The Relationship Between Solver Inputs & Outputs: Part 2

Hello, and welcome back to March issue of the Solver School Newsletter!

In last month’s issue, I started a 3-part series diving into how changes in solver inputs will impact the outputs. I introduced the series and examined changes in two of our variables: SPR and Board Textures.

In today's newsletter, I’ll continue with Part 2, where I will look at the next variables we can control: Bet Sizing Options.

A Reminder About the Value of Solvers

As I explained in last month’s issue, the most important thing to understand with solvers is how to manipulate inputs to measure the effects on the outputs. I believe that this is the most critical skill to develop when learning how to use a solver properly.

I’ve said many times in my content that a solver output is simply a snapshot of a game. In fact, it’s very unlikely that any individual solver situation will perfectly model actual gameplay. Instead, it’s a sophisticated model that’s based on our own biases and assumptions.

It’s critical for us to recognize that we won’t be perfect in building this model. And that’s okay!!

What the solver can tell us, which is insanely valuable, is how variations in inputs can impact the outputs. If you can start to understand how you are biased in your thinking process, forget about simply copying or trying to memorize the solver output, and begin charting the impact these variations in the inputs affect outputs, you’ll start to recognize the true power that solvers have.

Revisiting Our Baseline Example

In last month’s issue, I examined a scenario where the hijack (HJ) opens with a raise, and the big blind (BB) defends. We’ll go back to this baseline for this issue to look at the impacts of changes in bet sizing options and ranges.

As a reminder, the HJ has the following opening range:

And the BB has the following defense range:

The pot size is 10 big blinds, and the effective stack is 50 big blinds, making the SPR equal to 5.

Note: This may not be a realistic short-stacked situation. The point is to set a baseline from which we can look at deviations. It’s not to model and analyze an actual scenario we might face.

The board is the Kh-9d-5c:

On this K95 board, the HJ has an equity advantage, with 58.7% equity as compared to the BB’s 41.3%.

Finally, the bet size options for the HJ are as follows:

And the following bet size options for the BB:

With this setup, the BB’s PioSolver output generates an EV on the flop of 3.016 BB, checking 100% of hands:

This leaves an EV of 6.984 BB for the HJ. The HJ bets 94% of hands, checking back only 6%.

Now that we’re aligned on the baseline, let’s get into varying these inputs.

Bet Sizing Options: BB Perspective

For all other inputs, it’s easy to see how changing the inputs can impact the outputs. If we vary ranges, stack sizes, or the board, it might be obvious to most that the outputs will be different. But that’s not necessarily the case with bet sizing options.

Bet sizing options are part of the constraints we set on the game tree when we configure the solver. And as we’ll see, changing the betting options within the game tree will impact the overall solution.

Recall that in our baseline, the HJ had one option to bet on flops (50% pot) and two options to bet on turns and rivers (50% pot and 100% pot). Let’s take a look at a few different iterations:

• Variation #1: HJ with 3 flop bet sizing options (100% pot, 66% pot, 33% pot)

• Variation #2: HJ with 1 flop bet sizing option (33% pot)

• Variation #3: HJ with 1 flop bet sizing option (100% pot)

• Variation #4: HJ with 1 flop bet sizing option (200% pot)

• Variation #5: HJ with 1 bet sizing option on turns and rivers (50% pot)

I’m not going to show the outputs for the BB because the strategic frequencies don’t vary from the baseline. In all scenarios, the BB checks 100% of its range. However, the EV for the BB varies for each of these variations.

• Variation #1: HJ with 3 flop bet sizing options — EV = 3.005

• Variation #2: HJ with 33% bet sizing option — EV = 3.031

• Variation #3: HJ with 100% flop bet sizing option — EV = 3.117

• Variation #4: HJ with 200% flop bet sizing option — EV = 3.350

• Variation #5: HJ with 50% bet sizing option on turns and rivers — EV = 3.034

What we see here is somewhat interesting. The baseline EV with the option to bet 50% pot is 3.016 big blinds. When we increase the number of bet sizing options, the EV of the big blind decreases. I think this intuitively makes sense. If we give the solver more options, it can find a solution that utilizes all of those bet sizes that is closer to a true equilibrium.

If we change the single bet size, it could have a small impact on EV — as is in the case of the decrease in the bet size from 50% to 33% in variation #2. Or it could have a significant impact as in variations #3 and #4. As we reach #4, the solver is constrained by a clearly sub-optimal bet size of 200% pot. This leads to EV loss from the HJ and captured by the BB.

When we provide a single bet sizing option on turns in variation #5, the output is closer to the original baseline. This is due to a ripple effect — as the changes we make move further away from the node of the tree we are analyzing, the impact of those changes will be smaller overall.

Bet Sizing Options: HJ Perspective

When we look at the impact from the HJ perspective, there are two dimensions through which we can examine this. The first is through EV. This is fairly simple. Since it’s a 2-player game, EV of both the BB and HJ will add up to the pot size of 10. Any EV gained from the BB comes at the expense of the HJ. So the same takeaways we had from above apply.

Below are the EV outputs for all 5 variations for the HJ. Note that the HJ’s baseline EV was 6.984 big blinds.

• Variation #1: HJ with 3 flop bet sizing options — EV = 6.995

• Variation #2: HJ with 33% bet sizing option — EV = 6.969

• Variation #3: HJ with 100% flop bet sizing option — EV = 6.883

• Variation #4: HJ with 200% flop bet sizing option — EV = 6.650

• Variation #5: HJ with 50% bet sizing option on turns and rivers — EV = 6.966

The second dimension we can examine is through the HJ’s strategic frequencies. We saw that the BB doesn’t change strategic options and checks 100% of its range for all 5 variations. That is not the case for the HJ. Here are the outputs for the HJ below. Recall that at our baseline, the HJ checks 6% of hands and bets 94%.

As we can see here, there are big differences in outputs as we change options. In general, as smaller bet size options are included, the solver will bet a larger frequency of hands. Said simply, the solver can’t justify putting some of the lower equity hands into larger bet sizes, but it can at the smaller sizes.

Takeaways

As I have shown, there are nearly endless variations of bet sizing options that we can choose from, and even slight changes can sometimes significantly impact the output.

Unfortunately, even modern computers are not powerful enough and don’t have enough memory to let the solver optimize ranges from all the possible sizing options. It also wouldn’t be efficient for us to try to learn from those outputs.

There usually isn’t one “perfect” sizing choice — multiple sizing selections can yield similar EV outputs by constructing ranges to accommodate the sizing options.

It’s more important to understand sizing concepts and learn how to test different variations to find insights that can be recalled and implemented in-game.

Wrapping Up & Looking Ahead

That does it for today’s newsletter and part 2 of this 3-part series on the relationship between inputs and outputs.

In next month’s issue, I’ll finalize the series with Part 3 — a look at how changes in ranges (both input ranges and node-locked ranges) impact the outputs. I’ll then shift back to interpreting solver outputs in May’s issue.

For more free solver content, check out the rest of the Solver School website. It is filled with video and written content about solvers and analyzing poker with data.

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I appreciate you following along and reading the newsletter today.

Until next time.
Michael Lukich