Dimensionality: Analyzing Flop Data Using Various Characteristics

I teach a data analytics course at Georgetown University. It’s an entry-level analytics elective for graduate students pursuing a master's degree in public relations and communication. I designed the class to give students an overview of analytics and its real-world applications in marketing and communications. It’s a fun and rewarding side gig. I get to teach many smart young people at the start of their careers and introduce them to my field of work.

Since this is an introductory course, my students come into day one with varying levels of previous data experience. So, before diving into more advanced topics, I must first introduce foundational concepts applicable throughout the semester. I spend a few classes covering data visualizations and analysis techniques, but in that first class, I begin with two components of analytics that will be prominent in every analysis performed - metrics and dimensions.

I’ve talked about metrics in several of my previous posts. When I first began exploring my flop data set, I introduced metrics that PioSolver generates as its output from aggregation reports. I explored the various data points, like equity, expectation value, and equity realization, that help us gauge how our range performs against our opponent’s. In a second article, I looked at the rest of the solver output data, and explored the strategic options for each player and the corresponding betting or checking frequencies on each board. Both posts focus on various metrics that we can utilize to measure performance.

Metrics are a fairly simple thing to explain. It’s a measure of something and is represented as a number. Anything can be a metric for something. Here are a few examples:

• My cash rate in tournaments last week - 25%

• The number of cups of coffee I’ve had today - 3

• The time I spent negotiating with my 4-year-old last night to get her to eat her dinner - 20 minutes

All of these numbers are metrics for something. They may not always be useful — I’m sure no one is closely monitoring my coffee intake or my daughter’s eating habits. The point is that metrics can be used to quantify something. They are the base for all analyses as they help define what to measure.

Metrics alone can’t tell the whole story. While they help tell us what to measure, we need more specificity to understand the nuance in a data set. This is where dimensionality comes into play. Dimensions and metrics are complements that should always be used together in an analysis.

So, what is a dimension? Simply put, it is a way to group or cluster data points based on a characteristic. In marketing terminology, we often refer to this as segmentation, but there can be various ways we can demonstrate dimensionality. For example, I’m 38 years old. I lived in New Jersey for the first 18 years of my life and have lived in Maryland for the last 20. In this case, my age in years is a metric. The dimension is the state in which I currently reside. With this additional component, I can provide more detail about my overall lifespan.

I’m certain that everyone reading this blog has used these two foundational concepts at some point in their life, regardless of their data background. If you have played online poker, you probably use a hand database, such as Poker Tracker or Hold’em Manager. Within the software, there are a variety of metrics, including VPIP % (the rate at which you voluntarily put money into the pot), win rate, and dollars won. Those metrics can be analyzed in several different ways, one of which is position. When exploring these measures specific to the button, big blind, or any other position at the table, you’re examining metrics at a level of dimensionality — your position at the table.

Dimensions are critical for any descriptive data analysis. They provide various lenses through which we can explore data, helping to quantify various characteristics. This is important for us to do because we want to develop strategies that can be understood and implemented at the tables. Dimensions facilitate this process for us by helping us connect insights to action.

As I mentioned in my examination of formations, I like to start most analyses at the macro level to get a sense of the entire data set. While this is valuable initially, it’s never a place to stop. I can never get specific enough to develop an intricate strategy at this level of detail in an analysis.

At the other end of the spectrum, I could examine the data much more granularly at the flop level. However, this is time-consuming — there are 184 flops to explore, and going through them one by one is daunting. It’s often difficult to extrapolate concepts from individual boards that can be applied more broadly.

I can find a happy medium by choosing dimensions at which we can aggregate the data. I can get more granular with my insights than at the highest levels but stay macro enough to generate broader concepts that can apply to scenarios I might face more frequently at the table.

There’s no magic formula to determine which dimensions to explore. When doing so, I usually start with some general, descriptive characteristics that help me describe a data set. In this case, I want to select dimensions I can best remember and identify at the table. This will help me connect any takeaways from the data to in-game implementation. For today’s post, I’ve selected some initial dimensions of flops that I can explore throughout my analysis. While this post won’t be exhaustive — there are many ways to organize the data — it will introduce the foundational groupings we can use to describe flops.

