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)  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 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)  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% Thank You for Listening!