to do so.

NB! Before we move on I want to point out that I am not trying to put Brystmar[apostrophe]s low limit preflop defaults from 2007 in a negative light. His preflop recommendations for beginning NLHE players were very useful back in the day, and gave many new players an easy start. His strategies are best viewed as "training wheels" for staying out of trouble (and he no doubt saw them as such himself) and they were tailored towards the micro/low limit conditions that existed at the time. They will probably still work okay at the lowest micro limits, but I would not recommend anyone to play $25NL and higher with such tight and easily exploitable preflop strategies.

It[apostrophe]s clear for everyone who plays $25NL and higher these days that common NLHE strategy has developed in leaps and bounds since Brystmar[apostrophe]s 2007 recommendations. Light 3-betting was rare in the "old days", even at $100NL and $200NL. Today it[apostrophe]s common, even if you begin as low as $5NL.

The next step of the development of the average low limit NLGE regular back in the day was to add some light 3-bets in position (and Green Plastic[apostrophe]s 2006/2007 NLHE videos at Cardrunners inspired many to do so), call more raises and 3-bets in position, and in general get better at using positional advantage. A common mistake many aggressive players did was to 3-bet bluff with hands that were too strong to use as bluffs (for example, JTs). However, this did not cause man problems since most players defended poorly against 3-bets, particularly from out of position.

So the standard recipe in the good old days for an advanced low limit player who wanted to ramp up the aggression was to LAG it up in position. But not necessarily with balance in mind, and not necessarily with a good understanding of how to chose his value range, bluffing range and flatting range in a consistent manner. But this was not a big deal. He played tight out of position, opened a very wide range in position, and 3-bet something fierce in position against weak opponents. The 3-betting was very effective, since the raisers often did one of the following two mistakes:

- Folded a lot out of position and never 3-bet bluffed

- Called a lot with non-premium hands out of position

The first mistake lets the 3-bettor print money by giving him an opening to (in principle) 3-bet bluff any two cards. The second mistake occurs when the raiser tries to correct the first mistake, but he goes about it the wrong way. Defending against 3-bets by flatting weak hands out of position is ineffective, since the raiser now has to play postflop out of position in a scenario where it[apostrophe]s difficult for him to win without hitting the flop well. Playing weak starting hands well out of position against a good LAG player is hard, and often results in you losing more money postflop than if you had just folded to the 3-bet preflop.

The cure against light, positional 3-betting is of course to respond by 4-betting a correct value range (which follows from the size of our opening range), balanced with a correct amount of 4-bet bluffing. We have studied this in previous articles, and we have defined optimal strategies for the raiser from all positions. The 3-bettor uses similar thinking to design his 3-bet strategy so that the raiser can not 4bet bluff any two cards profitably. This way an equilibrium gets established.

This equilibrium is given by the optimal strategy pairs for the raiser and the 3-bettor defined in Part 1 and Part 2. We used mathematics to define these strategy pairs, but we can also think about them as a product of an evolutionary process.

The 3-better starts out by exploitative ant-two-cards 3-bet bluffing against a raiser that defends way too tight and folds too much. Then the raiser adjust by choosing a correct value range and introducing 4-bet bluffing. The 3-bettor responds by adjusting his value range and introducing 5-bet bluffing. To prevent the opponent from bluffing with any two cards anywhere, both players fine-tune their ranges until both are using an optimal set of ranges for 3/4/5-betting. "Optimal" here means that neither player can improve his EV by adjusting further. If one of them tries to do so, he is giving the other player an opportunity to increase [i]his[/i] EV by making and exploitative adjustment.

We will now illustrate such an evolutionary process using Pokerazor simulations:

[h2]2.2 Numerical testing of optimal heads-up 3/4/5-bet strategies[/h2]

We start with the following model:

[LIST]

[*]Alice (100bb) openraises to 3.5bb with her standard 25% range from CO

[*]Bob (100bb) is on the button and 3-bets to 12 bb or folds

[*]Alice defends against 3-betting by 4-betting to 25 bb or folding

[*]Bob defends against 4-betting by 5-betting all-in or folding

[*]Alice defends against 5-betting by calling all-in or folding

[*]The blinds always fold, no matter what Bob does

[/LIST]

So we are studying a scenario where Alice openraises, Bob 3-bets or folds, and the blinds never get involved. Alice then makes 1.5 bb (the blinds) per raise when Bob folds, which is a win rate of 150 bb/100. This is her baseline EV for the simulation.

