Examining The Relationships Among EV, Equity, and Equity Realization

This post was originally published on January 30, 2020, on my personal website, Lukich.io. I have since consolidated all of my poker-related content by reposting it onto Solver School.

Last week, I introduced the metrics we’ll utilize to measure success as we study flop play. I defined equity, expectation value, realization, and nut advantage in that post as our primary metrics. While I’m not quite at the point at which I can begin formally analyzing the complete data set, I plan to continue down the data exploration path today and understand these success metrics more.

I am making progress in the development of my flop database. I’ve identified the 33 different formations to include in this study. They represent common ways we arrive at the flop through both the offensive and defensive game trees. I will solve the 184-flop subset I introduced last week within PioSolver for each distinct formation. Over 6,000 solves will ultimately populate the data set that will be the foundation of this work.

Some of the formations are quicker to build than others. For example, 3-bet formations have smaller SPRs, and the players generally have narrower ranges. This limits strategic options and the ability to maneuver postflop, resulting in a quicker solve within Pio. By contrast, single-raise preflop formations yield larger SPRs because the pot size is smaller. When you combine that fact with generally wider ranges (e.g. blind vs blind formations), each individual solve in these formations takes significantly longer to complete.

After I complete the entire data set, I will derive additional metrics that we can utilize in this effort. But in the meantime, I will explore the 3 Pio calculated metrics in more detail today — equity, EV, and EQR. Nut advantage needs to be defined and tested, which I’ll do in a future post.

At the beginning of data analysis, it’s a good idea to understand the data set completely. Specifically, I hope to learn more about the relationship among these 3 metrics. I will utilize a portion of the partially completed database to do so. I will examine a common spot we all face — we raise preflop and are called by the big blind, arriving at the flop in position through the offensive tree.

I will explore the relationships between each metric and how they can be utilized to help us better identify our strategic actions on flops. Today’s portion will focus on the following:

• Our ultimate goal is to maximize EV, so I will examine how equity and EQR can be utilized to predict EV.

• I will study the competing relationship between equity and EQR.

• Finally, I’d like to understand the effect of position, so I’ll look at both an EP open and an LP open.

Solve Parameters

Here are the ranges I utilized for the analysis of my EP open and the BB defense:

• The EP range is my standard opening range in a normal game before I make any adjustments. As a default, I utilize a raise or fold strategy when opening preflop so that it will be linear. Because of the many players left to act and my poor position, I have constructed a stronger 15% range.

• The BB defense range is developed using Upswing Poker ranges as a baseline with some slight adjustments. I added half-combos around the edges to profile the average $2/$5 to $5/$10 player pool. While I don’t believe many players implement a mixed strategy consistently, this accounts for some variations when looking at the aggregate pool.

• The BB will have a condensed range. Because it can flat the raise and see a flop at a discounted price, the raising range should be tighter and more polarized. Considering this, I have discounted AA, KK, AK, and some of QQ/AQs from the BB calling range.

For my LP open vs the BB defense, here are the ranges I chose to use:

• The LP range is a blend between my standard CO and Button opening ranges before making any adjustments. I am not mixing combos in my ranges; the half-combos represent the hands I open from the Button but not the CO. As with the above, I still use a linear raise or fold strategy as a default. However, because of my positional advantage and the fewer players left to act, I open a wider range of 37% of hands.

• The BB defense range is again developed using Upswing Poker ranges as a baseline with some adjustments. The 3-bet range will expand slightly but will continue to be polarized. Thus, the BB calling range is still condensed. However, I added more hands. Most players will realize that because I am opening a wider range on the button, they will have to defend more widely from the BB. As a result, this range defends 39% of hands through a call.

For both the EP and LP solves, I utilized the following parameters:

• The open raise size is 4 BB, and only the BB calls — the starting pot size is 8.5 BB.

• Effective stacks are 196 BB. I am currently modeling all of my analyses using 200 BB deep pots. It’s harder to find much-existing work to study at the aggregate level for deeper stacked games — probably because of the time and resources to solve deeper spots. I also mainly play 200+ BB deep, which most applies to my development.

• I’m working to beat $5/$10, so I modeled the game to reflect this. As a result, I have multiplied the number of BBs by 10. This helps familiarize me with approximate bet sizing in dollars to recall and execute in-game more easily.

• I simplified the BB OOP strategy to reduce system resources. The OOP player can lead for 50% pot on any street. On the turn and river, I additionally give the OOP player an option to bet full pot. Given the equity disadvantage of the OOP range against a tight EP range and the positional formation, I wouldn’t expect much leading overall anyway.

• I provided our perspective — the IP player — with more strategic options. I can c-bet a large (2/3 pot) or small (1/3 pot) amount on the flop. On the turn and river, I can bet 1/2 pot, full pot, or a 1.5x pot overbet (as well as an all-in bet if certain conditions are met). Providing more strategic options for the solve should increase the accuracy of our overall equilibrium solution. The trade-off is increased complexity and mixing of individual combos. However, given that we’re not utilizing this for single-flop implementations, this is less of a concern.

• Finally, I solved each of the 184 flops using these inputs to an accuracy of 2% of the pot. For more micro-level analyses, I might want to solve down to 1% or less as an individual solve’s accuracy is important. However, for this mass data analysis, 2% accuracy is fine. Because we are aggregating 184 flops together, small margins of error within each flop are averaged out when rolled up.

Piosolver Summary Stats for Formations

Before looking at the relationships, I’d like to quickly show the summary stats of our success metrics from the solves.

• Unsurprisingly, we earn a greater EV with an EP range against a BB than with an LP range. Our EP range has 56% equity against the BB, resulting in us earning $56 of the $85 pot and over-realizing our equity by 18%.

