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ON THE BOREL AND VON NEUMANN POKER MODELS

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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

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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

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The Poker Models http://www.cs.virginia.edu/~mky7b/cs6501poker/ rng.html http://www.cs.virginia.edu/~mky7b/cs6501poker/ rng.html

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

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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

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La Relance

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When to bluff if P1 gets a number X

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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

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La Relance

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La Relance (Player 2 strategy)

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

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

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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?

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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?

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Von Neumann Betting tree:

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Von Neumann

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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.

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Von Neumann

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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

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Von Neumann

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

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

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

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Bellman & Blackwell

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FoldLow B High B Low B mLmL mHmH b1b1 b3b3 b2b2

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Bellman & Blackwell Where Or if

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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

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La Relance: (negative) Dependent hands

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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) ]

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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))

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

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

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