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# Lecture 13 - PowerPoint PPT Presentation

Lecture 13. 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 ) . Baby poker. There are 2 players, Player A and Player B.

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Presentation Transcript

### Lecture 13

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) 

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)

• Player A goes first.

• He either opens by putting \$1 into the pot,

• or passes by not betting.

• 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.

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.

• Bluffing: Opening on a losing hand (in this case a 1).

• Sandbagging: Passing on a winning hand (in this case a 3).

• 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

• 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

• 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.

“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

• 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)

• 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!

• 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

• 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

We removed obviously bad strategies to fit on the page.

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

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.

• 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?

• In Nash Equilibrium

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

• 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).