# Solve Nash equilibrium for poker

Heads up poker two bots with fixed betting. They can only bet 10, call 10, or fold pre flop. Alternate who is first (in position) each hand. In position has a 2 chip blind and out of position has a 1 chip blind. In position can bet, check/call, or check/fold. Out of position bet if checked to, call, or fold. I assume they would each come to the same hole cards for each of those 6 options.

There is so much variance in poker that unless you had a large number of chips one player is going to bust out before you come to a Nash equilibrium. If you are betting 10 then starting stack would be like 1000 in typical game. If you just let the player reload then has finite resources been violated.

How does a solver deal with variance. Even the very best hand (AA) will lose about 14% of the time.

Does a bot even need to know the rules of poker? Would you seed the bot with starting hands for those 4 decisions?

Take it to the next level where you bet the flop, turn, and river (streets). Bot needs to consider the board to decide how to bet.

It gets to be a lot of runs and this simple game does not allow for raises and ignores the blinds. Bot should even consider stack sizes in decisions.

• Two points. (1) Did you try estimating this empirically with reinforced learning? My first take would be to build a RNN that takes in all current and previous betting round states and returns bet/fold and if bet, raise amount. (2) For an easily computable mathematical answer, this question is almost certainly too complicated. Try solving a simpler one, like: "What's the Nash equilibrium in five-card draw with $n$ players and one round of betting?" – Neal Aug 1 '17 at 15:16
• @Neal I said take it to the next level. I am just starting with head up one round and no raises. Is RNN recurrent neural network. – paparazzo Aug 1 '17 at 15:31