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

– 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

Betting tree: outcomes for Player 1

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%

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