How Do Poker Solvers Work?

We use powerful software programs called solvers to compute GTO strategies for poker.

CFR: The most trusted solver algorithm

In 2007, Zinkevich et al. published a paper introducing Counterfactual Regret Minimization (CFR), and it has remained the gold standard of poker solver algorithms ever since. CFR is a simple yet effective iterative algorithm that is mathemtically guaranteed to converge to a Nash equilibrium in two-player zero-sum games.

The concept is straightforward:

1. Build a game tree that represents all possible actions in a game
2. Randomly initialize a strategy for each player
3. Simulate a hand and produce regret values for each action based on the EV loss
4. Update the strategy to increase frequency of low regret actions and decrease frequency of high regret actions
5. Repeat steps 3-4 for each player until the strategy converges to a Nash equilibrium

We measure convergence using exploitability. This is covered in more detail in the Solution Quality, dEV and EV Loss article.

CFR produces a lookup table of the frequencies for each available action for each available hand for every node in the game tree. The strategy at a node is often displayed as a strategy grid.

How many iterations is enough?

The more iterations a CFR solver runs, the closer the output strategy will be to a true Nash equilibrium. We can quantify the exploitability of a strategy by fixing one player's strategy and computing the best response for the other player to measure the EV difference. With this method, we can bound the error of the solution and run iterations until the exploitability is sufficiently small.

In general, solving to 0.5% dEV (exploitable for 0.5% of the pot) is considered standard practice, although proper commercial libraries sometimes set tighter thresholds. All GTO Genesis solutions are solved to either 0.2% or 0.1% dEV, depending on the flop type.

Limitations

CFR solvers provide absolute truth for the game tree they are given, but if the game tree itself is a poor representation of the actual game, the solution will not generalize well to actual poker.

• Hand abstraction: Some solvers group similar hands together to reduce computational complexity, causing the same strategy to be played for multiple hands.
• Bet sizing: Solvers can only optimize among the bet sizes they are given. In practice, one or two bet sizes is usually enough to capture a vast majority of the EV of a hand.
• Computational requirements: Solvers require significant computing power that scales with game tree size, range size, and exploitability threshold. You will rarely find precomputed solutions with many bet sizes and tight convergence because of the time and memory required to compute them.

Unexploitability in multiplayer games is simply not possible. While Nash equilibrium strategies exist for multiplayer games, opponents can exploit an equilibrium strategy by coordinating their strategies. This is a limitation for any kind of solver, not just CFR solvers.

The modern solver scene

There are many ways to source GTO poker solutions. These can be classified in 4 categories: personal CFR solvers, solver libraries, AI-assisted CFR solvers, and pure AI solvers.

Personal CFR solvers

This includes solvers like PioSolver, GTO+, HRC, and MonkerSolver. Using these solvers, you define ranges and game tree rules and wait minutes to hours while your own computer computes the solution to a single flop.

Pros

• Complete control over inputs and outputs
• Nodelocking

Cons

• Requires expensive hardware and a lot of time to compute solutions for complex scenarios
• Requires choosing the right bet sizes and abstractions
• Requires organizing and storing solutions for future reference

Solver libraries

This includes websites like GTO Genesis, GTO Wizard, and GTOBase.

Solver libaries compute millions of solutions using CFR solvers and store them in a giant database for users to browse. These solutions are precomputed and can be accessed instantly.

Pros

• Instant access to solutions
• Solutions can be solved with many bet sizes and tight convergence thresholds because the cost is shared among many users
• Better tooling to interpret and visualize solutions

Cons

• Limited to the configurations provided by the library
• Preflop ranges and bet sizes may not match your own preferences

AI-assisted CFR solvers

By using a neural network to "guess" EVs of deeper branches of the game tree, we can enhance existing CFR algorithms to compute solutions faster at the cost of precision. This is the approach used by solvers like GTO Wizard AI and DeepSolver to compute custom solutions in seconds instead of minutes or hours.

Pros

• Near instant access to custom solutions
• Control over most inputs
• Can benchmark exploitability on sample outputs to provide general exploitability distributions

Cons

• Exploitability can not be bounded because neural networks are not guaranteed to be accurate
• Neural network only as good as the data it was trained on, which may not be representative of your own game tree or bet sizes
• Nodelocking is not possible on future streets since deeper branches of the game tree are not actually simulated

Pure AI solvers

Some groups have trained neural networks to play poker without using CFR at all. Most of these are reinforcement learning agents that learn to play poker by playing against other agents millions of times.

Pros

• Near instant access to custom solutions
• Can respond and use any bet size

Cons

• There is no clear data that demonstrates pure AI solutions are sufficiently accurate

Key takeaways

The gold standard of poker solving trusted by the pros is still CFR, which is the most mathematically sound method for computing GTO strategies. AI exists to speed up solvers or produce solutions on the fly, but the precision and reliability of CFR solvers is currently unmatched, especially for specific nodelocked lines that may not appear in AI training data.