Before I jump in, I’d like to introduce the two data tables I’ll examine for each dimension. For consistency, I’ll also use a single formation throughout the post to examine each dimension. I’ve selected a late position open vs big blind defense range, with us as the LP opener.

The first data table shows the distribution of the 184 different flops by EV %. From my formation post, I introduced the EV % metric. It represents the percentage of the pot we win at equilibrium and is calculated by dividing our EV by the overall pot size. From the table below, it’s easy to see that the LP always earns at least 50% of the pot against the BB, even on the least favorable flops for its range, such as a JT9 two-tone board. The LP earns more than 60% of the EV on 114 184 flops or 62% of all flops measured. This is an important benchmark to remember, as I can compare the individual dimension values against this baseline to determine if a specific characteristic over- or under-indexes towards the higher EV % distribution.

The second data table should look somewhat familiar, as I’ve introduced this chart in previous posts. It shows our success metrics (equity, EV %, and EQR) and the frequencies with which we take various strategic actions for situations when it is checked to us and when the villain leads into us.

I’ll replicate both data tables for the LP vs BB formation at each level of dimensionality for further analysis below.

Suited Boards - Flushes

The first thing I think of when describing a flop is the suits. The suits of the cards allow us to make flushes. This is useful because it helps us split hands in our range at the individual combo level when constructing ranges. Often, we’ll play flush draw, or backdoor flush draw combos differently than we would with the same combo with different suits.

There are a few different configurations of suits on flops — each tends to play somewhat differently with various portions of our range:

• Monotone - all cards are of the same suit (As 4s 2s)

• Two-Tone - two of the cards are of the same suit (Jc Td 4c)

• Rainbow - all cards are different suits (Kc 8h 7d)

When looking at data aggregated at the different configurations above for the LP vs BB formation, I see some interesting things in the data.

The EV distribution chart below shows a clear difference between the board types in how they contribute to the overall EV in this formation. Rainbow boards over-index towards a higher EV %, with 76% of rainbow boards yielding greater than 60% of the pot for the LP player. The distribution of two-tone boards is more comparable to that of the entire data set, with 59% of boards yielding greater than 60% of the pot. Finally, monotone boards under-index to this dimension, with 93% of boards yielding less than 60% of the overall pot. From this data, I can determine that EV values will run closer to even within this formation as the flop becomes more suited.

Next, I can look at the success metrics and strategic decision frequencies in the second chart below. I’ve added some conditional formatting to make it easier. The boxes shaded in green represent values greater than one standard deviation above the group's average. The boxes highlighted in red represent values greater than one standard deviation below the group's average. I’ve also added shading in the frequency cells to visualize the percentage value. Finally, I’ve included some summary statistics that show the average value across all boards and the high vs low range to demonstrate the differences (or similarities) within the specific categories.

Equities run very close to one another across these boards. There’s only a 0.2% difference in the share of the pot when comparing the most favorable boards (rainbow boards) to the least favorable ones (monotone boards). It is clear, however, that the LP under-realizes its equity on monotone boards and over-realizes on rainbow boards. Consequently, the EV % gets lower as the board becomes more suited. Additionally, the equilibrium solution chooses more passive strategic options on monotone boards, checking back more frequently when checked to and raising less frequently when led into.

I will primarily examine the scenario when the villain checks since it accounts for 97+% of the time at equilibrium. The BB rarely leads across any dimensional configurations, so I can focus most of my studies within this formation, assuming that the opponent will check nearly his entire range.

As the board becomes more suited, the checking frequency increases. This isn’t surprising given the realization challenges above. However, what is interesting is that the shift comes from different betting regions in different configurations:

• As we shift from rainbow to two-tone boards, the increase in checking frequency mainly comes from the smaller bet region, and the size of the larger bet-size region stays consistent. On these boards, a portion of the hands the LP player chooses to bet for a smaller size on the flop now moves into a check back at equilibrium.

• As the shift continues from two-tone to monotone, the frequency at which the LP bets smaller at equilibrium stays relatively constant and the increase in checking back now comes from a decrease in larger bet size frequency.

