Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
if P1 bets, is it 1 chip, or any number of chips ? – Xoff Jan 18 '13 at 15:55
It's fixed at 1 bet. – Wesley Tansey Jan 19 '13 at 0:40

Many analytical solutions have been found, see

As for iterative methods, it has been proven that counterfactual regret minimization ". . . eliminates strictly dominated plays . . ." in games that are not two player zero-sum (see Richard Gibson's dissertation:

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.