Before we begin the simulations, let[apostrophe]s repeat the ranges and optimal strategy pairs we defined in Part 2:

[b]Alice[apostrophe]s 25% open-range from CO[/b]

[pre]22+

A2s+ A9o+

K9s+ KTo+

Q9s+ QTo+

J8s+ JTo

T8s+

97s+

87s

76s

65s

326 combos

25%[/pre]

The corresponding optimal strategy pair that[apostrophe]s being used when Bob 3-bets in position can be found from the summary of optimal strategy pairs in Part 2:

[img]http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-6/IP_3-bet_summary.png[/img]

Here is a download link for this document (right-click and choose "Save as"):

[url=http://dev.donkr.com/articles/bugs/v2/oppnlhe/oppnlhe-6/IP_3-bet_summary.doc]IP_3-bet_summary.doc[/url]

The optimal strategy pair is then:

[LIST]

[*][b]Bob:[/b]

3-bets {QQ+,AK,12 air} ={QQ+,AK,A5s-A3s} for value (including 5-bet-bluffing with Axs hands) and 70% av "IP 3-bet air list" as a bluff using a [url=http://www.random.org/]randomizer[/url]

[*][b]Alice:[/b]

4-bets {TT+,AQ+} for value and {AJ,AT,A9s-A7s} as a bluff

[/LIST]

[b]Pokerazor simulation 1 (Bob folds)[/b]

The baseline simulation is to let Bob fold 100%. Alice then picks up the blinds, and makes 1.5 bb each time (=150 bb/10):

- Alice openraises 25% from CO

- Bob folds

[pre]EV (baseline for Alice) =150 bb/100[/pre]

Bob now begins 3-betting so that Alice ends up with EV [lt] 150 bb/100. It[apostrophe]s obvious that Alice[apostrophe]s EV now becomes lower than the 150 bb/100 baseline, since she can not prevent Bob from making money by 3-betting only his best hands, for example his {QQ+,AK} default value range against her 25% CO openrange. Furthermore, it[apostrophe]s correct for Alice to never 4-bet bluff when Bob never 3-bet bluffs, so she has to fold the hands not strong enough to 4-bet for value against Bob[apostrophe]s tight value range and let Bob pick up the blinds.

We start by assuming that Bob never 3-bet bluffs and that Alice defends against the 3-bet by 4-betting the value range {KK+} after observing that Bob[apostrophe]s 3-betting range is {QQ+,AK}. This is correct since Alice does not want to get all-in preflop with QQ or AK against Bob[apostrophe]s {QQ+,AK} range (both are ~40% underdog) with only 3.5 bb invested.

[b]Pokerazor simulation 2 (Bob 3-bets only for value)[/b]

Bob 3-bets his value hands. We let him use the pure value hands {QQ+,AK} that he would have used in an optimal strategy against Alice[apostrophe]s 25% CO openrange, and the drops his 5-bet bluffs for now:

- Alice openraises 25% from CO

- Bob 3-bets {QQ+,AK} for value and 5-bets all-in against a 4-bet

- Alice 4-better {KK+} for value

And the rest follows automatically. We get:

[pre]EV (Alice; KK+) =141 bb/100[/pre]

If Alice also had 4-bet QQ/AK for value, the EV would have become:

[pre]EV (Alice; QQ+,AK) =139 bb/100[/pre]

So Alice[apostrophe]s choice of value range against Bob[apostrophe]s extremely tight 3-betting range is correct. Note that [i]Alice is now exploiting Bob by only 4-betting an extremely tight {KK+} value range![/i] What happens here is that Bob mostly leaves Alice alone so that she can pick up the blinds almost every time. Bob only pops up with a value 3-bet the 2.56% of the time he has {QQ+,AK}, and Alice then responds by folding everything but {KK+}.

