How to win at poker using game theory

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# How to win at poker using game theory - PowerPoint PPT Presentation

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

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)
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.
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
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
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.
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
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
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
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
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%