How to win at poker using game theory

Transcription

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
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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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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
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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)
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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
Fold
Ante £1
-£1
Player 1
Bet [B=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
• The highest card wins the pot
±£2
Fold
+£1
Player 2
– First both players ante £1
– Then the cards are ‘turned over’
Call
Émile Borel: ‘Applications aux
Jeux des Hazard’ (1938)
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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)
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New key assumption:
– Player 1 can now check
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New conclusions
– Player 1 should bluff with his worst hands
– The optimum bet is size of the pot
One Card Poker
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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
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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
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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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