How to win at poker using game theory
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How to win at poker using game theory. A review of the key papers in this field. The main papers on the issue. The first attempts Émile Borel : ‘Applications aux Jeux des Hazard’ (1938) John von Neumann and Oskar Morgenstern : ‘Theory of Games and Economic Behaviour’ (1944)

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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

    • ÉmileBorel: ‘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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Émile Borel: ‘Applications aux Jeux des Hazard’ (1938)

  • How the game is played

    • Two players

    • Two ‘cards’

      • Each card is given a independent uniform value between 0 and 1

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

    • No checking in this game

    • No raising or re-raising


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How the game is played

Betting tree: outcomes for Player 1

  • 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


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Émile Borel: ‘Applications aux Jeux des Hazard’ (1938)

  • Key assumptions

    • No checking

    • X≠Y (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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Émile Borel: ‘Applications aux Jeux des Hazard’ (1938)

Key Conclusions

  • Unique admissible optimal strategies exist for both players

    • Where no strategy does any better against one strategy of the opponent without doing worse against another – it’s the best way to take advantage of mistakes an opponent may make.

  • 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<0.11 where he should fold.

    • player 2 is to call unless Y<0.33 where he should fold

  • Player 1 can aim to capitalise on his opponents mistakes by bluffing


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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 cards have 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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