How Ranges Can Impact Solver Outputs

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

In February, I started a series on how input changes impact solver outputs. In that issue, I introduced a baseline example and examined the impact of changes in two variables — SPR and Board Textures — on the resulting solver outputs.

I continued that series earlier this month with a second part. This time, I adjusted the various Bet Size Options we can input when configuring game trees, again looking at how those variations resulted in different solver outputs.

I will wrap up the series today with a look into the most impactful of our inputs: Ranges. I’ll examine how changes in our starting ranges and flop frequencies through node-locking can result in drastically different strategies.

Revisiting Our Baseline Example

Let’s quickly ground ourselves on the scenario we’re studying.

In this example, the hijack (HJ) opens with a raise, and the big blind (BB) defends.

The HJ and the BB have the following ranges:

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 K-9-5 rainbow board, the HJ has an equity advantage, with 58.7% equity 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 the HJ with an EV of 6.984 BB. The HJ bets 94% of hands and checks back only 6%.

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

Ranges Are the Most Important Solver Input

Player ranges are the most critical in influencing the output of all the inputs required to build a game tree. Even slight variations in these ranges can lead to significant variations in the resulting strategies.

What makes things challenging is that this is where we’re at most risk of faulty assumptions. Many players will struggle to define their own ranges for a given situation. The process becomes a Herculean task when we extend that to trying to assign ranges for our opponents.

A simple Google search will yield plenty of suitable starting ranges available for free. Many sites also sell premium ranges. I sell my own set of ranges that can be loaded directly into solvers in the Solver School Starter Pack, which is included for free when you buy the Solver Masterclass.

Regardless of which ranges you choose, they should be a starting point. You should use them as baselines for analysis and adjust them according to other factors, such as game flow, player profiles, and player vs. player dynamics.

Let’s look at two examples of how changes in starting ranges can impact the resulting strategies.

Variation 1: Wider HJ Open Range

Suppose that instead of our HJ opening 17% of hands, he is a bit looser and opens the 28% range of hands below:

In this scenario, the BB still checks virtually 100% of its range on the flop (there’s a negligible amount of leading with 0.3% of the BB range), but the HJ strategy is a little bit different.

The HJ now only bets 65% of hands at equilibrium. Its overall range is somewhat weakened compared to the tighter 17% opening range from our baseline example. As a result, the HJ must now check some hands back or risk being exploited by counter-aggression from BB if it includes too many weak hands in the betting range.

The EV of the HJ also drops from 6.984 BB to 6.464 BB — a reduction of over 0.5 BB for the hand!

Let’s take a look at another example.

Variation 2: Wider BB Defense Range

For this example, let’s adjust the BB’s defense range instead. Returning to our 1.78% HJ opening range, we’ll use a BB defense range against a BTN open. This new range consists of more overall hands (~34% of hands as opposed to 22%), with more of the stronger top-end hands used as 3-bets and not retained in the calling range.

The BB still checks 100% of its range, but the HJ can now bet 100% of hands with impunity due to its massive equity advantage.

The HJ can bet everything in range, and there’s not much the BB can do about it. Better still, the HJ's EV goes up to 7.644 BB. He’s winning over 76% of the money in the pot with a simple “bet-everything” strategy because the BB plays a very capped, wide range.

With these two examples, you can see how starting ranges impact the outputs. Let’s see how this looks when we node-lock.

Node Locking

Node locking is the cheat code for using solvers. I will devote at least one future newsletter issue to exploring this topic, so I won’t explore the mechanics in depth here. However, I’ll share an example below that supports the abovementioned concepts.

Node locking can be challenging because it requires you to make specific assumptions about your own and your opponent’s range construction throughout the game tree. As with entering ranges, this gets easier over time with practice.

For this example, let’s look at the same formation as above but a different board. I want to return to the 9s-8h-7s board we examined in part 1 of this series.

After a BB check, the HJ bets 53% of its range at equilibrium. The BB’s response is to raise 12% of the hands in its range, call 50%, and fold 38%. This yields an overall EV for the BB of 4.4 BB when measuring from the root node.

We’ve all played against aggressive opponents who will likely bet too frequently. What if the HJ fits that profile in this situation?

We can answer this question by node locking the HJ range and forcing it to bet 100% of hands. We can then rerun the solver to find the optimal response against this strategy.

As you can see, there’s a lot more raising! The HJ is betting too many weak hands, leaving itself vulnerable to counter-exploitation. The BB now check-raises 41% (!!!) of hands at equilibrium, calling 28%, and folding 31%.

Many of these check-raises come from hands that flop a lot of equity: sets, straights, pair plus draws, straight draws, flush draws, etc.

The BB’s EV also increases to 4.77 BB in this situation.

This is just one example, but it shows how changes in an opponent’s range are not just applicable to starting ranges. It extends to how the range changes at different points of the game tree.

Takeaways

I feel like a broken record when I say this, but the most important thing to understand about solvers is how to manipulate inputs to measure their effects on outputs. This is the most critical skill to develop when properly learning to use a solver.

A solver output is simply a snapshot of a game. No singular solver output will perfectly model actual gameplay. We won’t be perfect in building this model, which is fine.

What the solver does great is help us understand how input variations 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 input variations affect outputs, you’ll be on the right path toward using solvers as a positive within your poker studies.

Wrapping Up & Looking Ahead

That concludes today’s newsletter and this 3-part series on the relationship between inputs and outputs.

I’m getting back on a 1.5-2x per month cadence, so expect a newsletter in your inbox around the beginning of April.

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.

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I appreciate you following along and reading the newsletter today.

Until next time.
Michael Lukich