Quantifying equity distributions (e.g. range advantage/nut advantage etc) for analysis reports?

You're asking a good question. Tombos might be the only person on this forum that could give you an accurate answer.

I'm not aware of the existence of anything like what you're talking about. It would be helpful if someone created something like that though, or maybe it already exists and I'm just not aware?

Practically speaking the closest thing to what you're talking about is the turn heat map feature that lists the equities for each possible turn card. That's very helpful to better visualize how equity is distributed across different run-outs.

This is a great question, Kler. And thanks for the vote of confidence, GreatWhiteFish!

Many coaches (myself included) have claimed that "range advantage" dictates betting frequency, while "nut advantage" dictates sizing. But this heuristic doesn't hold up to scrutiny.

What is "range advantage"?

The definition is debatable, but the simplest is just your range vs range equity. If your range has 60% equity, you have a "range advantage" (your line will be higher on the EQ distribution graph).

How well does range advantage predict betting frequency?

Poorly. One way to measure predictive power is by calculating the R² value (0 = no correlation, 1 = perfect correlation). Flopped equity has an R² of about 0.25, meaning there’s a weak relationship but nothing reliable. In other words, if I told you that you have 55% equity on the flop, could you accurately predict the GTO c-bet frequency? Probably not.

https://app.gtowizard.com/solutions?solution_type=gwiz&tmpId=1739680652254&soltab=strategy&gmfs_solution_tab=ai_sols&gametype=Cash6m500zSimple&depth=100&gmfft_sort_key=0&gmfft_sort_order=desc&history_spot=0

GTO Wizard

GTO Wizard

BTN vs BB SRP

SB vs CO 3BP

In short, "range advantage" turns out to be a terrible predictor of betting frequency. I actually go into more detail on this in a

https://app.gtowizard.com/coaching?video_id=e01a9649-1ad7-4091-99f5-e303cf5101c9&vid_coaches=tombos

GTO Wizard

GTO Wizard

The core issue is that range vs range equity by itself lacks crucial information—it doesn't account for range polarity or how well your range retains equity after forcing your opponent to fold weaker hands. These factors turn out to be essential for understanding GTO betting strategies.

What about nut advantage?

Nut advantage is poorly defined. My advice? Invent your own definition and test it until you find something that actually predicts bet sizing.

One method I’ve had some success with is what I call "equity squared buckets":

Details:

Spoiler

Show

Basically, take some equity buckets (like the ones in GTO Wizard), compute IP - OOP, and then multiply the difference by the square of the equity bucket’s midpoint.

Nut advantage = Σ ([IP% - OOP%] × EQ_Midpoint²) +… (repeat for all equity buckets)

The number you get is kind of meaningless in a vacuum, but taken as a comparison between other flops they seem to correlate nicely with betting volume.

Why square the equity? We want to weight nutted hands heavily and discount middling equity hands. In reality, the exponent should scale with SPR—deeper stacks → higher exponents. This is lightly inspired by some old-school poker theory that found that eq^2 better correlates with the real value of a hand.

That said, I haven’t tested this across all flops yet, so I can’t say for sure how well it predicts bet sizing globally. But anecdotally, it seems to correlate well with sizing and betting volume on a small subset of flops that I've tested.

Thanks for you reply. I'm still thinking it over, but wanted to confirm the formula/process for the equity buckets squared calculation.

I've written a function in python to take the equity values and midpoints as input, and calculate the output.

However, using the data from your example to test it, I'm not getting the same result +/- 1.75 as the final value.

So starting at the top (95% EQ midpoint):
We do, 1.9 - 1.7 = 0.2
then, 0.2 * .903 = .1806

then the next midpoint: 4.6 * .723 = 3.33

and so on...?
until finally having finished with the last EQ bucket, we just add all of our outputs together?

But here, this gives us a total of 5.36 (or .0536 if, for IP-OOP value, we convert as percentage, i.e. .002, .046, etc.) — so where am I going wrong?

This is really interesting. Is the 1.75 nut advantage calculated by multiplying only the top equity bucket? The decimal places only show 1.2 but I suspect that's the rounded 1.75?

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Oh I definitely messed up that nut advantage calculation, sorry guys! I was trying to recreate the process from memory. Let me clarify:

Here is a spreadsheet calculator you can...

Algorithm to compute NA:

Step 1) Calculate the eq bucket midpoints and square them
Step 2) Measure the amount of the OOP and IP range in each bucket (as a percentage)
Step 3) Compute the weighted average of Eq² buckets and IP range, and OOP range
Step 4) The difference between those ranges gives you the "nut advantage"

Using the "simple buckets" in the picture above as an example:

IP = (76.6% * 30.4%) + (39.1% * 41.8%) + (17.2% * 24.3%) + (2.7% * 3.5%) = 43.9%
OOP = (76.6% * 13.7%) + (39.1% * 10.5%) + (17.2% * 23.1%) + (2.7% * 52.8%) = 20.0%

IP Nut Advantage = 43.9% - 20% = 23.9%
OOP Nut Advantage = 20% - 43.9% = -23.9%

Note that you can use more buckets for better precision, but the 'advanced buckets' give me a similar numbers to the 'simple buckets' in most cases.

Also, I will add th

It is possible to do this with PioSolver as well, though it takes some extra work because Pio doesn't naturally come with an equity bucket function. Here's an example of how to do so.

I tested this method a bit more rigorously today on a bigger sample of flops. Unfortunately, I don't think Eq² buckets are actually very predictive...

Again I think the main problem is that "equity" doesn't carry enough information. The way your equity distributions on later streets is important. The way your hands equity changes as you narrow your opponent's range is important. There are important nuances that a raw equity calculation just can't capture.

This is BTN vs BB SRP, 100bb

I don't think you can retrieve the info you're looking for from the flop equity distribution. It has to do with other factors relating to it being an A high board (nut top pair already made, low urgency to bet for protection/denial) and 2 connected low cards (increased playability and nut potential in future streets for the BB). This particular combination means that a very small part of IP's range is interested in building the pot on the flop. So the reason IP bets so little even though it has a lot of equity is because of how the hand plays out on future streets, which I don't think can be quantified from a flop equity distribution.

The way you'd go about quantifying it would be
1. How urgent is it for top pair to get in money now
2. When money does go in, who is likely to become more nutted by the river

In a sense you could say OOP here is quite polarized, because if you bet your top pair you're only getting folds from hands that are virtually drawing dead (KQ,QJ,JT etc) and getting action from hands that can make your top pair miserable on a variety of future runouts.

Agreed. It would be cool to find a way to quantify these things though.

Yes, you would want to somehow calculate how much equity changes for ip/oop on turns/rivers and how often strong hands on the flop continue to be strong hands for turn and river.

If equities can change drastically over future streets and nut combos on the flop are rarely strong hands on turns/rivers (think flush draw boards or straight draw boards), then you would likely size down. Conversely, if strong hands on the flop are often strong hands vs. your opponents range over the aggregate of possible turns/rivers, then it is more likely large sizings (geometric) will be utilized.

These graphs showing the IP/OOP equities as their overall composition of their range are useful, but you'd somehow want to tie that into how those equities shift over possible future streets to get a better picture. If you were able to come up with a variable for accounting for this, then I think that would be a much better predictor of sizing and frequency. This is specifically related to "getmeoffs" (1) and (2) points.

I'm sure if tombos and acquaintances spent some time thinking about how to do this they could come up with something useful relatively quickly.