Quote:
Originally Posted by David Sklansky
[...] Could you now perhaps do the same calculation with one additional rule? There is still only one round of betting after the first card is dealt but when there is a showdown the active players get a second card which they add to their first one. Best total wins.
Below please find a (rounded) Nash equilibrium for this variant of the game.
Quote:
Originally Posted by David Sklansky
Also, I would like to see your solution when called pots are raked three dollars.
Do you mean the original game or the new variant with showdown?
Here is the solution for the variant with showdown (without rake):
HJ raises first in for cards >= 0.7941
CO raises first in for cards >= 0.7605
CO calls after [ raise ] for cards >= 0.8726
BTN raises first in for cards >= 0.7053
BTN calls after [ raise fold ] for cards >= 0.8672
BTN calls after [ fold raise ] for cards >= 0.8536
BTN calls after [ raise call ] for cards >= 0.9031
SB raises first in for cards >= 0.5358
SB calls after [ raise fold fold ] for cards >= 0.8416
SB calls after [ fold raise fold ] for cards >= 0.8264
SB calls after [ raise call fold ] for cards >= 0.8847
SB calls after [ fold fold raise ] for cards >= 0.8019
SB calls after [ raise fold call ] for cards >= 0.8834
SB calls after [ fold raise call ] for cards >= 0.8728
SB calls after [ raise call call ] for cards >= 0.9066
BB calls after [ raise fold fold fold ] for cards >= 0.8109
BB calls after [ fold raise fold fold ] for cards >= 0.7935
BB calls after [ raise call fold fold ] for cards >= 0.8638
BB calls after [ fold fold raise fold ] for cards >= 0.7648
BB calls after [ raise fold call fold ] for cards >= 0.8624
BB calls after [ fold raise call fold ] for cards >= 0.851
BB calls after [ raise call call fold ] for cards >= 0.8903
BB calls after [ fold fold fold raise ] for cards >= 0.6921
BB calls after [ raise fold fold call ] for cards >= 0.8627
BB calls after [ fold raise fold call ] for cards >= 0.8509
BB calls after [ raise call fold call ] for cards >= 0.8911
BB calls after [ fold fold raise call ] for cards >= 0.8318
BB calls after [ raise fold call call ] for cards >= 0.89
BB calls after [ fold raise call call ] for cards >= 0.881
BB calls after [ raise call call call ] for cards >= 0.9078
PS: I computed the solution by solving a system of integral equations numerically. I double checked that the players are indeed (nearly) indifferent at the computed thresholds by running a simulation (the absolute values of the expecations are < 0.0002$ using integration and <0.02$ using simulation). If you wish to reference this strategy please cite this post. Setting this up and coding it required some efforts. Again, I cannot guarantee that I didn't make a mistake. So if someone could verify this independently this would be great.