1. Introduction



This is Part 3 of the article series "PLO From Scratch". The target audience is micro and low limit players with some experience from limit or no-limit Hold'em, but little or no PLO experience. My goal with this series is to teach basic PLO strategy in a systematic and structured manner.

In Part 3 we will continue the discussion of principles for PLO preflop play. My original plan was to let this article be a combination of theory and practical guidelines, focusing on 3-betting, raising to isolate, and overlimping. But as the work progressed it became clear that there was too much material for only one article.

I therefore decided to let Part 3 be a purely theoretical article where we dive deeply into the concept of playability of PLO starting hands, both as a function of hand structure and as a function of how much money we put into the pot preflop. Then we will move on to more practical guidelines in Part 4, and use the theory from Part 3 as a tool.

We have previously talked about how PLO preflop and postflop play is closely knit together, and how the goal of our preflop play is to set ourselves up for profitable postflop situations. An extremely important concept that ties together preflop play and postflop play is flop equity distributions. This will be a useful tool for us, and we will study flop equity distributions thoroughly so that we can use it in our discussion of preflop play (and later also postflop play).

To use flop equity distributions quantitatively, we need some mathematics, and among other things we need to learn how to use numerical integration to extract useful data from flop equity distribution curves. But if this sounds Greek to you, don't panic!

Understanding the mathematical details is not necessary to understand what is going on, and the mathematical technicalities have been put in an Appendix. Those with an interest in mathematics can read the Appendix, the others can skip that part and simply use the numerical results.


2. An Introduction to Flop Equity Distributions



First, let's get an idea about what a flop equity distribution is. We go to ProPokerTools.com and enter KKxx vs AAxx as shown below (with KKxx at the top.

First, we calculate the equity for KKxx vs AAxx the usual way by clicking "Simulate":





Then we return to start and click "Graph" as shown below. This produces the flop equity distribution curve for KKxx vs AAxx:





(For the rest of the article: Click on all graphs to open them in full size in a separate browser window, so that you can see all the details.)

So what is this graph telling us? Simply put, it tells us how often KKxx has a certain minimum flop equity against AAxx (more about this in a moment). Furthermore, the total equity for KKxx vs AAxx equals the total area under the curve between 0 and 100% as shown below:



We will now formulate a series of questions and find their answers in order to illustrate how we will be using flop equity distributions in this article.


2.1 How often do we have at least x% equity on the flop?



For example, we can ask: How often does KKxx have at least 50% equity against AAxx? We find the answer by looking at the graph and finding the point where the graph has the value 50%, as shown below.:



We see that KKxx has minimum 50% equity on the top 19% of flops.


2.2 What is the total equity on top x% of flops?



For example, how much of the total equity that KKxx has against AAxx can be found between 0 and top 19% of flops? This is equivalent to finding the area below the graph between 0 and top 19% of flops, as shown below (the colored area under the curve). :



The answer is that we have 0.149 =14.9% total equity on the top 19% of flops, which is roughly half of our total 30.16% equity (we can verify this manually by recognizing that the colored area makes up about half of the total area under the graph).

We shall soon see why this number is useful for is, but first: How do we compute this area? ProPokerTools does not give us this number directly, and to compute it we have to resort to a mathematical technique called numerical integration.

I'm assuming the details of numerical integration will be a bit too technical for the majority of the readers, so I have put this material in an Appendix. Those who want to see how the calculations are done can read the Appendix and then return here. The rest can move on to the final question:


2.3 What is our average equity on the top x% of flops?



We have established that KKxx has minimum 50% equity against AAxx on the top 19% of flops, and that the top 19% of flops contains 14.9% equity in total. The next thing we want to know is: What is our average equity on the top 19% of flops? All flops in this region give us at least 50% equity, but how much equity do we have on average when we hit one of them?

The answer is simple:


The average equity on top x% of flops equals the total amount of equity that lies in this region (which is equal to the area under the curve in this region) divided by the width of the region (which is x).

So our average equity on the top 19% of flops for KKxx vs AAxx is equal to the area under the curve between 0 and top 19% of flops (which is 0.149) divided by the width of the region (which is 0.19 - 0 =0.19):

Average equity
=0.149/0.190
=0.784
=78.4%

This is also summarized on the graph below, and from here on we will use notation on this graph on all future graphs where we compute average equity on top x% of flops.