Paired Boards (or Trips) - Full Houses

The presence of multiple of the same card on the board can alter the strategic gameplay significantly. Often, the advantage will shift to the player with more of the paired card in his or her range. A range with a massive advantage on most boards against a different range could see its advantage shrink or even disappear on some paired boards. The configurations I will examine within this dimension are:

• Trips - all cards are the same value (Ac Ad Ah)

• Paired - two of the cards are the same value (8c 8s 6h)

• Unpaired - all cards are different values (Qs Jd 3h)

The EV distribution chart doesn’t yield much additional information about this formation in this dimension. The distribution of unpaired and paired boards is comparable to that of the entire set of 184 flops. The EV % is distributed slightly more towards the lower 50%-60% bucket on trips boards, but that’s somewhat irrelevant since these boards occur so infrequently, and we only have two instances within this sample.

The success metrics and strategic frequencies are below for these types of boards. The success metrics are mostly the same for the paired and unpaired boards, yielding almost identical equity, EV % and EQR values.

While the success metrics are comparable, the frequencies of the strategic actions at equilibrium are much different. The LP plays more of a realization strategy on unpaired boards, checking back 41% of the time compared to only 30% on paired boards. The solver also chooses the larger, more polarized bet sizing more frequently when it does bet on these unpaired boards. Almost half of the time it chooses to bet, the solver chooses a larger bet size at equilibrium on unpaired boards. On paired boards, the LP bets more frequently at equilibrium and chooses the small bet sizing much more often, at about a 5:1 ratio.

Connected Boards - Straights

People love to play straight draws. There are many straight draws in the game, each with varying quality. There aren’t any universally “standard” ways to examine this dimension. Michael Acevedo uses a categorization structure in his excellent book Modern Poker Theory that groups flops by how many straights can be flopped. While I think this is valuable to determine the degree of connectedness, many boards do not facilitate a flopped straight. As a result, I believe it loses a bit of nuance for most boards.

I decided to group this dimension by the types of straight draws (or straights) available on different boards.

• Straight - a flopped straight is possible (Qh Tc 8h)

• OESD - an open-ended straight draw is possible (9h 8d 3s)

• Gutshot - a gutshot straight draw is possible (Ks 9h 2d)

• None - no straight draw is present (Kh 7s 2d)

It’s important to acknowledge that there could be much-nuanced work to dig into this. For instance, both T9 and 65 are open-ended straight draws on an 872 board, but T9 is superior. As such, we could break down open-ended straight draws if we are on top of the board with T9, wrapped around the board with 96, or below the board with 65. A more effective categorization structure would likely combine my methodology with Acevedo’s to provide more detail into the “Straight” category.

The data may look somewhat surprising in looking at the EV distribution for the different types of boards within this dimension. While it seems intuitive that more dynamic boards should yield a lower EV % for the LP, that’s not necessarily the case. When an open-ended straight draw is present, the LP over-indexes towards the higher EV % distribution.

This is not necessarily consistent for all boards with draws, however. When a flopped straight is present, the LP slightly under-indexes towards the higher EV % distribution. When a gutshot is present, the LP massively under-indexes towards the higher EV % distribution.

The success metrics don’t vary much for any of the board configurations below. The equity, EV%, and EQR values are comparable for all four groupings.

The strategic frequencies are somewhat comparable for most of the boards as well. The big difference is on boards with no straight draw present. On these boards, the LP checks back less frequently. When it does bet, most range bets at the smaller bet size. This is likely because the board is static with few draws present, and the BB will struggle to find enough hands to defend — a similar concept we see on paired boards.

Grouping

Another way we can analyze a board is by the relative rank of its cards. When I started my flop work in 2019, I split cards into three groups — high, medium, and low. The high cards accounted for those T or above, medium cards were from 6 to 9, and low cards were from 2 to 6. I used this structure to group boards together for simplification. For example, a K83 board would be an HML — a king is high, an 8 is medium, and a 3 is low. A Q72 board would also be an HML. I found that this structure worked fairly well, but it did have one critical flaw — it didn’t account for the ace.

