How to win at poker using game theory

1. How to win at poker using game theory A review of the key papers in this field

2. 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)

3. 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.

4. 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

5. 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

6. 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.

7. 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

8. 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

9. 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

10. 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

11. 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%

12. Thank You for Listening!