Borel Card Games

8/2/2019 Borel Card Games

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How to win at poker using game

theory

A review of the key papers in this field


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The main papers on the issue

The first attempts mile Borel: Applications aux Jeux des Hazard

(1938)

John von Neumann and Oskar Morgenstern :Theory of Games and Economic Behaviour (1944)

Extensions on this early model Bellman and Blackwell (1949)

Nash and Shapley (1950) Kuhn (1950)

Jason Swanson: Game theory and poker (2005) Sundararaman (2009)


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Jargon buster

Fold: A Player gives up his/her hand. Pot: All the money involved in a hand.

Check: A bet of Zero. Call: Matching the bet of the previous player. Ante: Money put into the pot before any

cards have been dealt.


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mile Borel: Applications aux Jeuxdes Hazard (1938)

How the game is played Two players Two cards

Each card is given a independent uniform valuebetween 0 and 1

Player 1s card is X, Player 2s Card is Y

No checking in this game No raising or re-raising


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How the game is played

First both players ante 1 The pot is now 2

Player 1 starts first Either Bets or Fold Folding results in player 2 receiving 2 wins 1

Player 2 can either call or fold. Folding results in player 1 receiving 3 wins 1

Then the cards are turned over

The highest card wins the pot

Ante 1 Player 1

Fold -1

Bet [B=1] Player 2Call 2

Fold +1

Betting tree: outcomes for Player1


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mile Borel: Applications aux Jeuxdes Hazard (1938)

Key assumptions

No checking XY (Cannot have same cards) Money in the pot is an historic cost (sunk cost)

and plays no part in decision making.


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mile Borel: Applications aux Jeuxdes Hazard (1938)Key Conclusions

Unique admissible optimal strategies exist for bothplayers

Where no strategy does any better against one strategy ofthe opponent without doing worse against anotherits thebest way to take advantage of mistakes an opponent maymake.

The game favours Player 2 in the long run

The expected winnings of player 2 is 11% when B=1 The optimum strategies exists player 1 is to bet unless X


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John von Neumann and Oskar Morgenstern :

Theory of Games and Economic Behaviour (1944)

New key assumption: Player 1 can now check

New conclusions Player 1 should bluff with his worst hands The optimum bet is size of the pot


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One Card Poker

3 Cards in the Deck {Ace, Deuce, Trey} 2 Players One Card Each

Highest Card Wins Players have to put an initial bet (ante)

before they receive their card

A round of betting occurs after the cardshave been received The dealer always acts second


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One Card Poker

Assumptions Never fold with a trey Never call with the ace

Never check with the trey as the dealer Opener always checks with the deuce


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One Card Poker

Conclusions Dealer should call with the deuce 1/3 of the time Dealer should bluff with the ace 1/3 of the time If the dealer plays optimally the whole time, then

expected profit will be 5.56%


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Borel Card Games