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

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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

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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%