# The gameplan

In Poker NL Hold´em there exist about 10^165 different game states [1]. That's more than the number of atoms in the whole universe (about 10^80). As long as you don't have a machine with 10^49 yottabytes of RAM [2], you have to transform the natural game into an abstract game, if you want to calculate a GTO-Gameplan. This transformation includes several abstractions where some of them cost no EV, a bit EV, and significantly EV.

Every „GTO"-Gameplan you saw so far has significantly EV-Costing abstractions included. Under these boundary conditions, you still received a „GTO“-Solution, but how well is the quality of these strategies, when the boundary condition themselves cost you EV?

For the last three years, we were developing the GTO-Gameplan where only no EV-Costing abstractions were included. If you like, you can call this the "max. EV" GTO-Gameplan.

The strategy-differences are for some PF-Ranges, a few Flops, some Turns, and a lot of Rivers huge. The Differences are correlating mainly with the complexity of the situation which is correlating with the range size, stack size, and deepness in the game-tree.

A list of abstractions for Poker which is typically used on public solvers. The color describes the negative effect of the EV:

- Separating Pre/-Postflop Gameplan (red)

Separating Pre/-Postflop Gameplan reduces the complexity of the calculation.

To get a result for Preflop-Ranges, there has to be done assumptions for Pre/- and Postflop-Gameplan.

The Bet-Subset, amount of Flops, and Runouts have to be pre-defined by the User. These factors are highly correlating with the quality of the Preflop-Ranges. Every possible hand benefits from some specific actions.

For example: If you only allow a small 3Bet-Size on Preflop and big raise-sizes for Postflop, a Preflop-Solver starts to trap more often high equity hands like KK/AA vs. Open-Raises. Configuring optimal Bet-Subsets with respect to all possible Postflop scenarios is important. If these are suppressed by the configuration, the EV drops.

Calculate Flop + Turn -> River (in 2 Steps) (yellow to red)

Calculating the Flop and Turn at once and the River on demand leads to different River strategies related to if the whole Postflop-Gameplan would be calculated at once. The reason is, that for scenario 1 the Players have fixed ranges when they are in front of the River. In scenario 2 they aren't. This allows the Players to "choose" their strategies depending on the River action.

In fact, the Player who uses the Gameplan from scenario 2 has on average the higher Range-EV at the River.

Calculate Flop -> Turn - River (in 3 Steps) (red)

Calculating the Flop, then the Turn, and afterward the River on demand leads to different Turn and River strategies related to if the whole Postflop-Gameplan would be calculated at once. It is the same principle as above.

The Player who uses the Gameplan which was calculated in one step has on average the higher Range-EV at the Turn and River.

User has to define own Bet-Subset for Postflop (green to red)

Every hand has his highest EV in a specific action, but a lot of them are also fine in another one. This correlates how many trees in the future are possible, stack size, SPR, and range size.

Another fact is the gap between the possible sizings. Having a Bet-Subset: (40/55) and (40/80)% Potsizebet affects also the Check and 40% Betsize.

An optimal Bet-Subset is one where every possible hand of the range finds for every possible scenario his max. EV action[4].

Our Gameplan includes for every scenario these Bet-Subset which is in an interval from 1 to 6 actions at one node.

Removing one "red" abstraction follows into an advance of the required RAM to do the calculation.

For example: Connecting the Pre/-Postflop Gameplan with a simple Bet-Subset needs about 16.000.000 GB RAM.

This leads to the situation that it's not possible to receive a "max. EV"(= only green flags are allowed) GTO-Gameplan with a home pc or a virtual machine[3].

Poker-Scientist offers you this Gameplan.

Effective Simplification of the Gameplan

Eliminating non-relevant lines (nodes) makes the gameplan simpler and doesn't reduce the Range-EV. 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 neglectable nodes. Neglabe means that the Nash-Equilibrium didn't shift between the "complex" Gameplan and the simplified Gameplan.

Simple description

The algorithm counts the number of nodes "n" for the Gameplan. After it, he is removing "x" nodes from it until the first derivation of the function(n-x) = 0 and holding the condition of no shift at the Nash-Equilibrium.

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 max. EV GTO-Gameplan with the addon of efficient simplification.

Special thanks to the whole Team of Poker-Scientist, the University of Cologne, and our partner Google which supports us on our way.

Appendix:

[1]

[2]

[3]

[4]