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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

How to win at poker using game theory

A review of the key papers in this field

the main papers on the issue
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
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.
mile borel applications aux jeux des hazard 1938
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
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
mile borel applications aux jeux des hazard 19386
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.
mile borel applications aux jeux des hazard 19387
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
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
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 poker10
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 poker11
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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