So Alice exploits Bob[apostrophe]s squeaky tight 3-betting by not paying off his value hands with weaker hands. And since Bob never 3-bet bluffs, there is never any doubt about his range. Alice can then play perfectly against a 3-bet and drop all her (now unnecessary) 4-bet bluffs as well as her weakest value hands (and we remember that Alice[apostrophe]s optimal value range in CO is {TT+,AQ+}). Exploiting someone by folding a lot to his aggression is not the first thing that comes to mind when poker players think about exploitative play. But avoiding paying off a strong range unnecessarily is just as profitable as playing aggressively against players that fold too much.

If Alice had responded with her optimal 4-bet strategy which is {TT+,AQ} for value and {AJ-AT,A9s-A7s} as bluffs, we get:

[pre]EV (Alice; optimal strategy) =126 bb/100[/pre]

And we see that [i]Alice[apostrophe]s optimal 3/4/5-bet strategy out of position loses relative to the best exploitative strategy[/i] (which gave her 141 bb/100). This is an example of a general rule: If we have an opportunity to exploit someone, we will make more money from this than from continuing to use an optimal strategy. The reason is that Alice now sacrifices EV by defending against a non-existing threat. She has 4-bet bluffs in her 4-betting range to defend against Bob[apostrophe]s 3-bet bluffs, but Bob never 3-bet bluffs.

So even if Alice[apostrophe]s optimal strategy can not be exploited by any-two-cards 3-bet bluffing from Bob, [i]this defense is costing her EV relative to a strategy that exploits Bob[apostrophe]s very tight never-bluff 3-betting strategy[/i]. A good analogy would be a nations defense budget in peace time. There has been no warfare on Norwegian soil since 1945, but Norway has still had a defense budget in all the years since then. We are simply paying for a military defense so that other nations don[apostrophe]t get an opportunity to invade us without risk.

Now, let[apostrophe]s assume that Bob has been sitting there and 3-betting only {QQ+,AK} for value and folding everything else, while Alice has responded by 4-betting only {KK+} for value and folding everything else. Bob has observed that Alice almost always folds. It[apostrophe]s now easy for him to reach the conclusion that he could make a lot more money by throwing in some 3-bet bluffs, planning to fold then when Alice 4-bets. Since Bob does not have to worry about the blind players waking up with a hand, let[apostrophe]s allow him to 3-bet any two cards to maximally exploit Alice[apostrophe]s super tight defense against 3-bets:

[b]Pokerazor simulation (Bob 3-bets any two cards)[/b]

Bob uses his read on Alice, and changes his strategy to exploit her. He keeps 3-betting {QQ+,AK} for value, and then he 3-bets all other hands as a 3-bet bluff. Alice, not knowing that Bob suddenly has changed his strategy, keeps 4-betting {KK+} for value and folding everything else:

- Alice openraises 25% from CO

- Bob 3-bets {QQ+,AK} for value + any two cards as a 3-bet bluff

- Alice 4-bets {KK+} for value

[pre]EV (Alice) =-286 bb/100[/pre]

Alice now gets slaughtered by Bob[apostrophe]s any-two-cards 3-bet-bluffing [i]and she actually loses money from trying to steal the blinds from CO[/i]. Her first adjustment is to return to the optimal value range {TT+,AQ+} associated with her 25% CO openrange. This gives us:

[pre]EV (Alice; optimal 4-bet value range) =-39 bb/100[/pre]

It helps, but not enough. She will still lose money for every hand she openraises, unless she also starts 4-bet bluffing. Alice now returns to her complete optimal strategy in CO, designed to defend against any-two-cards 3-bet bluffing:

[pre]EV (Alice; optimal total 4-bet-range) =+179 bb/100[/pre]

Bingo! Alice[apostrophe]s optimal 4-bet-strategy not only prevents Bob from exploiting her by 3-bet bluffing with any two cards, it also punishes him for it. Alice now makes 179 - 150 =+29 bb/100 more than if Bob had simply folded every time and let her pick up the blinds.

