Lecture 13. Poker Two basic concepts: Poker is a game of skill, not luck. If you want to win at...

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Transcript of Lecture 13. Poker Two basic concepts: Poker is a game of skill, not luck. If you want to win at...
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. • 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)
Baby poker
• Player A goes first. – He either opens by putting $1 into the pot, – or passes by not betting.
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.
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.
Ploys
• Bluffing: Opening on a losing hand (in this case a J).
• Sandbagging: Passing on a winning hand (in this case a K).
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
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
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.
“Pure” strategies
• Possible Player A strategy (PF,PC,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
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)
Probabilistic strategies
• Probabilistic mixture of pure strategies.• Example Player A: – (PF,PF,PC) with probability 1/3 – (PF,PC,O) with probability 1/3 – (O,PF,PC) with probability 1/3
• Might be useful to bluff/sandbag sometimes but not all the time!
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 (PF,PC,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
Expected gain
• Cards are random; given both strategies we can compute expected gain for player A
• Example: • Strategies (PF,PC,O) vs (P/F, O/F,O/C)
– Expected gain for A = $1/6
cards (J, Q) (J, K) (Q, J) (Q, K) (K, J) (K, Q)
probability 1/6 1/6 1/6 1/6 1/6 1/6
A gains 1 1 +1 2 +1 +1
Abbreviated Game matrix
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
Optimal strategies
Player A: • Play
• (PF,PF,PC) with probability 1/3 • (PF,PC,O) with probability 1/2 • (O,PF,PC) 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.
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?
Value of game
• In Nash Equilibrium– Value of game: 1/18, i.e. Player A loses $1/18 per
game to player B.
``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 K and passfold with a J or Q).