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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.

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Presentation on theme: "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."— Presentation transcript:

1 ON THE BOREL AND VON NEUMANN POKER MODELS

2 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

3 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

4 The Poker Models  rng.html rng.html

5 La Relance  Who has the edge, P1 or P2? Why?  Betting tree:

6 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

7 La Relance

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11  When to bluff if P1 gets a number X

12 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

13 La Relance

14 La Relance (Player 2 strategy)

15  What can you infer from the properties of this function?  What if m ≈ 0? What if m ≈ 1?

16 La Relance (Player 1 response)  Player 1 does not have a good response strategy (why?)

17 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?

18 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?

19 Von Neumann  Betting tree:

20 Von Neumann

21  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.

22 Von Neumann

23  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

24 Von Neumann

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26 Von Neumann (Player 1 Strategy)  Find the maximum of the payoff function  a =  b =  What can we conclude here?

27 Von Neumann (Player 2 Response)  Player 2 does not have a good response strategy

28 Von Neumann (Player 2 Strategy)  This analysis is very similar to Borel Poker’s player 1 strategy, won’t go in depth here…  c =

29 Bellman & Blackwell

30 FoldLow B High B Low B mLmL mHmH b1b1 b3b3 b2b2

31 Bellman & Blackwell  Where Or if

32 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

33 La Relance: (negative) Dependent hands

34  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) ]

35 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))

36 Von Neumann: (negative) Dependent hands  Player 2 Optimal Strategy:  Player 1 Optimal Strategy:

37 Discussion / Thoughts / Questions  Is this a good model for poker?


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