2.4 Summary of the interpretation of data from flop equity distributions



We started with an example, KKxx vs AAxx, and learned how to read from the graph the top x% of flops where we have some minimum equity. For example, KKxx has minimum 50% equity against AAxx on the top 19% of flops.

Then we asked what the total amount of equity was on the top x% of flops. We learned that this was equivalent to finding the area under the flop equity distribution curve between 0 and x% of flops. For example, we found that we had a total amount of equity of 14.9% on the top 19% of flops for KKxx vs AAxx.

Finally, we asked what our average equity was on the top x% of flops. For example, we found that KKxx has 78.4% average equity against AAxx on the top 19% of flops.

So KKxx has minimum 50% equity against AAxx on the top 19% of flops, and the average equity on the top 19% of flops is 78.4%.

This is a calculation we will be using repeatedly throughout this article when studying the playability of various starting hands. So make sure you know what these numbers mean before you move on. For the rest of the article I will simply be presenting these results without detailed calculation, but I will include graphs where all the numerical data are written down, so that you can verify the calculations if you want to.


3. Modeling the Playability of Various Starting Hands Against AAxx



To practice using flop equity distribution data in strategic modeling, we will now do a simple study of the playability of 3 different types of PLO starting hands against AAxx. We will use a simple model where we assume that our opponent is totally committed with AAxx and that he will bet and raise at any opportunity until we are all-in.

The purpose of this study is:


  • To learn about the playability of various PLO starting hands as a function of pot size
  • To learn about the relation between flop equity distributions and playability
  • To learn specifically about playing AAxx and playing against AAxx
  • To learn how to use flop equity distributions in strategic modeling



3.1 Description of the model



We will study the playability of the following 3 starting hands against AAxx:

- 9 8 7 6 (double-suited perfect rundown)
- A K Q T (single-suited Broadway wrap)
- K K 7 2 (KK with worthless side cards)

We will let each of these hands meet AAxx in a raised pot, a 3-bet pot and a 4-bet pot with 100 BB starting stacks. Our opponent will be fully committed with AAxx, and he will bet and raise at every opportunity until we are all-in. For each case we will compute the EV of playing our hand from the moment we put the first chip into the pot preflop.

Villain's strategy
Villain's strategy is to bet and raise his AAxx hand at every opportunity until we are all-in.

Our strategy
We let 1, 2 or 3 pot-sized bets go into the pot preflop. Postflop we commit on the flop whenever we have the minimum equity necessary to do so. The minimum necessary equity is a function of our effective pot-odds for committing on the flop, which is given in the description for each scenario below.

Note that our model makes less sense when the preflop pot is small. Our strategy is to wait for flops good enough to commit fully on, and then we commit all our chips on the flop (meaning there is never any turn or river play). Strictly speaking this strategy only makes sense when there is only 1 or 2 bets left in the stack on the flop, so that we're in a commit-or-fold scenario.

But the purpose of this study is to investigate how different types of starting hands play against AAxx within the framework of a simple model, and we're not trying to model optimal postflop play. So even though our model is not completely realistic for small pots, it's good enough for our purpose.

The 3 scenarios we will study for each hand are:


Raised pot
- Villain raises pot (3.5 BB), we call.
- Pot on the flop: 8.5 BB
- Stack on the flop: 96.5 BB
- Effective pot-odds for committing on the flop: (8.5 + 96.5) : 96.5 =1.09 : 1
- Minimum equity for committing on the flop: 1/(1.09 + 1) =0.48 =48%

3-bet pot
- We raise pot (3.5 BB), Villain 3-bets pot (12 BB), we call
- Pot on the flop: 25.5 BB
- Stack on the flop: 88 BB
- Effective pot-odds for committing on the flop: (25.5 + 88) : 88 =1.29 : 1
- Minimum equity for committing on the flop: 1/(1.29 + 1) =0.44 =44%

4-bet pot
- Villain raises pot (3.5 BB), we 3-bet pot (12 BB), Villain 4-bets pot (37.5 BB), we call
- Pot on the flop: 76.5 BB
- Stack on the flop: 62.5 BB
- Effective pot-odds for committing on the flop: (76.5 + 62.5) : 62.5 =2.22 : 1
- Minimum equity for committing on the flop: 1/(2.22 + 1) =0.31 =31%

Calculating EV
We will be using the mathematical techniques we introduced in the discussion of flop equity distributions previously in this article. We start by running over to ProPokerTools.com to compute the flop equity distribution graphs for each hand match up:





For each scenario we know the minimum flop equity we need to commit on the flop (we computed these in the description of each scenario above), so we start by asking: On what top x% of flops do we have the minimum equity or better? The answer is top x% for some value of x that we read from the graph.