But is Alice[apostrophe]s optimal strategy the most profitable strategy against Bob[apostrophe]s any-two-cards 3-bet bluff strategy? No, and to find Alice[apostrophe]s most profitable strategy we use common sense. When Bob is 3-betting any two cards, but only continues with {QQ+,AK} after a 4-bet, he is extremely vulnerable for 4-bet bluffing. He only continues with 2.56% of his hands and folds the remaining 97.44% against Alice[apostrophe]s 4-bets

Alice can then maximally exploit Bob[apostrophe]s attempt to exploit her [i]by 4-bet bluffing him with any two cards[/i] (or rather, any two cards in her original opening 25% opening range). We let Alice continue with her optimal {TT+,AQ+} value range. Then she deviates from the optimal strategy by widening her 4-bet bluffing range to the rest of her 25% CO opening range. Bob, unaware that Alice has suddenly changed her strategy to exploit his strategy, keeps 3-bet bluffing any two cards:

[pre]EV (Alice; 4-bet bluff any two cards) =+1261 bb/100[/pre]

A huge increase in EV for Alice, [i]and Alice now makes about 8 times more than if Bob had folded every hand[/i]. This is an extremely clear illustration of the difference between optimal and exploitative play. Alice[apostrophe]s optimal defense strategy against Bob[apostrophe]s 3-bets guarantees that he can not exploit her by 3-bet bluffing any two cards, but he optimal defense did nothing to counter-exploit Bob extreme strategy. Alice made a little bit more than if Bob had folded every hand, but not a lot ((179 bb/100 vs 150 bb/100) .

But when Alice responds by choosing the strategy that exploits Bob[apostrophe]s any-two-cards-bluffing strategy maximally, her EV explodes. Still, this does not come without risk for her, since Bob can take his exploitative strategy to the next level and begin 5-bet bluffing any two cards to exploit Alice[apostrophe]s any-two-cards 4-bet bluffing. Alice must then make a new exploitative adjustment (for example, widening her value range dramatically and calling Bob[apostrophe]s 5-bets with a very wide range of value hands) to stay one step ahead of Bob.

We can view this exploit/counter-exploit process as "strategic ping-pong" where both players zig and zag, using extreme strategy changes in different directions to maximally exploit their opponent. When one of them has made a big strategy change to exploit her opponent, she also creates an opening that the opponent can exploit by making an adjustment of his own. Then the first player has to make another big strategy adjustment, and the process repeats itself ad infinitum.

When we are playing optimally, we are using a different mindset. Instead of trying to stay one step ahead of our opponents by sudden "gear changes", we can fall back on a strategy that performs more or less equally well no matter what our opponent does. If he uses an extreme strategy, our optimal strategy will win a bit from him, but maximizing our profit from his mistake(s) is not our main goal. Instead, we simply want to prevent him from exploiting us. In the simulations above we saw that this worked well for Alice, but not as well as the maximally exploitative strategy she can use against Bob[apostrophe]s any-two-cards 3-bet bluffing.

Okay Bob, now what? Bob can of course respond to Alice[apostrophe]s any-two-cards 4-bet bluffing by going to the next level and start 5-bet bluffing with any two cards. Alice then gets exploited for a while, until she realizes this and adjusts. But we will not take any-two-cards bluffing beyond what we did in the previous simulations, and we assume that Bob now adjusts by falling back on his optimal 3-betting strategy. Alice in turn falls back on her optimal 4-betting strategy:

[b]Pokerazor simulation 4 (both players use optimal strategies)[/b]

Both Alice and Bob now uses the optimal strategy. As we have seen in previous articles, this optimal strategy pair follows from Alice[apostrophe]s openrange. Both players are now protecting themselves against any-two-cards bluffing from their opponent in all phases of the 3/4/5-bet war.

Alice[apostrophe]s EV now becomes:

[pre]EV (Alice; both players 3/4/5-bet optimally) =+129 bb/100[/pre]

In the last round of simulations we will let one player stick to the optimal strategy while the other player is let "off the leash", free to try anything to increase her or his EV against the other player[apostrophe]s optimal strategy. Then we compare the resulting EV with EV when both players play optimally (129 bb/100 for Alice). We start out with Bob playing optimally, while Alice is testing out some deviations from her optimal strategy:

[b]Pokerazor simulation 5 (Bob plays optimally and Alice can do what she wants)[/b]

We remember that Alice makes 150 bb/100 when Bob doesn[apostrophe]t interfere, and Bob[apostrophe]s optimal 3-betting strategy reduces this to 129 bb/100 when Alice plays optimally too. We shall now see that Alice can[apostrophe]t do anything to increase her EV significantly when Bob sticks with his optimal strategy.