• The equity of our EP range is between 50% (T93 rainbow board) and 63% (A76 rainbow board). This is a reflection of our strong 15% opening range. Even on the worst flops in this particular formation, our overall range will have at least 50% equity.

• The EV of our EP range is between $47 (T98 two-tone board) and $67 (AK7 two-tone board). This means our EV is always greater than half the pot from this formation.

• The EQR values from the EP range are between 107% (T87 two-tone board) to 129% (622 rainbow board), meaning that we always over-realize our equity — sometimes by as much as 30%.

• By contrast, the LP range has 52% equity, resulting in $52 of the $85 pot, yielding an EQR of 118%.

• The equity of our LP range is between 48% (KQJ rainbow board) and 54% (732 rainbow board).

• The EV of our LP range is between $47 (QJ9 rainbow board) and $59 (622 rainbow board).

• The EQR values from the LP range are between 111% (A75 monotone board) to 131% (again, the 622 rainbow board, which appears to be one of the best boards for an IP raiser vs a BB).

At first glance, I find the distribution of values interesting. The metrics from our EP vs BB formation are distributed more widely than those from our LP vs BB formation, which are clustered.

Relationship Between Equity & EV

Now that we have a sense of the data sets, we can visualize our range’s equity as it relates to EV against our opponent within these formations across all flops.

The x-axis displays the equity of our range. We can see that our range has an overall equity advantage against the BB. Most values are above 50% equity, including all from the EP vs BB formation. This is expected — we will have top-of-range preflop hands, such as AA-QQ and AK, in our range, while the BB mostly won’t.

The y-axis displays the EV of the 184 different solves. These values range between 4.5 and 7 BB ($45 and $70). The EV for the most favorable boards vs the least favorable boards in the EP vs BB formation are wider than the deltas for the later position formations – we earn higher EV ceilings but lower EV lows. We will want to investigate why this might be the case. My initial gut tells me that our stronger and tighter range has greater potential on highly advantageous boards but can also completely miss some disadvantageous boards due to a lack of coverage.

It’s important to reiterate our point above. Our EV is never less than half the pot in this analysis within either formation, even though there are some boards where our range vs range equity is less than 50%. This again demonstrates the advantage position gives us against even a perfectly balanced opponent.

While there is a clear positive correlation between equity and EV, the fit of the trendline isn’t perfect. The R-squared value for these 2 plots are below. R-squared values measure the fit of a model. Values closer to 1 suggest a stronger relationship between the dependent (EV) and independent variable (equity).

• EP vs BB: 0.74

• LP vs BB: 0.66

While this shows a positive relationship between these two variables, the correlation is imperfect. Our equity advantage will not always translate to EV. We will over- or under-realize our equity share on certain boards.

Relationship Between Equity Realization & EV

To measure the impact our ability to realize equity has on EV, we should also look at EQR and its relationship with EV. I detailed how it is calculated last week’s post, but simply put, it will measure the amount we can over-realize our equity through EV as a percent of the current pot size.

As with EV, we can plot the EQR against EV for each of our 2 formations across all 184 flops. Below is this relationship shown visually:

The y-axis again shows the same EV values for all flops as modeled within the analysis. The x-axis of these graphs now shows the EQR for each solve. As we can see, EQR is also positively correlated to EV. The R-squared for these formations are shown below:

• EP vs BB: 0.57

• LP vs BB: 0.80

While there is a positive correlation for both, the EQR from an LP open is much more strongly related to EV than one from EP

The positive correlations between equity & EV and EQR & EV are unsurprising. The equation in last week’s post can be manipulated to be written as EV = (EQR x Equity) / Pot Size. This means that EQR and equity increases will lead to EV increases. We can also look at the equation and see that EQR and Equity are both linearly related to EV. This leads to the next question — Which one do we use?

Relationship Between Equity & EQR

Based on the above equation, equity, and EQR are inversely related. Therefore, I don’t expect to see the same level of correlation between the two variables. But understanding how these two metrics relate to one another is worthwhile to determine how they best predict EV.

As expected, the correlation between equity and EQR is poor. R-squared values are less than 0.10 for EP vs BB and 0.22 for LP vs BB. While there is a general upward trend — meaning that as our equity advantage increases, we should also expect to be able to over-realize our equity somewhat more often — there isn’t as strong of a relationship between the two metrics as with the other comparisons above. Thus, I can determine that while the two generally trend together, specific factors in flop textures can lead to choosing one over the other as an input into EV models.

Takeaways

I demonstrated above that both equity and EQR have direct positive relationships with EV. Additionally, while equity and EQR are loosely correlated to one another, there is a larger amount of noise in this data. Because equity and EQR have a weak correlation, the two variables will not always equally impact EV.

We can see this most clearly when looking at the relationship between EV and EQR compared to our relationship between EV and Equity through the position lens. Equity is more strongly correlated with EV when we are an EP raiser. As an LP raiser, EQR becomes a larger factor.

This yields one of my first conclusions from this analysis. I interpret these results by saying that when we open from an early position and are called by the blinds, the overall equity advantage of our range will be the most determinant factor when determining advantageous +EV situations. When we open from later positions and are called by the blinds, our ability to over-realize our equity is more important in determining advantageous +EV situations. This will be important for us to remember as we determine our strategic actions. As such, we may want to use different metrics for determining the types of boards that give us an advantage when we open from EP vs opening from LP. It’s also certainly something I plan to continue testing across additional formations.

I’ll spend my next post finalizing and exploring the rest of the database. This includes looking at strategic options detailed in the solver setup above. I’ll discuss the benefits and limitations of the selections and how they impact the data set.

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