Then we ask: What is the total amount of equity for our hand (e.g. the area under the curve) on the top x% of flops? We compute this number using numerical integration as described in the Appendix.

Finally, we ask: What's our average equity on the top x% of flops?. And we compute this number as previously described (total equity on top x% of flops divided by x).

We now have all the data necessary for computing our EV for playing out the scenario. The EV equation is:


EV =(1 - top_x)(-pf_bb) + top_x{av_equity(201.5) - 100}

where

- top_x =the top x% of flops with the minimum equity for committing
- pf_bb =the number of big blinds that goes into the pot preflop
- av_equity =our average equity on the top x% of flops

The interpretation of the EV equation goes like this:

There's a (1 - top_x) probability we will have to to check-fold the flop and lose our preflop investment pf_bb. The probability of hitting the flop well enough to commit is top_x, and in this case we have an average equity of av_equity in a 201.5 BB pot (our stack + Villains stack + the blinds) where we risk a total of 100 BB.

Now over to the simulations. For each scenario we include a graph with all the necessary numerical data written down (click on the graph to open it in full size in a separate browser window), and then we plug the numbers into the EV equation and compute our EV for the scenario. For each hand we find our EV in a raised, 3-bet and 4-bet pot. Finally, we do a summary where we draw conclusions about how the hand plays against AAxx as a function of pot size.


3.2 9 8 7 6 vs AAxx




Raised pot



top_x =42%
pf_bb =3.5
av_equity =68.9%

EV =(1 - 0.42)(-3.5) + 0.42{0.689(201.5) - 100} =+14.28 BB


3-bet pot



top_x =48%
pf_bb =12
av_equity =66.1%

EV =(1 - 0.48)(-12) + 0.48{0.661(201.5) - 100} =+9.66 BB


4-bet pot



top_x =65%
pf_bb =37.5
av_equity =58.5%

EV =(1 - 0.65)(-37.5) + 0.65{0.585(201.5) - 100} =-1.46 BB


Summary for 9 8 7 6 vs AAxx

EV (raised pot): +14.28 BB
EV (3-bet pot): +9.66 BB
EV (4-bet pot): -1.46 BB

Before looking at the EV results, we can note that 9 8 7 6 has a very smooth flop equity distribution curve with the equity evenly distributed over a wide range of flops.

This kind of curve indicates a hand that often hits the flop fairly well, which is what double-suited quality rundown hands like 9 8 7 6 do. We will often flop some combination of made hand + draw that is good enough to continue.

The smooth equity distribution of 9 8 7 6 means the hand plays well in big pots against AAxx, simply because we often hit the flop hard enough to commit on the flop and get a return on our preflop investment.

9 8 7 6 plays well in raised and 3-bet pots, and we have to go all the way to a 4-bet pot where 37.5% of the stack goes in preflop before the hand becomes unprofitable to play within the framework of our model (and even in this case we're barely below break even)

We will return to the topic of playing quality rundown hands against AAxx in Part 4, but at this point we clearly see why quality rundowns are good 3-betting hands. They play well in big pots, and even if we run into AAxx it's not the end of the world.



