How computers learned to play poker

In chess both players see the whole board. A program can look ahead from the position in front of it and pick the move that leads somewhere good. In poker you cannot see your opponent's cards, so you never know which position you are in. You know only the set of hands they could hold, their range.

Hidden information changes what a good strategy looks like. A player who always bets strong hands and always checks weak ones is easy to read and easy to beat. A strong strategy has to mix: bet some strong hands, check some, and bluff with some weak ones, each at the right rate. That is a mixed strategy, and finding the right rates for every hand in every spot is what made poker hard.

The target is a Nash equilibrium: a pair of strategies where neither player can gain by changing theirs alone. In heads-up zero-sum poker before rake, an equilibrium strategy cannot be exploited below the game's equilibrium value. It does not guarantee a profit after fees or over a finite run. Players call it GTO, for game theory optimal.

The breakthrough was a method called counterfactual regret minimization, or CFR, published in 2007 by researchers at the University of Alberta. A program plays the game against itself, over and over. After each hand it asks, for every decision it made, how much better each other action would have done. That shortfall is its regret.

Next time it reaches the same spot, it plays each action in proportion to how much it regrets not having played it. Actions that keep looking better get played more, and actions that keep looking worse fade out. In finite two-player zero-sum games with perfect recall, the average strategy approaches an equilibrium as regret falls.

CFR is still the core of most solvers. The trainer's own preflop charts come from a CFR solve of the six-handed preflop game, as how the charts are made explains.

Limit hold'em fell first because its bets are fixed, which keeps the game small enough to solve almost completely. No-limit is far bigger, since any bet size is allowed, so DeepStack and Libratus solved smaller versions of the game as the hand went on instead of storing an answer for every spot.

In January 2017 Libratus played 120,000 hands of heads-up no-limit against four of the best professionals at Rivers Casino in Pittsburgh and finished well ahead. Two years later Pluribus beat professionals at a six-handed table, where equilibrium theory gives no guarantee. Its starting strategy was trained in eight days on a single 64-core server, which its makers put at under $150 of cloud computing.

The same methods now run on a laptop. A solver takes one spot, two ranges, the stacks and a short list of allowed bet sizes, and computes an equilibrium strategy for that game: how often each hand bets, checks, calls or folds, and what each choice is worth.

Those answers are only as good as the game they solve. A solver that may only bet a third of the pot or the whole pot gives the best strategy with those two sizes, not with every size. It also takes time: in this trainer's own tests, an exact solve of one flop took from half a minute to over two minutes on a laptop, across three flops in one spot.

That is fine for study and too slow for a drill, which is why the trainer ships exact solutions for a fixed set of flops and uses an engine for the rest.

Solved Hand ships exact solutions for 735 flops across 17 heads-up spots. The packs are numerical solutions for fixed ranges and bet menus, solved to a tolerance rather than with zero error. On a supported flop without a stored answer, Turnstone searches the spot on demand. It plays the flop out for 30 rounds and, at the turn, asks a network trained on 100,000 exact turn solves what each hand is worth. With a copy of the engine already running, it delivers flop strategy in seconds; the first answer after a quiet spell is slower. The turn and river use numerical solves with restricted bet menus and a configured stopping tolerance.

A search is close to an exact solve, not equal to one, so its grade is marked as an engine estimate. The published accuracy benchmark covers 575 stored flops across 13 spots, measured on 2026-10-13. That benchmark is a subset of the current library; the four additional spots do not inherit its accuracy claim:

Average loss

0.3% of pot

over 575 flops in 13 spots

Highest spot average

0.79% of pot

3-bet pots at 40 big blinds

Time per flop

Seconds

with the engine already running

In chips, on an ordinary flop pot:

Those averages are not error bounds for an individual hand or action. A close decision can change under an approximate answer, so every grade at the table says where it came from: exact, engine estimate with its measured loss when available, approximate, or rule of thumb.

• It protects against exploitation in the solved heads-up game. Against a player who folds too often, a best response wins more.
• Its answer belongs to the game that was solved. Change the stacks, the bet sizes or the ranges and the answer can change.
• It covers the spots that were solved. The stored library includes raised pots, 3-bet pots, a 4-bet pot and a limped blind battle. The accuracy figures above cover the measured raised and 3-bet spots. Hands outside supported coverage receive approximate or rule-of-thumb grades, with the source shown.

A solver shows what a balanced strategy does within those limits. Studying the answer means working out why it chooses those actions.