For example, let[apostrophe]s assume that Alice drops the weakest hands TT/AQ from her value range when she sees that Bob uses the strong value range {QQ+,AK} (and remember that he also 3-bets/5-bets with his 5-bet bluffs A5s-A3s). Alice then chooses to 4-bet TT/AQ as before, but she folds them when Bob 5-bets all-in:

[pre]EV (Alice; folds TT/AQ to 5-bet) =+128 bb/100[/pre]

Alice[apostrophe]s EV drops a little, and she can[apostrophe]t increase her EV by playing a tighter value range against Bob[apostrophe]s strong value range. The reason is that Bob has put exactly so many 5-bet bluffs in his 5-betting range that Alice becomes indifferent to folding or calling with her weakest value hands. When Alice tries to avoid paying off Bob[apostrophe]s better value hands, his 5-bet bluffs makes more money.

Can Alice increase her EV by 4-bet-bluffing more than optimally? We test this by letting her 4-bet bluff any two cards against Bob[apostrophe]s optimal strategy, keeping everything else optimal:

[pre]EV (Alice; 4-bet-bluffs any two cards) =+139 bb/100[/pre]

A small EV increase, but nothing comparable to the results of the extreme exploitative any-two-cards adjustments in previous simulations. Note that a perfectly optimal strategy for Bob should make it impossible for Alice to increase her EV, but our optimal strategy implementations probably contain some "numerical noise", since we have made some approximations and rounding along the way.

At any rate, Alice[apostrophe]s attempt to exploit Bob[apostrophe]s optimal strategy with any-two-cards 4-bet bluffing only results in a small EV change. Bob[apostrophe]s optimal strategy therefore protects him well against exploitative 4-bet bluffing from Alice. And if Bob wanted to, he could probably fine-tune his strategy (for example, remove a couple of 3-bet bluff combos) to eliminate Alice[apostrophe]s small EV increase completely.

So we have seen that Bob[apostrophe]s side of the optimal 3/4/5-bet strategy pair seems robust against Alice[apostrophe]s extreme adjustments. Let[apostrophe]s now turn to Alice, and let her play the optimal strategy while Bob is allowed to do whatever he wants:

[b]Pokerazor simulation 6 (Alice plays optimally, while Bob can do what he wants)[/b]

We saw previously that Bob[apostrophe]s attempts to exploit Alice[apostrophe]s optimal strategy by 3-bet bluffing any two cards didn[apostrophe]t work for him. He lost against her optimal strategy (her EV increased from the baseline 150 bb/100 to 179 bb/100), and he lost a lot when Alice counter-exploited him by 4-bet bluffing any two cards (her EV increased to 1261 bb/100). We shall now repeat this simulation by using Bob[apostrophe]s optimal strategy as a starting point, and then we make the adjustment that he 3-bet bluffs any two cards on top of that. All other ranges for value and 5-bet bluffing are as in the optimal strategy:

[pre]EV (Alice; Bob 3-bet-bluffs any two cards) =+179 bb/100[/pre]

Alice[apostrophe]s EV increases with +50 bb/100 (from 129 bb/100) relative to Bob[apostrophe]s optimal strategy, and +29 bb/100 relative to the baseline EV when Bob always folds (150 bb/100). And we remember that if Alice wants to, she can exploit Bob hard by 4-bet bluffing any two cards and pocket more than 1200 bb/100 until Bob adjusts back. So Bob can[apostrophe]t increase his EV by aggressive 3-bet bluffing, which is what we expected.

Then we let Bob 3-bet bluff any two cards and also 5-bet bluff any two cards. In other words, we let him play like a complete maniac, where he 3-bets any two cards and then 5-bets any two cards if he gets 4-bet.

[pre]EV (Alice; Bob 3-bet-bluffs/5-bet-bluffs any two cards)

=+300 bb/100[/pre]

This causes Alice[apostrophe]s EV to more than double relative to the 129 bb/100 she has against Bob[apostrophe]s optimal strategy. We conclude that Alice[apostrophe]s optimal strategy is waterproof against any-two-cards 3-bet bluffing, and that Bob only hurts himself if he tries.