3.3 A K Q T vs AAxx




Raised pot



top_x =22%
pf_bb =3.5
av_equity =68.6%

EV =(1 - 0.22)(-3.5) + 0.22{0.686(201.5) - 100} =+5.67 BB


3-bet pot



top_x =26%
pf_bb =12
av_equity =64.9%

EV =(1 - 0.26)(-12) + 0.26{0.649(201.5) - 100} =-0.90 BB


4-bet pot



top_x =43%
pf_bb =37.5
av_equity =53.7%

EV =(1 - 0.43)(-37.5) + 0.43{0.537(201.5) - 100} =-17.88 BB


Summary for A K Q T vs AAxx

EV (raised pot): +5.67 BB
EV (3-bet pot): -0.90 BB
EV (4-bet pot): -17.88 BB

A K Q T has a fairly smooth flop equity distribution curve, but compared to 8 7 6 more of the equity is "crammed together" on fewer flops (to the left of the curve). So A K Q T is more of an "either-or" hand than 9 8 7 6

This hand plays fairly well against AAxx in a raised pot, but we run into problems when the pot grows big. The reason for this is that we're effectively playing a "dangler hand" with only 3 playable cards when we're up against AAxx, since the ace on our hand does not do much for us in this case.

The "dangler" in our hand makes it harder to hit flops well enough to continue postflop, and this has consequences as the preflop pot grows bigger. The more chips we put into the pot preflop, the more important it is to hit a lot of flops, so that we won't have to check-fold often and lose our preflop investment. This problem becomes very clear in a 4-bet pot where we lose 18 BB.

From this we can draw an important conclusion: When we're playing an ace high Broadway wrap (4 Broadway cards headed by an ace) and get 4-bet preflop, we have to fold with a 100 BB stack if we suspect we're up against AAxx. To call, we need to be playing much deeper stacks, so that the AAxx hand can not simply push the last pot-sized bet all-in on any flop and be done with the hand. Our hand simply doesn't hit enough flops hard enough to make it profitable for us to call the 4-bet against AAxx in the hands of an opponent who will put in the last bet on any flop.



3.4 K K 7 2 vs AAxx




Raised pot



top_x =15%
pf_bb =3.5
av_equity =80.2%

EV =(1 - 0.15)(-3.5) + 0.15{0.802(201.5) - 100} =+6.26 BB


3-bet pot



top_x =15%
pf_bb =12
av_equity =80.2%

EV =(1 - 0.15)(-12) + 0.15{0.802(201.5) - 100} =-1.04 BB



4-bet pot



top_x =15%
pf_bb =37.5
av_equity =80.2%

EV =(1 - 0.15)(-37.5) + 0.15{0.802(201.5) - 100} =-22.93 BB


Summary for K K 7 2 vs AAxx

EV (raised pot): +6.26 BB
EV (3-bet pot): -1.04 BB
EV (4-bet pot): -22.93 BB

K K 7 2 has a very "polarized" flop equity distribution curve with most of the equity "crammed together" on a small number of flops. These are mostly the flops where we hit top set, plus the flops where we hit two pair, trips of better with the side cards.

This has two immediate consequences:

- There are few flops where we will be able to commit
- But when we can commit, we will have very good equity

So K K 7 2 is a hand that plays better in small pots where we have good implied odds. When the pot grows, our problem is that we won't find enough flops to commit on, and having to check-fold most of the flops will hurt us more and more (if you study the graphs you will see that we commit on top 15% of flops in all cases, which means that we don't find any more flops to commit on as we move from a raised pot to a 4-bet pot).

The problem becomes extremely clear in a 4-bet pot where we lose 23 BB. We also lose in a 3-bet pot, but only barely (note that within the framework of our model we have excellent implied odds for set-mining against AAxx, since our opponent will always pay us off when we flop top set).


3.5 Summary of our model study of starting hand playability against AAxx



Our main conclusions from the model study were:


  • 9 8 7 6 plays well against AAxx in raised and 3-bet pots, and is a slight loser in a 4-bet pot
  • A K Q T and K K 7 2 play well against AAxx in raised pots, are slight losers in 3-bet pots, and are big losers in 4-bet pots.


When drawing conclusions it's important to keep in mind that we have been working with a model of reality, not reality itself. In particular, it's important to keep in mind that we calculated EV from the point where the first chip went into the pot preflop. We started by assuming we would be playing against AAxx, so our EV is the EV for choosing to play the hand when we know that we will play it against AAxx in a raised, 3-bet or 4-bet pot.

But in practice we don't know this when the hand starts. For example, if we 3-bet a raiser with 9 8 7 6 , and we get 4-bet, this does not necessarily mean we will lose money by calling, even if we assume we're always up against AAxx. Because now we have to estimate our EV from the point we're calling the 4-bet, assuming we're up against AAxx, not from the point where we put the first chip into the pot (where our opponent will have a much wider range).