The ace is a special card in poker. It’s the most powerful card in the game, with the highest rank. It also has versatility and can be played as a low card in making the wheel. Because of this, we have to treat the ace differently than the rest of the high cards. After realizing this, I adjusted my grouping towards the end of 2019 to give the ace its category while maintaining the rest of the HML concept. This structure yields the following potential values:

• AAA - As Ac Ad

• AAH - Ah Ad Ks

• AAM - Ah Ac 8h

• AAL - As Ad 2h

• AHH - As Qc Qd

• AHM - As Jh 9d

• AHL - Ac Th 4s

• AMM - As 9h 8h

• AML - Ad 6s 5s

• ALL - Ac 4s 2c

• HHH - Kc Qh Td

• HHM - Ks Kh 8h

• HHL - Qh Js 2c

• HMM - Ts 9s 8c

• HML - Ks 7s 3h

• HLL - Qc 5s 3s

• MMM - 9c 9s 7c

• MML - 8s 6c 2s

• MLL - 7c 5s 5c

• LLL - 5s 3c 2s

There are many categories within this dimension, so I may want to simplify this somewhat in the future. However, it’s fairly easy to identify at the table with little effort. Below is the EV distribution for these groups. While I won’t review each group, there are a few things to call out.

Boards that over-index towards the higher EV % bucket are the ALL, HLL, MMM, MML, MLL, and LLL boards. A good rule of thumb is that if there are 2 low cards with a high card or if the high card is a 9 or lower, the LP is more likely to earn a higher percentage of the pot. Boards that under-index towards the higher EV % bucket are AHH, AHM, AHL, HHH, HHM, and HMM boards. The LP will likely earn closer to 50% of the pot at these formations.

When looking at the data within the success metrics and strategic frequency table below, there’s a lot to unpack:

There are board groups where the LP player appears to have a significant advantage at equilibrium. The most notable of these boards are those with 2 low cards on the flop — HLL, MLL, and LLL. The LP range has a higher equity advantage against a BB defense range on these boards than others. The LP player also over-realizes its equity more effectively.

On these boards, the LP range doesn’t check back very frequently, only checking on average 21% of the time across all 3 groups after the BB checks. When the LP range does choose to bet on these boards, the equilibrium solution favors a smaller bet size at a ratio of approximately 2:1.

Another insight I can see is that the ace is not a particularly good card on the board in this formation, contrary to what many might think. I covered this in an analysis I did in a regression model I previously built to quantify characteristic values within this formation. The LP range performs less favorably against a BB defense on ace-high boards than the other board types. This is presumably because the big blind will defend with many Ax in his range, including a percentage of many offsuit combos. On these boards, the LP checks back more frequently than in the other groups at equilibrium, ranging from ~40% of the time on the AAL and ALL boards to ~60% on the AHL and AMM boards.

The ace's presence is also why ALL boards, such as Ah 4c 2s, aren’t included in the more favorable group with the two low cards that I described above. While the LP does retain an equity advantage on these boards, it doesn’t over-realize its equity as favorably due to the Ax coverage from the BB. So even though the ALL board is the best of the A-high boards, it’s still clearly less favorable than the other ones with two low cards.

Conclusion

I’m barely scratching the surface with this analysis at a dimensional level. There are other dimensions that I could choose to study using other factors or combining these. For example, Acevedo combines the flush and full house groups to create one to analyze board texture. This helps to demonstrate the interaction between the two dimensions and show the nuance between some boards, such as the paired rainbow vs unpaired rainbow ones.

My goal within today’s analysis is not to uncover every insight across all 32 formations for every dimension. That would be a much longer post. That information could fill an entire book. Instead, I want to demonstrate the types of ways in which we should be thinking when examining this data to translate it into actionable insights.

As I continue analyzing this (or any) data set, the investigation of dimensions becomes somewhat of an iterative process. I’ll typically find myself digging into the various data variables and noting anything interesting. If there appears to be some nuance within a particular data area, I may try to identify additional ways to aggregate or examine it to determine any concrete findings.

As I progress with my work, I’ll explore the data at this level of granularity myself for a larger set of formations. From there, I hope to develop more concrete findings that I can extrapolate into protocols to help guide the development of an implementable flop strategy.

If you have any comments or thoughts, please feel free to leave any comments below. You can also contact me at [email protected] or on Twitter or YouTube through the links in the footer below.

Thanks for reading.

-Lukich