Consider G = betting_game(n) .Let T be is its Breath First tree....Also let say we have m possible different
hand. This game:
3n+3 nodes
n+2 decision nodes:
First 2 and last 2 decision nodes has 2 choices. n-2 other ones have 3 choices
Also, for each player define 'Max Number Of Bet Target', MNBT as: MNBT = 2R+B
For now let say n=12: Strategy can be seen as:
For each possible holding hand we should decide:
As OP: What do you want your last raise make the MNBT ? ( assuming opponent is coming!) What is your action in
case opponent passed your MNBT? call or fold?
Therefore, for each possible holding hand for OP there [(n/2)+1]*2=n+2 choices.
As IP: Facing Bet: Go for Max 2R =< n/2. Call or Fold in case opponent passed your Max 2*R
As IP: Facing Check: Go for Max 2R+B =< n/2 . all or Fold in case opponent passed your Max 2*R+B
Therefore for each possible holding hand for IP there [(n/2)+1]*2=n+2 choices.
So Total Number Of Pure Strategies for each position Is: (n+2)^m
For n=2,m=10: 10^6 -- For n=4,m=10: 6*10^7 -- For n=6,m=10: 10^9
Note that this is all pure strats, even crazy ones and dominated ones
Important Question How can we consider non-dominated ones?
Probabilistic weighted combo of Pure Strategies
Important question: Each Pure Strategy ps: (Pstate, Hand) ---> Action. i.e:
Finally, How should we define Strats pure or behavioral or mixed? We could construct them as a tree of table, and work with them. Which one is better? One idea is to consider behavioral strats as tree to fully use tree structure of the game Can we do everything in terms of matrix and np arrays?
See your action as Fold/Check, Bet/Call, Raise cut-offs at each public nodes C(5,3)=10 -- C(8,3)=56 -- C(10,3)=120 -- C(20,3)=1140 -- Monte Carlo Tree Search: See your action as Fold/Check, Bet/Call, Raise cut-offs
Number Of Public Nodes is n+2
Number of Information States is (n+2)*m
Number of World States is (n+2)*m^2