Lecture 13

1. Lecture 13

2. Poker Two basic concepts: • Poker is a game of skill, not luck. • If you want to win at poker, • make sure you are very skilled at the game, and • always play with somebody worse than you (and who doesn’t cheat) 

3. Baby poker There are 2 players, Player A and Player B. • The deck consists of only 3 cards, (J, Q, and K). • The players begin by putting $1 into the pot. Each player is then dealt 1 card(one card is left in the deck)

4. Baby poker • Player A goes first. • He either opens by putting $1 into the pot, • or passes by not betting.

5. Baby poker • Player B then plays. • If Player A passed, Player B may • also open by putting $1 into the pot, • or pass • If Player A opened, then Player B • must either call by also putting $1 into the pot, • or fold. • If player B opened, then Player A (who passed the first time) must either call Player B or fold.

6. Baby poker This ends the play. • If either player folds, then the other player gets everything that was put into the pot to that point. • If neither player folds, then the players com- pare cards. The player with the highest card gets everything in the pot.

7. Ploys • Bluffing: Opening on a losing hand (in this case a 1). • Sandbagging: Passing on a winning hand (in this case a 3).

8. Game play • First need to describe strategy for the entire game • This is a list of what the player would do in all possible situations • Player actions for Player A • O: open initially (no further decisions for A) • PC: pass initially, then call if B opened • PF: pass initially, then fold if B opened

9. strategies • Player actions for Player B • O/C: open if Player A passes and call if Player A opens • O/F: open if Player A passes and fold if Player A opens • P/C: pass if Player A passes and call if Player A opens • P/F: pass if Player A passes and fold if Player A opens

10. Using Strategies • Each of these decisions must be made based only on the value of the card seen by the respective player. • Thus the above strategies must occur in triples, indicating what specific plays must be made upon being dealt J, Q, or K.

11. “Pure” strategies • Possible Player A strategy (P-F,P-C,O) • Pass then fold if J • Pass then call if Q • Open if K • Possible Player B strategy (P/F, O/F,O/C) • Pass or fold if J • Open or fold if Q • Open or call if K

12. Ploys • The bluffing strategies are the ones where an O appears in the 1 slot. • The sandbagging strategies are the ones where a P appears in the 3 slot. • Example: • Strategy (O,P − C,P − C) for player A is both bluffing and sandbagging • Strategy (O/F, P/C, O/C) for B is bluffing (sandbagging is ineffectual for B in this simple game)

13. Probabilistic strategies • Probabilistic mixture of pure strategies. • Example Player A: • (P-F,P-F,P-C) with probability 1/3 • (P-F,P-C,O) with probability 1/3 • (O,P-F,P-C) with probability 1/3 • Might be useful to bluff/sandbag sometimes but not all the time!

14. Assessing Strategies • The outcome of a particular round of poker depends upon • the strategies chosen by each of the players, • and the cards dealt to each of the players (this component is random) • Example: • Strategies (P-F,P-C,O) vs(P/F, O/F,O/C) • Cards dealt Q vs K • Game will develop as Pass, Open, Call • The pot will be $4 at the end, noone folded • B wins $2, equivalently A wins -$2

15. Expected gain • Cards are random; given both strategies we can compute expected gain for player A • Example: • Strategies (P-F,P-C,O) vs (P/F, O/F,O/C) • Expected gain for A = -$1/6

16. Abbreviated Game matrix We removed obviously bad strategies to fit on the page.

17. Recall maximin • Select a strategy S • Here a probabilistic mixture of pure strategies • If the opponent knew my strategy, what is the worst they can do to me? • This will be one of the pure strategies (Why?) • Select the probabilistic strategy that maximizes this worst case scenario

18. Optimal strategies Player A: • Play • (P-F,P-F,P-C) with probability 1/3 • (P-F,P-C,O) with probability 1/2 • (O,P-F,P-C) with probability 1/6 • In layman’s terms : • holding a 1: open 1/6 of the time, and pass and fold 5/6 of the time • holding a 2: always pass initially, and then call 1/2 of the time and fold the other 1/2 • holding a 3: open 1/2 of the time and pass and call the other 1/2.

19. Optimal strategies Player B: • Play • (P/F,P/F,O/C) with probability 2/3 • (O/F,P/C,O/C) with probability 1/3 • In layman’s terms • holding a 1: always fold if B opens, but open 1/3 of the time, and pass 2/3 of the time if A passes. • holding a 2: always pass if A passes, but call 1/3 of the time and fold 2/3 of the time if A opens • holding a 3: always open or call. • Optimal strategy is not unique! Can you find another one?

20. Value of game • In Nash Equilibrium • Value of game: -1/18, i.e. Player A loses $1/18 per game to player B.

21. ``Reward’’ for being honest • If Player A cannot bluff or sandbag: • Value of game is -1/9: Player A now loses $1/9 to Player B • If Player B cannot bluff: • Value of game is 1/18: Player B now loses $1/18 to Player A • If neither player can bluff or sandbag: • Value of game is 0: it is an even game (and a pretty boring one; both players always open with a 3 and pass-fold with a 1 or 2).