Let's say we extend the popular half-street Kuhn poker variant to 3 players. The rules would be as follows:
Rules of 3-Player Half-Street Kuhn Poker
- 3 Players
- 4 cards in the deck (J, Q, K, A; or 0-3 if you prefer)
- All players ante 1 chip
- P1 can bet or check
- If P1 checks, P2 and P3 must check. Showndown occurs.
- If P1 bets, P2 and P3 can each call or fold.
This is effectively an extensive form game, where you have 24 possible combinations of hands:
P1=A P2=K P3=Q
P1=A P2=K P3=J
P1=A P2=Q P3=K
P1=A P2=Q P3=J
P1=A P2=J P3=K
P1=A P2=J P3=Q
... (18 more)
Each combination starts a subtree with one decision node each for P1 and P2, and two nodes for P3:
Prob(P1 bets)
Prob(P2 calls | P1 bets)
Prob(P3 calls | P1 bets and P2 calls)
Prob(P3 calls | P1 bets and P2 folds)
Each subtree results in 5 leaf nodes: c, bcc, bcf, bfc, bcf. This means we have a game tree of $24*5=120$ leaf nodes.
My question is two-fold. First, this game seems small enough to solve analytically-- how would that work? Second, are there any iterative methods guaranteed to converge to an optimal solution for > 2 player games? I know Counterfactual Regret Minimization and Fictitious-Play are only assured for 2-player games.