Solution Quality, dEV, and EV Loss

Not all GTO solutions are created equal. This article goes over what makes a solution high quality and how you can tell whether it is good enough for you to use.

Approximate equilibria

We can asymptotically approach Nash equilibrium, but we can never reach it exactly. There is no such thing as a perfect GTO solution, but it is possible to get close enough that it does not matter for even the most sensitive use cases. The question is: how close is close enough?

Exploitability

We measure the closeness to equilibrium using exploitability (also known as dEV or delta EV). Exploitability is measured as a percentage of the pot, and represents the amount a perfectly countering opponent could theoretically exploit the strategy for.

Less exploitable solutions also tend to be cleaner and easier to execute, since dominated sizes disappear as solutions converge.

In general, tighter convergence is strictly better, but computation time grows exponentially as thresholds are tightened. Use your own judgment when deciding whether solution data is acceptable for your intended use case.

GTO Genesis solutions for SRP spots are guaranteed to be below 0.2% dEV and all other spots below 0.1% dEV, making GTO Genesis an industry leader in low exploitability.

EV loss from simplification

If we build a game tree that is too simple, our solutions may not be representative of the actual game.

Consider a BTN vs. BB SRP on AK5 rainbow, as shown in the image in the section above. Our solutions converged on a BTN c-bet strategy using strictly B125 with an EV for the BTN of 3.08bb. If we only allow B33 or check instead of our usual suite of sizes and holding all else equal, the BTN EV drops to 3bb.

We show the EV difference from different simplifications in the table below. Note that all solutions are converged to below 0.2% dEV.

In this case, the B125 size is the "optimal" size for this flop out of the sizes we tested. The optimal size depends on a variety of factors including board texture, ranges, and SPR. Sometimes the optimal strategy is mixing between mutliple sizes. By giving the solver as many sizes as we can, we maximize the solver's ability to find the best strategy for the spot.

The takeaway: we must provide the solver with the bet sizes it needs to accurately represent the real game of poker.

Signs of poor convergence

How can we tell if solutions are sufficiently converged? Any reputable provider will disclose the exploitability of their solutions, but there are other signs to look for as well.

1. Using sizes we know are not optimal for the spot. It is generally known that in a BTN vs. BB SRP, AK5 rainbow is an overbet board. Strategies that have not converged on a pure overbet in this spot are likely not converged.
2. Mixing between actions that are clearly dominated. The mixing of actions for a given hand should only occur when each action has the same EV. This mainly applies in the main lines – nodes that involve a zero-frequency action may ignore this rule because the solver is not optimizing for those nodes.
3. Taking strange lines. This is harder to determine since it is subjective, but as an example it is commonly known that on AK5 rainbow, the BTN should be checking back AA because it blocks the hands that can call an overbet. If a solution is overbetting AA in this spot, we probably shouldn't trust the solution.

Key takeaways

At the end of the day, studying high-quality solutions is what matters. Low exploitability is always better than high exploitability, but there are diminishing returns as you approach zero. We recommend using the best solutions you can reasonably access, and use common sense when evaluating convergence metrics and the construction of the game tree.