ON THE BOREL AND VON NEUMANN POKER MODELS

Comparison with Real Poker  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  rng.html rng.html

La Relance  Who has the edge, P1 or P2? 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

 When to bluff if P1 gets a number X<c?  Intuitively, P1 bluffs with c 2 c and folds with X<c 2.  Why? If P2 is playing with the optimal strategy, how to choose when to bluff is not relevant. This penalizes when P2 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)

 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, P1 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

 Since P1 can check,  now he gets positive value out of the game  P1 now bluff with the worst hand. Why? On the bluff part, it’s irrelevant to choose which section of (0,a) to use if P2 calls (P2 calls only when Y>c) On the check part, it’s relevant because results are compared right away.

Von Neumann

 What if player / opponent is suboptimal?  Assumed Strategy  Player 1 Bet if X b, Check otherwise  Player 2 Call if Y > c, fold otherwise  If c is known, Player 1 wants to keep a c

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

FoldLow B High B Low B mLmL mHmH b1b1 b3b3 b2b2

Bellman & Blackwell  Where Or if

La Relance: Non-identical Distribution  Still follows the similar pattern  Where F and G are distributions of P1 and P2, c is still the threshold point for P2. π is still the probability that P1 bets when he has X<c.  What if ?

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 c) ]  + 2 * [ P(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?