And then we have a profitable call with a lot of chips already in the pot, and a hand that is easy to play well in big pots postflop. Our model told us that choosing to play a 4-bet pot with 9 8 7 6 against AAxx was a slightly losing play, but in practice we will profit from calling the 4-bet, planning to commit on the flops whenever we hit well enough. We will discuss this scenario in more detail in Part 4.

However, if we get 4-bet with A K Q T or K K 7 2 we should fold with 100 BB stacks. These hands were big losers in 4-bet pots within our model, and they will also be losers if we calculate the EV from the point where we get 4-bet.

From the work done in this model study we can conclude that when we get involved in big heads-up pots, we prefer hands that hit a lot of flops fairly well, and not hands that hit a small number of flops hard. The latter category are implied odds hands, and they play much better when we keep the preflop pot small and preserve implied odds.


3.6 An important observation concerning the play of AAxx



We will end our model study with an observation concerning the play of AAxx in big pots, and this topic is important enough to warrant its own section:

If you're playing a heads-up pot with a 100 BB stack and get the opportunity to 4-bet pot after a pot-sized raise and a pot-sized 3-bet, you can then push the remainder of your stack blindly all-in on any flop, and there is nothing your opponent can do to exploit it

The Villain in our model used this exact strategy, and we saw that all our hands lost against this strategy in a 4-bet pot, even when we played perfectly against him (remember that we committed perfectly on any flop where it was correct for us to do so).

Even the premium hand 9 8 7 6 was a loser in a 4-bet pot within the framework of our model, and this is one of the hands that performs best against AAxx! (The hands that perform best against AAxx are double-suited premium rundowns without gaps)

So we have tested the "4-bet and push any flop" strategy against one of the worst hands AAxx can be up against, and with perfect play to boot, and but still Villain could push any flop blindly with his AAxx without losing money.

Based on this model study we can therefore conclude:

If you can get at least 1/3 of your starting stack into the pot preflop in a heads-up scenario (so that you have at most 1 pot-sized bet left in the stack on the flop), you can push all-in on any flop if you want to, and there is nothing your opponent can do to exploit it.

For example, let's say you start with 100 BB and raise pot with A A 7 2 out of position. Your opponent 3-bets pot, you 4-bet pot, and Villain calls. The flop comes hideous and coordinated T 9 8 and you're not sure if you want to push all-in.

You can push anyway. We have shown that pushing the last bet all-in on any flop can not hurt you. There may be a better way to play the hand, but pushing blindly all-in heads-up in a 4-bet pot with a 100 BB starting stack can not be exploited by your opponent.

This is a strategy most PLO players learn early (mostly by hearing about it, and then starting to use it), but now we have used hard methods to demonstrate why this is a viable strategy, and by doing so we have made this strategy "our own".

And this is an important point with regards to the modeling work we have done in Part 3. Our focus has been on developing methods for doing interesting theoretical work using flop equity distributions, and we have not tried to do new and interesting theoretical studies right off the bat. The conclusions drawn from our modeling work are well known to most experienced PLO players, but it's important to note that we reached these conclusions by employing a method and not by guessing, using intuition or relying on experience.

The gist of it is that if we have a method that can confirm results that are already known, then we have reason to believe that the method will be reliable also when we move into uncharted territory and explore the unknown. So the theoretical work in this article has given us a set of tools to be used in future work, and we take these tools with us and move on to Part 4.


4. Summary



In this article we started with a thorough study of flop equity distribution curves, and we learned how to extract useful data from them.

Then we did a modeling study where we used flop equity distributions to draw conclusions about the playability of various starting hand types against AAxx. As a result of this we also reached a useful and interesting conclusion concerning the play of AAxx in 4-bet heads-up pots.

The work done in this article was purely theoretical and rather "heavy", but I hope you found it interesting, and that you saw the value of building these theoretical tools for use in strategic modeling. We will use a "lighter" approach in Part 4, and my plan is to discuss concrete guidelines for 3-betting and 4-betting preflop, as well as isolation raising and overlimping. The theoretical tools we built in Part 3 will be included in our arsenal for future use, and we will probably use some of them in Part 4.

Good luck!
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