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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 – Émile 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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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 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 £1Player 1Fold-£1Bet [B=1]Player 2Call±£2Fold+£1 Betting tree: outcomes for Player 1

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Émile Borel: Applications aux Jeux des 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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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 – its 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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