The Relationship Between Solver Inputs & Outputs: Part 1

Hello, and welcome back to the Solver School Newsletter!

After a bit of a break, I’m back with this February issue in our ongoing journey towards poker solver mastery.

In today's newsletter, I’ll continue to dig deeper into the relationship between inputs and outputs when using a poker solver.

As I covered in prior newsletters, a poker solver is a computer program that uses complex game theory algorithms to analyze poker scenarios. We input certain parameters about a poker situation, including factors like board details, bet sizing options, and player ranges. The solver then outputs approximations of game theory optimal strategies for both players.

But how do all those inputs actually affect the strategic outputs? That's what we'll explore over the next three newsletter issues.

Examining How Inputs Impact the Outputs

Understanding how inputs and outputs are related — and more importantly, how to manipulate inputs to measure the effects on the outputs — is the most critical skill to develop when learning how to use a solver properly

A solver output is simply a snapshot of a game modeled on a discrete set of inputs based on our own biases and assumptions. Recognizing that we’ll never be perfect in modeling those inputs is important. To get good at using solvers, we must learn to recognize that the outputs will not be perfect.

By training yourself to acknowledge these implicit biases and moving towards identifying the effect that variations in inputs have on the outputs, you’ll start to unleash the power solvers can have in teaching us about gameplay.

I can tell you all the ways that changes in the inputs affect the outputs, but the best way to do so is with examples.

Setting Our Baseline Example

Over the course of the next few newsletters, I will examine the following situation. A hijack (HJ) opens with a raise, and the big blind (BB) defends. For these examples, I’ll assume that the HJ has the following range:

I’ll also assume that 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:

With this board, the HJ has a massive equity advantage, with 58.7% equity as compared to the BB’s 41.3%.

I set this up in PioSolver using the following bet size options for the HJ:

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

This will be our baseline scenario from which we will compare all future outputs.

Stack-to-Pot Ratio (SPR)

The ratio between the effective stacks and the pot size, or SPR, greatly affects the value of position. When SPRs are lower, the value of the HJ’s positional advantage decreases. With shallow stacks, the hijack can't threaten the big blind with multiple future bets.

We can see this by looking at the output when the SPR drops to 1. If we change nothing else in the solution except for the effective stacks and decrease them to 10, the outputs change.

The BB’s EV increases to 3.464 and develops a small leading range of 6% of hands:

The HJ’s EV decreases to 6.536. He also checks back much more significantly when the BB checks, only betting 63% of hands:

When SPRs are higher, around 4-5 and beyond, the value of position increases. The hijack can leverage multiple future bets to push equity against draws and weak holdings. More speculative hands gain value as implied odds open up. The in-position hijack will over-realize equity compared to the out-of-position big blind.

However, SPR does not have a direct linear relationship to expectation value and frequencies. The effects plateau around an SPR of 4-5 before leveling out at higher ratios.

Board Cards / Texture

Different boards interact with ranges differently. For example, on our K95 rainbow baseline at 5 SPR, the hijack has the nuts advantage with overpairs, top sets, and strong Kx.

But if we change the board to 987 with two spades, the big blind connects extremely well, with many straights, pairs, draws, and combo hands. As a result, its EV now jumps to 4.406. The BB will also lead more frequently, betting 13% of hands.

The HJ EV drops significantly to 5.594 and has to check more frequently (only 53% betting range) on this board to realize equity.

Boards will impact ranges in different ways. There are 22,100 unique flops in hold’em, of which 1,755 are strategically different – it’s incredibly inefficient to study each board individually.

Instead, it’s more effective to categorize boards into segments as a way to simplify and group like-flops together. The best segmentations are those with heuristics that can be easily recalled and implemented at the table.

For flops, I recommend using board characteristics, such as Trips/Paired/Unpaired to measure the ways that cards interact with each other or Rainbow/Two-Tone/Monotone to measure the ways that suits work with one another. Similar groupings can be created for turns and rivers (e.g. cards that pair the top card, flush completing cards, bricks, etc.).

The specifics don’t matter from person to person. The most important thing is to find a way to group boards together that works for you and that you can remember at the table.

Wrapping Up & Looking Ahead

That does it for today’s newsletter!

Quick housekeeping note — going forward, I’m going to drop to 1 newsletter per month. I’m exploring other projects this year. To ensure that I don’t burn myself out or impact the quality of the work that I publish, I’m reducing the overall quantity.

In next month’s issue, I’m going to continue this post with a further look into the relationship between inputs and outputs. I’ll stick with the K95 board and vary bet sizing options to show how changes in those inputs impact the outputs. I’ll then finish up the series in April with a look at how ranges impact outputs.

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.

I have several courses for sale. If you want to take the next step in using solvers, check out my flagship product, The Solver Masterclass. It contains everything you need to know to become an expert at using solvers to analyze poker data.

You can also follow me on Twitter and YouTube, where I share more solver-focused and poker strategy insights.

I appreciate you following along and reading the newsletter today.

Until next time.
Michael Lukich