Managing Editor
GTO Wizard AI has made some important progress toward its next roadmap milestone of adding full multiway postflop functionality to its solver. The app now supports custom solutions for three-handed situations on later streets, with or without a rake.
The team behind GTO Wizard AI is impressively transparent about its efforts. In a blog post earlier this week, the team published benchmarks for the new feature. For river situations, the app can provide less than a 0.1% margin of error within seconds. Earlier streets are more difficult, though GTO Wizard AI uses a neural network to speed those calculations up. As a result, the app still only takes seconds to solve flop and turn situations, but the margin of error increases to around 0.2% to 0.3% for most hands, with rare outliers up to 0.7%.
Based on the team’s description, that neural network seems conceptually similar Google’s approach with AlphaGo. That superhuman Go-playing algorithm uses two neural networks. One suggests moves, and the other guesses at the probability of victory from the resulting positions. Similarly to that second AlphaGo module, GTO Wizard “intuits” what a hand is worth on later streets based on the experience stored in its neural network, without needing to work out optimal play for those streets.
The “margin of error” here is Nash distance, which translates to maximum theoretical exploitability as a percentage of pot size. For instance, if the pot is 10 big blinds (BB), a 0.1% Nash distance equates to a maximum loss of 1 BB per hundred hands relative to perfect play.
That may sound like a lot, given that many players win only single digits of BB per hundred hands. However, perfect solvers don’t exist for No-Limit games, and these sorts of Nash distances are better than a human player can achieve.
Solving for Three Players is Much Harder Than Two
The differences between one poker situation and another loom much larger for machines than they do for humans. That’s because we rely on heuristics (a fancy word for “rules of thumb”) instead of brute force calculation.
Fixed-limit, heads-up Texas Hold’em was fully solved years ago. There were even machines allowing players to play rake-free against a computer, because the strategy was provably unbeatable.
However, the difference between limit and no-limit games is that the no-limit format makes every action a choice between tens or hundreds of options (depending on stack depth and bet increments) instead of just two or three.
The size of a game tree is a function of the number of choices at each “node,” raised to the power of the number of decisions until the end of the game.
So, if we expect two players to each make three decisions (six total) until we reach a showdown, and the options are only fold, check/call, or bet/raise, then the game tree has approximately 36, or 729 branches.
But if you give the players 100 possible bet sizes, then the number becomes more like 1006, which is one quadrillion. That’s why solvers don’t look for exact solutions in no-limit poker, but instead use various techniques to arrive at an approximate solution, while trying to minimize that Nash distance.
Adding a third player in this particular example changes the exponent from six to nine (three decisions times three players), which also makes the solution another million times harder.
In other words, stack depth and player count compound each other in an exponential way when it comes to the solving difficulty.
The Truth About Multiway Poker: There Is No Solution
Just looking at the combinatorial difficulty of a solution understates the challenge of multiway poker, however. The reality is that no solution exists, due to a fundamental limitation of game theory. At least, not in the same sense that solutions exist in a two-person game.
The problem is that players can gang up on one another, intentionally or not.
Game theory assumes that all players know every other player’s strategy and looks for a situation where no individual player can change their current strategy without hurting themselves. That’s called an “equilibrium.”
Facing one opponent, a player using an equilibrium strategy is unexploitable. If the opponent deviates, the opponent will do worse. It’s possible that the player could do even better by also adjusting to exploit the opponent. However, they will never suffer a loss by sticking with the equilibrium strategy.
That’s not true in a multi-way situation if two or more opponents change their strategies simultaneously. There can be multiple equilibria in a multi-way game—even infinitely many, if it’s a game of imperfect information, like poker. Some of them will be more advantageous for a given player than another.
So, if two players simultaneously switch from one equilibrium to another, it ends up being the third player who has, effectively, “deviated” and is getting a less-than-perfect result. Furthermore, even if that player adjusts, the new equilibrium might be worse for them than the previous one.
That’s an important caveat to keep in mind when “solving” multiway spots.
What’s Next on the Roadmap?
Multiway situations are one of two special cases the GTO Wizard AI team is currently working on. The other work in progress is Progressive Knockout Bounties. Although much of the functionality for PKO analysis is already in place, the company says that a more detailed version will be complex enough that it intends to spin it off into a separate subscription for players who specialize in that popular format.
There is a lot more work to be done on the postflop multiway solver, such as handling situations with more than three players and integrating tournament ICM considerations. Intergrating each new feature of the app with every previous advancement compounds the complexity of the task.
Later steps on the roadmap include:
- Pot-Limit Omaha
- Translating solver outputs
- Skill-adaptive solutions
That last seems to mean settings to accept small compromises in EV to simplify the game and create good-enough lines for lower-level players.
Additionally, the app’s pricing plan lists a number of features as “coming soon,” such as puzzles, stats & reports, and “Opponent Leak Finder.”
