In no-limit Texas Holdem Poker, more than 10^165 different game states are possible . This is more than the total number of atoms in the whole universe (about 10^80). To calculate a GTO gameplan of this size would require a machine with 10^49 yottabytes of RAM . To solve this problem with the computing power available to us, various abstractions are applied to transform the natural game into a simpler one. These abstractions have varying effects on the difference between the gameplan’s EV and the maximum possible EV, known as the EV loss. Some abstractions cause no EV loss, some a small amount of EV loss, and some significantly affect EV loss.
Every GTO gameplan so far has been calculated with abstractions that cost significant EV. Under these boundary conditions, a GTO solution was still produced - but how high is the quality of these strategies, when the boundary condition themselves cost EV?
Over the last three years, we have developed a GTO gameplan that includes no EV-costing abstractions. The strategy differences are particularly noticeable in the preflop ranges, some flop and turn solutions, and many river solutions. The differences correlate with the complexity of the situation, which in turn is related to the range size, stack size, and depth within the game tree.
Let’s take a look at the most common abstractions for poker typically used by solvers. The color describes the effect on the EV compared to optimal solutions.
Separating pre and postflop gameplan – High EV loss
Separating the calculation of the pre and postflop gameplan reduces the complexity of the calculation. To calculate the preflop ranges requires various assumptions to be made. The bet size subset, number of flops, and runouts have to be pre-defined by the user. These factors are highly connected with the quality of the preflop ranges. The EV of every hand is influenced by these decisions.
For example: If you only allow a small preflop 3-bet size and large postflop raise sizes, a preflop solver chooses to trap more often vs. open raises with high equity hands in its range, like KK/AA. Configuring optimal bet sizing subsets with respect to all possible postflop scenarios is important. If this is not done, the EV cost is significant.
Calculating flop, turn, and river in 2 steps – Medium-high EV loss
Calculating the flop and turn at the same time, but calculating river solutions as required leads to different river strategies when compared to when the whole postflop gameplan is calculated at once. The reason is that when calculating the river independently of the flop and turn, the players have fixed ranges when they reach the river. When the full gameplan is calculated from start to finish, the ranges are not fixed at the river. When ranges are not fixed, this allows the players to "choose" their strategies depending on the river action. Overall, the player who uses a gameplan calculated from flop to river in one step has on average a higher range EV at the river.
Calculating flop, turn, and river in 3 Steps – High EV loss
Calculating the flop, then the turn, and finally, the river in 3 independent steps leads to different turn and river strategies when compared to when the whole postflop gameplan is calculated at once. It is the same principle as above. The player who uses a gameplan calculated from flop to river in one step has on average a higher range EV at the turn and river.
User defines their own bet size subset for postflop – Low to high EV loss
Every hand has its highest EV for a specific action, but a lot of hands are also fine EV-wise when taken into a different action. This directly relates to how many trees are possible in the future, as well as stack size, SPR, and range size.
Another factor is the gap between each possible bet sizing. Having a bet sizing subset of 40%/55% (gap of 15) of pot or 40%/80% (gap of 40) of pot also affects the check and 40% bet ranges. An optimal bet subset is one where every possible hand of the range finds its max. EV action for every possible scenario . Our gameplan includes these bet subsets for every scenario, choosing between 1 to 6 actions at each node.
Removing one high EV loss abstraction dramatically increases the amount of computer RAM required to perform the calculation. For example, connecting the pre/postflop gameplan with a simple bet sizing subset needs about 16,000,000 GB RAM. This leads to a situation in which it is not possible to calculate a "maximum EV" GTO gameplan (low-EV loss for all abstractions) with a home pc or a normal virtual machine .
Poker Scientist offers this gameplan.
Effective Simplification of the Gameplan:
Eliminating non-relevant lines (nodes) makes the gameplan simpler and doesn't reduce the players‘ range EVs. The Nash equilibrium approach, used by solvers, doesn't care about the number of lines it produces - but the player does.
We added an additional algorithm to the gameplan which eliminates negligible nodes. Negligible means that the nash equilibrium didn't shift between the "complex" gameplan and the simplified gameplan.
The algorithm counts the number of nodes "n" for the gameplan. It then removes "x" nodes from it until the first derivation of the function(n-x) = 0 and when the condition of no shift at the nash equilibrium is true.
The result is that the gameplan was further simplified on average by over 41%, as measured by the equation:
#nodes(simplified gameplan) / #nodes(complex gameplan):
Poker Scientist presents the highest EV GTO gameplan with the addition of efficient simplification.
Special thanks to the whole Poker Scientist team and the University of Cologne.