ON THE BOREL AND VON NEUMANN POKER MODELS
ON THE BOREL AND VON NEUMANN POKER MODELS
Comparison with Real Poker: � Around 2. 6 million possible hands for 5 card stud � Hands somewhat independent for Texas Hold ‘em Let’s assume probability of hands comes from a uniform distribution in [0, 1] Assume probabilities are independent
The Poker Models La Relance Rules: � Each player puts in 1 ante before seeing his number � Each player then sees his/her number � Player 1 chooses to bet B/fold � Player 2 chooses to call/fold � Whoever has the largest number wins. von Neumann Rules: � Player 1 chooses to bet B/check immediately � Everything else same as La Relance
The Poker Models http: //www. cs. virginia. edu/~mky 7 b/cs 6501 pok er/rng. html
La Relance Who has the edge, P 1 or P 2? Why? Betting tree:
La Relance The optimal strategy and value of the game: � Consider the optimal strategy for player 2 first. It’s no reason for player 2 to bluff/slow roll. � Assume the optimal strategy for player 2 is: Bet when Y>c Fold when Y<c � Nash’s Equilibrium
La Relance
La Relance
La Relance
La Relance
La Relance When to bluff if P 1 gets a number X<c? � Intuitively, P 1 bluffs with c 2<X<c, (best hand not betting), bets with X>c and folds with X<c 2. � Why? If P 2 is playing with the optimal strategy, how to choose when to bluff is not relevant. This penalizes when P 2 is not following the optimal strategy.
La Relance What if player / opponent is suboptimal? Assumed Strategy � player 1 should always bet if X > m, fold otherwise � player 2 should always call if Y > n, fold otherwise, Also call if n > m is known (why? ) Assume decisions are not random beyond cards dealt Alternate Derivations Follow
La Relance
La Relance (Player 2 strategy)
La Relance (Player 2 strategy) What can you infer from the properties of this function? What if m ≈ 0? What if m ≈ 1?
La Relance (Player 1 response) Player 1 does not have a good response strategy (why? )
La Relance (Player 1 Strategy) Let’s assume player 2 doesn’t always bet when n > m This function is always increasing, is zero at n = β / (β + 2) � What should player 1 do?
La Relance (Player 1 Strategy) If n is large enough, P 1 should always bet (why? ) If n is small however, bet when m > What if n = β / (β + 2) exactly?
Von Neumann Betting tree:
Von Neumann
Von Neumann Since P 1 can check, � now he gets positive value out of the game � P 1 now bluff with the worst hand. Why? On the bluff part, it’s irrelevant to choose which section of (0, a) to use if P 2 calls (P 2 calls only when Y>c) On the check part, it’s relevant because results are compared right away.
Von Neumann
Von Neumann What if player / opponent is suboptimal? Assumed Strategy � Player 1 Bet if X < a or X > b, Check otherwise � Player 2 Call if Y > c, fold otherwise � If c is known, Player 1 wants to keep a < c and b > c
Von Neumann
Von Neumann
Von Neumann (Player 1 Strategy) Find the maximum of the payoff function a = b = What can we conclude here?
Von Neumann (Player 2 Response) Player 2 does not have a good response strategy
Von Neumann (Player 2 Strategy) This analysis is very similar to Borel Poker’s player 1 strategy, won’t go in depth here… c =
Bellman & Blackwell
Bellman & Blackwell m. L Fold m. H b 1 Low B High B b 3 Low B b 2 High B
Bellman & Blackwell Or if Where
La Relance: Non-identical Distribution Still follows the similar pattern Where F and G are distributions of P 1 and P 2, c is still the threshold point for P 2. π is still the probability that P 1 bets when he has X<c. What if ?
La Relance: (negative) Dependent hands
La Relance: (negative) Dependent hands Player 1 bets when X > l � P(Y < c | X = l) = B / (B + 2) Player 2 bets when Y > c � (2*B + 2)*P(X > c|Y = c) = (B + 2)*P(X > l|Y = c) Game Value: � P(X > Y) – P(Y < X) � + B * [ P(c < Y < X) – P(l < X < Y AND Y > c) ] � + 2 * [ P(X < Y < c AND X > l) – P(Y < X < l) ]
Von Neumann: Non-identical Distribution Also similar to before (just substitute the distribution functions) � a | (B + 2) * G(c) = 2 * G(a) + B � b | 2 * G(b) = G(c) + 1 � c | (B + 2) * F(a) = B * (1 – F(b))
Von Neumann: (negative) Dependent hands Player 2 Optimal Strategy: Player 1 Optimal Strategy:
Discussion / Thoughts / Questions Is this a good model for poker?
- Slides: 37