The results from the last two simulations are worth noticing, since they can be uncomfortable scenarios to play when you don[apostrophe]t know whether or not you are defending correctly. But as we have seen, even against a total maniac who 3-bets you from position at every opportunity, you don[apostrophe]t have to do anything else than respond with your memorized optimal 4-bet strategy. If you do this, he will loose money relative to playing an optimal strategy himself, and he will probably end up losing money overall (so that 3-betting is worse for him than folding).

Note that an aggressive 3-bettor can still reduce our EV relative to the baseline EV (i.e. we will make less when he sometimes 3-bets than when he always folds), which is of course intuitively obvious since he has a lot of strong hands in his 3-betting range as well. For example, he could choose to 3-bet only AA, and we could not do anything to prevent him from making money in this situation and reduce our EV. ). So if we always respond with our optimal strategy, we can[apostrophe]t deny the 3-bettor some +EV.

For example, if Bob deviates from his optimal strategy by increasing his 3-bet bluffing from 70% of "IP 3-bet air list" to 100% of the list, we get:

[pre]EV (Alice; Bob 3-bet-bluffs all of "IP 3-bet air list")

=+130 bb/100[/pre]

We still make a little bit more (129 bb/100 --[gt] 130 bb/100) when Bob increases his 3-bet bluff percentage beyond the optimal percentage, but he still reduces our EV relative to our baseline EV when he always folds (150 bb/100 --[gt] 130 bb/100). We can[apostrophe]t prevent Bob from making some money in this situation, and we just have to accept that a player in position has the right to make money by 3-betting us. Of course, our openraise will still be nicely profitable overall, just less profitable than if he had always folded behind us.

As we saw previously, we can exploit a complete maniac by deviating from optimal play to take advantage of the gaping holes in his strategy, particularly if he folds too much to 4-bets. If he lets us exploit him, we can make [i]more[/i] money from an exploitative strategy than from our optimal strategy. But then we have to play guessing games with him, and we also run the risk of offering big openings to the other players at the table (they can deviate from optimal play to exploit our non-optimal play). Since an optimal strategy will protect us (and then some) from getting exploited by a wild 3-bettor, this trade-off might not be worth it

A couple of obvious adjustments we can use to exploit a very aggressive with position on us 3-bettor are:

[LIST]

[*]4-bet bluff more, if he folds easily to 4-bets (in other words, he defends his loose 3-betting range to tightly)

[*]Drop 4-bet bluffing, but 4-bet more hands like AJ, AT, 99, 88, etc. for value, if he folds too little to 4-bets and calls and 5-bets a lot with weak hands

[/LIST]

But we don[apostrophe]t have to make these adjustments to defend out of position against overly aggressive 3-betting. Our optimal strategy is more than enough. It might [i]feel[/i] like we[apostrophe]re getting exploited, and some of the reason for that is that a strategy where we fold a lot (70% in the optimal strategy, as explained in Part 1) feels "weak". But the reality is that a maniac 3-bettor in position ends up costing himself if he starts 3-betting any two cards against our optimal strategy. Keep this in mind every time you feel exploited by a 3-bettor in position.

[h1]3. Summary[/h1]

We have tested optimal strategy pairs for heads-up 3/4/5-betting using the analysis software Pokerazor. We started with a discussion of ABC preflop strategies without 3-bet bluffing or 4-bet bluffing. We then used simulations to show how ineffective and vulnerable these strategies are against players who are capable of reraising as a bluff with any two cards. As a part of this simulation we looked at exploitative adjustments we can make against players with big leaks in their 3/4/5-bet strategies.

Then we tested the robustness of the optimal 3/4/5-bet strategies we defined in previous articles, with the raiser out of position. We concluded that both the raiser[apostrophe]s and the 3-bettor[apostrophe]s optimal strategies were robust, and that they did not give the opponent openings he could exploit by bluffing with any two cards.

In Part 7 we[apostrophe]ll do numerical simulations for flatting heads-up in position. Among other things we[apostrophe]ll compare EV for flatting versus 3-betting for value with hands that are in between clear value hands and clear flatting hands (for example QQ against a tight UTG raiser). In the last half of Part 7 we[apostrophe]ll adjust our heads-up 3/4/5-bet strategies for blind vs blind scenarios.

Good luck!

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